LU20113 is being considered for publication in Physical Review Letters.

Breaking of Lorentz invariance caused by the interplay between spin-orbit interaction and
transverse phonon modes in quantum wires
by D. V. Efremov, Weyner Ccuiro, Luis E. F. Foà Torres, et al.

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 Breaking of Lorentz invariance caused by the interplay between spin-orbit interaction and
transverse phonon modes in quantum wires

D. V. Efremov,1 Weyner Ccuiro,2, 3 Luis E. F. Foa Torres,4 and M. N. Kiselev3

1
IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany
2
International School for Advanced Studies (SISSA), via Bonomea 265, I-34136 Trieste, Italy
3
The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34151, Trieste, Italy
4
Departamento de Fısica, Facultad de Ciencias Fısicas y Matematicas, Universidad de Chile, Santiago, 837.0415, Chile
(Dated: July 24, 2024)
We investigate Lorentz invariance breaking in quantum wires due to Rashba spin-orbit interaction and trans-
verse phonons. Using bosonization, we derive an effective action coupling electronic and mechanical degrees
of freedom. Stikingly, at a quantum phase transition between straight and bent wire states, we find a gapped
phonon mode and a gapless mode with quadratic dispersion, signaling the breaking of Lorentz invariance. We
explore stability conditions for general potentials and propose nano-mechanical back-action as a sensitive tool
for detecting this transition, with implications for Sliding Luttinger Liquids and dimensional crossover studies.

Introduction and overview of key findings.– Physics shows

a surprising hierarchy of alternating Lorentz and Galilean
invariance, a fundamental principle governing the behavior
of objects moving at relativistic speeds. As we descend to
non-relativistic energies, the Lorentz invariance gives way
to Galilean invariance. Intriguingly, in condensed matter
physics, lower energy excitations once again obey Lorentz
invariance. A prime example is the quasiparticles in one-
dimensional systems [1, 2]. Here, electrons are described by Figure 1. Schematic representation of the experimental setup. The
the Luttinger liquid model [3–5] whose excitations are bosons, orange beam represents a suspended double clamped nanowire with
obeying Lorentz invariance. Similarly, the mechanical de- a displacement y(x). Green supports can also be used as electric
grees of freedom, described by phonons, also demonstrate contacts.
Lorentz invariance.
Nano-electromechanical systems have captivated re- nal magnetic field with the elastic modes of the wire was first
searchers for decades [6], offering a wide array of potential considered by Ahn and co-authors [28–30]. In this model, the
applications and research avenues. Their small size and high coupling of the electric charge Luttinger liquid with the elas-
sensitivity make them invaluable tools in novel microelec- tic modes of the nanowire occurs due to the Lorentz force as
tronics [7–9]. At the cutting edge are ultrathin membranes a result of the perpendicular displacements of the wire in the
and nanowires, fabricated from diverse materials including magnetic field. Here we show that the Rashba spin-orbit inter-
silicon [10, 11], metals [12], graphene [13], carbon nanotubes action couples the charge to the elastic modes of the wire via
[14], and many others [8]. These systems now operate in a spin current, which occurs due to transverse displacements.
the quantum-mechanical regime, where the behavior of the As a result of the interaction of the two corresponding Gold-
electronic component, when considered in isolation, is well stone modes, which correspond to the Luttinger liquid and the
understood. The most pronounced quantum-mechanical elastic modes, the charge transforms into a Higgs boson and
effects manifest in lower-dimensional systems, particularly a Goldstone mode with quadratic dispersion. This interplay
nanowires. It is well established that many one-dimensional leads to intriguing physical phenomena, including the break-
systems are accurately described by the Luttinger liquid ing of Lorentz invariance under specific conditions. In the
model [1, 2, 15, 16]. Theoretical [17–20] and experimental following we present our Hamiltonian model, derive an effec-
studies [21, 22] also point out Luttinger liquid physics in tive action using bosonization techniques, and explore the re-
dimensions larger than one, with recent experiments in sulting mode structure and stability conditions under various
twisted bilayer tungsten ditelluride (tW T e2 ) [23] which potential parameters.
inspired further theory efforts [24]. However, to further Hamiltonian model.– The key component of the model con-
advance these devices, it is crucial to comprehend the quan- sidered here is the Rashba spin-orbit interaction arising from
tum effects arising from the coupling between mechanical the external potential. It is described by the following Hamil-
motion and electric current. Recent interest has also focused tonian for a single electron:
on the interactions between electrons and bosonic modes
in cavities [25–27], expanding our understanding of these p̂2x
Ĥ = + αso (p⃗ˆ × ⃗σ ) · ∇U (y, z) + U (y, z), (1)
systems. 2m
The coupling of the electrons of the nanowire in an exter- where p̂x is the momentum operator, m stands for the electron
 2

