Weak-coupling theory for partial condensation of mobile excitons

Igor V. Blinov1, ∗
1
On leave from Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
(Dated: April 15, 2024)
We studied formation of charge density wave between valleys in a system with double-well-like
dispersive valence band which captures some of the features of rhombohedral graphene trilayer.
In a regime with 2 Fermi surfaces: electron- (with radius pi ) and hole-like (po ) – an instability
in particle-hole channel appears at q = qc + δq, where qc = po − pi . In a weak coupling regime
(x/ϵF ≪ 1) presence of an additional energy scale ∝ mqc δq gives rise to several regimes with
distinct spectrum and transport properties: in a regime with small order parameter x ⪅ m⟨pF ⟩δq
Fermi arcs show up and change conductance qualitatively. At larger values of the order parameter
arXiv:2509.11304v1 [cond-mat.str-el] 14 Sep 2025

Fermi arcs are gapped out. Regimes are also distinguished by different effective exponents ζ in
ζ
conductance correction σ ∝ τD where τD is scattering time off disorder and 1 ≤ ζ ≤ 2.

I. INTRODUCTION system with quadratically dispersive bands and screened

interactions in 2D does not change shape of the bands
and therefore does not change the resistance according
When electrical current fixed by the external condi- to Drude formula. Then there are several alternatives:
tions, dissipated heat goes as ∝ j 2 /σ, so that in order
to minimize external influence1 , most of the things at 1. Drude formula is inaccurate due to large quantum
low temperatures conduct well. It is then reasonable to corrections;
expect finding unusual physics in areas where highly re-
2. long-range nature of interaction is important;
sistive phases are present.
Insulators (σ = 0) usually appear because of very severe 3. a phase realized in nature is not intervalley coher-
restrictions, such as statistics2 , particle conservation3 or ent.
charge of particles effectively being zero4 .
Besides true insulators, highly resistive phases can be of Mean-free path in the sample lD ≈ 1µm10 , which makes
interest (known as failed insulators5 ). We may expect the first alternative less plausible, since quantum correc-
resistance per quasiparticle being high in systems where tion decays as a power of lD pF . And, while the band
two interacting subsystems present: one with charged curvature induced by the momentum-dependence of in-
carriers, and the other without such. One of the exam- teraction could be important, it is known that in many
ples of such phase is high-temperature superconductors, cases most of the relevant physics can be modeled with
where both spinons and chargons are present6 . a properly chosen value of the contact interaction in the
Another system of this kind would consist of ex- limit of small density12 or small screening length. As a
citons (compound particles with zero charge) and consequence, in this, as well as in the previous paper13
holes/electrons. we have explored the last path.
When a system consists of several layers with energy hy- Namely, we suggest as a candidate a phase with val-
bridisation γ doped away from the charge neutrality, it ley and spatial symmetries broken simultaneously: e.g.
is reasonable to expect that at some displacement field an intervalley coherence established at a vector Q =
D > γ, but not D ≫ γ a hole-like Fermi surface will be lo- K − K ′ + q, where q ≈ po − pi ≪ Q, which would cor-
calized in the top-most layer, and the electron-like Fermi respond to formation of an incommensurate large-period
surface will be largely in a bottom-most layer, given the crystal lattice in the second order of the order parameter
field direction is positive. ABC-stacked (or rhomboedral) or, another words, to an incommensurate charge density
graphene has three layers and in sufficiently strong elec- wave. In the language of the band theory, the interval-
trical field (of order 100 mV/nm) perpendicular to the ley potential flattens bands, and as a result, the effective
plane seem to satisfy a condition with two types of carri- mass changes. This phase explains qualitatively suppres-
ers present. In this regime, two Fermi surfaces7 , electron- sion of conductivity at low temperatures. In this case,
and hole-like for each spin and valley are present. as long as the state is metallic, correction should go as
There, a number of correlated phases were observed8 . A δσ ∝ τD , where τD is the initial scattering time.
resistive phase, called partially isospin polarized (PIP) However, because of presence of two Fermi surfaces in
is mostly consistent with an intervalley coherence8,9 . the system: inner, electron-like, with a smaller Fermi
Similarly to twisted graphene bilayer, adjacent phase momentum pi and outer, hole-like, with a large Fermi
is superconducting10 and appears in a regime with two momentum po , local gaps may open and, consequently,
Fermi surfaces. Besides the C6 -symmetric intervalley co- system may enter non-metallic regime of conductance.
herent phase9 , a phase with additionally broken rota- As a result, correction to the conductance contradicts
tional symmetry11 was suggested to explain the experi- Drude formula. Conductance correction depends on q
ment. However, it is known that Stoner transition in a and has, as a function of it, few regimes, with a leading
 2

term going as some power of τD (the scattering time due x, y, z at a momentum q in analogy to the ferromagnetism
to disorder), and me is the effective mass of electrons: labels, have it in the form:
 α
xq
δσ ∝ (me τD )ζ , (1) ρq = ρ0q /2 + zq /2 + xq /2 + iyq /2, (4)
me

where effective ζ is from 1 to 2 (in the experimentally rel- where the last two correspond to an order parameter that
evant regime it has a value around 3/2). To shed some we call intervalley coherence at finite q. Note here that
light on these regimes, in this paper we start from a sim- each component of pseudospin has both K − K ′ + q and
ple fermionic model (Sec III) that can be of relevance −K +K ′ +q terms, which will give rise to charge modula-
both for ABC-graphene14–16 , and systems with Rashba tion at momentum q. Assume that intervalley coherence
electrons17 , then build an effective bosonic theory valid at a finite q established, say in x-direction: xq ̸= 0 and
in the low-temperature regime, evaluating all the rele- there are at least 2 reciprocal lattice vectors q. In real
vant coefficients (Sec V) microscopically, and calculate space, expectation value of the density is:
the correction to the conductance due to formation of
the intervalley order in Sec VI both in the second order X X
in xq and, extending perturbative in xq τD expansion, ob- ρ(r) = ⟨c†σ (r)cσ (r)⟩ = ⟨c†σ (k + q)cσ (k)⟩eiq·r , (5)
tain a general formula. σ σ,k,q
To conclude, we comment on the difference with related
theories18,19 , problems with the current theory and its
application to transport calculations, as well as future meaning that in the second order in xq there will
prospects. be long-range variations of density with wavevector
K − K ′ + q − (K − K ′ − q) = 2q ≪ |K − K ′ | with
large period π/|κ|, with κ = mini,j (qi + qj ) which is not
II. ORDER PARAMETER generically commensurate with the period of the density
wave appearing in the first order, which is roughly equal
to 2π/|K − K ′ |. In what follows, I denote a SO(3) order
To clearly define an intervalley order at finite q, we first
parameter as Mq = (xq , yq , zq ) = (X̄q , zq ).
revise a definition of the intervalley coherence. A knowl-
Displacement field breaks z → −z symmetry and
edgeable reader may consider skipping this section.20
opens a gap. When the gap is sufficiently large in
Charge density order (CDW) is defined through the
comparison to other energy scales present in the model
Fourier components of electron density at non-zero q:
(kinetic, coupling between the layers, interaction),
X the low energy description may involve a single band.
ρq = ⟨c†σ (k + q)cσ (k)⟩, (2)
Sufficiently strong displacement field D∥ẑ distributes
k,σ
electrons between the layers. At very small momentum
where summation over quasimomentum k goes over the valence band of the trilayer is mainly located within
whole Brillouin zone. For single layer graphene, effective the conduction band of the B-sublattice of the bottom
low-energy description involves 4 flavors of fermions: val- layer, while at larger momentum it restores the original
ley and spin, with valley being an area in k-space close form, which is delocalized in the sublattice space. As a
to one of ′ ′ result, Fermi surface has a property of having positive
√ energy-extrema at K and K points: K/K = mass for small momentum and negative mass for large
(±4π/3 3, 0)/a0 . Then Fermi creation/annihilation op-
erators can be represented as a sum of two operators momentum. Hence a regime with 2 Fermi surfaces
defined in different subregions of k-space: cσ (k) = (annular Fermi surface) can be established, with bands
cσ,K (k)δk,K + cσ,K ′ (k)δk,K ′ , δk,K = θ(−|k − K| + kδ ) being electron-like for inner circle, and hole-like for outer.
are the Heaviside functions with cutoff kδ < |K − K ′ |/2.
′ Because of presence of 2 different Fermi surfaces
Introducing additionally p = k − K /K and Pauli matri-
within each electron flavor, the order parameter
ces τ 0,x,y,z acting in the valley space, one can rewrite the
Mq at sufficiently large momentum will have two
CDW-order in the form
components qualitatively different from each other:
X Mq = Mm,q + Mex,q , where Mm,q is a metallic compo-
ρq = ⟨c†σ,a (p + q)cσ,b (p)⟩δq,K−K ′ (τab
0 z
+ τab )/2 nent and corresponds to coherence established between
p,σ
X alike Fermi surface (inner-inner, outer-outer). We call
y
+ ⟨c†σ,a (p + q)cσ,b (p)⟩(1 − δq,K−K ′ )(τab
x
+ iτab )/2, it metallic for it does not lead to a gap opening. Mex,q
p,σ is an excitonic component that establishes coherence
(3) between different Fermi surfaces (inner-outer, outer-
inner). Having in mind presence of both component
with arguments of creation/annihilation operators may help to understand both the critical behavior and
cσ,a (p)/c†σ,a (p) being by modulus of K. Or, finally, in- electromagnetic response. We will see later that the
troducing pseudomagnetization vector with projections excitonic component is related to Fermi arcs.
 3

III. BAND MODEL polarization function can be evaluated through:

X na (p + q) − na′ (p)
Within the tight-binding model21,22 electronic states in Πaa′ (q) = = Ia (q) + Ia′ (q),
ξa (p + q) − ξa′ (p) + iδ
graphene trilayer are described by 2 × 3 spinors in each p
(valley and spin) flavor. For ABC-stacked graphene, the (7)
largest interlayer hybridization (γ1 ) is between B1 and where na (p + q) ≡ n(ξa (p + q)), ξa (p + q) ≡
∂2 2
A2 , and A3 and B2 . Then in the regime γ1 ≫ vD pF ≈ ∂p2 ϵ(pa )(p − p2a ), pa characterizes Fermi surface
D ≫ γ2 ≫ γ3...6 , where (γ2...6 are the remaining tun- (a = i, o) and Πa(a′ ) (q) simply to denote terms with
neling constants) a valence band can be described by single density na (p) in the numerator.
double-well dispersive band13 with trigonal warping (a
term that breaks continous rotational symmetry down
to C3 ):

ϵ(p) = (mp2 + λp4 ) + ∆p3 cos(3θp ), (6) A. Zero temperature response

For q < 2pi/o response of approximately quadratically

where λ < 0 and m > 0. The role of the last term is to
dispersive particles in 2D changes slowly – for q < pf :
lift the SU (2) degeneracy in the valley space, choosing
valley coherent state (x − y) instead of valley polarized 1
(z). However, when anisotropy is small ∆/me ≪ 1, we Πhom = − , (8)
2|me |
may safely make ∆ = 0 (corresponds to the next-nearest
vertical neighbor tunneling, coupling between A1 and B3 , where an effective inverse mass of the quasiparticles
γ2 → 0) and postulate x ordering. In this work, we fix ∂2
ϵ(p ) ≡ ±m = ±
p
m2 + 4λµ for inner/ outer Fermi
∂p 2 a e
∆ = 0 for simplicity, to make ϵ(p) a function of p2 only.
surfaces. When q no longer can connect two points on
We claim, however, that most of the conclusions of the
Fermi surface,
paper will still survive finite ∆.
Then Fermi momentums for inner- (i) and outer- 
1
q 
1
Πhom = − 1 − 2 2
q − (2pi ) . (9)
Fermi (o) surfaces are given by p2i/o = −m/(2λ) ±
p 2 2me
(m/2λ)2 + µ/λ. Presence of 2 Fermi surfaces with
Fermi momentums po and pi implies that an instability Because p2i = (me − m)/2λ the system enters the second
at qc ≈ po − pi should be present. regime for electrons located at the inner Fermi surface
for µ < µu = (1 − (4/5)2 )m2 /4λ.

For hetero-processes, static response vanishes below

IV. RESPONSE qbound = qc − δqb , with δqb ≪ qc ≡ po − pi , since it will
be impossible to satisfy energy conservation condition.

In a trilayer, phase transition happens as a function To calculate the hetero-response, we note that quadratic
of hole density at a certain critical nc . Usually it can be mass approximation is valid whenever the quartic part is
attributed to increase of the density of states ν at the small: −δ < ξ(p + q) < δ and −δ < ξ(p) < δ, so that the
Fermi level. Usual epistemology of phase transitions says integration limits −δ/me + p2s < p2 < δ/me + p2s
that since electron susceptibility to an inhomogenous and −2δ/me + p2s′ − p2s − q 2 < 2ps q cos(θ) <
pseudomagnetization xq is a function of q, a phase at q 2δ/me − p2s + p2s′ − q 2 , and δ = me (me /λ)δd with
that minimizes non-interacting susceptibility is realized δd ≪ 1, which we estimate later in this section.
at nc .
Dividing the hetero-response into the inner- and outer-
Within RPA23 for contact interaction, susceptibility parts, we get an integral with ϵo (p + q) − ϵ(p) in the
(or response) can be expressed through components of denominator. Each difference defines angle θc,i/o for
non-interacting polarization operator Πa = Tr(Π̂σa )/2: which energy conservation is satisfied together with elec-
Πaq = Πaq /(1 + λΠaq ). In negligence of the trigonal warp- tron/hole being close to the Fermi surface. Clearly, such
ing the model is rotationally invariant in the pseudospin integral is maximized when θc,i equal to one of the in-
space, and hence Πaq = Πq . tegration limits, so that the integrand does not change
At low temperatures, only points close to the Fermi sign. Not surprisingly, we will get an expression analo-
energy should contribute to the response. We then gous to response to the Peierls instability for quasi-one-
divide Πx into 4 parts: between alike Fermi surfaces dimensional electrons constrained to angle θc,i/o 24 :
in different valleys Πii/oo (homo-part), and between  
different Fermi Πio/oi (hetero-part). Such division is θc,i pi ϵc,i
Ia = log , (10)
legitimate for q 2 < (p2o − p2i )/2. A component of the 2me π pi + po ⟨ϵF ⟩
 4

with cutoffs dictated by the kinetic energy of relative The former energy is to represent transversal energy
motion of a hole and an electron. As such, ⟨ϵF ⟩ is the scale: energy increases as we increase angle of the mo-
maximum kinetic energy of relative motion that satisfies mentum from 0 to θc,i quadratically for fixed q = qc .
energy conservation: ⟨ϵF ⟩ = me (pi + po )2 /2 , ϵc,i plays At vicinity of zero temperature, response has a usual for
the role of the low-energy cutoff: ϵc,i = me qc2 θc,i2
, and an Fermi gases1 quadratic temperature-dependence. In neg-
25
analogous expression for Πo . For small δq, ligence of the temperature-dependence of the homo-part,

2δqpo/i δq 2 (p2o + p2i )
 π 2 θc,i/o T2
2
θc,i/o ≡ ± . (11) Πlow,i/o (T ) ≈ Πi/o + q 2 , (16)
pi/o qc qc pi/o (po + pi )2 12me (1 + 2pi/o ) TL,i/o

In the regime of interest, small characteristic temperature

Angle θf depends on cutoff δ. Then small momentum TL,i/o ≈ 10−4 − 10−5 m ≈ 10−2 − 10−1 meV is an order
offset δq also is a function of the cutoff, and the equality of magnitude smaller than the Fermi temperature. If the
of two angles is what determines δq. Clearly, good choice phase transition happens within this regime, the critical
of δ will make the sum Πaa′ closest to the exact response temperature is
with quartic dispersion. One source of inaccuracy is due
 m 1/2
to the incompleteness of the integration range over ener- e
Tc,L = TL (−V Π0 − 1)1/2 , (17)
gies, and goes as ∆Prange ∝ (δ − δ1/2 )/(me δ) where δ1/2 V
is such that (p2o − p2i )/2 = δ1/2 /me . The other source where
of mistake is due to the approximation to the quartic s
dispersion ∆Papp ∝ 2λ2 δ 4 /m3e δ 2 , which is minimized at π θc,i θc,o
δ = δ1/2 |me /2λ| = |me /2λ|2 ≪ 1, and then: TL−1 ≡ 1/2 T −2 + T −2 (18)
12 (1 + q/2pi ) L,i (1 + q/2po ) L,o
|me /λ|2 A prefactor −me Π0 − me Vc−1 ∝ ∂(me Π)/∂µ(µ − µc )
δq 2 =  , (12)
po +pi p2o +p2i should be less than 1 to satisfy Tc,L < TL,i/o con-
2 qc + (po +pi )2
dition. Since ∂(me Π)/∂µ(µ = µc ) ≈ 103 the con-
We now choose the interaction constant in order to match dition is satisfied in the range µ − µc ≈ 10−3 meV ,
the density of the first transition nc ≈ 1.2 × 1012 cm−2 which lies within the experimentally relevant regime.
(which for single band model corresponds to µc = 7 × See Fig. 1 A for plot of the low-temperature regime.
2
10−4 m): With TL,i/o ∝ me pi/o qθc,i/o ∝ me δq we conclude that
−1/2 1/4
 1/2 ! TL ∝ θc ∝ qc , so that Tc non-analytically vanishes
1 1 ϵc,i ϵc,o at qc = 0 point.
−1 + ⟨θc ⟩ log ≈ −me Vc−1 , (13) At higher temperatures, T ≫ T1 , momentum difference
2π e2 ϵF,i ϵF,o
between two Fermi surfaces becomes smeared: difference
with ⟨θc ⟩ = (θc,i + θc,o )/2, so that Vc ≈ 0.45m. Note that between pi and po can be neglected. Another words, the
the hetero-part of the response does not diverge and of system becomes effectively indistinguishable from a sys-
the same order that the homo-part. Therefore it gives tem with an electron-like and a hole-like Fermi surfaces
significant increase of critical temperature as well as value with the same Fermi momentum (e.g. BCS or excitonic
of the interaction at which the phase can be stabilized. condensate). As a consequence, for TL < T < TF,i/o , di-
rect calculation shows that the temperature-dependence
follows the logarithmic law:
B. Critical temperature
TL,i/o (1 + Γ0 (1))
∆Πhigh,i/o (q, Ω, T ) ≈
Normally, in an electron liquid the only relevant 2m2 pi q sin(θc,i )
  e 
temperature scale is the Fermi energy µ = p2f /2m. For
 
TL,i/o T 1 TL,i/o
+ log − 1 − ,
low temperatures static linear responses of Fermi liquid 2m2e pi q sin(θi ) TL,i/o 2 T
at finite q are at best quadratic functions of (T /µ)2 and (19)
hence can be considered nearly temperature independent
for T ≪ µ. We thus look at the hetero-Fermi response which using (14) could be interpreted as temperature-
only. There will be one additional characteristic energy dependent part of the polarization operator of light elec-
for each Πi/o trons with inverse mass mef f = me /θc,i and agrees with
earlier numerical studies13 . Note here that because TF,i/o
2
and TL,i/o are different by a factor of 10-100, tempera-
me pi qc θc,i/o ture dependence in this regime is nearly linear. In this
TL,i/o = qc , (14)
1+ 2pi/o
regime, the critical temperature goes as
−1 m2 q m2
eq −1 me
) ⟨Te⟩
Tc,H = ⟨TL ⟩g e−(Πo +V a e1+f ⟨T ⟩a ≈ ⟨TL ⟩g e−(Πo +V ) θc e,
TF,i/o = 2me p2i (15) (20)
 5

FIG. 1. Behavior of the critical temperature (A) as a function of the chemical potential and an order parameter (B) for infrared
cutoff Ω = TL for different values of the interaction constant V = 0.17, ..., 0.49. Here, parameter λ = −240m. Plot A gives an
estimate of the critical temperature in the range 0.1-0.5 K.

