SCIENCE CHINA

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. Invited Review . May 2020 Vol. 63 No. 5: 250001

https://doi.org/10.1007/s11433-019-1477-7

Holographic topological semimetals

Karl Landsteiner1* , Yan Liu2,3* , and Ya-Wen Sun4,5
1 Instituto
de Fı́sica Teórica UAM/CSIC, C/ Nicolás Cabrera 13-15, Campus Cantoblanco, Cantoblanco 28049, Spain;
2 Center for Gravitational Physics, Department of Space Science, Beihang University, Beijing 100191, China;
3 Key Laboratory of Space Environment Monitoring and Information Processing, Ministry of Industry and Information Technology,

Beijing 100191, China;

4 School of Physics & CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences,

Beijing 100049, China;

5 Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received October 14, 2019; accepted November 8, 2019; published online February 25, 2020

The holographic duality allows to construct and study models of strongly coupled quantum matter via dual gravitational theories.
In general such models are characterized by the absence of quasiparticles, hydrodynamic behavior and Planckian dissipation
times. One particular interesting class of quantum materials are ungapped topological semimetals which have many interesting
properties from Hall transport to topologically protected edge states. We review the application of the holographic duality to
this type of quantum matter including the construction of holographic Weyl semimetals, nodal line semimetals, quantum phase
transition to trivial states (ungapped and gapped), the holographic dual of Fermi arcs and how new unexpected transport properties,
such as Hall viscosities arise. The holographic models promise to lead to new insights into the properties of this type of quantum
matter.
gauge/gravity duality, topological semimetal, Weyl semimetal, anomaly
PACS number(s): 11.15.-q, 04.62.+v, 11.30.Rd, 03.65.Vf, 67.55.Hc

Citation: K. Landsteiner, Y. Liu, and Y.-W. Sun, Holographic topological semimetals, Sci. China-Phys. Mech. Astron. 63, 250001 (2020),
https://doi.org/10.1007/s11433-019-1477-7

Contents

1 Introduction 250001-2

2 A short review on the holographic duality 250001-3

3 The holographic Weyl semimetal 250001-4

3.1 Weyl semimetal from Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-4
3.2 Holographic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-5
3.3 Anomalous Hall conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-7

*Corresponding authors (Yan Liu, email: yanliu@buaa.edu.cn; Karl Landsteiner, email: karl.landsteiner@csic.es)

⃝
c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 phys.scichina.com link.springer.com
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-2

3.4 Universality of the quantum phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-7

3.5 Holographic Fermi arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-8
3.6 Fermionic probes and the topological invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-9
3.7 Finite temperature, conductivities and viscosities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-10
3.8 Axial Hall conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-13
3.9 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-14
3.10 AC conductivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-14
3.11 Butterfly velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-15

4 Weyl semimetal/Chern insulator transition 250001-15

5 Holographic nodal line semimetals 250001-18

5.1 Quantum field theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-18
5.2 Holographic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-19
5.3 A generic framework for topological states from hologrphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-21
5.4 Fermionic probe on the holographic nodal line semimetal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-21
5.5 Topological invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250001-22