mass, αso denotes the Rashba spin-orbit coupling strength, where v = vF /g is the electronic velocity and vF is Fermi
⃗σ represents the Pauli matrices which describe the electron’s velocity. For short-range interaction V (x − x′ ) = V δ(x − x′ ),
spin, and U (y, z) is the spatially dependent confining po- g is given by g = (1 + V /(πvF ))−1/2 . For simplicity we
tential. It’s important to note that the electric field E ⃗ = assume g = 1. The full expression can be easily recovered.
−∇U (y, z), representing the gradient of the potential, is non- The mechanical degrees of freedom is given by the elastic
vanishing due to the absence of inversion symmetry in the sys- response of the wire on the transversal strain. We study a one-
tem. dimensional suspended double clamped beam of mass density
Following [31] we assume the external potential as ρm and stiffness T extending along the x direction. We as-
sume that y0 (x) = z0 (x) = 0 corresponds to equilibrium.
U (y, z) = by y 2 /2 + bz z 2 /2 + cyz. (2)
The mechanical part is related to the strain in the beam, which
By a special choice of the potential U (y, z) it is possible to is associated with small transverse displacements along the y
bring the single-particle Hamiltonian Eq. (1) into a quadratic and z axes from the equilibrium position: uy (x) = y(x) − y0
form, similar to the model of N electrons moving in a syn- and uz (x) = z(x)−z0 . The corresponding energy depends on
thetic magnetic field and described by the Hamiltonian h = the gradient of the displacements, and its behavior is captured
1/(2m)(p̂x − Ax )2 . One should note, however, that this syn- by the Hamiltonian:
thetic magnetic field is not breaking time-reversal symmetry
2 !
and therefore preserves Kramers’ degeneracy. In our problem, πζ2
X Z 
T ∂uζ
it is realized with the following conditions Hu = dx + (6)
2ρm 2 ∂x
p ζ=y,z
2 −1 2 )−2 − 4c2 ,
2by,z = (mαso ) ± (mαso (3)
where c is kept as a free parameter. Under this condition the Here πζ (x) are the conjugated momenta of uζ (x), ζ =
Hamiltonian takes the following form: y, z, satisfying the commutation relations [uζ (x), πζ ′ (x′ )] =
iδ(x − x′ )δζζ ′ .
1 2
Ĥ = (p̂x + mαso [(bz z+cy)σy −(by y+cz)σz ]) . (4) Effective action.– Using the canonical transformation we
2m get the action from the Hamiltonian. It takes the form:
To demonstrate the basic properties of the dynamical model
with a synthetic magnetic field, we choose √ by = bz = c = S = SLL + Su + SLL−u
2 −1
(2mα√ so ) . Rotation in the spin (σ y − σz ) 2 →√ σz , (σy + Z ∞
v X β
Z
σz ) 2 → 2 → y, dx v −2 (∂τ ϕα )2 +(∂x ϕα )2


σ
√ y and the coordinate space (z + y)/ SLL = dτ
(y − z)/ 2 → z simplify the equation to the form: 2πg α=± 0 −∞
2 Z β Z ∞
1