p
where ⟨TL ⟩g = TL,i TL,o is the geometric average of of 3 2-dimensional momentums as well as frequencies. It
two characteristic temperatures, ⟨T ⟩a = TL,i /(2pi θi ) + is one of the coefficients that distinguish the partial exci-
TL,o /(2po θo ), ∆⟨T ⟩a = −TL,i /(2pi θi )+TL,o /(2po θo ), and tonic condensation from LO-phase and lead to the forma-
f = ∆⟨T ⟩a /(m2e q) log(TL,o /TL,i ) ≈ 026 . tion of the C6 rotationally invariant intervalley crystal.
After the summation over the Matsubara frequencies U
can be represented as a sum of 4 terms U1 +U2 +U3 +U4 ,
V. LOW ENERGY THEORY each with 3 energy differences in the denominator in the
form ∆ϵi = ϵ(p + qi ) − ϵ(p).
We now construct a low-energy theory through a Integrands are most peaked when all differences are close
Hubbard-Stratonovich transformation. We proceed to zero. For generic density and finite q it is not possible
23 to do that for 3 arbitrary q, therefore generically coeffi-
through usual steps : decouple interaction in particle-
particle channel, then integrate out fermion modes, ex- cients may diverge at worst as δθδpΩ−2 , where δθδp is the
pand the logarithm in effective action in powers of Mi . region where 2 out of 3 ∆ϵi ≈ Ω. Generically, the mea-
To justify the next step, the order parameter (Mi ) must sure of this region is Ω2 which makes the coefficient finite.
be small in comparison to an inverse mass of a fermion For qi = −qj one of the energy differences vanishes, and
(∝ ν −1 , where ν is the density of states) or, alternatively, then Ui (qi , 0, qk ) ∝ Ω−3 δθdδp, where δθδp ∝ Ω2 is the
temperature. Truncating the expansion at the 4-th order, measure of the area where two differences vanish, thus
we obtain a standard Landau-Ginzburg (LG) theory:27 making the coefficient divergent at most as Ω−1 .
X The most divergent configuration is qi = qk = −qj ,
Mi (q) χ(q) + λ−1 Mi (q) since δθδp ∝ Ω and therefore at worst the integral is


f=
i divergent as Ω−2 . Since terms in the sum have opposite
X uncompensated frequency sum is less divergent than
+ U (qi , qi +qj , qi +qj +qk )Ml (qi )Ml (qj )Mn (qk )Mn (ql ),
individual terms (see Appendix C for details of the
i
computation) and for the choice of Ωi = −Ωj = −Ωk
(21)
U (qi , 0, qi ) ∝ Ω−3/2 . Similarly, divergence for the total
The energy difference between the valley-polarized state sum U gi ̸= gk is less and goes as δθδpΩ−2 , which in
2 2
and the valley-coherent state goes as ∆ xq /µ ∝ ∆ /µ, 3 2 the worst case scenario results in Ω−1/2 divergence. For
which justifies ∆ = 0 limit discussed earlier for ∆2 /µ ≪ Ω ∝ Tc we then can ignore all other terms and use only
1. Without loss of generality then, we take Mi (q) = the most divergent term in the free energy expansion.
(xq , 0, 0), where |q| is a momentum that we treat as a
parameter of the theory. As we discussed before13 , for The most divergent term can be simplified to:
very weak interaction λ → 0 the state is fully SU (2)
symmetric xq = 0. However, when λ increases fully sym- Z
n(ξp )(ξp+gi − ξp )3
metric state becomes unstable towards formation of the U (g i , 0, g i ) = −4
(Ω2i + (ξp+gi − ξp )2 )3
intervalley coherence at a finite q = po − pi + δq.
∂2
Z
n(ξp )
= −2 2 Re (1 + O(Ω1 )) (22)
∂Ω ξp+gi − ξp + iΩi
A. Higher order terms
An equilibrium state should correspond to the combina-
To find the order parameter behavior, we calculate the tion of Ωi -s such that the term is the least divergent:
higher-order terms in the expansion of the free-energy Ωi ≈ −Ωk , Ωi ≈ −Ωj . Just like with the response, the
(21) U (qi , qi + qj , qi + qj + qk ), which are the functions integral can be divided into the homo- and hetero-parts.
 6

The latter (22) can be rewritten in the form: VI. RESISTANCE CHANGE

Z
∂ δ(ξp ) When external perturbation does not change topol-
U (gi , 0, gi ) = 2Re ogy of the Fermi surface, number of charge species
i∂Ω iΩi + ξp+gi − ξp
XZ dθps stays the same, correction to the conductivity may
≈ −2 , only originate through the change of band velocity or
Ωi + ms (ps + q + 2qps cos(θ) − p2s′ )
2 2 2 2
s,s′ quasiparticle wave functions. Naturally, given SU (2)
symmetry, presence of homogenous potential (such as
where s, s′ = i/o. For the homo-part (s = s′ ) the most pseudo-magnetization x(q = 0)) in 2D should not change
divergent part: neither mass for nearly quadratic bands nor the matrix
elements of τD , thus leaving the conductance unchanged.
 m 1/2 21/2 π
e
U4,hom = − (23) On the other hand, spatially varying intervalley
Ω (me q)3
potential x(q) obstructs movement of charges, which
The divergence becomes Ω−3/2 at a special point q = 2pa , can be expressed through the change of their velocities
and Ω−1/2 otherwise. Since the Landau-Ginsburg coeffi- and wavefunctions. We should then expect the change
cients can be discontinuous, we do not worry about that of conductance in the perturbative regime to be of
feature at qc = 2pi and take Ω−1/2 behavior to be valid the form δσ ∝ x2q τD /ϵj , where ϵj is the smallest finite
everywhere. characteristic energy for one of the carrier species and
Then in the leading order contribution to V4 is logarith- summation assumed over j.
mically divergent:
In a regime with week external potential xq /Σ ≪ 1,
1 pi (v/(2ri qc ))2
2
correction from the hetero-processes has a form of x2q τD
V4,het,i = 2 − ivΩ )3/2
log(αi Ω), (24) 29
– similarly to superconductivity , it appears because
2me pi + po (θio ri q c
of the coupling of electron- and hole-like bands at the
verge of the gap opening, and the smallest energy scale
where v is an inverse average Fermi velocity,
p of the unperturbed system is disorder self-energy Σ:
v = 1/me (po + pi ) and ri,1/2 = 2(poi/o +pi ) are coeffi-
cients of unclear utility. If δqc = 0, the divergence, in δσ ∝ −x2q τD
2
. (26)
full analogy to the homo-part, becomes Ω−3/2 log(Ω).
Because here electron- and hole-like bands have different
It follows then from the minimum energy argument that Fermi velocities, the total correction, unlike its counter-
δq should be nonzero28 . The presence of homo-part of part in BCS30 , contributions from two different bands do
scattering amplitude contrasts the presented phase from not cancel each other. We see that because of different
Larkin-Ovchinnikov18 , where, naturally, only hetero-part nature of Fermi surfaces present and formation of the
is present. charge density wave we have non-metallic behavior of
conductance. While it seems to be rare, such cases are
In the condensed phase, excitations have a gap known31 .
∝ x2q that fixes the infrared divergence. To compensate
for its absence in the perturbation theory, we choose a Once the potential increases, for δq ̸= 0 coupling
regularization through a sensible cutoff. For simplicity, of different Fermi-surfaces leads to the formation
we pick Ω = TL . Then, ignoring difference of θc,i from of Fermi arcs in the hetero-channel, thus increasing
θc,o , we have for the order parameter far away from the the number of species. It happens in the regime
1/2 2
critical point: xq /(2κio/oi me ) < 2pi/o qθc,i/o , where θc,i/o is the central
 1/2 angle for the inner(outer) Fermi-surface. Along the
TL radial direction quasiparticle composing Fermi-arcs
Mi2 (q) ∝ q 3 m2e (25) are light, while in orbital bands are nearly flat, with
me
mass ∝ (me sin(θc,i/o ))−1 . Contribution of heavy and
For δq weakly dependent on qc (large qc ) we conclude effectively 1D linearly dispersive quasiparticles to the
that Mi (q) ∝ q 25/16 which vanishes non-analytically at conductance is
1/2
q = 0. In small qc limit δq ∝ qc and as a result the −1
δσ1,1 ∝ −xq τD (θc,i −1
+ θc,o ), (27)
power changes to 53/32.
Interestingly, it can be larger by its absolute value than
Since U (gi , 0, gi ) is P
dominant, and the condensa- one present in the purely partially insulating phase: e.g.
tion energy is ∝ − i ϵq (ϵq /U (gi , 0, gi )), the most ∝ x2q τD /me . .
energetically beneficial is a phase with the largest One way to interpret this non-negligible correction to
number of g-vectors, therefore it is the C6 phase that is the conduction is through formation of the countercur-
established (Fig.1). rent: e.g. negative cross terms between electrons and
 7

counting powers of Σ, we should have Σ−4 and the area

dpdθ ∝ Σ2 , hence giving τD 2
-dependence. Because of the
smallness of δq (or, corollary, because of the fact that the
area dpdθ ∝ (sin(θc,i/o ))−1 is large), hetero-process dom-
inate and enter with a different sign. In the limit of large
scattering length (τD ≫ m−1 e ), the leading correction is

c |xq |2 X
σxx ∝− cot(θc,s ) + o(Σ/me )
FIG. 2. Corrections in the leading order in intervalley pseudo-
Σ2 s
magnetization xq to the electron part of the conductivity of 2
 1/2 X
2D system with double-well dispersion. Wavy line within the 3 me TL
∝ −q 2 cot(θc,s ), (30)
fermion loop denotes propagator ⟨xq x−q ⟩. On the symmetry Σ me s
broken side, we take it simply the order parameter squared
x2qi for q = qi and vanishingly small frequency. Corrections where c1 = O(1) and θc ≈ 0 and we used (23) (see Ap-
can be divided into a correction to the density of states (a) pendix D for details of the calculation). Interestingly,
and a correction to the vertex (b). terms O(Σ−1 ) are effectively absent, since Σ/me is van-
ishingly small, while contributions from DOS and vertex
2
correction that go as τD do not cancel each other. Be-
holes in the expression for current. cause of this term correction to the conductance may not
be vanishingly small even in case xq /Σ < 1, xq /me ≪ 1.
In a regime with xq ≈ Σ, previously linearly dis-
persive bands, when smeared by the energy scale
Σ, better approximated by quadratically dispersive
B. Series resummation
touching bands, thus leading to
xq 3/2 −1 −1 Deeper inside the partial condensate phase, higher or-
δσ1,2 ∝ − τ (θc,i
1/2 D
+ θc,o ), (28)
me der terms in the expansion in xq τD become relevant and
formula (30) is no longer valid. Note that in the 4-
where the power of 3/2 comes from the divergent density th order, the correction to the conductance are propor-
of states. Naturally, as one may guess, we will see below tional to the coefficients of the Landau-Ginsburg theory
that the power 3/2 is, in fact, approximate (see Fig. A.7). U (q1 , q2 , q3 ), and hence we assume that the main contri-
Finally, once potential gaps out a Fermi arc, all bution comes from terms without momentum transfer.
electron-hole pairs in that region of k-space become con- This subset of diagrams is possible to sum up to an infi-
densed. For τD me ≫ 1, major contribution to δσ nite order.
comes from homo-process, negative, and has normal form Perturbative expansion, for example, can be obtained
∝ −xq τD , as well as correction due to the change of the through a functional integral. Since log(1+G−1 (A+xq ))
electron density ∝ x2q τD /me . Having briefly discussed is the functional to expand, all terms come with the
conductance change semi-qualitatively, we now are to same combinatorial coefficient, and hence the correc-
find more rigorous arguments through direct computa- tion
R should have a form of−1a simple ratio of the form
tion. As such, we will see that the correction due to x2q vp vp/p+q 1/(x2q + gp−1 gp+q ), where gp = Gp (iω =
the Fermi arcs is, in fact, unimportant and another term i0+). Indeed, one can show32 that total correction to
−1 −1
2
∝ x2q τD (θc,i + θc,o ) dominates. conductance δσ can be divided into three physically dis-
tinct terms
A. Second order X |xq |2 Z 2
δσodd = vp vp+q Im (Dq (p)) (31)
q
2π
For small values of ratio xq /Σ and q = qc + δq it can
be shown perturbatively that the power is 2: δσ ∝ x2q τD
2
. for processes where an odd number of excitons is ab-
The correction to the fermion part of the conductance is sorbed/emitted between current vertices, and
given by the sum of two diagrams (see Fig. 2) can be
simplified down to: X |xq |2 Z
−1
vp2 Im gp+q


x
δσeven,2 = Dq (p) Im (gp Dq (p))
x
vp+q x

x
+ vp vp+q + (ξp + ξp+q )
∂vp+q π
3xq x−q Σ2
Z q
(Σ) ∂px
σxx = 2 (32)
2π (ξp2 + Σ2 )(ξp+q + Σ2 )2
(29)
Situation is similar to scattering of excitons that enter X |xq |4 Z
U4 coefficients, except since the σ ∝ G2p G2p+q , character- δσeven,4 = − vp2 Im (gp Dq (p))
2
(33)
istic time τD ∝ 1/Σ enters in a doubled power. Indeed, q
2π
 8

for processes with even number of exciton emis-

sion/absorption between current vertices, and Dq (p)
−1
stands for Dq (p) = (gp−1 gp+q − x2q )−1 and summation is
performed over all reciprocal lattice vectors q that form
a crystal lattice33 . Formula (32) simply takes into ac-
count change of the electron Green function δG11 34 with
corrected vertex.
We distinguish three dimensionless parameters:
ϵx = xq /me , dimensionless energy of an excitation
1/2
across the Fermi arc ϵQ = pi/o qc sin(θc,i/o )/κio/oi
as well as ϵdis = Σ/me for the problem. We now
briefly describe limiting cases classified by the values of
these parameters (there is a plot in appendix – Fig. A.7). FIG. 3. Fermi arcs (dashed, white) in the regime with small
1/2 2
exciton gap xq /(me κio ) < pi/o qc θc,i/o within the conduction
Ratio ϵx /ϵQ determines whether scaterring of carriers band (left) and valence bands (right)
off excitons is energetically allowed. For ϵx /ϵQ ≪ 1 cor-
rection to conductance has a form δσ1,1,scat + δσ1,1,cond ,
where ductivity. It also appears because of the presence of two
 2  2 Fermi surfaces (electron- and hole-like) with different ra-
x q vi 1 x q vo 1
δσ1,1,scat ∝ − − dius (pi and po ) of a band model, and leads to oscillations
Σme pi qc θc,i Σme po qc θc,o of density with period ≈ (po −pi )−1 much larger than the
(34) lattice constant.
vi2 θc,i v 2 θc,o We showed that unlike the LO-phase in 3D, minimum of
δσ1,1,cond ∝ − − o , (35) energy at least within the weak-coupling theory, corre-
Σme Σme
sponds to a phase with C6 symmetric order parameter
and the first term corresponds to scattering off excitons, and not to C2 . In contrast to LO-phase, reciprocal lat-
while the second, order parameter independent term tice vectors of such intervalley crystal are not exactly
simply subtracts condensed fraction from electron equal to po − pi , but with a small correction δq because
density, and we also assumed ϵx ≫ ϵdis for simplicity of the anharmonicity of bands. This fact modifies both
here. order parameter behavior and transport properties qual-
itatively.
For large values of the ratio ϵx /ϵQ > 1, scattering As such, coefficients U4 , corresponding to scattering of 2
off excitons is forbidden, and hence the whole correction intervalley excitons, diverge as Ω−1/2 and not Ω−1 , which
comes from the fact that part of the system is in the leads to a larger value of the order parameter xq . Inter-
condensate: estingly, critical temperature of the phase ∝ xq has two
v2 regimes: when Tc ≫ T1 , the behavior is of MacMillan
δσ3,1,cond ∝ − i . (36) me
Σme type Tc ∝ TL e− θc λ , while at weakly interacting regime
So that we see that in agreement to naive picture13 the it has a square root dependence on interaction strength
conductance is reduced by fraction of fermions that form Tc,L ∝ TL (−V Π0 − 1)1/2 .
excitons. For ϵx /ϵQ < 1 there is also a conductance cor- Order has two components, metallic (coming from the
rection (it appears both in series δσodd and δσeven,2 ) lin- contribution between alike Fermi surfaces) and insulat-
ear in xq Σ−1 associated with conversion of electrons into ing (coming from the contribution between the different
holes and countercurrent generation: Fermi surfaces).
  Because for small xq the phase has Fermi arcs, Fourier
xq vi (vi − vo ) vo (vo − vi ) transform of quantum oscillations at very low frequencies
δσ2,1 ∝ − + , (37)
Σm2e pi qθc,i po qc θc,o should have a maximum, which is consistent to the ex-
for xq τD ≫ 1. periment. However, the latter is also consistent with par-
tially isospin polarized11 order. What seems to be incon-
Once ϵdis approaches ϵx as a generic rule q we would sistent, is that resistance by a factor of 1.5−2 larger than
the resistance of the paramagnetic phase. It is, however,
need to multiply each term linear in xq by xq / x2q + Σ2 straightforward to explain with partial condensation of
or a square of it (see Appendix V for details). excitons, since some of electron-hole pairs condense and
do not contribute to the transport on the mean-field level.
In the regime where emission of an exciton is permitted
VII. CONCLUSION by energy conservation ϵo (p + q) − ϵi (p) = ϵx (q), charge
carriers acquire new mechanism to relax energy with a
A phase with partial intervalley exciton condensate is characteristic time scale ∝ τX . Then ignoring other
similar to Larkin-Ovchinnikov (LO) phase of supercon- changes induced by the presence of the order parameter,
 9

by Matthiessen’s rule, the correction to the conductivity adjacent layers, enhancement of tunneling conductivity
in z-direction can be seen at the point of the phase
τD 2
transition.
δσ ∝ − τD ∝ −τD /τX (38) As for quantum correction to the conductance, they
τD /τX + 1
decay with increase of the scattering length, and thus
are not important in clean pf ls ≫ 1 limit. We also note
for τX ≫ τD . This is exactly what we obtained in small that since excitons have a weak coupling to electrical
2
xq limit. Remnants of this τD behavior can be observed field ∝ qc and their response is going to be proportional
at any doping before Fermi arcs disappear. to their density at momentum other than q = qc + δq,
In this region, change of the resistance, dependent on we expect that some descendants of Aslamasov-Larkin
relative strength of the order parameter xq , inverse scat- (AL) corrections35 should be less important at low
−1
tering time τD , and characteristic transversal kinetic temperatures in the regime of interest for hetero-part of
2 2
energy of excitons me qpi θc,i /me qpo θc,o have several δσ. We do not exclude a possibility, however, that AL-
n
power regimes δσ ∝ (τD xq ) with n effectively ranging type of corrections to total conductance will cancel term
from 2 to 1. It is possible, however, that some of this (xq τD )2 in (34) – we leave this question to future studies.
region is masked by the superconducting phase. Once
a system doped below this regime, scattering of charges Finally, it is necessary to remark that classical fluctu-
to neutral collective waves is no longer permitted, and ations of the order parameter may change the phase
most of the change in the conductance is due to change transition character19 from second to the first, however
of carrier concentration, and ratio δσ/σ0 ∝ −x2q /no is since pjump of the order parameter is of the order
disorder independent. We described this regime before13 . Tc /me ϵ∆ /me ≪ xq /me ∝ Tc /me , with ϵ∆ being the
rotonic minima, most of conclusions of this paper should
Another feature of the phase with C6 symmetric not be refuted by this fact.
order parameter is that in the second order of the VIII. ACKNOWLEDGEMENTS
order parameter density should
′
have a correction
′
of the
form δρ(r) ∝ x2q ei(K−K −q)r e−i(K−K +q)r ρ(q)ρ∗ (−q)
R
I.V.B. thanks Daniil Antonenko, Allan MacDonald,
which generates large-period oscillations with periods Andrey Semenov, Nemin Wei and other collegues for
10 − 100nm visible in STM microscopy for sufficiently valuable suggestions and comments, as well as hospi-
large systems or Bragg spectroscopy. Third, because tality and friendly environment of Yale University, Ts-
electron- and hole-like Fermi surfaces for large values of inghua University and A. Alikhanyan National Labora-
the displacement field D > γ1 , where γ1 is the energy tory (Yerevan Physics Institute) where some parts of this
scale of the hybridization between A and B sublattices of work have been done.

∗ 12
blinov@utexas.edu E. M. Lifshitz and L. P. Pitaevskii, Statistical physics: the-
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N. F. Mott, Reviews of Modern Physics 40, 677 (1968). arXiv preprint arXiv:2303.17350 (2024).
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N. D. Mermin, Physical Review 176, 250 (1968). H. Zhou, L. Holleis, Y. Saito, L. Cohen, W. Huynh, C. L.
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J. Frenkel, Physical Review 37, 17 (1931). Patterson, F. Yang, T. Taniguchi, K. Watanabe, and A. F.
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H. Zhou, T. Xie, A. Ghazaryan, T. Holder, J. R. Ehrets, 762 (1965).
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M. Serbyn, et al., Nature 598, 429 (2021). In short, it is charge density wave at a momentum close but
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S. Chatterjee, T. Wang, E. Berg, and M. P. Zaletel, arXiv not equal to the momentum connecting two valleys. An au-
preprint arXiv:2109.00002 (2021). thor received a lot of repetitive questions during seminars
10
H. Zhou, T. Xie, T. Taniguchi, K. Watanabe, and A. F. that showed that some sort of clarification of the latter is
Young, Nature 598, 434 (2021). necessary. A reader familiar with graphene may consider
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C. Huang, T. Wolf, W. Qin, N. Wei, I. Blinov, and A. Mac- skipping this section.
21
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Rinehart and Winston) Appendix C 1 (1976).
 10

22 33
P. R. Wallace, Physical review 71, 622 (1947). Note that the origin of asymmetry in δσDOS between p
23
A. Altland and B. D. Simons, Condensed matter field the- and p + q is a convention of how we label current vertex.
34
ory (Cambridge university press, 2010). B. L. Altshuler, A. G. Aronov, and P. Lee, Physical Review
24
Alternatively, we could have talked about 1D particles with Letters 44, 1288 (1980).
35
large inverse effective mass mef f = me /θc,i . L. Aslamasov and A. Larkin, Physics Letters A 26, 238
25
See Appendix B. (1968).
26 36
While deriving this formula, we neglected Γ0 (1) ≈ 0.21. It R. E. Peierls, Quantum theory of solids (Clarendon Press,
is possible, in fact, to prove, that this term is a result of 1996).
37
the approximation. It seems that α = 1, thus making the difference between
27
L. D. Landau and V. L. Ginzburg, Zh. Eksp. Teor. Fiz. 20, V4,c and V4,seo only quantative: two divergences of Ω−1
1064 (1950). instead of one.
28 38
In fact, a stricter inequality connecting sin(θc ) and the A. Migdal, Qualitative methods in quantum theory.
39
cutoff should hold: sin(θc ) <. It is possible that δq that G. D. Mahan, Many-particle physics (Springer Science &
corresponds to the minima of energy is, in fact larger, than Business Media, 2013).
40
the one that corresponds to the minimum of the intervalley There must be a part that cancels against the diamagnetic
reponse. term: absent here.
29 41
N. A. Stepanov and M. A. Skvortsov, Physical Review B Can be proven by induction, since upon addition of new
97, 144517 (2018). x2q , two new distances will be added, one would correspond
30
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(2001). to odd.
31 42
P. A. Lee, Physical review letters 71, 1887 (1993). The latter apparently expresses an energy conservation be-
32
See Appendix E for proof. tween electron-hole pairs and an exciton formation.
43
There could be a factor of 21/2−3/2 mistake here.
44
May happen because of additional terms in gi we neglected.