6 Alternative approaches 250001-22

7 Summary and outlook on further research 250001-23

1 Introduction as the effect of non-vanishing Berry flux through the Fermi

surface, and chirality manifests itself as monopole like singu-
Weyl semimetals are a very interesting new form of gapless larity of the Berry connection [8]. These notions are of course
topological quantum matter [1-5]. What makes them special bound to the validity of a quasiparticle picture in which inter-
is that the electronic excitations behave in a very unusual way. actions play a subordinate role. The question arises then if
The electronic quasiparticles can be described by the Weyl the salient features of Weyl semimetals do persist in a strong
equation known from high energy physics. In the massless coupling context in which there are no clear quasiparticle ex-
limit the Dirac equation can be decomposed into two irre- citations.
ducible parts, called Weyl equations which differ by chiral- The holographic duality (also known as AdS/CFT corre-
ity. In this way the electronics of Weyl semimetals is gov- spondence and gauge/gravity duality) has been used over the
erned by the laws of relativistic physics. One of the corner- last decade and a half as a tool to investigate precisely this
stones of relativistic quantum field theory is that the concept type of questions. No attempt will be made here to give
of chirality is sometimes incompatible with quantum theory. a concise account of the workings of the holographic dual-
Whereas classically the number of left- and right-handed chi- ity. There are excellent reviews available in refs. [9-13]. In
ral fermions is separately conserved, quantum mechanically fact the holographic duality has provided already outstand-
this is no longer true. This is the so-called chiral anomaly ing results. The modern understanding of hydrodynamics as
[6, 7]. Since the electronics of Weyl semimetals is governed a derivative expansion and its validity is put to an all new
by the chiral Weyl equation the chiral anomaly also has pro- and sound footing using insights from holography [14, 15].
found consequences on the physics of these materials. Many Anomaly induced transport properties [16, 17] such as the
of the most exotic and interesting phenomena such as the the chiral magnetic and chiral vortical effects and their relations
appearance of surface states (Fermi-arcs) or exotic transport to anomalies can be most easily understood with the means
phenomena such as the Hall effect are directly linked to the of holographic duality [18, 19]. The reader is reminded of
chiral anomaly. Much of the physics of Weyl semimetals can the conceptual difficulties in the interpretation of the formula
be understood by analyzing the properties of the one-particle for chiral magnetic effect. This is in sharp contrast with the
wave function. For example the anomaly can be understood clarity of its theory in holography. It is by now well-known
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-3

that the chiral magnetic effect vanishes in equilibrium but it is symmetric gauge theory with gauge group S U(Nc ) in four di-
probably not universally acknowledged that this has been cal- mensions [24]. String theory needs ten dimensions and that is
culated first in holographic models [20]. Another example is why there is the compact five sphere. The isometry group of
the direct connection between the temperature dependence of this five dimensional sphere is SO(6). In the dual field theory
the chiral vortical effect and the gravitational contribution to this is the internal global symmetry of the N = 4 gauge the-
the chiral anomaly [21, 22]. Therefore holographic duality is ory. The metric of the five dimensional anti de-Sitter (AdS)
not only interesting because of its inherently strongly coupled space is
nature but also because it gives valuable insight via the holo-
r2 ( 2 ) L2
graphic perspective on conceptually difficult problems such ds2 = −dt + dx 2
+ 2 dr2 , (1)
as anomaly induced transport. This provides more than suffi- L2 r
cient theoretical motivation for studying holographic models where 1/L2 is a measure for the curvature of the AdS space.
of Weyl semimetals. It is also noteworthy that hydrodynam- The N = 4 supersymmetric gauge theory with gauge group
ics and thus strongly coupling behavior has been reported for S U(Nc ) in four dimensions is characterized by two physical
the Weyl semimetal WP2 [23]. It is possible that holographic parameters the Yang-Mills coupling gYM and the rank Nc of
models can serve as models for such type of materials. As we S U(Nc ). On the dual string theory side, there are two param-
will review here, also in the case of Weyl semimetals holog- eters, the fundamental length scale ls and the string coupling
raphy is able to provide new perspectives, leading to new di- gs (the amplitude for a string to split in two). The AdS5 ge-
rections of research and even allowing the discovery of new, ometry has curvature R = −20/L2 where L is an AdS radius
possibly unexpected transport phenomena. scale. These parameters are related in the AdS/CFT corre-
The review is organized as follows. In sect. 2 we give a spondence as the following way:
flash review of the holographic duality. Then we introduce
the holographic model of a (time reversal symmetry break- L4
g2YM Nc ∝ , (2)
ing) Weyl semimetal in sect. 3. The most important results ls4
stemming from working with this model are also reviewed. 1/Nc ∝ gs . (3)
These include the existence of a quantum phase transition be-
tween the Weyl semimetal and a topological trivial state, cal- From the above relations, we can see that the AdS/CFT cor-
culation of the Hall conductivity, the calculation of topologi- respondence is a strong weak duality. From eq. (2), for weak
cal invariants via fermionic holographic spectral functions, fi- curvature on the string theory side the AdS radius L is large
nite temperature and viscosities, in particular the appearance which indicates a large ’t-Hooft coupling constant on the field
of Hall viscosity at the critical point of the quantum phase theory. In this parameter regime we can neglect the stringy ef-
transition, the calculation of the axial Hall conductivity, the fects and use type IIB supergravity to approximate the string
effects of disorder and the properties of quantum chaos across theory. If we further take the rank Nc of the gauge group to
the quantum phase transition. be very large, i.e., the large Nc limit, for the string theory gs
All these results are obtained in models that show a tran- is very small, we can ignore the quantum loop effects and end
sition between the Weyl semimetal and a trivial semimetal. up with the theory of classcal supergravity!
In sect. 4 a new model is introduced in which the transition From the above analysis, we found that the classical
is between the Weyl semimetal and an insulating state. In (super-)gravity on AdSd+1 space is the infinite coupling and
sect. 5 a generalization to holographic nodal line semimet- infinite rank limit of a gauge theory in d dimensions. This
als is discussed. We briefly point to alternative approaches is known as the AdS/CFT correspondence in its most useful
of applications of the holographic duality to the physics of form for applications to quantum many body physics. Here
Weyl semimetals in sect. 6. In sect. 7 we first briefly summa- we have allowed ourselves to be already a bit more general.
rize and then give an outlook on possible interesting future Once we have understood the original example based on the
directions of research. maximally supersymmetric four dimensional field theory, we
conjecture that every gravitational theory with some addi-
tional suitably chosen matter fields on AdSd+1 is dual to a cer-
2 A short review on the holographic duality tain d dimensional quantum field theory. We take this point
of view in the applications of the AdS/CFT correspondence
We now give a flash review on the holographic duality [9-12]. to quantum many body systems. The additional matter fields
The origin of the holographic duality lies in string theory. In chosen on the gravity side is according to a particular sym-
its original form it states that a certain type of string theory metric property of the underlying quantum field theoretical
(type IIB) on the space AdS5 × S5 is dual to N = 4 super- system that one is interested in.
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-4