1 T X
dx vs−2 (∂τ uη )2 + (∂x uη )2


Ĥ = p̂x + σz y . (5) Su = dτ
2m αso 2 η=y,z 0 −∞
Z ∞
iαso −1 β
Z
With this the electronic part of the Hamiltonian can be writ-
SLL−u = − dτ dx(∂τ ϕ+ − ∂τ ϕ− )uy .
ten as π 0 −∞
Z  2 (7)
1 X 1
He = dxΨ†α (x) p̂x + uy σαz Ψα (x)
2m αso
α,β=± vs is the phonon sound velocity and β = 1/T . Here we
Z
1 would like to note that the last term corresponds to the cou-
+ dxdx′ V (x − x′ )ρ(x)ρ(x′ ),
2 pling of the transverse displacement to the spinor current
e
where α = ±1 are the eigenstate indexes of the diagonal- J = mcα so
(∂t ϕ+ − ∂t ϕ− ), which emerges due to deviation
ized Hamiltonian and uy (x) = y(x) − y0 is y component of the beam from the equilibrium. It is worth to introduce new
of the transverse displacements from the equilibrium posi- fields ϕc + ϕs :
tion and y0 is the equilibrium position (see Fig. 1). The
1 1
second term
P describes the density-density interaction. Here ϕc = √ (ϕ+ + ϕ− ), ϕs = √ (ϕ+ − ϕ− )
ρ(x) = α=± Ψ†α (x)Ψα (x) and V (x − x′ ) is the electron- 2 2
electron interaction. We note that there are other possible
consequences of spin-orbit interaction and its interplay with which describe the charge and spin degrees of freedom respec-
electron-electron interactions such as those considered in [32] tively.
which are beyond our present scope. Using the Fourier transformations of ϕα (x, τ ) and uζ (x, τ )
After linearization of the spectra close to the Fermi-level
and the standard procedure of bosonization the Hamilto- 1 X ikx−iωn τ
ϕα (x, τ ) = √ e ϕknα (8)
nian [1, 2] takes the following form: βL k,n
X v Z
−1
1 X ikx−iωn τ
σz uy )2 + g −1 (∂x ϕs )2 uζ (x, τ ) = √


He = dx g(∂x θs − αso e uknζ (9)
2π βL k,n
s=±
 3

5
to the following conditions for stability of the system:
4
2
ω+
by − mαso (b2y + c2 ) > 0, 2
bz − mαso (b2z + c2 ) > 0
3
2
(by + bz )))2 < (bz − mαso
2
(b2z + c2 )) (12)
ω(k)
(c(1 − mαso
2 ω = vs k 2
ω=vk ×(by − mαso (b2y + c2 ))
1 ω-

0
We introduce the small transverse displacements as z−z0 =
0 1 2 3 4 r cos φ and y − y0 = r sin φ. Rotation in spin space we get
k
1 2
Ĥ = (p̂x + Λrσz ]) + Ũ (13)
Figure 2. Eigenmodes ω± (k) (solid lines). The dashed lines denote 2m
the bosonic mode for electron ω = vk, phonon ω = vs k.
with

Λ(φ) 1/2
= (by sin φ + c cos φ)2 + (bz cos φ + c sin φ)2

the Euclidean action can be written as .
(mαso )
X X X v  ω2 
S= n
2
+ k ϕ∗nk,α ϕnk,α
2
The effective potential is
2πg v
k ωn α=c,s
X X X T  ω2 r2

+ n
+ k u∗nk,ζ unk,ζ
2
(10) Ũ = (by sin2 φ + bz cos2 φ + c sin(2φ))
2 v 2
s
2 (14)
k ωn ζ=y,z
√ r2 Λ2 (φ) K(φ)
X X X 2α−1 ωn − = r2
so
unk,y ϕ∗nk,s − u∗nk,y ϕnk,s 2m 2


+
ω
2π
k n ζ=y,z
with
Within this action we have four modes. Two modes are  !2 1/2
decoupled. The decoupled modes are the transverse phonon b̃z + b̃y  b̃z − b̃y
K(φ) = + + c̃2  cos(2φ − φ0 )
mode along z - direction with dispersion ω = vs k and the 2 2
charge mode of the Luttinger liquid with dispersion ω =
vk. The spin mode of the Luttinger liquid couples with − c2 mαso
2
,
the transverse phonon mode along the y direction. It gives (15)
2
a gapped transverse phonon mode with dispersion ω+ =
q 2c̃
1 2 2 2 2