CONTENTS

I. Introduction 1

II. Order parameter 2

III. Band model 3

IV. Response 3
A. Zero temperature response 3
B. Critical temperature 4

V. Low energy theory 5

A. Higher order terms 5

VI. Resistance change 6

A. Second order 7
B. Series resummation 7

VII. Conclusion 8

VIII. Acknowledgements 9

References 9

A. Band structure 11
1. Bounds 11

B. Response 13
1. Homo-part 13
2. Hetero-part 13
3. Critical temperature 17

C. 4-th order terms 18

1. Estimation of divergences in V4 -term 20
 11

2. Calculation of V4,∥ 22

D. Conductance up to O(x2q ) 23
1. Vertex correction 23
2. DOS correction 25
3. Explicit expression for π2 (Ω, q) 26
4. Estimate 26
5. Accurate calculation 28

E. Conductance up to O(xnq ) 29
1. 4-th order 29
2. 6-th order 30
3. Combinatorics 31
4. Total correction to conductance 32
a. Self-energy correction 32
b. DOS-correction 37
c. Doubly corrected DOS 47
d. Vertex correction 49

Appendix A: Band structure

In the following 3 sections, we embark on a journey to evaluate all the coefficients of the Landau-Ginsburg theory
for the intervalley charge density wave at an incommensurate wavevector q = po − pi + δq using microscopic fermionic
theory. First, we are to simplify quartic band-structure to quadratic band-structure with two species of fermions.
Let me first derive some relations for the band structure. The dispersion is

ϵ(p) = mp2 + λp4 − µ, (A1)

hence Fermi momentums are

r
m m 2 µ
p2i,o =− ± + (A2)
2λ 2λ λ
So that once
m2
> µ, (A3)
22 λ
we have opening of the inner (electron-like) Fermi-surface.

1. Bounds

Most of the integrals acquire their value very close to one of the Fermi circles. Hence we can divide the integral into
4 parts: when p, |p + q| ≈ pi , p, |p + q| ≈ po (homo-part) and p ≈ pi , |p + q| ≈ po , p ≈ po , |p + q| ≈ pi (hetero-part).
More specifically, most of the contribution comes from the region:

δ δ
− < p2 − p2s < , (A4)
me me

δ δ
− < (p + q)2 − p2s′ < . (A5)
me me

Hence integration over angle happens in the range

δ 1 p2′ − p2s − q 2 p2′ − p2s − q 2 δ 1

− + s < cos(θ) < s + . (A6)
me ps q 2ps q 2ps q me ps q
 12

Here, we will be interested in the value of the transferred momentum close to qc : q = qc +δq. To establish a connection
between δq, δ we expand in powers of δq. For concreteness, start with s′ = o, s = i. Expand in vicinity of θc,i such
that cos(θc,i ) = (p2o − p2i − (qc + δq)2 )/(2pi (qc + δq)):

δi 1 δi 1
− < − sin(θc,i )(θ − θc,i ) < + . (A7)
me pi q me pi q

which gives:

2 δi 1
θc,i = . (A8)
me pi q

On the other hand, using the definition above for the central angle θc,i :

2  2 ! 
θc,i δq 2 po + pi
  
δq δq 2qc + δq δqpo
1− = 1− + 1 − δq =1− + 2 , (A9)
2 qc qc 2pi qc qc pi qc 2pi

meaning that

δq 2
 
δi 1 δqpo po + pi
= − 2 . (A10)
2me pi q qc pi qc 22 pi

Similarly, we get for the outer Fermi surface:

2 δo 1
θc,o = . (A11)
me po q

and from an analogous definition of the outer Fermi surface we can obtain:

δq 2 po + pi
 
δo 1 δqpi
= − 2 . (A12)
2me po q qc po qc 22 po

Note that from the requirement δqo = δqi follows

δi /p2o ≈ δo /p2i ≡ δ ′ . (A13)

Introducing their arithmetic average δ we get δ = (δi + δo )/2 = δ ′ (p2o + p2i )/2 = δ ′ |m/λ|. As a result:
2
2 δ ′ po/i 2δqpo/i δq 2 ⟨p⟩
θc,i/o = = − 2 , (A14)
me qc pi/o qc pi/o qc pi/o

so that δq ∝ ⟨p⟩δ ′ /me . Then we choose δ such that it minimizes the mistake of the quadratic approximation:
X n(ξ(p + q)) − n(ξ(p)) X n(ξa (p + q)) − n(ξa′ (p))
∆I = − , (A15)
p
ξ(p + q) − ξ(p) + iδ ′
ξa (p + q) − ξa′ (p) + iδ
p,a,a

where ξ(p) = ξa (p) + λ(p2 − p2a )2 . Since the energy range δ, ϵ(p + q) − ϵ(p) ≈ δ, then mistake from the range should
scale, by the order of magnitude, as −(δ − δ1/2 )/me δ (δ1/2 is the division of momentum range exactly in the middle
between pi and pRo ) and the mistake from the approximation comes in the second order expansion in λ, hence should
be around ∝ C n(me (p2 − p2a ))λ(p2 − p2a )2 /m2e (p2 − p2a )2 ∝ Cλ log(δ)/me , where we ignored the mistake coming
from the restriction over the angle range, since it should take care of double-counting. Then the best value should be
given by, approximately,

δ1/2 λ
− + C = 0, (A16)
me δ me

where C is of order 1. It should be then that δ ≈ (me /λ)δ1/2 . With δ1/2 ≈ (p2o − p2i )/2 = me /λ, so that δ ≈ (me /λ)2 .
 13

Appendix B: Response

The generic expression for response is

XZ
Π(q, iΩ) = d2 pG(p, iωn )G(p + q, i(ωn + Ω)), (B1)
n

or, after summation over Matsubara frequencies, is:

n(ξp+q ) − n(ξp )
Z
Π(q, iΩ) = d2 p . (B2)
iΩ + ξp+q − ξp

1. Homo-part

Homo-part has the form normal for metallic response at finite q:

n(ξp+q ) − n(ξp )
Z
Π(q, iΩ) = dθdpp . (B3)
iΩ + me (2pq cos(θ) + q 2 )

a. Zero temperature value For small frequencies, response is peaked at ξp+q ≈ ξp , which for homo-part at tem-
peratures far below Fermi is equivalent to ξp ≈ 0 and hence corrections due to constrain of the integration range over
angle discussed in section A 1 should be O(Ω). We then extend integration range to −π and π and represent the
response on the inner Fermi surface in the form:
Z Z Z 
dθdppn(ξp ) dθdppn(ξp ) dθdppn(ξp )
Π(q, iΩ) = − − = −2Re
−iΩ + me (2pq cos(θ) + q 2 ) iΩ + me (2pq cos(θ) + q 2 ) iΩ + me q(2p cos(θ) + q)
(B4)
Which gets us
 
2 !1/2 2 !1/2
π  i22 Ω i22 Ω
 
2Ω 2Ω
Πii (q < 2pi , iΩ) = − − +i − + 
me q me me q me me q
 
2 !1/2 2 !1/2
π  2 i22 Ω 2
 
2Ω i2 Ω 2Ω
+ q + − − i (2pi )2 − q 2 − +  (B5)
me q me me q me me 2

and
 
2
 2 !1/2 2
 2 !1/2
π  2 2 iΩ 2Ω i2 Ω 2Ω
Πii (q > 2pi , iΩ) = − q − (2pi )2 + − − q2 + − . (B6)
me q me me q me me q

And identical expressions for the outer-part, except for pi → po . For finite temperatures, we get quadratic dependence
on temperature with a characteristic scale of ϵF,i .

2. Hetero-part

a. Zero temperature value The hetero-response can be written in the form

Z Z
2 n(ξp,i ) n(ξp,o )
Π(q, iΩ) = − d p + d2 p
−iΩ − ξp,i + ξp+q,o −iΩ − ξp+q,i + ξp,o
Z Z
2 n(ξp,o ) n(ξp,i )
+ d p − d2 p
iΩ + ξp,o − ξp+q,i iΩ + ξp+q,o − ξp,i
≡ −2Re(Ii (q, iΩ)) + 2Re(Io (q, iΩ)) (B7)
 14

where the first term responsible for electrons on the inner Fermi-surface, while the second – on the outer Fermi-surface.
The integral over the inner part is
d2 p d2 p
Z Z
n(ξp,i ) 1 n(ξp,i )
Ii (q, Ω) = 2
= iΩ
(B8)
(2π) iΩ + ξp+q,o − ξp,i me (2π)2 m − ((p + q) 2 − p2 ) − (p2 − p2 )
o i
e

The integral over p can be taken either through a substitution δp = p2 − p2i and linearization in δp, or directly as an
integral over p. In zero temperature limit,
1. Exact integral over p
Z pi
1 dθdp p
Ii (q, Ω) = iΩ
(B9)
me 0 (2π)2 me − 2p2 − 2pq cos(θ) + p2o + p2i − q 2

which has roots at pi,1/2 = − 2q cos(θ)± 21 (q cos(θ))2 + 2(p2o + p2i − q 2 + iΩ/me ). Note here that since the angle
p

is close to 0, we can expand in the vicinity of θ = 0:

q 1
q
pi,1/2 = − (1 − θ2 /2) ± (q(1 − θ2 /2))2 + 2(p2o + p2i − q 2 + iΩ/me )
2 2
q 1
q
= − (1 − θ2 /2) ± q 2 (1 − θ2 ) + 2(p2o + p2i − q 2 + iΩ/me )
2 2
q 1
q
= − (1 − θ2 /2) ± 2(p2o + p2i ) − q 2
2 2
iΩ/2me q2 θ2
± ∓ = κi,1/2 + qθ2 ri,1/2 ± iαΩ, (B10)
(2p2o + 2p2i − q 2 )3/2
p
22 2p2o + 2p2i − q 2
which defines coefficient ri,1/2 as well as α:
pi/o
ri,1/2 = (B11)
2(po + pi )
The angle at which the real part of pi − pi,1/2 vanishes we call the central angle

δq 2 (p2o + p2i )
 
2 pi − κi,1 2δqpo
θc,i ≡ ≈ + , (B12)
qri,1 pi qc qc pi (po + pi )2
The integral now can be represented in the form:
Z pi Z pi  
1 dθdp p 1 dθdp pi,1 pi,2 1
Ii (q, Ω) = − =− − (B13)
2me 0 (2π)2 (p − pi,1 )(p − pi,2 ) 2me 0 (2π)2 p − pi,1 p − pi,2 pi,1 − pi,2
Integration over p gives:
Z θc,i ! ! !
1 dθ 1 (pi − p̄i,1 )2 + α2 Ω2 (pi − p̄i,2 )2 + α2 Ω2
Ii (q, Ω) ≈ 2 pi,1 log − pi,2 log + iπpi,1 ,
2 me 0 2
(2π) pi,1 − pi,2 p̄2i,1 + α2 Ω2 p̄2i,2 + α2 Ω2
(B14)
where pi,1/2 = Re(p̄i,1/2 ). Introducing a critical angle:

δq 2 (p2o + p2i )
 
2 pi − κi,1 2δqpo/i
θc,i/o ≡ ≈ ± (B15)
qri,1 pi/o qc qc po/i (po + pi )2
or, for small δq,

2 2δqpo/i
θc,i/o = . (B16)
pi/o qc
we can rewrite the response in the form:
! ! !
2
− θ2 )2 + (α/qri )2 Ω2
Z θc,i
1 dθ 1 (θc,i (pi − pi,2 )2 + α2 Ω2
Ii (q, Ω) = 2 pi,1 log −pi,2 log +iπpi,1 .
2 me 0 (2π)2 pi,1 − pi,2 (κi /qri + θ2 )2 + (α/qri )2 Ω2 p2i,2 + α2 Ω2
(B17)
 15

which reaches its maximal value for θc,i = θf and after integration, is equal to
 
θc,i κ1 ϵc
Ii (q = qc + δq, Ω = 0) ≈ 2 log , (B18)
2 me κ1 − κ2 ϵF,i e2

where ϵc = (θc qc )2 me and ϵF = me (pi + po )2 /22 , where ϵc plays the role of infra-red cutoff for the relative degree
of freedom of an exciton, has a form similar to the density-density response of the 1D atom chain at q and is
related to the Peierls distortion36 .
When frequency kept finite,

κ1 θf
log((θf2 − θc2 )2 + (αΩ)2 ) − 22


Ihet,i =
κ1 − κ2
κ1 θc′
log((θf − θc )2 + (αΩ)2 ) − log(θc2 + (αΩ)2 ) − log((θc + θf )2 + (αΩ)2 ) + log(θc2 + (αΩ)2 ) .


− (B19)
κ1 − κ2

b. Finite temperature correction: inner FS contribution Inner part of the correction due to the finiteness of
the temperature can be written as

pi
n(me (p2 − p2i )) − 1
Z
1
∆Ii (q, Ω, T ) = dppdθ
me 0 2p2 + q 2 + 2pq cos(θ) − m∆e + iΩ
me
pΛ
n(me (p2 − p2i ))
Z
1
+ dppdθ , (B20)
me pi 2p2 + q2 + 2pq cos(θ) − m∆e + iΩ
me

where ∆ = me (p2o + p2i ). Changing integration variable to y = βme (p2 − p2i ), I get for the correction:

0
n̄(y) − 1
Z
1
∆Ii (q, Ω, T ) = dydθ 2y y ∆ iΩ
2m2e β −βme p2i βme + +2p2i q2 + 2( βm + p2i )1/2 q cos(θ) − me + me
e
Z βme p2Λ
1 n̄(y)
+ dydθ 2y y ∆ iΩ
, (B21)
2m2e β 0 βme + 2p2i + q2 + 2( βm + p2i )1/2 q cos(θ) − me + me
e

where n̄(y) = (1 + ey )−1 which can be rewritten as

Z βme p2i
1 n̄(y)
∆Ii (q, Ω, T ) = − dydθ 2y 2 + q 2 + 2(− y + p2 )1/2 q cos(θ) − ∆ + iΩ
2m2e β 0 − βm + 2p i βme i me me
e
Z βme p2Λ
1 n̄(y)
+ dydθ 2y y ∆ iΩ
, (B22)
2m2e β 0 2 2
βm + 2pi + q + 2( βm + pi )
2 1/2 q cos(θ) −
me + me
e e

At a fixed y, the denominator, as a function of θ, has two regimes: when the integral has a root at some θ, and the
regime of analyticity. Let us define θi to be an angle where the denominator vanishes at y = 0:

p2o − p2i − q 2
cos(θi ) = . (B23)
2pi q

Expanding in the vicinity of θ = θc,i , I can rewrite the integrals in the form:

Z βme p2i
1 n̄(y)
∆Ii (q, Ω, T ) = − 2 dydθ 2y q
2me β 0 − βm e
(1 + 2pi cos(θi )) − 2(1 − 2βmye p2 )pi q sin(θi )(θ − θi ) + iΩ
me
i
Z βme p2Λ
1 n̄(y)
+ dydθ 2y q , (B24)
2m2e β 0 βme (1 + 2pi cos(θi )) − 2(1 + 2βmye p2 )pi q sin(θi )(θ − θi ) + iΩ
me
i
 16

which after integration gives

Z Λ
1 dyn̄(y)
∆Ii (q, Ω, T ) = y
2m2e β 0 2(1 − 2βme p2i
)pi q sin(θi )
 2  2 !  2  2 ! !
2y q Ω 2y q y Ω
log (1 + cos(θi )) + −log (1 + cos(θi )) − 2(1 − )pi q sin(θi )θi +
βme 2pi me βme 2pi 2βme p2i me
Z Λ
1 n̄(y)
− dy
2m2e β 0 2(1 + 2βmye p2 )pi q sin(θi )
i
 2  2 !  2  2 ! !
2y q Ω 2y q y Ω
log (1 + cos(θi )) + −log (1 + cos(θi )) + 2(1 + )pi q sin(θi )θi +
βme 2pi me βme 2pi 2βme p2i me
Z Z
πi dyn(y) πi dyn(y)
+ − , (B25)
2m2e βpi q sin(θi ) 1 − 2βmye p2 2m2e βpi q sin(θi ) 1 + 2βmye p2
i i

The imaginary part ∝ T 2 /TF,i

2
here will not contribute to the action, and therefore we omit it in what follows. Clearly,
there are 2 characteristic temperatures present here. One corresponds to the Fermi scale: TF,i = me p2i , while the
other corresponds to mixing of electron-hole excitations:
2
me pi qθc,i me po δq
TL,i = q q ≈ . (B26)
1+ 2pi cos(θi ) + 2pi sin(θc,i )θc,i 1 + 2pq i

a) For low temperatures, T ≪ TL,i , we neglect terms O((T /TF,i )2 ),

Z Λ  2  2 !
1 dyn̄(y) y q q 2 Ω
∆Ii (q, Ω, T ) = − 2 2 log − (1 + cos(θi ) + θ )+1 +
2 me β 0 pi q sin(θi ) βpi qθi2 2pi 2pi c,i pi qθi2
Z Λ  2  2 !
1 n̄(y) y q q 2 Ω
+ 2 2 dy log (1 + cos(θi ) + θ )+1 +
2 me β 0 pi q sin(θi ) βpi qθi2 2pi 2pi c,i pi qθi2
Z Z
πi dyn(y) πi dyn(y)
+ − , (B27)
2m2e βpi q sin(θi ) 1 − 2βmye p2 2m2e βpi q sin(θi ) 1 + 2βmye p2
i i

and expand the logarithms in powers of yT /TL,i :

!
π2 T2 π 2 θi T2
 
1
∆Ii (q, Ω, T ) = 2
≈ q 2 . (B28)
12me pi q sin(θi ) TL,i 12me (1 + 2pi ) TL,i

b) At higher temperatures, TL,i < T ≪ TF,i the integrand is peaked at a value of y where the argument of the
logarithm vanishes, which is

βpi qθi2 TL,i

y1,< = q q 2 = , (B29)
1+ 2pi cos(θi ) + 2pi θc,i T
βpi qθi2 TL,i
y2,< = − q q 2 =− . (B30)
1+ 2pi cos(θi ) + 2pi θc,i T

for the corresponding terms. We then write the integral in the form:

Z Λ  2  2 ! !
1 dyn̄(y) TL,i ΩTκ
∆Ii (q, Ω, T ) = − 2 2 log y− +
2 me β 0 pi q sin(θi ) T 2T me
Z Λ  2  2 ! !
1 n̄(y) TL,i ΩTκ
+ 2 2 dy log y+ + , (B31)
2 me β 0 pi q sin(θi ) T 2T me
 17

where I additionally introduced Tκ = me /(1 + 2pq i cos(θi )). We rescale the integration variable y → yTL,i /T and
extend the range to ∞, after the division of the integration range we get:

Z Λ  2 !  2 ! !
TL,i dyn̄(y) 2 ΩTκ 2 ΩTκ
∆Ii (q, Ω, T ) = − 2 2 log (y − 1) + − log (y + 1) +
2 me 0 pi q sin(θi ) 2T1 me 2T1 me
Z 1 Z ∞
TL,i y TL,i 1
= 2 + 2 (B32)
me pi q sin(θc ) 0 1 + eyTL,i /T me pi q sin(θc ) 1 y(1 + eyTL,i /T )

which after division of the latter integral into 2 over ranges (1, T /TL,i ) and (T /TL,i , ∞):
  
TL,i TL,i 1 TL,i
∆Ii (q, Ω, T ) ≈ (1 + Γ0 (1)) + log(T /TL,i ) − 1− . (B33)
2m2e pi q sin(θc,i ) 2m2e pi q sin(θc,i ) 2 T

clearly, the linear behavior here may only arise as an approximation to log(T /TL,i ): for example while doing an
expansion close to 1.

c. Finite temperature correction: outer FS contribution The expression for the outer contribution is

d2 p n(−me (p2 − p2o )) d2 p n(−me (p2 − p2o ))

Z Z
1 1
Io (q, Ω) = iΩ
= iΩ ∆
.
me (2π)2 m + 2p2 + q 2 − p2i − p2o + 2pq cos(θ) me (2π)2 m + 2p2 + q 2 + 2pq cos(θ) − me
e e
(B34)
And therefore calculation goes over the same lines for the outer contribution, hence we simply change pi → po in the
expressions from the previous section. Define

βpo qθo2 TL,o

q q 2
≡ (B35)
1+ 2po cos(θc,o ) + 2po θc,o T

βpo qθo2 TL,o

q q 2
≡− (B36)
1+ 2po cos(θc,o ) + 2po θc,o T

a) T ≪ TL,o :
!
π2 T2 π 2 θo T2
 
1
∆Io (q, Ω, T ) = 2
≈ q 2 . (B37)
12me po q sin(θo ) TL,o 12me (1 + 2po ) TL,o

b) TL,o < T < TF,o :

  
TL,o TL,o 1 TL,o
∆Io (q, Ω, T ) ≈ (1 + Γ0 (1)) + log(T /TL,o ) − 1− . (B38)
2m2e po q sin(θc,o ) 2m2e po q sin(θc,o ) 2 T

3. Critical temperature

On the mean-field level, the transition temperature given by the solution of the equation

Π0 + ∆Π(Tc ) + V −1 = 0. (B39)

The first characteristic temperature, TL,i ≈ me po δq/(1 + q/2pi ) is of order 10−5 m ≈ 10−1 K, which, judging by
the appearance of the peak in resistance (Fig S610 ) is of order of critical temperature for PIP phase. For critical
temperatures below TL,i/o , we get:
v
u 12me
u π2 (−Π0 − V −1 )
Tc,low =u 2
t θi /TL,i 2
θo /TL,o
. (B40)
1+ q + 1+ q
2pi 2po
 18

FIG. A.4. Graphic representation of the 4-th order coefficient.

which in the range of interest gives 10−4 −10−5 m. For higher temperatures, the formula instead resembles MacMillan:
αi αo
−1
e−Π0 −Π̃−V
α +αo α +αo
Tc = TL,ii TL,oi (B41)

where
TL,i
αi = , (B42)
2m2e pi q sin(θc,i )

TL,o
αo = 2
, (B43)
2me po q sin(θc,o )

and
TL,o TL,i
Π̃ = (1 + Γ0 (1)) + (1 + Γ0 (1)) . (B44)
2m2e po q sin(θc,o ) 2m2e pi q sin(θc,i )

Due to the fact that we can separate two components: electronic-coherent and bosonic condensates, for CDW at
q = qc + δq, we may talk about two temperature-induced phase transitions on the mean-field level. Indeed, for the
hetero-part the critical temperature is the same as (B40) except with a change Π0 → Πhet
v
u 12me
u π2 (−Πhet − V −1 )
Tc,het,low =u 2
t θi /TL,i 2
θo /TL,o
, (B45)
1+ q + 1+ q
2pi 2po

while for the homo-part the temperature dependence correction to the response function is (2/3)(1 −
(q/pi/o )2 )−3/2 π 2 (T /TF,i/o )2 (1/me ).
r
12me
Tc,hom,low = Thom (−Πhom − V −1 ). (B46)
π2
2
where Thom = (1 − (q/pi )2 )3/2 TF,i
2
+ (1 − (q/po )2 )3/2 TF,o
2
. Clearly, Tc,hom,low ≫ Tc,het,low .