The dual field theory lives in the four dimensional space- the equations of motion for fields. The generating functional
time parametrized by (t, x), where x denotes the vector of log Z can be simplified and computed by the classical action
spatial coordinates (x, y, z). It is sometimes said that the dual evaluated on the classical solution. In this case in the asymp-
field theory lives on the the boundary of AdS space when tak- totic expansion eq. (4) the coefficient Φ1 (x) is the vacuum
ing the limit r → ∞. But this is not really true, all of the bulk expectation value of the dual operator sourced by Φ0 :
has a field theory interpretation. The best way of thinking
⟨O(x)⟩ ∝ Φ1 (x). (7)
about the “holographic” direction is as an energy scale. The
high energy limit of the theory is given by r → ∞ and vice This scheme applies to all fields in AdS, also to the met-
versa the infrared limit is r → 0. This allows a direct geo- ric itself. The operator that corresponds to the metric is
metric interpretation of the renormalization group flow of a the energy-momentum tensor. In the same way the opera-
holographic theory from the ultraviolet (UV) to the infrared tor that corresponds to a gauge field in AdS is a current. The
(IR). essentials of the holographic dictionary are summarized in
The other important ingredient of the holographic dictio- Table 1.
nary, the rules that allow us to extract field theory information One can use this dictionary to generate new solutions that
from gravitational physics is the identification between fields are deformations of the simple AdS space (eq. (1)) by switch-
in AdS and operators and couplings of the dual field theory. ing on certain couplings. In practice this means that one de-
If we consider the solution to a (second-order) field equation mands specific boundary conditions on suitably chosen AdS
in AdS space it allows an expansion for large r of the form: fields that represent couplings in the dual field theory. Let us
( now explain how this strategy can be implemented to obtain
1 ( 1 )) 1
( ( 1 ))
a holographic version of a Weyl semimetal.
Φ = ∆ Φ0 (x) + O 2 + ∆ Φ1 (x) + O 2 . (4)
r − r r + r

We assume there that ∆± ≥ 0 and ∆− < ∆+ . Φ0 (x) is the 3 The holographic Weyl semimetal
boundary value (non-normalizable mode) of the field Φ(r, x)
in AdS space and at the same time it is interpreted as a cou- To find the holographic background solution we first must
pling or source for an operator in the dual field theory. When identify what kind of deformations we need to introduce
we do the path integration over the fields in AdS, we have in AdS space to mimic the essential features of a Weyl
to keep the boundary values Φ0 (x) fixed. What we obtained semimetal. In order to do so we first review quickly a quan-
finally is a functional Z[J] of the form: tum field theoretical model of a Weyl semimetal.
∫
Z[J] = [dΦ] exp(−iS [Φ]), (5) 3.1 Weyl semimetal from Dirac equation
J(x)=Φ0