2 where tan φ0 = and
2 (vs + v )k + Λ + ((vs2 + v 2 )k 2 + Λ2 ) − vs2 v 2 k 4 , b̃z −b̃y
−2 2
where Λ = 2πvF ραso and a gapless mode ω− = 2 2
1

q
2
b̃z = bz (1 − mαso bz ), b̃y = by (1 − mαso by )
2 2 2 2 (16)
2 (vs + v )k + Λ − ((vs2 + v 2 )k 2 + Λ2 ) − vs2 v 2 k 4 . 2
The gapless mode reflects the breaking of the Lorentz invari- c̃z = c(1 − mαso (bz + by ))
ance of the bosonic mode of the Luttinger liquid. Instead of
We choose the new axes along the direction 2φ = φ0 and
the linear dispersion, which corresponds to the Lorentz invari-
2φ = φ0 + π and y ′ and z ′ . The minimum of the effective
ance of the Luttinger liquid, the dispersion is ω− ∼ k 2 , which
potential corresponds to the direction of z ′ -axis. The effec-
corresponds to the Galilean invariance. The spectrum of the
tive external potential Eq.(14) depends on the transverse dis-
obtained quasiparticles are shown in Fig. 2
placements. It leads to the renormalization of the action of the
General case.– Previously, we examined an external poten-
mechanical degrees of the beam. If K(φ0 /2+π/2) = 0 is the
tial subjected to the constraint Eq. (3). Now we explore the
quantum critical point (see Fig. 3). The condition Eq. (3) cor-
effect the case without the constraint.
responds to K(φ0 /2+π/2) < 0. For K(φ0 /2+π/2) > 0 the
1 2
vacuum is trivial while for K(φ0 /2 + π/2) < 0 one needs to
Ĥ = (p̂x + mαso [(bz z+cy)σy −(by y+cz)σz ]) + Ũ (y, z) rederive the effective action with non-trivial vacuum uy0 ̸= 0.
2m
Z β Z ∞
where T X
Su = dτ dx
2 −∞
2
mαso ζ=y ′ ,z ′ 0
(17)
Ũ (y, z) = U (y, z) − [(by y + cz)2 + (bz z + cy)2 ] (11) 
K(φζ ) 2

2 × vs−2 (∂τ uz′ )2 + (∂x uz′ )2 + uz′ ,
T
Negative Ũ (y, z) leads to the shift of the position of the equi-
librium of the beam, uy,z (0) ̸= 0. A simple calculation lead where φy′ = φ0 /2 and φz′ = φ0 /2 + π/2.
 4

Using the standard procedure we come to the action

X X X v  ω2 
S= n
2
+ k 2
ϕ∗nk,α ϕnk,α
2πg v
k ωn α=c,s
X X X T  ω2 K(φζ )

+ n
2
+ k 2
+ u∗nk,ζ unk,ζ (18)
ω ′ ′
2 v s T
k n ζ=y ,z
X X X Λ(φζ )ωn
unk,ζ ϕ∗nk,s − h.c .


+
ω ′ ′
2π
k n ζ=y ,z

The potential strongly renormalizes the phonon modes.