Appendix C: 4-th order terms

Coefficients of the Landau-Ginsburg theory of the 4-th order are, generically, divergent. They are given by the
diagrams on Fig.A.4 and can be written as:
Z
U (gi , gi +gj , gi +gj +gk ) = d2 pG(p, iωn )G(p+gi , iωn +iΩi )G(p+gi +gj , iωn +iΩi +iΩj )G(p+gi +gj +gk , iω+iΩi +iΩj +iΩk )

Using an identity
 
1 1 1 1 1
= − , (C1)
iωn − ϵp i(ωn + Ω) − ϵp′ iΩ − (ϵp′ − ϵp ) iωn − ϵp i(ωn + Ω) − ϵp′
 19

we can rewrite the expression above as

 
1 1 1
−
iΩi − (ϵp+gi − ϵp ) iωn − ϵp i(ωn + Ω) − ϵp+gi
 
1 1 1
− , (C2)
iΩk − (ϵp+gi,i,j − ϵp+gi,j ) i(ωn + Ωi,j ) − ϵp+gi,j i(ωn + Ωi,j,k ) − ϵp+gi,j,k

where I denoted gi,j = gi + gj , gi,j,k = gi + gj + gk and similarly for bosonic frequencies. The latter consists of 4
terms:

1 1 1 1 1 1
−
iΩi − (ϵp+gi − ϵp ) iΩk − (ϵp+gi,i,j − ϵp+gi,j ) iωn − ϵp i(ωn + Ωi,j ) − ϵp+gi,j iωn − ϵp i(ωn + Ωi,j,k ) − ϵp+gi,j,k
!
1 1 1 1
− + , (C3)
i(ωn + Ωi ) − ϵp+gi i(ωn + Ωi,j ) − ϵp+gi,j i(ωn + Ωi ) − ϵp+gi i(ωn + Ωi,j,k ) − ϵp+gi,j,k

which can be finally reexpressed as a sum over 4 single-fermion Green’s functions with multipliers

1 1
iΩi − (ϵp+gi − ϵp ) iΩk − (ϵp+gi,j,k − ϵp+gi,j )
 
1 1 1
−
iΩi,j − (ϵp+gi,j − ϵp ) iωn − ϵp i(ωn + Ωi,j ) − ϵp+gi,j
 
1 1 1
− −
iΩi,j,k − (ϵp+gi,j,k − ϵp ) iωn − ϵp i(ωn + Ωi,j,k ) − ϵp+gi,j,k
 
1 1 1
− −
iΩj − (ϵp+gi,j − ϵp+gi ) i(ωn + Ωi ) − ϵp+gi i(ωn + Ωi,j ) − ϵp+gi,j
 !
1 1 1
+ − , (C4)
iΩj,k − (ϵp+gi,j,k − ϵp+gi ) i(ωn + Ωi ) − ϵp+gi i(ωn + Ωi,j,k ) − ϵp+gi,j,k

or, explicitly,

1 1 1  1 1 
−
iΩi − (ϵp+gi − ϵp ) iΩk − (ϵp+gi,j,k − ϵp+gi,j ) iωn − ϵp iΩi,j − (ϵp+gi,j − ϵp ) iΩi,j,k − (ϵp+gi,j,k − ϵp )
 
1 1 1
+ −
i(ωn + Ωi,j ) − ϵp+gi,j iΩj − (ϵp+gi,j − ϵp+gi ) iΩi,j − (ϵp+gi,j − ϵp )
 
1 1 1
+ −
i(ωn + Ωi,j,k ) − ϵp+gi,j,k iΩi,j,k − (ϵp+gi,j,k − ϵp ) iΩj,k − (ϵp+gi,j,k − ϵp+gi )
 !
1 1 1
+ − (C5)
i(ωn + Ωi ) − ϵp+gi iΩj,k − (ϵp+gi,j,k − ϵp+gi ) iΩj − (ϵp+gi,j − ϵp+gi )

Each term here corresponds to an excitation, that started at one of the p + gn points and hopping around over. Minus
would correspond to an opposite direction of hop on the lattice in k-space. After shifting in k-space by one of the
g-vectors and performing Matsubara summation, we have 4 integrals in k-space:

U1 = U (gi,j , gk , −gi,j,k , gi , iΩi,j , iΩk,−i,−j , iΩ−2k,−i,−j , iΩ2k,2i,j )

Z
n(ϵp ) 1  1 1 
= −
iΩi − (ϵp+gi − ϵp ) iΩk − (ϵp+gi,j,k − ϵp+gi,j ) iΩi,j − (ϵp+gi,j − ϵp ) iΩi,j,k − (ϵp+gi,j,k − ϵp )
iΩk − (ϵp+gi,j,k − ϵp+gi,j )
Z
n(ϵp ) 1
=−
iΩi − (ϵp+gi − ϵp ) iΩk − (ϵp+gi,j,k − ϵp+gi,j ) (iΩi,j − (ϵp+gi,j − ϵp ))(−iΩi,j,k − (ϵp − ϵp+gi,j,k ))
= V4 (gi , gij , gi,j,k |iΩi , iΩi,j , iΩi,j,k ), (C6)
 20

in the first line, the first term apparently corresponds to hoppings p → p + gi → p → p + gi + gj → p + gi + gj + gk → p.

It seems that for this process to have non-zero contribution one would need to demand that both gi + gj and
gi + gj + gk belong to the first shell of g-vectors. Note also that the combination gi + gj + gk should not sum to
zero, since that would require to have x(0) as one of intervalley fields. Also note that the third argument in V4 is
redundant and the reciprocal momentum gi and corresponding frequency are uncompensated.

Finally, the second integral is

Z  
n(ϵp ) 1 1 1
U2 (gi , gj , gk ) = −
iΩi − (ϵp − ϵp−gi ) iΩk − (ϵp+gj,k − ϵp+gj ) iΩj,k − (ϵp+gj,k − ϵp ) iΩj − (ϵp+gj − ϵp )
= V4 (−gi , gj,k , gj , −iΩi , iΩj,k , iΩj ), (C7)
So that |gj,k | ≤ g to make physical both the first and the second terms

The second terms corresponds to p → p + gi → p → p + gi + gj + gk → p + gi + gj → p, which implies

that gj + gk and gi + gj also should belong to the first shell for it to be nonzero. Alternatively, we can view it as
a 5-fermion loop with an additional vertex that goes as iΩk − (ϵp+gi,j,k − ϵp+gi,j ) →g→0 iΩk − v̄p · ḡk . To preserve
the change of the valley property, such boson should correspond do either σz or σ0 components. Also note that we
omitted frequencies in notation for V4 integrals and the third component, here it is −gi that is uncompensated, thus
cancelling it from the 1st term.

For the third integral,

Z  
n(ϵp ) 1 1 1
U3 (gi , gj , gk ) = −
iΩi − (ϵp−gj − ϵp−gi −gj ) iΩk − (ϵp+gk − ϵp ) iΩj − (ϵp − ϵp−gj ) iΩi,j − (ϵp − ϵp−gi,j )
iΩi − (ϵp−gj − ϵp−gij )
Z
n(ϵp ) 1 1
=
iΩi − (ϵp−gj − ϵp−gi,j ) iΩk − (ϵp+gk − ϵp ) iΩj − (ϵp − ϵp−gj ) iΩi,j − (ϵp − ϵp−gi,j )
= V4 (gk , −gj , −gi,j |iΩk , −iΩj , −iΩij ) (C8)
so that for the first it is neccessary that |gi + gj | ≤ g and same for the second, and gk is uncompensated.

The fourth integral, in turn:

Z  
n(ϵp ) 1 1 1
U4 (gi , gj , gk ) = −
iΩi − (ϵp−gj,k − ϵp−gi,j,k ) iΩk − (ϵp − ϵp−gk ) iΩi,j,k − (ϵp − ϵp−gi,j,k ) iΩj,k − (ϵp − ϵp−gj,k )
= V4 (−gk , −gi,j,k , −gj,k | − iΩk , −iΩi,j,k , −iΩjk ), (C9)
1st term of which would require that |gi,j,k | ≤ g, while the second necessitates |gj,k | ≤ g. So that it also compensates
extra gk coming from the third term. It is clear that up to the restrictions and bosonic frequency change all 4 terms
are equivalent. Let us now evaluate all V4 terms.

1. Estimation of divergences in V4 -term

First, we analyze divergencies of V4 for different combinations of arguments:

Z
n(ϵp ) 1
V4 (gi , gi,j , gi,j,k ) = . (C10)
iΩi − (ϵp+gi − ϵp ) (iΩi,j − (ϵp+gi,j − ϵp ))(iΩi,j,k − (ϵp+gi,j,k − ϵp ))
For low temperatures, the integral is taken over two disjoint regions:
pi
d2 p
Z
1
V4 (gi , gi,j , gi,j,k ) =
0 iΩi − (ϵp+gi − ϵp ) (iΩi,j − (ϵp+gi,j − ϵp ))(iΩi,j,k − (ϵp+gi,j,k − ϵp ))
Z pΛ
d2 p 1
+ , (C11)
po iΩ i − (ϵ p+g i − ϵ p ) (iΩ i,j − (ϵ p+g i,j − ϵ p ))(iΩ i,j,k − (ϵp+gi,j,k − ϵp ))
 21

In such an integral, there can be ∝ Ω2 area where the integrand diverges as Ω−3 : for that one would need to require:

ϵp = ϵp+gi,j , (C12)
ϵp = ϵp+gi,j,k , (C13)
ϵp = ϵp+gi , (C14)

which is not possible to satisfy without one of g-s being zero, and one possible choice is |gij |, while the other, for C6
symmetric case, is gi,j,k = 0. First consider |gij | = 0 alternative.
Without loss of generality, pick x axis parallel to gi . Based on the roots of the denominator, there could be 3
separate situations:
1. 0 = θk , so that gk is parallel to gi when V4 is the largest (V4,∥ ),

2. 0 ̸= θk , but there is a region in momentum space of width δp ≈ Ωα , when yi,l (p) = yk,m (p) (V4,c )37 ,
3. 0 ̸= θk and the two roots of denominators are always separable (V4,sep ).
Simple power counting shows that

V4,∥ ∝ Ω−3 δxδp ≈ Ω−2 , (C15)

since areas for the most divergent part grows δxδp ∝ Ω

V4,c ∝ Ω−3 δxδp ≈ Ω−2+α , (C16)

and

V4,sep ∝ Ω−2 δx ≈ Ω−1 . (C17)

However, because integrals with uncompensated frequencies come in pairs, there could be mutual cancellation making
the divergence slower. In the equilibrium, by minimum energy argument V4 we should pick the combination of
frequencies that gives the coefficient the smallest possible value. Since the leading term comes from the choice
gi = −gj = gk , we choose frequencies to minimize this term.

Ω2
V4 (gi , 0, gi ) ∝ − v(gi , 0, gi ), (C18)
iΩi Ωi,j Ωi,j,k

A similar term from the 2nd integral will read:

Ω2
V4 (−gi , 0, −gi ) ∝ v(−gi , 0, −gi ), (C19)
iΩi Ωj Ωj,k

for Ω → 0. If reversal symmetry is not broken, the v(−gi , 0, −gi ) = v(gi , 0, gi ), and so by choice Ωij Ωi,j,k = Ωj Ωj,k
we can cancel the leading divergence. It corresponds to

Ωj = −Ωi + δΩ′ (C20)

Ωk = Ωi − δΩ′ + δΩ, (C21)

where δΩ, δΩ′ ≪ Ω. For the third and the fourth terms, we have:

Ω2
V4 (gi , gi , 0) ∝ v(gi , gi , 0), (C22)
iΩk Ωj Ωi,j

Ω2 v(−gi , −g, 0)
V4 (−gi , 0, −gi ) ∝ − , (C23)
iΩk Ωijk Ωjk

For which choice (C20) also works. Let us find, however, combinations that belongs to the second and the third
groups. Equation ϵ(p) = ϵ has two roots as a function of p (absolute value of momentum). So that there will be two
combinations of roots that define the second group.
 22

The first is given by (p, gi ) = (p, gk ) and hence, for the choice gi = gx̂

cos(θp ) = cos(θp ) cos(θk ) + sin(θp ) sin(θk ), (C24)

which has, as solutions

sin(θk )2
cos(θp )2 = ≤1 (C25)
2(1 − cos(θk ))
and the requirment that cos(θp ) has the same sign as cos(θk − θp ). Available values are θk = πn/3 (with n = 0
excluded) satisfy the inequality. However, it is clear the gk should be in the the same half plane as gi , and hence only
π/3 and 5π/3 will satisfy the equality.

Another solution corresponds to the second root of the dispersion relation (p + gi )2 = (p + gk )2 + ∆2p , where
p
∆2p = 2 (m/2λ)2 + (µ + ϵ)/λ. So that

cos(θp ) = cos(θp − θk ) + ∆2p /(2pg), (C26)

hence for θk = π can be easily satisfied on the shell p = po

cos(θp ) = ∆2p (4po g) = ⟨pF ⟩/(2po ) (C27)

similarly, for θk = 2π/3 we get

√
3 3
cos(θp ) − sin(θp ) = ⟨pF ⟩/(2po ), (C28)
2 2
the l.h.s. is bounded by 9/4 + 3/4 = 3, which is straightforward to satisfy, since r.h.s is less than 1. It is also clear
that once we can satisfy for 2π/3 it is also satisfied for −2π/3.

Thus, finally, all 5 vectors belongs to the second group, with 2 corresponding to homo-processes, and 3 – to
hetero.

Finally, it is clear that is impossible to have Ω2 divergence for gijk = 0, since simultaneous satisfaction of
(C12) implies that ϵp−gk = ϵp+gi and since gk ̸= gi , impossible to satisfy for all p, rather we have divergence of the
second class.

2. Calculation of V4,∥

Once we established the 4-th order diagram is maximized at gi = gk = −gj , we are now to evaluate it. For the
choice (C20) we write the sum of the 4 in the form:

V4,Σ =
(ξp+gi − ξp )2iδΩ′ + (δΩ′ − Ω)(Ω + δΩ − δΩ′ ) + Ω(Ω + δΩ)
Z  
1
− n(ξp )
iδΩ′ (iΩ − (ξp+gi − ξp ))(i(Ω + δΩ − δΩ′ ) − (ξp+gi − ξp ))(i(Ω + δΩ) − (ξp+gi − ξp ))(i(−Ω + δΩ′ ) + (ξp+gi − ξp ))
(ξp+gi − ξp )2iδΩ − (δΩ + Ω)(Ω + δΩ − δΩ′ ) + Ω(Ω − δΩ′ )
Z  
1
− n(ξp )
iδΩ (iΩ + (ξp+gi − ξp ))(i(Ω + δΩ − δΩ′ ) + (ξp+gi − ξp ))(i(Ω + δΩ) + (ξp+gi − ξp ))(−i(−Ω + δΩ′ ) − (ξp+gi − ξp ))
(C29)

Once O(δΩ2 ) terms neglected, the 4-th order coefficient can be represented as

∂2
 3 
n(ϵp )(ϵp+g − ϵp + iΩ) n(ϵp )(ϵp+g − ϵp − iΩ)
Z Z
4Ω ∂
V4,Σ = 2 + 2 = −2 Re (I) + Im I , (C30)
(iΩ − (ϵp+g − ϵp ))4 (iΩ + (ϵp+g − ϵp ))4 ∂Ω2 3 i∂Ω3
where I is the part of the particle-hole response
Z
n(ξp )
I= (C31)
iΩ + ϵp+g − ϵp
 23

FIG. A.5. Order parameter dependence on the Fermi energy for ΩIR = TL (solid line) and ΩIR = Tc for interaction constant
V = 0.45. Order parameter behavior is nearly the same for both choices.

calculated in Section B. For the homo-part of the response :

1/2 1/2 !
iΩ22 22 Ω2 iΩ22 2 2 Ω2
 
π π
Iii (q < 2pi ) = − − 2 2 −i − + 2 2 + fii ∝ − (1 − i)(a′1 − a′2 ) + fii , (C32)
me q me me q me me q me q

where the last part is weakly dependent on frequency Ω and a′ = 2eiπ/4 (Ω/me + i(Ω/me q)2 )1/2 = a′1 + ia′2 , so that
in the leading order V4,hom = −21/2 π(me /Ω)1/2 (me q)−3 . For the hetero-part, the response is given by (B19):

1 κ1 θf 1 κ1 θc′
log((θf2 − θc2 )2 + (αΩ)2 ) − log((θf − θc )2 + (αΩ)2 ) + fio ,



Iio (q < 2pi ) = (C33)
2me κ1 − κ2 2me κ1 − κ2
q
where fio is part that depends on Ω weakly, and θc′ = θc2 − rivΩ i qc
. Then in the leading order contribution to V4 is
logarithmically divergent:

1 pi (v/ri qc )2 1 po (v/ro qc )2
V4,het = ivΩ
log(αi Ω) + log(αo Ω) (C34)
2
2me pi + po (θio − r q ) 3/2 2me pi + po (θoi − rivΩ
2
q )
3/2
i c o c

Choosing cutoff to be Ω ∝ Tc we should be able to neglect the contribution of the hetero-part whenever
θc3 me qc−2 ≫ Tc log(Tc )2 , which gives a characteristic temperature of order 0.1me thus by far exceeding actual
Tc ∝ me 10−4 in regime of interest.