where the source J(x) is the boundary field Φ0 (x). This In Weyl semimetal, the physics around the nodal points can
source J(x) couples to a (gauge invariant) operator O(x) be described by a quantum field theoretical model which
with conformal dimension ∆+ in the field theory1) . Perform- takes the following form of a “Lorentz breaking” Dirac equa-
ing functional differentiation of eq. (5) with respect to the tion [26, 27]:
( )
sources, we can obtain the connected correlation functions of i∂/ − eV/ − γ5 γ · b + M ψ = 0 , (8)
the gauge invariant operators O(x) in the quantum field the-
where X/ = γµ Xµ with Xµ ∈ {∂µ , Vµ }, Vµ is the electromagnetic
ory:
gauge potential, γµ is the Dirac matrices, and γ5 = iγ0 γ1 γ2 γ3 .
δn log Z We can define left- or right-handed spinors via (1 ± γ5 )ψ =
⟨O1 (x1 ) · · · On (xn )⟩ = . (6)
δJ1 (x1 ) · · · δJn (xn ) ψL,R . The axial gauge field b breaks the time reversal

These are of course rather formal expressions. In general one

does not know how to do this type of path integral including Table 1 The essentials of the holographic dictionary

the metric degrees of freedom or even the proper string the- Field in AdS Dual operator

ory dual. In the large Nc and large coupling g2YM Nc limit, Metric gµν Energy-momentum tensor T µν

the gravitational theory becomes classical. The path inte- Gauge field Aµ Current J µ
Scalar field Φ Scalar operator O
gral eq. (5) now is dominated by the classical solutions from
∫
1) This choice is known as standard quantization. When d2 ≤ ∆+ ≤ d2 + 1, for the dual field theory we could add a double trace deformation dd xO(x)2
which is irrelevant close to the fixed point, to generate a flow to a new fixed point. In this case ∆+ and ∆− exchange their roles, i.e., Φ1 (x) is now interpreted as
the source J(x) which couples to an operator O(x) of conformal dimension ∆− . This is known as alternative quantization of the bulk theory [25].
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-5

symmetry and is introduced to separate the Weyl points in as the order parameter2) of this special topological phase tran-
the momentum space as we will show from the perspectives sition [34]. In more general case, additional massless Dirac
of energy spectrum. For simplicity we take b = bez . M is the fermions might show up, and the topologically trivial phase
mass of the Dirac field. might be a semimetal instead of a gapped trivial phase. Then
The energy spectrum of eq. (8) is shown in Figure 1. When this quantum phase transition goes from a topologically non-
|b| > |M| the spectrum is ungapped. There is a band inversion trivial semimetal to a trivial semimetal. This will be exactly
in the spectrum and at the crossing points the wave function the case of our holographic model in the next subsection. In
is described by the one of Weyl fermions. The separation
√ of sect. 4 we will improve on this and discuss a holographic
the Weyl points in momentum space is given by 2 b2 − M 2 model with a phase transition to a Chern insulator.
along the direction indicated by the vector b. At low energies The anomalous Hall effect (eq. (10)) in the quantum field
it is described by the the effective theory with√the Lagrangian theory is obtained from a one-loop contribution to the polar-
of the form eq. (8) with Meff = 0 and beff = b2 − M 2 ez . For ization tensor. However, there are infamous regularization
|b| < |M| the system is gapped with gap 2Meff = 2(|M| − |b|). ambiguities [35] in the quantum field theory. There are some
The axial anomaly ways to resolve the ambiguity, e.g. by considering anomaly
1 µνρλ cancellation arising from chiral edge states at the boundaries
∂µ J5µ = ε Fµν Fρλ + 2M ψ̄γ5 ψ (9) (Fermi arcs) [36] or by matching to a tight-binding model
16π2
[26, 33].
indicates there is an anomalous Hall effect in the Weyl
What can we learn from this for building a holographic
semimetal phase [28-33]
model? First we see that there are two U(1) symmetries at
1 play. One of them, the axial one, is anomalous and explic-
J= beff × E . (10)
2π2 itly broken by the mass term in the Dirac equation. The
Thus by tuning M/b, from the band structure we see that anomaly√gives rise to the quantum Hall effect eq. (10) as
there is a quantum phase transition from topologically non- long as b2 − M 2 > 0. The mass term can be identified as
trivial Weyl semimetal phase to a trivial insulating phase. a source for the operator ψ̄ψ. We can take the mass to be the
This phase transition is beyond the Landau classification and expectation value of a complex classical scalar field that is
is an example of a topological phase transition. In both charged under the axial U(1) symmetry. Because of this an
phases of the system, the same symmetries of the underlying expectation value breaks the axial symmetry already on the
theory are explicitly broken by the the couplings M, b. Due classical level. Under a chiral rotation ψ → iαγ5 ψ the op-
to the fact that in the topologically nontrivial phase there is erator ψ̄ψ transforms into ψ̄ψ → 2iαψ̄γ5 ψ. Furthermore the
a nontrivial Hall effect while in the topological trivial phase parameter b or more generally bµ couples to the axial current
there is trivial Hall effect, the Hall conductivity can be taken J5µ = ψ̄γ5 γµ ψ and can therefore be understood as the back-
ground value of an axial gauge field. These considerations
(a) (b) (c)
give us the ingredients we need to implement in the holo-
graphic model.