Negative K(φ) is attributed to a non-trivial vacuum of the
system while the vacuum for positive K(φ) is trivial. In both
cases, away from the quantum critical point, Lorentz invari-
ance is preserved and the spectrum of the lowest energy Gold-
Figure 3. Schematic phase diagram of possible coefficients b = bz,y
stone mode ω ∝ k. The Lorentz invariance is broken at the vs c. The critical region is in blue. The outside grey area corresponds
quantum critical point resulting Galilean invariant Goldstone to the region of instability. The inset represents the same transition
mode being ω ∝ k 2 . as a function of λ = K(ϕ0 /2 + π/2). Here vG is a sound velocity
Correlation functions To see the consequences of of the Goldstone mode ω− which vanishes at the quantum critical
the coupling between the electronic and mechanical de- point.
grees of freedom, we calculate the correlation functions
⟨uz (q, ωn )uz (q, ωn )⟩ and ⟨ϕc (q, ωn )ϕc (q, ωn )⟩. To focus on
the most important features of the coupling assuming that the The mechanical back action information is important as the
stiffness of the mechanical degrees of freedom is strongly Rashba spin-orbit interaction can be responsible for opening
anisotropic, leading to the survival of only one of the mechan- a gap in the charge bosonic correlation function which in turn
ical modes. This assumption results in a linear polarization of will influence the elastic Goldstone modes. Yet another inter-
the mechanical motion. The general case will be considered esting questions related to behavior of either driven or excited
elsewhere. mechanical system are associated with effects of controllable
Using the standard procedure of integration of complimen- non-linearity of elastic modes (interaction effects), possibility
tary degrees of freedom we get of parametric amplification and dissipative dynamics associ-
ated with mechanical quality factor (friction).
1 (ωn2 + v 2 k 2 ) Final remarks and conclusions.– The solution of a single-
⟨u∗kωn ξ ukωn ξ ⟩ = (19) wire model represents a very important step in understanding
ρm D
of the behavior of array of quantum wires coupled either by
and tunneling or by interaction (mechanically driven Sliding Lut-
tinger Liquids (SLL). We summarize below the key findings,
πvF (ωn2 + vs2 k 2 + K/ρm )
⟨ϕ∗kωn ,s ϕk,ωn ,s ⟩ = (20) theoretical and experimental challenges and relevance of the
D prediction to the state-of-art nano-mechanical and quantum
where transport experiments.
Possible experimental realization. The InSb and InAs
D = ((ωn2 + v 2 k 2 )(ωn2 + vs2 k 2 + K/ρm ) + ω̃ 2 ωn2 ) (21) high mobility nanowires [21, 22] are some favorable mate-
rials for proposed experiment. The bulk g-factor |g| ∼ 50
and ω̃ 2 = Λ2 vF /πρm . in InSb is a hallmark for strong Rashba spin-orbit interac-
Performing analytical continuation we get the correspon- tion. The ”state-of-art” for InSb fabrication [21] is ∼ 10-
dent response functions. The imaginary parts of the response microns long wires of 80-100 nm diameter. The large bulk
functions are shown in Fig. 4. electron’s mobility of 7.7 × 104 cm2 /V S at T = 300K
Mechanical back action as a tool for quantum spectroscopy and large Rashba parameter αR = 0.45 − 0.64eV Å makes
of the Luttinger liquids. One sees by comparing plots of me- it a very promising material for the nano-mechanical setup.
chanical correlation functions on Fig. 4 at (a), b) and away While the g-factor of bulk InAs |g| ∼ 10 and typical mobil-
(e), f) the QCP that the character of the Goldstone mode spec- ity ∼ 1 − 2 × 104 cm2 /V S [22] is not as high as of InSb,
trum (quadratic vs linear) and the re-distribution of the low this material can also be used for proposed experiments.
frequency spectral weight allows to identify the position of From a single wire to few coupled quantum wires: the next
the QPT (see Fig. 3 insert) with a high accuracy even without crucial step towards Sliding Luttinger Liquids (SLL) and di-
measuring the quantum transport correlation functions (c),g) mensional crossover from 1D to quasi-2D quantum field the-
and (d),h) which provide complementary information about ories involves solving the two-wire problem coupled by tun-
corresponding Goldstone spectra and the spectral weights. neling or interaction. A particularly intriguing question, both
 5

(a) (b) (c) (d)

(d
)

(e) (f) (g) (h)

( (
e f
) )

Figure 4. Correlations functions a) Im⟨u∗y (k, ω)uy (k, ω)⟩; b) ω 2 Im⟨u∗y (k, ω)uy (k, ω)⟩; c) Im⟨ϕ∗s (k, ω)ϕs (k, ω)⟩; d)
ω 2 Im⟨ϕ∗s (k, ω)ϕs (k, ω)⟩ at the the position of the quantum critical point. Plots e) - h) show the same correlations functions as in a)-c)
away from the quantum critical point (see the phase diagram on Fig. 3). The parameters are vs = 1, vF = 2, ρ = 1, Λ = 1. Parameter K = 0
for a) - d) and K = 4 for e) - h).