Notice again that homo-contribution to the 4-th order scattering amplitude dominates over the hetero-, thus
contrasting this case to superconductivity at finite q 18 . As expected for bosonic theories38 4-th order coefficient
diverges at Ω → 0. We try now to choose a sensible frequency cutoff ΩIR . In the symmetry broken phase, this
problem is fixed through opening of the gap ∆ ∝ xq in the excitation spectrum. In a system without a global gap,
−1/4
ΩIR ∝ Tc . Because xq ∝ ΩIR , physical results are not sensitive to the error in the choice of ΩIR (see Fig. A.5). We
q r   −1
121/2 θc,i θc,o
then pick ΩIR = TL = π T 2 + T 2 for its convenience.
L,i L,o

Appendix D: Conductance up to O(x2q )

1. Vertex correction

In this variant, I follow discussion in Mahan39 . It leads to the absence of Ω0 term in the imaginary part. Vertex
correction to the response has a form:
Z
c
χv (q, iΩ) = xq (Ω1 )x−q (−Ω1 ) vp vp+q Gp (iωn )Gp (i(ωn + Ω))Gp+q (i(ωn + Ω + Ω1 ))Gp+q (i(ωn + Ω1 )) (D1)
 24

When the excitonic fields xq are taken to be classical (Ω1 = 0), the summation over the frequencies can be made into
the integration over the three contours. Then
Z
dz
χcv (q, iΩ) = xq x−q vp vp+q n(z)Gp (z)Gp+q (z)Gp (z + iΩ)Gp+q (z + iΩ)
Cb1 2πi
Z
dz
+ xq x−q vp vp+q n(z)Gp (z)Gp+q (z)Gp (z + iΩ)Gp+q (z + iΩ) (D2)
Cb2 2πi
With contour Cb1 going from −∞ + iδ to ∞ + iδ, and on the lower part of the cut from −∞ − iδ to ∞ − iδ, while
Cb2 going from −∞ − iΩ + iδ to ∞ − iΩ + iδ, and on the lower part of the cut from −∞ − iΩ − iδ to ∞ − iΩ − iδ.
After changing a variable in the second part, I get
Z ∞
c dϵ
χv (q, iΩ) = xq x−q vp vp+q n(z)Gp (z + iΩ)Gp+q (z + iΩ)
−∞ 2πi
(Gp (z + iδ)Gp+q (z + iδ) − Gp (z − iδ)Gp+q (z − iδ))
Z ∞
dϵ
+ xq x−q vp vp+q n(z)Gp (z − iΩ)Gp+q (z − iΩ) (Gp (z + iδ)Gp+q (z + iδ) − Gp (z − iδ)Gp+q (z − iδ)) , (D3)
−∞ 2πi
which becomes
Z ∞
dϵ
χcv (q, iΩ) = xq x−q vp vp+q n(z)(Gp (z+iΩ)Gp+q (z+iΩ)+Gp (z−iΩ)Gp+q (z−iΩ))(Gp (z+iδ)Gp+q (z+iδ)−Gp (z−iδ)Gp+q (
−∞ 2πi
(2)
An expression (Gp (z + iδ)Gp+q (z + iδ) − Gp (z − iδ)Gp+q (z − iδ)) is analogous to a spectral function Ap,p+q (z) =
Im(Gp (z + iδ)Gp+q (z + iδ)) = (Ap (z)rp+q (z) + Ap+q (z)rp (z)), where Gp (z + iδ) = rp (z) + iAp (z). We now make an
analytical continutation iΩ → Ω + iδ and since it is an imaginary part that contributes to the conductivity40 ,
Z ∞
dϵ (2) (2) (2)
πv (q, Ω) = 2xq x−q vp vp+q n(ϵ)(Ap,p+q (ϵ + Ω) − Ap,p+q (ϵ − Ω))Ap,p+q (ϵ)
−∞ 2π
Z ∞
dϵ (2) (2)
= 2xq x−q vp vp+q (n(ϵ) − n(ϵ + Ω))Ap,p+q (ϵ + Ω)Ap,p+q (ϵ) (D4)
−∞ 2π
Stationary response Ω → 0 is then equal to
Z
Ω (2) (2)
πv (q, Ω) = 2xq x−q d2 pvp vp+q Ap,p+q (Ω)Ap,p+q (0) =
2π
Z
Ω
2xq x−q lim d2 pvp vp+q (Ap (0)rp+q (0) + Ap+q (0)rp (0))(Ap (Ω)rp+q (Ω) + Ap+q (Ω)rp (Ω))
2π Ω→0
Z
Ω
= 2xq x−q d2 pvp vp+q (2Ap (0)rp+q (0)Ap+q (0)rp (0) + A2p (0)rp+q
2
+ A2p+q (0)rp2 ) (D5)
2π
which should give a positive or negative contribution to the response dependent on whether the homo- (same kind
(2)
of Fermi surface) or hetero-part is dominant. Unless Ap,p+q (Ω) has a divergent as Ω3 derivative, there is no Ω0 term
in the imaginary part. The real part has a non-zero coefficient of Ω−1 term. Let us simplify the expression for the
vertex correction to the response now. Note that
(ξp + iΣ)(ξp+q + iΣ) − ξp ξp+q + Σ2 (ξp − iΣ)(ξp+q − iΣ) − ξp ξp+q + Σ2
   
(2) Σ(ξp + ξp+q )
Ap,p+q = 2 2 = 2 = − 2
(ξp + Σ2 )(ξp+q + Σ2 ) 2i(ξp2 + Σ2 )(ξp+q + Σ2 ) 2i(ξp2 + Σ2 )(ξp+q + Σ2 )
(D6)
which after substitution in (D5) leads to
(Σ2 − ξp ξp+q )2
Z  
Ω 1
πv (q, Ω) = xq x−q d2 pvp vp+q 2 − 2 (D7)
4π (ξp2 + Σ2 )(ξp+q + Σ2 ) (ξp2 + Σ2 )2 (ξp+q + Σ 2 )2
Note that it gives either positive or negative correction dependent on whether the homo- (same Fermi surface) or the
hetero-part dominates. The vertex correction, alternatively, can be expressed in the form:
Z
Ω
π2,2 (q, iΩ) = |xq |2 d2 pvp vp+q Im(Gp Gp+q )Im(Gp Gp+q ), (D8)
2π
which form suggestive of being positive for vp vp+q > 0.
 25

FIG. A.6. Corrections in the leading order in intervalley pseudo-magnetization xq to the electron part of the conductivity of 2D
system with double-well dispersion. Wavy line within the fermion loop denotes propagator ⟨xq x−q ⟩. On the symmetry broken
side, we take it simply the order parameter squared x2qi for q = qi and vanishingly small frequency. Corrections can be divided
into a correction to the density of states (a) and a correction to the vertex (b).

2. DOS correction

The diagram (a) reads

Z
χcDOS (q, iΩ) = xq x−q vp vp Gp (iωn )Gp+q (i(ωn ))Gp (i(ωn ))Gp (i(ωn + Ω)), (D9)

which is equivalent to the expression with integration over z

Z Z
dz dz
χcDOS (q, iΩ) = xq x−q vp vp Gp (z)Gp+q (z)Gp (z)Gp (z + iΩ) + vp vp Gp (z)Gp+q (z)Gp (z)Gp (z + iΩ)
C+ 2πi C+− 2πi
Z 
dz
+ vp vp Gp (z)Gp+q (z)Gp (z)Gp (z + iΩ) , (D10)
C− 2πi

where the counterclockwise C+/− is the upper (and lower) half-plane bounded by (−∞ + iδ, ∞ + iδ) (and (∞ − iδ −
iΩ, −∞ − iδ − iΩ)), while C+− is the contour over the middle rectangualar:

Z ∞+iδ Z ∞−iδ
dϵ dϵ
χcDOS (q, iΩ) = xq x−q vp vp n(z)Gp (z)Gp+q (z)Gp (z)Gp (z+iΩ)− vp vp n(z)Gp (z)Gp+q (z)Gp (z)Gp (z+iΩ)
−∞+iδ 2πi −∞−iδ 2πi
Z ∞−iΩ Z ∞−iΩ−iδ 
dϵ dϵ
+ vp vp n(z)Gp (z)Gp+q (z)Gp (z)Gp (z + iΩ) − vp vp n(z)Gp (z)Gp+q (z)Gp (z)Gp (z + iΩ) ,
−∞−iΩ 2πi −∞−iΩ−iδ 2πi
(D11)
Shifting the contour in the first terms by ±iδ, in the last terms by iΩ ± δ, we obtain:
Z ∞
c dϵ
χDOS (q, iΩ) = xq x−q vp vp n(z)Gp (z+iΩ)(Gp+q (z+iδ)Gp (z+iδ)Gp (z+iδ)−Gp+q (z−iδ)Gp (z−iδ)Gp (z−iδ))
−∞ 2πi
Z ∞ 
dϵ
+ vp vp n(z)Gp (z − iΩ)Gp (z − iΩ)Gp+q (z − iΩ)(Gp (z + iδ) − Gp (z − iδ)) , (D12)
−∞ 2πi

In the second term do the analytic continuation, and shift integration variable by Ω
Z ∞
c dϵ
χDOS (q, iΩ) = 2xq x−q vp vp n(ϵ)Gp (ϵ + Ω + iδ)Im (Gp+q (z)Gp (z)Gp (z))
−∞ 2π
Z ∞ 
dϵ
+ vp vp n(ϵ + Ω)Gp (z − iδ)Gp (z − iδ)Gp+q (z − iδ)Im (Gp (z + Ω)) , (D13)
−∞ 2π

Taking the imaginary part of the latter,

Z ∞
dϵ
πDOS (q, iΩ) = 2xq x−q vp vp n(ϵ)Im (Gp+q (z)Gp (z)Gp (z)) Im (Gp (z + Ω))
−∞ 2π
Z ∞ 
dϵ
− vp vp n(ϵ + Ω)Im (Gp+q (z)Gp (z)Gp (z)) Im (Gp (z + Ω)) , (D14)
−∞ 2π
 26

Which becomes in the stationary case Ω → 0

Z
Ω
πDOS (q, Ω → 0) = 2xq x−q d2 pvp2 Im (Gp+q (0)Gp (0)Gp (0)) Im (Gp (Ω)) (D15)
2π
whose sign is, generally speaking, undetermined.

3. Explicit expression for π2 (Ω, q)

We also see that the total correction, given by the sum of the DOS- and vertex-corrections, is:

π(q, Ω → 0) = πDOS (q, Ω) + πv (q, Ω) (D16)

From (D15) it follows that:

2x2q Σ2 vp2 (ξp2 − Σ2 + 2ξp+q ξp ) 2x2q ΩΣ2 vp2 (2ξp2 + 2ξp+q ξp − (ξp2 + Σ2 ))
Z Z
2 2
σDOS (q, Ω) = d p 2 2 = d p 2
π (ξp + Σ2 )3 (ξp+q + Σ2 ) π (ξp2 + Σ2 )3 (ξp+q + Σ2 )
2 2 2 Z 2 2 2 Z
2 xq ΩΣ vp ξp (ξp + ξp+q ) 2xq ΩΣ vp2 (ξp2 + Σ2 )
= d2 p 2 2 − d 2
p 2 (D17)
π (ξp + Σ2 )3 (ξp+q + Σ2 ) π (ξp2 + Σ2 )3 (ξp+q + Σ2 )

while from (D8)

x2q Σ2 vp vp+q (ξp + ξp+q )2

Z
σv (q, Ω) = d2 p 2 (D18)
π (ξp2+ Σ2 )2 (ξp+q + Σ2 )2

for α, γ = 1. For the x − x response, vp = ∂ξ/∂px = cos(θ)∂ξ/∂p. Integrating by parts, we can rewrite the first part
of the DOS-part as:

vp (ξp + ξp+q )(2ξp+q vp+q ) vp′ (ξp + ξp+q ) + vp (vp + vp+q )

Z Z  
dpx
Σ2 dpy − 2 + 2 (D19)
(ξp2 + (αΣ)2 )2 (ξp+q + (γΣ)2 )2 (ξp+q + (γΣ)2 )

After the resummation of 2 terms we have:

 ′
Σ2 x2q d2 p vx,p (ξp + ξp+q ) + vp (−vp + vp+q )
Z 
σ(q, Ω → 0) = 2 . (D20)
π (ξp2 + (αΣ)2 )2 (ξp+q + (γΣ)2 )

The latter can be represented as a derivative over parameter α:

 ′
x2q 1 ∂ d2 p vx,p (ξp + ξp+q ) + vp (−vp + vp+q )
Z 
σ(q, Ω → 0) = − 2 . (D21)
π 2α ∂α ξp2 + (αΣ)2 ξp+q + (γΣ)2

4. Estimate

For Σ ≪ me , most of the contribution comes from the vicinity of the Fermi surface. Hence we can divide the
integral into 4 parts: when p, |p + q| ≈ pi , p, |p + q| ≈ po (homo-part) and p ≈ pi , |p + q| ≈ po , p ≈ po , |p + q| ≈ pi
(hetero-part). More specifically, the most of the contribution comes from region:
Σ Σ
− < p2 − p2s < , (D22)
me me

Σ Σ
− < (p + q)2 − p2s′ < . (D23)
me me
Hence the value is acquired in the range

Σ 1 p2′ − p2s − q 2 p2′ − p2s − q 2 Σ 1

− + s < cos(θ) < s + . (D24)
me ps q 2ps q 2ps q me ps q
 27

a) For the homo-part, s′ = s, the area over the angle is

Σ 1 q q Σ 1
− − < cos(θ) < − + . (D25)
me ps q 2ps 2ps me ps q

Hence homo-part of the response contributes mostly at smaller hole concentration, when qc = po − pi < 2ps .
Define angle cos(θs ) = −q/2ps , then when such angle exist,

Σ 1 Σ 1
− < − sin(θs )δθ < . (D26)
me ps q me ps q

And hence we can estimate the homo-part (q > 2pf ) of the response to be

x2q
σhomo ≈ (cot(θs,i ) + cot(θs,o )). (D27)
2πΣ2
For small q (q < 2pf ) case,

x2q
σhomo ≈ (cot(θs,i ) + cot(θs,o )). (D28)
2πQ2ii Σme

b) For the hetero-part instead, the angle region is

Σ 1 p2′ − p2s − (p2o + p2i − 2po pi + 2δqqc ) p2′ − p2s − (p2o + p2i − 2po pi + 2δqqc ) Σ 1
− + s < cos(θ) < s + , (D29)
me ps q 2ps q 2ps q me ps q

and hence for the inner-outer (s′ = o, s = i) processes, we have the angle close to 0

po − pi − δqqc /pi δq δq
cos(θio ) = =1− − (D30)
qc + δq qc pi

for the outer-inner processes (s′ = i, s = o), instead, angle is close to π:

po pi − p2o − δqqc δq δq
cos(θoi ) = = −1 − + (D31)
po (qc + δq) po qc

In both cases then, independently on the doping, for any finite δq, angles θio and π − θio are small:
q
sin(θio ) ≈ 2δq(qc−1 + p−1
i ) (D32)
q
sin(θoi ) ≈ 2δq(qc−1 − p−1
o ) (D33)

Hence the ”value acquisition” area should be larger than the one of the homo-processes apart from a special
point 2pi = qc :

Σ Σ
− < sin(θio )δθ < , (D34)
me pi qc me pi qc
Σ Σ
− < sin(θoi )δθ < . (D35)
me po qc me po qc

The corresponding contributions to the conductivity correction is

x2q
σhet ≈ − ⟨pf ⟩ (cot(θc,i ) + cot(θc,o )) , (D36)
πqΣ2

We then expect that the correction to the conductance goes as τd2 and the processes with scattering between different
−1
Fermi surfaces (hetero-processes) dominate by factor of θc,i ∝ (δq/qc )−1/2 ≈ 10.
 28

5. Accurate calculation

To obtain a better approximation for conductance, let us look once again at the total expression:
 ′
Σ2 x2q d2 p vx,p (ξp + ξp+q ) + vp (−vp + vp+q )
Z 
σxx (q, Ω → 0) = 2 . (D37)
π (ξp2 + (αΣ)2 )2 (ξp+q + (γΣ)2 )

The latter can be represented through the derivative:

′
x2q ∂ d2 p vx,p (ξp + ξp+q ) + vp (−vp + vp+q )
Z  
σxx (q, Ω → 0) = − 2 . (D38)
2πα ∂α ξp2 + (αΣ)2 ξp+q + (γΣ)2

a) Hetero-processes
For particles residing on different Fermi-surfaces, we get, in δp-approximation:
δp δp
x2q ∂ cos(θ)(δp(1 − κio ) − Q) − 2(pi + 2p ) cos(θ)(q + 2(pi + 2pi ) cos(θ))
Z
dδpdθ i
σio =− αΣ 2  2 (D39)
απm2e ∂α 2
δp + ( me ) (δpκ + Q)2 + γΣ
io me

where variable Q ≡ p2i + q 2 − p2o + 2pi q cos(θ). Integration over δp can be performed through the residues, which
are situated at δp3/4 = ±iαΣ/me , δp1/2 = −Q/κio ± iγΣ/(me κio ):

2 2 iΣα q
x2q ∂ dθ −Q − 2pi q − 2 pi cos(θ) + me (1 − κio − pi − 4 cos(θ)) cos(θ)
Z
σio =−  2
αme ∂α αΣ γΣ
(Q + iΣα 2
me κio ) + me
  
2 2 iγΣ Q q
x2q κio ∂ dθ −Q − 2pi q − 2 pi cos(θ) + me κio − κio 1 − κio − − 4 cos(θ) cos(θ)
Z
pi
−  2 . (D40)
αme ∂α γΣ γΣ
(Q − iΣα 2
me κio ) + me

After linearization Q = −2pi q sin(θc,i )(θ −θc,i ) we can change integration variable to Q. We can shift integration
contour above to iγΣ/me in the second term without crossing the poles. In the first term, however, contour shift
(1)
below to −iακio Σ/me leads to crossing the pole at Q1 = −iΣακio /me + iγΣ/me . Doing subsequent variable
changes in both terms we move integration limits to their initial values. Then the integral has the form:

q
x2q ∂
Z
dQ′ −Q − 2pi q − 22 p2i cos(θ) + iΣα
me (1 − κio − pi − 4 cos(θ)) cos(θ)
σio = −  2
2αme ∂α
αΣpi q sin(θc,i ) γΣ
(Q′ )2 + m e
  
x2q κio ∂
Z
dQ′′ −Q − 2pi q − 22 p2i cos(θ) + me κio − κQio
iγΣ
1 − κio − pqi − 4 cos(θ) cos(θ)
−  2 . (D41)
2αme ∂α γΣpi q sin(θc,i ) γΣ
(Q′′ )2 + m e

(1)
where Q1 = −iαΣκio /me + iγΣ/me . Which implies that the correction to the conductivity in the leading
order goes as

x2q π⟨pf ⟩ cot (θc,i )

σio = − + O(x2q /(me Σ)), (D42)
qΣ2
and an analogous contribution from outer-inner processes.
b) Homo-processes

For particles residing on the same Fermi surface,

δp
x2q ∂ (δp(1 + κii ) + Qii ) + 2(pi + 2p ) cos(θ)q
Z
dδpdθ i
σii = − αΣ 2  2 , (D43)
απm2e ∂α 2
δp + ( me ) (δpκ + Q )2 + γΣ
ii ii me
 29

where Qii = 2pi q cos(θ) + q 2 . Integration over residues

iαΣ
δp1,2 = ± , (D44)
me
Qii iγΣ
δp3,4 =− ± . (D45)
κii me
gives:

iαΣ 1 iαΣ
x2q dθ ( me (1 + κii ) + Qii ) + 2(pi + 2pi me ) cos(θ)q
Z
∂
σii = −  2
2αme Σ ∂α α γΣ
( iαΣ 2
me κii + Qii ) + me
1
x2q dθ (δp3 (1 + κii ) + Qii ) + 2(pi + 2pi δp3 ) cos(θ)q
Z
∂
− κii 2 . (D46)
2αme Σ ∂α γ

iγΣκii 2 αΣκii
(Qii − me ) + me

In the regime q > 2pi no pole present near the real axis (up to terms of order Σ/me ), thus making the contribution
∝ x2q (me Σ)−1 . Poles are at:

Q0
(1) iΣ −α(1 − 2p2i ) ± γ
Q± = iαΣ
, (D47)
me 1 + 2m p2 e i

and
 
(2) iΣ Q0 (γ ± α)
Q± = 1− 2 , (D48)
me 2pi 1 − 2miΣe p2 (γ ± α)
i

where Q0 = p2i − p2o + q 2 = −2pi qc + 2qc δq. For

iαΣ 1 iαΣ
x2q dθ ( me (1 + κii ) + Q) + 2(pi + 2pi me ) cos(θ)q
Z
∂
σii = − (1) (1)
2αme Σ ∂α α iαΣ 2
(1 + 2m 2 ) (Q − Q+ )(Q − Q− )
e pi

x2q ∂
Z
dθ (δp3 (1 + κii ) + Qii ) + 2(pi + 2p1 i δp3 ) cos(θ)q
− κii (2) (2)
. (D49)
2αme Σ ∂α γ (1 − i(γ−α)Σ
2me p2
)(1 − i(γ+α)Σ
2me p2
)(Q − Q+ )(Q − Q− )
i i

In the leading in Σ approximation then we get from the pole integration:

πx2q q
σii ≈ cot(θii )(1 + cos(θii )), (D50)
2Σ2 pi

which is smaller than the hetero-contribution since cos(θii ) = q/(2pi ) < 0.5 in the regime of interest (See Fig.
A.10) by a factor of 5-10.

Appendix E: Conductance up to O(xn

q)

In this section, we obtain an expression for conductance up to an infinite order in xq through series summation.

1. 4-th order

In the 4-th order, there will be 3 types of diagrams. An analog of the vertex correction is:
Z
4
π2,4 (q, Ω → 0) = |xq | vp vp+q Gp (iωn )Gp+q (iωn )Gp+q (i(ωn + Ω))Gp (i(ωn + Ω))Gp+q (i(ωn + Ω))Gp (i(ωn + Ω)) (E1)
 30

and shows up when only one x-s is one of the sides. There are 2 such diagrams. Then there is an analog of the
DOS-correction (with all x-lines on one side) that reads:
Z
4
π1,4 (q, Ω → 0) = |xq | vp2 Gp (iωn )Gp (i(ωn + Ω))Gp+q (i(ωn + Ω))Gp (i(ωn + Ω))Gp+q (i(ωn + Ω))Gp (i(ωn + Ω)) (E2)

and an additional diagram

Z
4
π3,4 (q, Ω → 0) = |xq | vp2 Gp (iωn )Gp+q (iωn )Gp+q (i(ωn + Ω))Gp (i(ωn + Ω))Gp+q (i(ωn + Ω))Gp (i(ωn + Ω)), (E3)

that has a two legs on each side, both come with a factor of 3. An explicit expression for the vertex correction is:
XZ
4
π2,4 (q, Ω → 0) = |xq | vp vp+q Gp (iωn )Gp+q (iωn )Gp+q (iωn + iΩ)Gp (iωn + iΩ)Gp+q (iωn + iΩ)Gp (iωn + iΩ). (E4)

After doing the same manipulations as for the 2-nd order correction, we get for the vertex correction:
XZ
4 Ω
π2,4 (q, Ω → 0) = |xq | vp vp+q Im((Gp Gp+q )2 )Im(Gp Gp+q ). (E5)
2π
The DOS-correction analogously can be written as:
|xq |4 Ω
Z Z
π1,4 (q, Ω → 0) = |xq |4 vp2 Gp Gp Gp+q Gp Gp+q Gp = vp2 Im(Gp,> )Im(Gp,> Gp+q,> Gp,> Gp+q,> Gp,> ) (E6)
2π
and, similarly,
|xq |4 Ω
Z
π3,4 (q, Ω → 0) = vp2 Im(Gp,> Gp+q,> Gp,> )Im(Gp,> Gp+q,> Gp,> ), (E7)
2π
where for shortness Gp,> without the frequency argument denotes Gp,> (iωn → 0).

2. 6-th order

In the 6-th order, there will be 4 separate diagrams. Among them, there will be 2 vertex-like corrections:
Z
6
π2,6 (q, Ω → 0) = |xq | vp vp+q Gp (iωn )Gp+q (iωn )Gp+q (iωn + iΩ)Gp (iωn + iΩ)

|xq |6 Ω
Z
Gp+q (iωn + iΩ)Gp (iωn + iΩ)Gp+q (iωn + iΩ)Gp (iωn + iΩ) = vp vp+q Im(Gp Gp+q )Im((Gp+q Gp )3 ) (E8)
2π
and
Z
π4,6 (q, Ω → 0) = |xq |6 vp vp+q Gp (iωn )Gp+q (iωn )Gp (iωn )Gp+q (iωn )

Gp+q (iωn + iΩ)Gp (iωn + iΩ)Gp+q (iωn + iΩ)Gp (iωn + iΩ)

|xq |6 Ω
Z
= vp vp+q Im((Gp Gp+q )2 )Im((Gp+q Gp )2 )), (E9)
2π
while DOS correction is:
Z
π1,6 (q, Ω → 0) = |xq |6 vp2 Gp (iωn )Gp (iωn + iΩ)Gp+q (iωn + iΩ)Gp (iωn + iΩ)

Gp+q (iωn + iΩ)Gp (iωn + iΩ)Gp+q (iωn + iΩ)Gp (iωn + iΩ)

|xq |6 Ω
Z
= vp2 Im(Gp,> )Im((Gp,> Gp+q,> )3 Gp,> )), (E10)
2π
and
Z
π3,6 (q, Ω → 0) = |xq |6 vp2 Gp (iωn )Gp+q (iωn )Gp (iωn )Gp (iωn + iΩ)

Gp+q (iωn + iΩ)Gp (iωn + iΩ)Gp+q (iωn + iΩ)Gp (iωn + iΩ)

|xq |6 Ω
Z
= vp2 Im(Gp,> Gp+q,> Gp,> )Im(Gp,> Gp+q,> Gp,> Gp+q,> Gp,> )). (E11)
2π
 31

3. Combinatorics

All terms originate from the expansion of the effective action of the form
1
Scoup = Tr(log(1 + G−1 (A + xq ))) = ... + Tr((G−1 (A + xq ))2k ) + ... (E12)
2k
Beside the linear coefficient coming from the Taylor expansion of the logarithm, each type of the correction (e.g. (E8))
in a given order comes with combinatorial coefficient.Term that corresponds to the order 2k has xq in power of 2n
and A in power of 2. In the second order, resulting terms are

Tr((G−1 (A + xq ))2 )e.m = Tr((Axq )2 ) + Tr((xq A)2 ) + Tr(Ax2q A) + Tr(A2 x2q ) + Tr(x2q A2 ) + Tr(xq A2 xq )
= 2Tr((Axq )2 ) + 4Tr(A2 x2q ), (E13)

where for convenience G−1 are implicit on the r.h.s.. Presence of the trace merges terms distinguished by the cyclic
permutations. Another words, we can define an equivalence relation between 2 permutations ai ∼1 aj through a
cyclic permutation. Moreover, permutations inside the elements of the same kind (x or A) are equivalent, which
defines another equivalence relation. The quantity invariant under the cyclic permutation is the minimal distance
between the vector potentials. The last sum then represents the sum of the elements of the quotient group of a
permutation group P2n+2 of 2n + 2 elements with respect to the cyclic permutations (PC ) as well as permutations
inside the class of elements (P2n , P2 ).
In order 2n, there are n + 1 elements of the quotient group . Let us pick a single element vi of the group P2n+2 that
belongs to the equivalence class i. All elements of the equivalence class make an orbit of an element vi w/r to C.

Then this class will have 2n + 2 − r elements, where r is a number of solutions of the equations of the type:

C k vi = vi , (E14)

where C k is the cyclic permutation of the order k. The cyclic permutation can be represented as a matrix 2n+2×2n+2
with ones at the upper sub-diagonal.

0 1 ...
 
 0 0 1 ...
C= (E15)
... 
1 0 0 ...

Let us represent each term in (E13) by vector ui : every 1 will denote A and 0 denotes xq . Then equation (E14)
will be equivalent to C kPvi ui = vi ui = uj : r then is equal to the number of eigenvectors of C k=1,... that belong to
the set of vectors with j uj = 2. Any vector should return to itself after doing 2n + 2 cyclic permutations, hence
all eigenvalues are of the form λj=1...n+1 = ±ei2πj/(n+1) . For k = 1, there is a single eigenvector of value 1, the one
filled with 1-s. The only chance to have a vector with 1 ones invariant under some power of C is to have ones at a
distance of n + 1, such that C n+1 vinv = vinv . Obviously, there will be n+1 such states.
Hence there will be a single equivalence class with r ̸= 0 with n + 1 elements.
We have then n classes with 2n + 2 elements and one with n + 1.

For even n, there are n/2 classes with odd distance (”vertex”-correction) and n/2 + 1 with even distance
(”DOS”-) between current vertices41 . Hence vertex correction will come with combinatorial coefficient

NDOS,even = (2n + 2)n/2 + n + 1 = (n + 1)2 , (E16)

Nvertex,even = (2n + 2)n/2 = n(n + 1). (E17)

For ”DOS” correction, except for the symmetric term, all other terms can be divided into two (distance k and
2n − k − 1), each then coming with coefficient n + 1. Or, taking into account 1/2(k + 1) of logarithm, we get 1/2 for
each term in the sum.
Similarly for n odd, there are (n − 1)/2 + 1 classes with (”vertex”-) correction, and (n + 1)/2 with ”DOS”-correction.

NDOS,odd = (n + 1)2 , (E18)

Nvertex,odd = n(n + 1). (E19)
 32

By the same argument, there is a coefficient of 1/2 in front of each term of ”vertex”-correction.

Overall, there must be (n + 1)2 - ”DOS”-like elements, and n(n + 1) ”vertex”-like terms.

4. Total correction to conductance

It is possible to write an equation for the response function, for vertex-like (vertices at different momentum) and
DOS-like (vertices at the same momentum) terms separately. Namely, we can do series resummation for two parts
separately without solving Bethe-Solpeter. Since part of the correction comes from the change in the spectrum of the
quasiparticles, it is instructive to start with finding corrected Green function through the Dyson equation. As before,
we implicitly take q equal to qc + δq by the absolute value.

a. Self-energy correction

In the calculation of higher-order correction to the conductance, we neglected terms such that q brought by the
excitonic fields is different, since those, analogously to the 4-th order correction, are negligibly small. Equivalently,
we consider correction to the conductance being composed out of n equal terms, where n is the number of reciprocal
lattice vectors of the order parameter xq . Spectrum of the electrons that contribute to the change in the conductance
can be modeled by the solution of the Dyson equation with a single q vector. It has a form of a 2 × 2 matrix equation:

G(p, p′ ) = G0 (p) + G0 (p)Σq G(p + q, p′ ) (E20)

G(p + q, p′ ) = G0 (p + q) + G0 (p + q)Σ−q G(p, p′ ) (E21)

whose solution is

G(p, p′ ) = (1 − G0 (p)Σq G0 (p + q)Σ−q )−1 (G0 (p) + G0 (p)Σp G0 (p + q)) , (E22)

where Σq is a matrix in k and valley spaces. For valley symmetric system [G0 (p), Σq ] = 0 and hence

G(p, p′ ) = ((G0 (p + q)G0 (p))−1 − Σ2q )−1 G−1

0 (p + q) + Σq . (E23)

Using explicit form of the intervalley self-energy and Green’s function, I obtain:

G−1


′ 0 (p + q) + xq
G(p, p ) = . (E24)
(G0 (p + q)G0 (p))−1 − x2q

Then the correction to the Green’s function is equal to

x2q G−1
0 (p + q)
δG11 (p, p′ ) ≡ G11 (p, p′ ) − G0 (p) = , (E25)
(G0 (p + q)G0 (p))−1 − x2q

xq
δG12 (p, p′ ) = . (E26)
(G0 (p + q)G0 (p))−1 − x2q

Thus the quasiparticle spectrum is given by

r
ξp + ξp+q (ξp − ξp+q )2
λ± (p, q) = ± + x2q ≡ ⟨ϵp ⟩ ± ∆(p) (E27)
2 22
where dispersion of the original particles ξp is allowed to have a finite imaginary part. As before, we divide contributions
from hetero-processes (when G0 (p + q) and G0 (p) are on different Fermi surfaces) and homo-processes (same Fermi
surface).
 33

−3/2
FIG. A.7. Scaled correction (each point is mmltiplied by Σ3/2 = τD ) to conductance as a function of the chemical potential for
3 different self-energies in the experimentally relevant range. Inset: effective power ζ ≡ Σ∂ log(δσ)/∂Σ. Note that Qc < Qmax
region (0.7m ⪅ µ ⪅ 0.75m) is dominated by δσeven,4 terms, so that countercurrent part that dominates δσeven,2 , δσodd is
effectively absent.

a. Hetero-processes For quasiparticles originating from mixing different Fermi-surfaces, there is a gap opening
around nesting sites. Indeed, let us find a equation of the Fermi surface around these. Using ξp = me (p2 − p2i ),
ξp+q = −me (p2 − p2i + 2(p − pi )q cos(θ) + Q), which from ⟨ϵp ⟩ = ∆(p)42 gets us
 
2
m2e (Q + 2p̃q cos(θ))2 = (2xq )2 + m2e (2p̃(p + pi ) + 2p̃q cos(θ) + Q) , (E28)

which after resolving quadratic equation:

v !2
u
Q u Q x2q
p − pi = p̃ ≈ − q ±t q − , (E29)
4pi (1 + pi cos(θ)) 4pi (1 + pi cos(θ)) 4m2e p2i (1 + pqi cos(θ))

defines a closed curve in (p, θ)-space where a constant energy surface intersects 0.
 34

FIG. A.8. Band structure of quasiparticles originating from the inner-inner processes (θ = cos−1 (−q/2pi )) and inner-outer
processes (θ ≈ 0). µ = 7.1 × 10−4 m here. We see here that the contribution from the inner-outer processes to the conductance
is, in fact, of insulator for large gap xq ≫ Σ, while for smaller gap xq ≈ Σ it should be analogous to the semi-metal. Gray zone
close to the Fermi energy (ϵ = 0) is to denote smearing induced by the disorder self-energy (Σ = 5 × 10−5 m here).

Clearly, it has solutions only for some θ, namely whenever

Q xq
√ > . (E30)
2 κio me

Because Q ≤ Q(0), a regime with Fermi arcs exists only for:

2
pi qθc,i xq
≧ , (E31)
2 me
so that the hole has enough transversal energy to absorb an exciton during a one-dimensional jump from θ = 0 → θc,i .
Since for small θc,i , Fermi arcs will be concentrated in small region of k-space, for Σ ≈ me Q the quasiparticles will
behave essentially as one-dimensional.
Additionally, note that with respect to the momentum perpendicular to the Fermi surface p⊥ ∝ pi θ the dispersion is
linear with the density of states
1 1
νhet = νθ ∝ , (E32)
me 2pi q sin(θc,i )
−1
and therefore has large (by a factor of ∝ θc,i ≈ 10) contribution to transport processes in comparison to the homo-
processes.
For large values of xq , contribution from the hetero-processes is gapped for all θ.
b. Homo-processes For the same quasiparticles originating from alike Fermi-surfaces, the quasiparticles are metal-
lic, since they always intersect the zero-energy level. Using ξp = me (p2 −p2i ), ξp+q = me (p2 −p2i +2(p−pi )q cos(θ)+Qii ),
where Qii = 2pi q cos(θ) + q 2 so that for ⟨ϵp ⟩ = ∆(p) we get:

m2e
 2 
2 2

2 me 2 2
2p − 2p i + 2(p − pi )q cos(θ) + Qii = (Q ii + 2(p − pi )q cos(θ)) + x q , (E33)
22 22

which is different from by the sign change x2q → −x2q , hence there always be two closed Fermi surface given by
v !2
u
Qii u Qii x2q
p − pi = p̃ ≈ − q ± t
q + (E34)
4pi (1 + pi cos(θ)) 4pi (1 + pi cos(θ)) 4me pi (1 + pqi cos(θ))
2 2

1/2
which close to nesting Qii ≈ 0 simply results into p − pi ≈ ±xq /(2me pi κio ) ≪ pi , so that we conclude that the mass
of the quasiparticles composed out of particles residing on the same surfaces remains nearly the same
1
νhom ∝ . (E35)
me
35
 36

FIG. A.10. Parameters κio,min = 1−(q/pi ) as well as (κio,max +1)/2 as function of density. Densities at which these parameters
can have zeros are irrelevant for the present study.

FIG. A.11. Points where change of regime happens: for Fermi energy large than µc (µc = 0.778×10−3 for outer-inner processes,
and µc = 0.87 × 10−3 for inner-outer processes) Fermi arcs disappear.
 37

b. DOS-correction

Unlike the vertex-correction, DOS-correction has only odd powers of Green function and forms the expansion of
the form:

|xq |2
Z
σeven = vp2 Im(Gp,> )Im(Gp,> Gp+q,> Gp,> )
π
|xq |4 |xq |4
Z Z
2 2
+ vp Im(Gp,> )Im((Gp,> Gp+q,> ) Gp,> ) + vp2 Im(Gp,> Gp,> Gp+q,> )Im(Gp,> Gp+q,> Gp,> )+
π 2π
|xq |6 |xq |6
Z Z
2 3
vp Im(Gp,> )Im((Gp,> Gp+q,> ) Gp,> ) + vp2 Im(Gp,> Gp+q,> Gp,> )Im((Gp,> Gp+q,> )2 Gp,> )
π π
|xq |2
Z
= vp2 Im(Gp )Im(Gp Gp+q Gp (1 − x2q (Gp Gp+q ))−1 )
2π
|xq |2
Z
+ vp2 Im(Gp Gp+q Gp )Im(Gp (1 − x2q (Gp Gp+q ))−1 )
2π
|xq |4
Z
+ vp2 Im((Gp Gp+q )2 Gp )Im(Gp (1 − x2q (Gp Gp+q ))−1 ) + ...
2π
|xq |2 |xq |2
Z Z
= vp2 Im(Gp (1−x2q (Gp Gp+q ))−1 )Im(Gp Gp+q Gp (1−x2q (Gp Gp+q ))−1 )+ vp2 Im(Gp )Im(Gp Gp+q Gp (1−x2q (Gp Gp+q ))−1
2π 2π
|xq |2
Z
vp2 Im(Gp (1 − x2q (Gp Gp+q ))−1 ) + Im(Gp ) Im(Gp Gp+q Gp (1 − x2q (Gp Gp+q ))−1 )


=
2π
|xq |2
Z
vp2 Im((2Gp − Gp Gp+q Gp x2q )(1 − x2q (Gp Gp+q ))−1 ) Im(Gp Gp+q Gp (1 − x2q (Gp Gp+q ))−1 )


=
2π
|xq |2
Z
= vp2 Im(Gp (1 − x2q (Gp Gp+q ))−1 )Im(Gp Gp+q Gp (1 − x2q (Gp Gp+q ))−1 )
π
|xq |4
Z
− vp2 Im(Gp Gp+q Gp (1 − x2q (Gp Gp+q ))−1 )Im(Gp Gp+q Gp (1 − x2q (Gp Gp+q ))−1 )
2π
|xq |2
Z
= vp2 Im(G−1 p+q ((Gp Gp+q )
−1
− x2q )−1 )Im(Gp ((Gp Gp+q )−1 − x2q )−1 )
π
|xq |4
Z
− vp2 Im(Gp ((Gp Gp+q )−1 − x2q )−1 )Im(Gp ((Gp Gp+q )−1 − x2q )−1 ) (E36)
2π
It is also clear that it has a meaning of the conductance calculated with δG1,1 (E25) and a corrected single
vertex(twice), plus additional term with both vertex corrected and bare Green’s functions. Let me call the former
σDOS and the latter σv,2 . We now calculate σDOS :

Explicit form is
|xq |2 Σ2 (ξp+q ξp − |xq |2 − Σ2 ) − (ξp+q + ξp )ξp+q (ξp+q ξp − |xq |2 − Σ2 ) + (ξp+q + ξp )ξp
Z   
1
σDOS = − vp2
π (ξp+q ξp − |xq |2 − Σ2 )2 + Σ2 (ξp+q + ξp )2 (ξp+q ξp − |xq |2 − Σ2 )2 + Σ2 (ξp+q + ξp )2 ξp2 + Σ2
(E37)
or, introducing parameters α and γ
!
|xq |2 Σ2 (ξp+q ξp − |xq |2 − Σ2 )2 − (ξp+q + ξp )2 ξp ξp+q − (ξp+q ξp − |xq |2 − Σ2 )(ξp+q 2
− ξp2 )
Z
1
σDOS =− vp2
π (γ 2 (ξp+q ξp − |xq |2 − Σ2 )2 + α2 Σ2 (ξp+q + ξp )2 )2 ξp2 + Σ2
∂ |xq |2 Σ2
Z
1 1
= vp2 α2 2
γ∂γ 2πγ 2 2 2 2
(ξp+q ξp − |xq | − Σ ) + γ 2 Σ (ξp+q + ξp ) p 2 ξ 2 + Σ2

2 Z
∂ |xq | ξp ξp+q 1
− vp2 α2 2
α∂α 2πγ 2 2 2 2
(ξp+q ξp − |xq | − Σ ) + γ 2 Σ (ξp+q + ξp ) p 2 ξ 2 + Σ2

2
|xq | Σ2 ∂ 1
2 (ξp+q − ξp2 )γ 2
Z
2 1
+ v p α2 2
(E38)
π 2xq ∂xq 2γ 2 2 2 2
(ξp+q ξp − |xq | − Σ ) + γ 2 Σ (ξp+q + ξp ) p 2 ξ 2 + Σ2
 38

FIG. A.12. Different regions in the small xq case: Qio (0) < Qc,io .

We then define three generating functions gDOS,i (p̄ = (γ, xq , α)) such that:
∂
σDOS = (xq Σ)2 gDOS,i (p̄), (E39)
2pi ∂pi
or
vp2 fi (p, θ)
Z
1 1
gDOS,i (p̄) = α2 2
(E40)
πγ 2 (ξp+q ξp − |xq |2 − Σ2 )2 + γ 2 Σ (ξp+q + ξp )2 ξp2 + Σ2
2
where f1 = 1, f2 = (ξp+q − ξp2 )/2, f3 = −ξp ξp+q /Σ2 .
a. Hetero-contribution We start with the i-o process since these are the processes that by a factor of sin(θc )−1
larger. Clearly, the poles are the same as before (See E 4 d) except for α → α/γ and
Z
1 fi (δp) 1
gDOS,i = vp2 Σ2 |xq |2
(E41)
πγ 2 δp + m2 (−(κio δp + Q)δp − m2 − m2 )2 + γα22m
2 Σ 2 Σ2
2 (δp − κio (δp + Q))
2
e e e e

the integrand has additional poles at δp3/4 = ±iΣ/me , as well as poles common to the vertex correction:
s
iΣαq 2
Q + γm cos(θ) Σ2 |xq |2

e pi
1 iΣαq iαΣQ
δp+,1/2 = − ± 2
Q+ cos(θ) − 2 − 2 − (E42)
2κio (2κio ) me pi γ me κio me κio me γκio
Because xq /me ≪ pi q, with latter being a scale of Q, the integrand is still peaked close to θc,i , hence we neglect
angle dependence of κio : cos(θ) ≈ cos(θc,i ) (the regime with high density κio = 0 can be a topic of separate study,
but it is irrelevant for experimentally accessed regime, see Fig. A.10). We still can expand Q in vicinity of it.