theoretically and experimentally, relates to the quantum noise coupled to the chiral mode at the edge.
associated with injecting charge or spin current in one wire In summary, we have analyzed a nano-mechanical system
and measuring the noise in the second. This effect is ex- consisting of a double-clamped suspended metallic nano-wire
pected to be analogous to the quantum noise observed in subject to an external dc electric field. The transverse flexu-
co-propagating edge modes in Integer Quantum Hall (IQH) ral elastic modes of the wire couple to the electronic charge
ν = 2 devices [33]. By adjusting the electric circuit, one can degrees of freedom, with the coupling strength controlled by
also model counter-propagating modes. The two-wire model the Rashba spin-orbit interaction in the wire. We derived the
serves as a minimal system for studying quantum interference hydrodynamic action of the model and demonstrated that it
effects controllable by interaction. is described by bosonic Gaussian theory. Our analysis re-
Nano-mechanically coupled Sliding Luttinger Liquids of- veals a quantum phase transition between two gapless phases
fer a promising avenue for investigating the interplay be- with trivial and non-trivial vacua, corresponding to straight
tween electronic and elastic degrees of freedom in nano- and bent equilibrium positions of the wire, respectively. We
mechanically driven Integer and Fractional Topological Insu- showed that the dispersion of the softest Goldstone mode tran-
lators. This model enables the study of 2D metal-insulator sitions from sound-like to quadratic at the quantum phase
quantum phase transitions (QPTs) in bulk and the emergence transition point. The breaking of Lorentz invariance at the
of gapless chiral modes at the edge. The simplest model de- QPT is attributed to the current-displacement character of the
scribing a transition to the Integer Topological Insulator state, boson-boson interaction. Our findings suggest that measur-
analogous to the transition to the IQH regime in a synthetic ing the back-action of the nano-mechanical system provides a
magnetic field [15], can be formulated by accounting for tun- highly sensitive and efficient spectroscopic tool for detecting
neling between neighboring wires and adjusting the chemical the QPT position, complementing quantum transport experi-
potential to the position of the bulk gap [31]. In the Frac- ments. We hope that our proposal may trigger new experi-
tional Topological Insulator state, the bulk gap opens due to ments with suspended nanowires.
inter-wire interaction effects [31]. This allows for the explo- We thank B.L. Altshuler, Ya. Blanter, A.Chubukov,
ration of various bulk instabilities driven by phonons and their Y.Gefen, Th. Giamarchi, D. Maslov, C. Mora, F. von Op-
influence on chiral edge modes. Additionally, elastic modes pen, S. Sachdev and J. Schmalian for fruitful discussions. The
associated with vibrations of a membrane in a Corbino disk work of M.N.K is conducted within the framework of the Tri-
geometry provide access to modeling 2D transverse phonons este Institute for Theoretical Quantum Technologies (TQT)
 6

and supported in part by the NSF under Grants No. NSF of the Elastic Properties and Intrinsic Strength of Monolayer
PHY-1748958 and No. PHY-2309135. M.N.K. acknowledges Graphene,” Science 321, 385 (2008).
support of the IHP (UAR 839 CNRS-Sorbonne Universite) [14] S. Raychaudhuri and E. T. Yu, “Critical dimensions in coher-
and LabEx CARMIN (ANR-10LABX-59-01) L.E.F.F.T. ac- ently strained coaxial nanowire heterostructures,” Journal of
knowledges financial support by ANID FONDECYT (Chile) Applied Physics 99, 114308 (2006).
[15] C. L. Kane and M. P. A. Fisher, “Transport in a one-channel
through grant number 1211038, The Abdus Salam Interna- luttinger liquid,” Phys. Rev. Lett. 68, 1220 (1992).
tional Center for Theoretical Physics and the Simons Founda- [16] H. J. Schulz, G. Cuniberti, and P. Pieri, “Fermi Liquids and Lut-
tion, and by the EU Horizon 2020 research and innovation tinger Liquids,” in Field Theories for Low-Dimensional Con-
program under the Marie-Sklodowska-Curie Grant Agree- densed Matter Systems (Springer, 2000).
ment No. 873028 (HYDROTRONICS Project). D.V.E. ac- [17] P. W. Anderson, “Hall effect in the two-dimensional Luttinger
knowledges the financial support of DFG (grant numbers liquid,” Physical Review Letters 67, 2092 (1991).
529677299, 449494427) and hospitality of Abdus Salam In- [18] X. G. Wen, “Metallic non-Fermi-liquid fixed point in two and
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