Then there will be an anoother critical angle θcrit , at which sign of the phase changes:
1/2
2xq κio
Qc,io = , (E43)
me
expansion of Qio in the vicinity of the (first) critical angle gets us Qio (θ) = −2pi q sin(θc,i )(θ −θc,i ). Assuming Σ ≪ xq ,
there will be 2 distinct regimes Qio (0) < Qc,io , Qc,io > Qio .
Then the integral over δp:
iΣ
vp2 fi ( m )
Z
1 e
1
gDOS,i = 2 6 Σ |xq |2
γ me iΣ iΣ Σ2 2 α2 Σ2 iΣ iΣ
me (−(κio m e
+ Q) m e
− m2e − m2e ) + γ 2 m2e ( me − κio ( m e
+ Q))2
vp2
Z  
2i fi (δp+,2 ) 1 fi (δp−,1 ) 1
+ −
γ 2 κ2io m6e δp+,2 − δp−,1 Σ2 (δp
δp2+,2 + m 2
Σ2
+,2 − δp+,1 )(δp+,2 − δp−,2 ) δp−,1 + m2 (δp−,1 − δp+,1 )(δp−,1 − δp−,2 )
2
e e
(E44)
 39

For convenience, define two auxiliary functions ⟨fi ⟩ and δfi such that fi (δp+,2 ) = ⟨fi ⟩ + δfi and fi (δp−,1 ) = ⟨fi ⟩ − δfi ,
then the whole generating function can be written as a sum of 3 qualitatively different terms:

iΣ
vp2 fi ( m )
Z
1 e
1
gDOS,i = 2Σ |xq |2
πγ 2 m6e iΣ
(−(κio m + iΣ
Q) m − − Σ2 2 α2 Σ2 iΣ iΣ
− κio ( m + Q))2
me e e m2e m2e ) + γ 2 m2e ( me e

vp2 ⟨fi ⟩
Z  
i 1 1 1 1
+ −
πγ 2 κ2io m6e δp+,2 − δp−,1 δp2+,2 + Σ2
m2e
(δp+,2 − δp+,1 )(δp+,2 − δp−,2 ) δp2−,1 + Σ2
m2e
(δp−,1 − δp+,1 )(δp−,1 − δp−,2 )
vp2 δfi
Z  
i 1 1 1 1
+ 2 2 6 Σ2
+ Σ2
,
πγ κio me δp+,2 − δp−,1 2
δp+,2 + m2e
(δp+,2 − δp+,1 )(δp+,2 − δp−,2 ) δp2−,1 + m2e
(δp−,1 − δp+,1 )(δp−,1 − δp−,2 )

where the most significant difference from the vertex part is the presence of δfi ∝ O(Σ/me ) part. For completeness,
write new functions in terms of poles:
⟨f1 ⟩ = 1 (E45)
δf1 = 0 (E46)
m2e
(κ2io − 1)(δp2+,2 + δp2−,1 ) + 2Qκio (δp+,2 + δp−,1 ) + Q2


⟨f2 ⟩ = (E47)
2
m2e
(κ2io − 1)(δp+,2 + δp−,1 ) + 2Qκio (δp+,2 − δp−,1 )


δf2 = (E48)
2
m2
⟨f3 ⟩ = e2 κio δp2+,2 + δp2−,1 + Q(δp+,2 + δp−,1 )



(E49)
2Σ
m2
δf3 = e2 (κio (δp+,2 + δp−,1 ) + Q) (δp+,2 − δp−,1 ) (E50)
2Σ
We now take each of the 5 integrals over angle in three regimes, distinct by the behavior of the poles. To define the
boundaries, let us write the expression for the poles in the form that clearly separates phase and the absolute value:
iΣαq
Q+ m e pi γ cos(θ)
δp+,1/2 = −
2κio
2 1/4
 !2  
 2  2 2 
1 Σ q α 2Σ Σ α Q q
± Q2 − Q2c,io − cos(θ) − κio + 1+ cos(θ)  eiϕ(θ)/2 ,
(2κio )2 me pi γ me me γ κio 2pi
(E51)
where phase ϕ:
   
2Σ α q
− me γ Q 1 + 2pi cos(θ)
ϕ(θ) = tan−1  . (E52)
 
 2  2
2 2 Σ q α 2Σ
Q − Qc,io − me pi γ cos(θ) − me κio

As discussed before, based on the phase behavior, we distinguish 3 different regimes: positive real part, negligible
imaginary part, region close to the Qc,io , negative real part, negligible imaginary part. Given the disorder is not too
strong (Qc,io ≥ Σ/me ) we set boundaries to:
2ΣQ 2ΣQc,io
Q2 − Q2c,io ≥ ≈ (E53)
me me
for the first region,
2ΣQc 2ΣQc,io
− < Q2 − Q2c,io < (E54)
me me
for the second, and
2ΣQc,io
Q2 − Q2c,io < − (E55)
me
for the third.
 40

FIG. A.13. Region 1: comparison of the exact poles (E42) as a function of θ with approximate expression (E56) in the region
1 with linear expansion of Q (A, dotted), quadratic expansion of Q (A, dashed) in the vicinity of θ = θc,i and expansion near
θ = 0 (B dotted, semi-transparent). We use linear approximation in the following calculation. Vertical line to denote the end
of the region 1.

1) First, for xq /me < Q(0)/(2κio ) poles exhibit again all 3 possible behaviors (see E 4 d)

(a) As such, in the first region the poles are

i(κio + 1) αΣQ
  !
1 iΣαq 1
1/2 me γ
δp+,1/2 =− Q+ cos(θ) ± Q2 − Q2c 1− , (E56)
2κio me pi γ 2κio Q2 − Q2c

meaning that poles present in the upper half-plane are δp+,2 and δp−,1 . Additionally, denote

1
Σ2
= ⟨s⟩ + δs, (E57)
δp2+,2 + m2e
1
Σ2
= ⟨s⟩ − δs. (E58)
δp2−,1 + m2e

Note right away a derivative with respect to α or γ of these functions

∂ ∂ ∂p+,2 ∂ ∂p−,1
(⟨s⟩/δs) = (⟨s⟩/δs) + (⟨s⟩/δs) (E59)
∂(α/γ) ∂δp ∂(α/γ) ∂δp ∂(α/γ)
δp=δp+,2 δp=δp−,1

will add at least a power of ∝ Σ/(Q2 − Q2c )1/2 ∝ Σ1/2 . A derivative w/r to xq does not change the power of
Σ. By an identical argument, ∂⟨fi ⟩(δfi )/(∂α(γ)) also adds at least a power of Σ1/2 . We then conclude that
the dominant contribution in the small Σ/me limit comes from derivative w/r to α of the ratio difference
or, if vanishing, derivatives of fi /s w/r to α:

iΣ
vp2 fi ( m )
Z
1 e
1
gDOS,i,1 = 2Σ |xq |2
πγ 2 m6e iΣ
(−(κio m + iΣ
Q) m − − Σ2 2 α2 Σ2 iΣ iΣ
− κio ( m + Q))2
me e e m2e m2e ) + γ 2 ( me e

vp2 ⟨fi ⟩⟨s⟩

Z  
1 1 1
+ −
2πγ 2 κ2io m6e δp+,2 − δp−,1 (δp+,2 − δp+,1 )Im(δp+,2 ) Im(δp−,1 )(δp−,1 − δp−,2 )
vp2 ⟨fi ⟩δs
Z  
1 1 1
+ +
2πγ 2 κ2io m6e δp+,2 − δp−,1 (δp+,2 − δp+,1 )Im(δp+,2 ) (δp−,1 − δp+,1 )Im(δp−,1 )
vp2 δfi ⟨s⟩
Z  
1 1 1
+ +
2πγ κ2io m6e
2 δp+,2 − δp−,1 (δp+,2 − δp+,1 )Im(δp+,2 ) Im(δp−,1 )(δp−,1 − δp−,2 )
vp2 δfi δs
Z  
1 1 1
+ − .
2πγ 2 κ2io m6e δp+,2 − δp−,1 (δp+,2 − δp+,1 )Im(δp+,2 ) Im(δp−,1 )(δp−,1 − δp−,2 )
 41

FIG. A.14. Plot of the order parameter as a function of the chemical potential µ in units of mass, normalized to pi qc

Substituting approximate expressions for poles now, we get

iΣ α (κ√io +1)Q q
vp2 (⟨fi ⟩⟨s⟩ + δsδfi ) (κio + 1)Q + cos(θ)
22 κ3io
Z me γ κio Q2 −Q2 pi
c 1
gDOS,i,1 = dθ p iΣ α q (κio +1)2 Q2 2
γm5e Σα Q2 − Q2c + ( pqi

me γ pi cos(θ) − cos(θ))2 αΣ(κio +1)Q
Q2 −Q2c Q2 − Q2c + √
γme κio Q2 −Q2c
2 2
iΣα (κio +1) Q q
Q2 − Q2c pqi cos(θ) +
p
22 κ3io
Z
vp2 (⟨fi ⟩δs + ⟨s⟩δfi ) me γ κio (Q2 −Q2c ) pi cos(θ) 1
+ dθ p iΣ α q (κio +1)2 Q2 2 .
γm5e Σα Q2 − Q2c + − ( pqi cos(θ))2

m e γ pi cos(θ) αΣ(κio +1)Q
Q2 −Q2c Q2 − Q2c + √
γme κio Q2 −Q2c

where as before Q = p2i − p2o + q 2 + 2pi q cos(θ). For small Qc ≪ 2pi qc (see Fig. A.14) it should be still true
that the integral is peaked in vicinity of θc,i , and hence we can linearize Q = −2pi q sin(θc,i )(θ−θc,i ) (see Fig.
(A.13) for comparison of θ-dependence of poles in linear approximation of Q and other approximations).
Since the minimum possible value of the difference Q2c − Q2 ∝ 2Qc Σ/me , for Σ ⪅ Qc (pi /qc ) ≈ 10−4 , it
should be possible to approximate the expression above by

22 κ3io vp2 (⟨fi ⟩⟨s⟩ + δsδfi ) 1

Z
1 dQ
gDOS,i,1 = 5
p
γme Σα κio + 1 pi qc sin(θc,i ) Q2 − Q2c Q
2 3 vp2 (⟨fi ⟩δs + ⟨s⟩δfi ) qc
Z
2 κio 1 dQ
+ cos(θc,i ). (E60)
γm5e Σα (κio + 1)2 pi qc sin(θc,i ) Q2 pi
In the numerator, we neglect Q-dependence and simply use Q ≈ Qc and also concentrate on i = 3 (derivative
w/r to α). Then δp3,− ≈ δp4,+ ≈ −Qc /(2κio ), and hence

m2e κio
⟨f3 ⟩⟨s⟩(Q = Qc ) = − . (E61)
Σ2
Then

2κ3io v 2 ⟨fi ⟩⟨s⟩ 24 κ4io pi (cos(θc,i ))2 1

Z Z
1 dQ 1 1 dQ
gDOS,i,1 = pp ≈ √ √
γm5e Σα κio + 1 pi qc sin(θc,i ) Q2 − Q2c γme Σ3 α κio + 1 2Qc qc sin(θc,i ) Q − Qc Q
√ !
25 κ4io 1 1 pi cos(θc,i )2 Q 0 − Q c
= √ tan−1 . (E62)
γαme Σ3 κio + 1 2Qc qc sin(θc,i ) Qc
1/2

As a consequence, conductance in this region is:

7/2 √ !
23 xq κio vi vo Q0 − Qc
σDOS,het,1 ≈ tan−1 , (E63)
Σm2e κio + 1 pi q sin(θc,i ) 1/2
Qc
 42

FIG. A.15. Region 2a: comparison of the exact poles (E42) for regions 2a as a function of θ with approximate expression (E64).
Vertical lines to denote boundaries of the region 2a or 2b (gray – Qio,lin + δQ1 , with linear approximation of Q, black – Qio,lin ,
linear Q, red – exact Qio on A, and an analogous labeling used for B) and C). Disorder self-energy is Σ = 10−6 on A), and
Σ = 10−5 for B) and C). On C), we used linear approximation for Q: Q = 2qpi cos(θc,i )(θ − θc,i ).

which, in fact, is comparable to the quadratic correction in the range xq /Σ ≈ 1. Note that correction has
a positive sign, which is somewhat counterintuitive. It is also clear that in the real part of the correction
there are no terms O(Σ/me ).
Therefore, the sign change as a function of Σ/me should come from first term, with poles coming from
the original quasiparticles unaffected by the potential. We expand on the sign change at the end of this
section.
(b) The middle region, just as before, can be divided into two subregions: left-vicinity of Q = Qc (a) and right
vicinity of this point.
a) In the left-middle region, the imaginary part of the pole is no longer negligible in comparison to the
real part. For concreteness, we pick a region of size Σ defined through 0 < Q2 − Q2c < 2A1 Qc,io (Σ/me ).
In this region, we approximate roots by:
iΣαq
Qc + pi γme cos(θ)
δp+,1/2,iia = −
2κio
 1/4
 1/2  2  2  2 !2
1 Σ  αQc (κio + 1) + Σ q α
± κio + cos(θc,i )  e−iπsign(Σ)/4 . (E64)
κio me 2γ me pi 2γ

and take A1 = 1/4. For convenience, let me denote:

 −1/2
 2  2  2 !2
αQc Σ q α
τC = m−1
e
 (κio + 1) + κio + cos(θc,i ) 
2γ me pi 2γ
 −1/2
 2  2  2 !2
α 1 q α
= + κio + cos(θc,i )  , (E65)
γτX τD pi 2γ

so that we obtain an expression

iΣαq
Qc + pi γme cos(θ) 1 −1/2 − iπ
δp+,1/2,iia = − ± (τC τD ) e 4 sign(Σ) . (E66)
2κio κio me
b) In the right vicinity of θcrit,i (Qc ), defined by −2A1 Qc mΣe < Q2 − Q2c < 0, roots are
iΣαq 1/2 iQc Σ
!1/4
Q+ cos(θ) 1− me (κio + 1)

pi γme αΣ κio + 1 δQ π
δp+,1/2,iib ≈ − ± Qc e−i 2 sign(Σ) . (E67)
2κio γme 2κ2io 1+ iQc
δQ
Σ
me (κio + 1)
The problem, however, is that δQ/Qc of order 1 and we cannot expand the bracket. We then use
constant approximation (See Fig.A.15, A.17) in the whole region close to Qc .
The generating function then is
 
27/2 πκ4io pi τD (τD τC )3/2 1
gDOS,het,2 = A1  . (E68)
 
2
γ 2 qc sin(θc,i )

q α τC
2− pi γ cos(θc,i ) τD
 43

FIG. A.16. Region 2b: comparison of the exact poles (E42) for regions 2b as a function of θ with approximate expression (E64).
Vertical lines to denote boundaries of the region 2a or 2b (red – Qio − δQ1 (dark red with linear approximation of Q), black –
Qio (gray for linear approximation) on A, and an analogous labeling used for B) and C). Disorder self-energy is Σ = 10−6 on
A), and Σ = 10−5 for B) and C). On C), we used linear approximation for Q: Q = 2qpi cos(θc,i )(θ − θc,i ).

FIG. A.17. Region 3: comparison of the exact poles ((E42), blue, solid) for region 3 as a function of θ with approximate
expression ((E71), dashed lines). Vertical lines to denote boundaries of the region 3 (gray (black) – Qio,lin − δQ1 in linear
approximation of Q (Qio − δQ1 ), and pink (red) to denote angle corresponding to Q = 0. We use the linear upper boundary
in the calculation. Σ = 10−6 here.

the correction to conductance is

   
2ατC ∂τC α2
5/2
2 A1 π pi κ4io x2q (τD τC ) 1/2
 3 ∂τC τC τD + ∂α τD

q

σDOS,het,2 ≈ + cos(θc,i )2 

2 2
sin(θc,i ) q γ2 m2e 2 ∂α pi γ
  
q α τC q α τC
2− pi γ cos(θc,i ) τD 2 − pi γ cos(θc,i ) τD
(E69)

where the derivative of τC can be written as:

 2 !
∂τC α 1 1 q
= −τC3 2 2 + τ2 cos(θ) , (E70)
∂α γ τX D pi

1/2
where we also recalled that Qc = 2xq κio /me .

(c) In the region to the far right from Qc , roots are:

 
s
Q+ iΣq α
m e pi γ cos(θ) Qc
2 
Q
2
 (κio + 1) mαΣQ 2
e γκio

δp+,1/2,iii = − ∓i − 1 − i  2  sign(Σ).
 (E71)
2κio 2κio 2κio 
Q |xq |2
4 2κio − m2e κio

With properties as in region I:

Re(δp+,1/2,iii ) = Re(δp−,1/2,iii ), (E72)

Im(δp+,1/2,iii ) = −Im(δp−,1/2,iii ). (E73)
 44

After some simplifications, generation function in the third region becomes:

24 π vp2 ⟨fi ⟩⟨s⟩κ3io (Q2c − Q2 )

Z
gi ≈ 2 6 dθ
γ me (κio + 1)Q + piq cos(θ)(Q2c − Q2 )1/2
i
iq
picos(θ)(Q2c − Q2 ) + (κio + 1)Q(Q2c − Q2 )1/2
  2   . (E74)
αΣQc
2 2 2
(Qc − Q ) + (κio + 1) γme Q2c − Q2 − ( mΣe pqi αγ cos(θ))2


real part of which, in negligence of (Σ/me )(q/pi ) cos(θ))2 in comparison to Qc δQ takes the form
5/2  1/2
25 πτD κ4io (pi /q) 1 2γ
gi ≈ 3/2 3/2 (κ + 1)1/2
, (E75)
2
γ me sin(θc,i )Qc io α
where we extended integration range from 0 to ∞, which should be a reasonable approximation as long as
Qc and Σ/me (δQ and Σ/me ) are well separated. So that the resulting contribution to the conductance is
13/4
1/2 κio (pi /q) 1
σ3 ≈ −22 πx1/2
q τD . (E76)
sin(θc,i ) (κio + 1)1/2
b. Homo-contribution Close to the Fermi surface, for homo-processes (without loss of generality we look at the
inner-inner case) dispersion can be written in the form ξp+q = me (κδp + Qii (θ)) and ξp = me δp, where Qii (θ) ≡
q 2 + 2pi q cos(θ) and, as before, κ ≡ 1 + (q/pi ) cos(θ). The generating function is then:
Z 2
1 vp fi (α, γ, xq , δp) 1
gi ≡ 2 6 Σ2 2  2 . (E77)
πγ me 2 2 2

δp + m2 Σ +x
δp(κii δp + Qii ) − m2 q + mΣα (δp(κ + 1) + Q )
2
e
eγ
ii ii
e

The poles then are

v !2
iΣα iΣα
u
Qii − me γ (κii + 1) u Qii −
me γ (κii + 1) x2q + Σ2 iΣα
δp+,3/4 = − ±t + + Qii . (E78)
2κii 2κii m2e κii me γκii

Note that since here (effectively) angular kinetic energy ∝ Q2ii /(2κii ) enters with the same sign as the gap x2q /m2e ,
there is no point where the real part of the expression under the root changes sign. We then anticipate, based on the
discussion in the hetero-contribution, that the homo-contribution will have linear τD behavior. Additionally, in the
regime Q2 + Q2c ≫ 2κii Σ2 /m2e we can expand:
s  
iΣα 2  2 iΣ α κii −1
Qii − m (κii + 1) Qii Q 2 2
2 me γ κii Q ii
eγ
δp+,3/4 ≈ − ± + 1 +  2  . (E79)
 
2 
2κii 2κii 2κii Qii Qc
2κii + 2κii

After integrating over the residues we have, naturally, the same expression:
iΣ
vp2 fi (α, γ, xq , m )
Z
1
gi ≡ 2 5 iΣ iΣ iΣ
e
iΣ
γ me Σ κ2 ( m e
− δp+,3 )( m e
− δp−,3 )( m e
− δp+,4 )( m e
− δp−,4 )
Z 2
vp (⟨fi ⟩⟨s⟩ + δfi δs)
 
2i 1 1 1 1
+ 2 6 −
γ me κ2 (δp+,3 − δp+,4 ) δp+,3 − δp−,3 δp+,3 − δp−,4 δp+,4 − δp−,3 δp+,4 − δp−,4
Z 2
vp (⟨fi ⟩δs + δfi ⟨s⟩)
 
2i 1 1 1 1
+ 2 6 + , (E80)
γ me κ2 (δp+,3 − δp+,4 ) δp+,3 − δp−,3 δp+,3 − δp−,4 δp+,4 − δp−,3 δp+,4 − δp−,4
where averages ⟨f ⟩, ⟨s⟩ and differences δf , δs have the same meaning as before, but with new expressions for poles.
Since ∝ ⟨f ⟩⟨s⟩ is by a factor of τD larger than everything else, we approximate the whole expression by it. After
doing substitution, it becomes:
 
Z 2
2π v ⟨fi ⟩⟨s⟩κ  1 1 1 1
gi ≈ pp + ,

5 Q(κ−1)α/γ
p iΣ α Q(κ−1)α/γ
p iΣ α
γαme Σ Q + Qc κ + 1 + √ 2 2
2 2 Q + Qc + me γ (κ + 1) κ + 1 − √ 2 2
2 2 2 2

Q + Qc − m e γ
(κ + 1)
Q +Qc Q +Qc
(E81)
 45

which in the limit of small q/pi can be represented as

Z 
2π 1 1
gi ≈ vp2 ⟨fi ⟩⟨s⟩κ
γαm5e Σ
p α
p iΣ α
(κ + 1) Q2 2 2 2
+ Qc + Q(κ − 1) γ Q + Qc + m e γ
(κ + 1)

1 1
+ p α
p iΣ α
(E82)
(κ + 1) 2 2 2 2
Q + Qc − Q(κ − 1) γ Q + Qc − m (κ + 1)
e γ

p
As before, we expect major contribution to come from ⟨f3 ⟩. In negligence of Σ/me w/r to Q2 + Q2c ⟨f3 ⟩ can be
approximated by

m2e Q2c
⟨fi ⟩ = − , (E83)
(2Σ)2 κii

(2κii )2 2(Q/Qc )2 + 1
⟨s⟩ = . (E84)
Q2c

Then correction to conductivity is

2x2q vi2 (2κii )3/2 (2κii )1/2 Qmax 23/2 xq vi2 (κii )5/2 (2κii )1/2 Qmax
   
−1 −1
δσDOS,hom = tan = tan , (E85)
Qc Σm3e Jii (κii + 1)Qc Σm2e Jii (κii + 1)Qc
r  2
where Jii = 2pi q sin(θi,i ) = 2pi q 1 − 2pq i , which is also positive, surprisingly43 . Note also that for large Σ (and
small Qmax ) Qc = (22 κio /me )(x2q + Σ2 )1/2 can be a better approximation, which will give a larger power effectively.
To take it more rigorously, we make a substitution z = eiθ :

κ3ii pi Q2c z 2 /(pi qc )

I  
2 dz
g3 ≈ − 2− . (E86)
γαme Σ3 κii + 1 q iz (q/pi z + (z 2 + 1))2 + z 2 (Qc /pi qc )2

The first term has poles at z1 = 0, the second has non-vanishing poles at
s 2
q 2 + iQc q 2 + iQc q 2 + iQc 1
z1/2,+ = − ± −1≡− ± D, (E87)
2pi qc 2pi qc 2pi qc 2pi qc
s 2
q 2 − iQc q 2 − iQc q 2 − iQc 1
z1/2,− = − ± −1≡− ± D∗ , (E88)
2pi qc 2pi qc 2pi qc 2pi qc

among which |z1,+ | < 1 and |z1,− | < 1. Taking the integral over residues, we obtain:

(22 π) κ3ii pi Q2c

  
z1,+ z1,−
g3 ≈ − 2− − . (E89)
γαme Σ3 κii + 1 q z1,+ − z1,− (z1,+ − z2,− )(z1,+ − z2,+ ) (z1,− − z2,− )(z1,− − z2,+ )

Or, after using (E87):

(22 π) κ3ii pi Q2c /2 −q 2 − iQc + D −q 2 + iQc + D∗

  
g3 ≈ − 2− − .
γαme Σ3 κii + 1 q −iQc + (D − D∗ )/2 D(−iQc + (D + D∗ )/2) D∗ (iQc + (D + D∗ )/2)
(E90)
c. Sign-change of the DOS-correction As the scattering lifetime decreases and becomes comparable to xq , first
term coming from the pole δp = iΣ/me can no longer be neglected. Again, we divide correction into hetero- and
homo-contributions. Let us perform an expansion into powers of Qme τ2 , where τ2 = (x2q + Σ2 )−1/2 , or, equivalently,
Qme τC
(a) Hetero-contribution The generating function in zeroth order is
(0)
vp2 fi
Z
(0) 1
gDOS,i→o = , (E91)
me Σ γ 2 (x2q + Σ2 (1 − κio ))2 − α2 Σ4 (1 − κio )2
 46

(0) (0) (0)

where f1 = 1, f2 = (Σ2 /2)(1 − κ2io ), f3 = −κio , which gives contributions to conductivity

(0) Σvi2 θc,i (x2q + Σ2 (1 − κio ))2

δσDOS,1 = − , (E92)
me x2q (x2q + 2Σ2 (1 − κio ))2

(0) Σ3 vi2 θc,i 2 (x2q + Σ2 (1 − κio ))

δσDOS,2 = (κio − 1) , (E93)
me x2q (x2q + 2Σ2 (1 − κio ))2

(0) Σ5 vi2 θc,i κio (1 − κio )2

δσDOS,3 = − , (E94)
me x2q (x2q + 2Σ2 (1 − κio ))2

which in Σ/xq → ∞ limit takes the form

(0) vi2 θc,i Σ

δσDOS,i→o (Σ/xq → ∞) = − (1 + κio ), (E95)
2me x2q

and me Σ−1 term is absent from the expansion. The opposite limit reads

(0) Σvi2 θc,i

δσDOS,i→o (xq /Σ → ∞) = − . (E96)
me x4q

Given in small Σ ≪ Qmax limit the correction has a form of ∝ 22 π(xq /Σ)(1/θc,i )pi /qc , sign change of the i-o
part of the conductance happens close to the point
−1
Σcrit,i−o ≈ (x3/2 1/2
q /me )(pi θc,i ) , (E97)

which is close to 10−5 in the range of the interest.

(b) Homo-contribution Without loss of generality, we take an integral for the inner-inner part of the correction:

vp2 fi
Z
1
gi,DOS,i→i = (E98)
Σ2 +x2 2
2  2
Σm5e
  
iΣ iΣ
me (κii me + Qii ) − m2 q + mΣα eγ
iΣ
me (κ ii + 1) + Q
e

  
iΣ iΣ
with auxiliary function being f1 = 1, f2 = m2e /2 (κii − 1) m e
+ Q (κii + 1) me + Q , f3 =
 
iΣ
−(ime /Σ) m e
κii + Q . In small Qc /Σ limit (for region 1) or Q/Σ for all other regions, we expand into
powers of Q ≡ p2i + 2pi q cos(θ) to observe the sign change of conductivity correction. Additionally, for we are
interested in regime q/pi < 1, we may expand in κii − 1. In the 0-th order then I have:
(0)
(0) 2π vp2 ⟨fi ⟩
gi,DOS,i→i = , (E99)
Σme 2γΣ2 + γx2 2 − Σ4 (2α)2


q

(0) R (0)
where averaging is performed over the angle ⟨fi ⟩ = 1/(2π) dθfi (θ). Then its contributions conductivity
are:

2
(0) 2πΣ vp2 Σ2 2 + x2q
δσ1,DOS,i→i =− (E100)
me x2q 22 Σ2 + x2 2


q

(0)
δσ2,DOS,i→i = 0 (E101)

(0) 2πΣ vp2 Σ4 22

δσ3,DOS,i→i = (E102)
me x2q 22 Σ2 + x2 2


q
 47

which gives zero upon resummation in Σ → ∞ limit. Expanding both expressions in xq /Σ:
 !2 
2 2 2 2
(0) πv i Σ x
 1+ q − q
x x q πvi2
δσDOS,i→i = − 2 2 2 2
− (1 − 2 2 )2  = − , (E103)
me xq 2 2Σ 2 Σ 2 Σ 2me Σ

suggesting that there is no sign change of the homo-part of conductance as a function of Σ/xq 44 . Sign change
point is
xq ∝ 2me qpi θi,i /π, (E104)
above which DOS-part of the homo-part of the conductance correction becomes positive.

c. Doubly corrected DOS

The conductivity correction corresponding to both Green function dressed, is:

|xq |4
Z
σv,2 = − vp2 Im(Gp ((Gp Gp+q )−1 − x2q )−1 )Im(Gp ((Gp Gp+q )−1 − x2q )−1 ). (E105)
2π
When written explicitly,
!2
Σ2 |xq |4 ξp (ξp + ξp+q ) + (ξp ξp+q − x2q − Σ2 )
Z
1
σv,2 =− vp2 . (E106)
2π ξp2 + Σ2 γ 2 (ξp ξp+q − x2q − Σ2 )2 + α2 Σ2 (ξp + ξp+q )2

We can rewrite it in terms of generating function gv,2 as three derivatives:

x4q Σ2 ∂ vp2
Z
1
σv,2 =
22 πγ ∂γ (ξp + Σ ) γ (ξp ξp+q − xq − Σ )2 + α2 Σ2 (ξp + ξp+q )2
2 2 2 2 2 2

x4q ∂ vp2 ξp2

Z
+ 2
2 πα ∂α (ξp2 + Σ2 )2 γ 2 (ξp ξp+q − x2q − Σ2 )2 + α2 Σ2 (ξp + ξp+q )2
x3q Σ2 ∂
Z
ξp (ξp + ξp+q ) 1
− 2 2 vp2 . (E107)
2 πγ ∂xq (ξp2 + Σ2 )2 γ 2 (ξp ξp+q − x2q − Σ2 )2 + α2 Σ2 (ξp + ξp+q )2

Since term goes has a multiplier (xq /Qmax )2 ∝ (Qc /Qmax )2 , it is of no importance for small value of this parameter
(regime of Fermi arcs). In this range, two first terms are about of the same value for (Qc /Qmax )4 ≈ Σ/me the second
goes at worst as ∝ x4q /(m2e Q4max Σ2 ), and the first is ∝ me /Σ for small Σ. For large value of (Qc /Qmax )2 , the term
dominates over DOS- and vertex-corrections. In this regime, pole at δp = iΣ/me gives the largest contribution, and
as a result, the first term with a derivative over γ should dominate. Note, however, that generating function can be
written (for all corrections) as
gv,2 (xq , α, γ) = γ −2 Gv,2 (xq , α/γ), (E108)
therefore
∂
gv,2 (xq , α, γ) = γ −2 G′v,2 (xq , α/γ)(1/γ), (E109)
∂α

∂
gv,2 (xq , α, γ) = −2γ −3 Gv,2 (xq , α/γ) − γ −4 G′v,2 (xq , α/γ)α, (E110)
∂γ
and so
∂ ∂
gv,2 (xq , α, γ) = −2γ −3 Gv,2 (xq , α/γ) − gv,2 (xq , α, γ)αγ −2 . (E111)
∂γ ∂α
We now proceed to calculation of hetero-part.
 48

a. Hetero-contribution In variables θ, δp, the integral reads:

x4q Σ2 vp2
Z
∂ 1
σv,2,het,2 = dθdδp Σ2 2 x2q
, (E112)
23 πγ ∂γ (δp2 + m2e ) γ 2 (δp(κio δp + Q) + + Σ2
)2 + (αΣ)2
− κio δp − Q)2
m2e m2e m2e (δp

where Q = p2o − p2i + q 2 − 2pi q cos(θ). Up to a factor, contribution of quasiparticle poles (eigenstates of mean-field
Hamiltonian) should be the same as in DOS. In large Qc /Qmax limit, it is δp1,2 = iΣ/me pole that dominates, hence
we write:
x4q dθvi2
Z
∂
δσv,2,het,3 = 4 x2
. (E113)
2 γm5e Σ ∂γ γ 2 ( iΣ (κio iΣ + Q) + q2 + Σ 2 )2 + (αΣ)
2 2
2 ((1 − κio )
iΣ
− Q)2
me me me me me me

The denominator has roots at:

x2q Σ
Q4,+ = − α − (1 − κio ), (E114)
me (1 + γ )iΣ ime

x2q Σ
Q4,− = − α − (1 − κio ). (E115)
me (1 − γ )iΣ ime
Q4,− (1 − α/γ) →α→γ −x2q /(ime Σ). The whole thing can be rewritten in the form

x2q i dθvp2 (Q − Q∗4,+ ) x4q dθvp2 (Q − Q∗4,+ )2 dθvp2 Q(Q − Q∗4,+ )2

Z Z Z
1
δσv,2,het,3 = − 3 + − 5
2 (Σme )2 Q2 + |Q4,+ |2 27 (Σme )3 (Q2 − |Q4,+ |2 )2 2 Σme (Q2 − |Q4,+ |2 )
(E116)
which for small Qmax /|Q4,+ | goes as
(xq vi )2 Qmax x4q vi2 Qmax
δσv,2,het,3 ≈ − − , (E117)
2 JΣme (xq + 2Σ (κio − 1)) 2 Σme γ J (xq + 2Σ2 (κio − 1))2
3 2 2 5 2 2

which changes effective power from Σ−1 to Σ−3 . In the opposite limit |Q4,+ |/Qmax ≪ 1 there is a term independent
on the order parameter:
2
vi2 Qmax

π x q vi
δσv,2,het,3 ≈ − − . (E118)
2J 22 Σme 25 Σme J
b. Homo-contribution
x4q Σ2 ∂ vi2
Z
1
σv,2,hom = 3 dθdδp Σ2 2 2 x2q
, (E119)
2 πγm6e ∂γ (δp2 + m 2 ) γ (δp(κii δp + Q) − − Σ2
)2 + (αΣ)2
+ κii )δp + Q)2
e m2e m2e m2e ((1

Q = 2pi q cos(θ) + q 2 . As in the hetero-contribution, it is plausible that the dominant contribution comes from
δp = iΣ/me pole. Hence we write:
x4q vi2
Z
∂
σv,2,hom ≈ 4 dθ 2  2  2 , (E120)
2 γm3e Σ ∂γ x2q

iΣ iΣ Σ2 αΣ iΣ
γ2 m e
(κii me + Q ii ) − m 2 − m2 + me me (1 + κ ii ) + Q ii
e e

which can be rewritten in the form:

x2q i vp2 x4q vp2 vp2 Q
Z Z Z
1
σv,2,hom = − 3 2 2 dθ + 7 3 3 dθ 2
− 5 3
dθ , (E121)
2 Σ me Q − Q4,+ 2 Σ me (Q − Q4,+ ) 2 Σ me Q − Q4,+
that goes in the Qmax /|Q4,+ | ≪ 1 limit as
x2q vi2 Qmax
σv,2,hom,3 ∝ − , (E122)
23 Jme Σ (x2q + Σ2 (1 + κii )2)
while in the large Qmax /|Q4,+ | ≫ 1 limit it follows the law
x2q vi2 π v 2 Qmax
σv,2,hom,3 ∝ − 5 2 2
− 5i , (E123)
2 Σ me J 2 Σme J
so that the whole difference between homo- and hetero-corrections comes from the Jacobian and Qmax .
 49

d. Vertex correction

Based on E 1 and E 2 it is clear that the R perturbative expansion for the correction with vertices having different
momentum consists of terms of the form vp vp+q Im((Gp Gp+q )m )Im((Gp Gp+q )n ) to allow for the momentum change
in the vertex. The sum

|xq |2
Z
σv = vp vp+q Im(Gp Gp+q )(Im(Gp Gp+q ) + |xq |2 Im((Gp Gp+q )2 + |xq |4 Im((Gp Gp+q )3 + ...)
2π
|xq |4
Z
+ vp vp+q Im((Gp Gp+q )2 )(|xq |2 Im((Gp Gp+q )2 + |xq |4 Im((Gp Gp+q )3 + ...) + ... (E124)
2π
can be rewritten as
! !
|xq |2
Z
1 1
σv = vp vp+q Im Im . (E125)
2π G−1 −1
p Gp+q − |xq |
2 G−1 −1
p Gp+q − |xq |
2

Evaluation can be done along the same steps: we divide the integration into 4 regions and use approximate values
for the poles. Note, however, that up to a factor and additional poles the generating function is the same. We also
saw that additional poles for DOS correction, as well as additional terms (with derivatives of the generating) are not
important at least in the limit of large mean-free path. We then simply use expressions from the previous section
with corrected multipliers.

Using now general expressions for Green’s functions we can rewrite (E125) in the form:

x2q 1 ∂ (xq Σ)2

Z
vp vp+q
σv = − 2 α 2 = gv , (E126)
2π 2αγ ∂α (ξp ξp+q − Σ2 − x2q )2 + γ 2 Σ2 (ξp + ξp+q )2 2α

where generating function is nearly identical to that we used in the DOS-section:

Z
1 vp vp+q
gv = − 2 2 , (E127)
2π(γΣ) (ξp ξp+q − Σ − x2q )2 + αγ 2 Σ2 (ξp + ξp+q )2
2

following the notation of the previous section, that would correspond to a term with numerator ⟨f ⟩⟨s⟩ = −m2e /Σ2 .
Within the constant velocity approximation, we can simply write answers using results of the previous section.
a. Hetero-contribution We then immediately fill the gaps for the hetero-contribution. For concreteness, start
with i-o proccess. In the hetero case, ξo,p+q = −me (p2 + 2pq cos(θ) + q 2 − p2o ) and ξi,p = me (p2 − p2i ). In terms of a
variable δp = p2 − p2i it can be written ξo,p+q = −me (δp(1 + pqi cos(θ)) + 2pi q cos(θ) + p2i + q 2 − p2o ) ≡ −me (κio δp + Qio )
and ξi,p = me δp. Since θc,i ≈ 0 we approximate vp+q ≈ pi + q. Now we use results of the previous section.

Also, for xq /me < Q(0)/(2κio ) exhibits all 3 possible behaviors, we consider only that.
Σκio
a) In the far-left region, Q > Qc + 4me

5/2 √ !
xq κio vi vo Q0 − Qc
σv,het,1 ≈ tan−1 , (E128)
Σm2e κio + 1 pi q sin(θc,i ) Qc
1/2

Σκio Σκio
b) In the middle-region, Qc − 4me < Q < Qc + 4me , within the constant poles approximation
   2   
q τD ∂
2 9/2 2α cos(θc,i ) + 2τ τ
2 ∂α C
xq τD κio (po /q) (2τC /τD )1/2  1 ∂ γpi C 
σv,het,2 ≈ − 2   2τC ∂α τC +
 2 
22 γ 2 sin(θc,i )
   
αq τD τD αq

− γpi cos(θc,i ) + 2τ C 2τC − γpi cos(θc,i )

(E129)
 2  2 1/4  2
α q
where again Σ
me κ2io,1 + Q2c α
γ κ2io,2 = (me τC )−1/2 , and κio,1 = 22 κio + cos(θc,i )2 , κio,2 =
γ pi
2q


22 κio + pi cos(θc,i ). And the derivative ∂τC /∂α = −τC3 αm2e /γ 2 Q2c κ2io,2 + (q/pi )2 cos(θc,i
2
)/(me τD )2
 50

Σκio
c) For the far-right region, Qc − 4me > Q:

13/4
1/2 κio (po /q) 1
σv,het,3 ≈ 2πx1/2
q τD . (E130)
sin(θc,i ) (κio + 1)1/2

b. Homo-contribution Changes in the story of homo-contribution are more intricate, since the angle where the
thing is peaked is not small generically. Using result of the previous section (E82), we write:

vp vp+q ⟨fv ⟩
Z
1 κii
gi ≈ − 5
. (E131)
2γαme Σ κii + 1 Q2 + Q2c

. For Qc ≪ Qmax we still should be able to use linear angle approximation. Then we immediately get:

x2q x2q κii vpi vp+q

i
Z
κii vpF vpF +q −1
 
σii ≈ − 2 =− tan Qmax / Q̃c
2γαm3e Σ κii + 1 Q2 + Q̃c 2γαm3e Σ κii + 1 Q̃c Jii
1/2
xq κii i i −1
 
≈− v p v p+q tan Qmax / Q̃c . (E132)
22 γαm2e ΣJii κii + 1

where Q̃c = 2(κii x2q /m2e + κii Σ2 /m2e )1/2 . Clearly, this correction is going to be ∝ τD . Or, without the approximation,
using an explicit form for velocities, I get:
 
I 2 2 2 2
2 κii dz  (z + 1) /q 2(z + 1)z/(pi q)
gi ≈ − 2 + qz  (E133)
γ αme Σ3 κii + 1 zi ( qz + (z 2 + 1)2 )2 + Q2c z22 ( + (z
2 2
2 + 1)2 )2 + Qc z
2
pi (pi q) pi (pi q)

. where the poles the same as in DOS section ((E87)). After substitution, I get:

22 π κii 1 π κii 1
gV = − − 2
γαme Σ3 κii + 1 q 2 2q γαme Σ3 κii + 1 −iQc + (D − D∗ )/2
((q 2 + iQc − D)2 + (2pi q)2 )2 ((q 2 − iQc − D∗ )2 + (2pi q)2 )2
 
−
D(−iQc + 21 (D + D∗ ))(−q 2 − iQc + D) D∗ (iQc + 12 (D + D∗ ))(−q 2 + iQc + D∗ )
 2 !
2 (q + iQc − D)2 + (2pi q)2 (q 2 − iQc − D)2 + (2pi q)2
+ (2q) − ,
D(−iQc + (D + D∗ )/2) D(iQc + (D + D∗ )/2)

so that the contribution to the conductance is

2πx2q κii 1 πx2q κii 1

σhom,v = +
γαme Σ κii + 1 q 2 (2q) γαme Σ κii + 1 −iQc + (D − D∗ )/2
2

((q 2 + iQc − D)2 + (2pi q)2 )2 ((q 2 − iQc − D∗ )2 + (2pi q)2 )2

 
−
D(−iQc + 21 (D + D∗ ))(−q 2 − iQc + D) D∗ (iQc + 12 (D + D∗ ))(−q 2 + iQc + D∗ )
 2 !
2 (q + iQc − D)2 + (2pi q)2 (q 2 − iQc − D)2 + (2pi q)2
+ (2q) − .
D(−iQc + (D + D∗ )/2) D(iQc + (D + D∗ )/2)