3.2 Holographic model

A holographic action allowing the implementation of the

above symmetries and breaking pattern is
∫ [ ( )
√ 1 12 1 1
S = d5 x −g 2 R + 2 − F 2 − F 2
2κ L 4 4
(α( ) )
+ ϵ abcde Aa Fbc Fde + 3Fbc Fde + ζR fgbc Rgf de
3
]
∗
Figure 1 (Color online) Phases of the Dirac equation (eq. (8)). The fig-
− (Da Φ) (D Φ) − V(Φ) ,
a
(11)
ure shows the dispersion relation as a function of p⊥ and p3 . (a) Deep in
the Weyl semimetal phase; (b) the critical point in between the two; (c) the where κ2 is the Newton constant, L is the AdS radius and
gapped phase. α, ζ are the Chern-Simons coupling constants3) . In field the-

2) It is not a traditional order parameter but Hall effect is known to serve as signature of topologically non-trivial Fermi surfaces [28].
√
3) Note that ϵabcde = −gεabcde with ε0123r = 1. Our conventions for indexes are as follows: latin indexes from the beginning of the alphabet {a, b, . . . } are
five dimensional ones, greek indexes are four dimensional ones and latin indexes from the middle of the alphabet {i, j, m, n} are purely spatial indexes.
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-6

ory, we have conserved electromagnetic current and non- We take the following ansatz for the zero temperature solu-
conserved axial current. As shown in Table 1, the conserved tion:
currents in the field theory are dual to gauge fields in AdS dr2
space. The electromagnetic U(1) current is dual to the bulk ds2 = u(−dt2 + dx2 + dy2 ) + + hdz2 ,
u (16)
gauge field Va in AdS with field strength F = dV. The axial A = Az dz , Φ = ϕ ,
U(1) current is dual to the gauge field Aa in AdS with field
where u, h, Az , ϕ are functions of r. In this case M/b is the
strength F = dA. Since the axial symmetry is anomalous in
only tunable parameter of the system due to the conformal
the field theory and in the bulk the anomaly is characterized
symmetry. We set 2κ2 = L = 1.
by the Chern-Simons part of the action (eq. (11)) with cou-
pling constants α and ζ. The gauge invariant regularization Critical solution The following Lifshitz solution is an
corresponds to this choice of Chern-Simons term with which exact solution of the system:
the electromagnetic U(1) symmetry remains non-anomalous. dr2
ds2 = u0 r2 (−dt2 + dx2 + dy2 ) + + h0 r2β dz2 ,
The anomaly arises in a gauge variation of the axial gauge u0 r 2 (17)
field δAa = ∂a θ as a boundary term: β
A z = r , ϕ = ϕ0 .

A= It exists an anisotropic Lifshitz symmetry (t, x, y, r−1 ) →

∫ (α ( ) ) s(t, x, y, r−1 ) and z → sβ z. The irrelevant deformations can
√ β
d4 x −gθ ϵ µνρλ Fµν Fρλ + 3Fµν Fρλ + ζRαβµν R αρλ . be introduced to flow it to UV with the boundary conditions
3
(12) eq. (15). The four constants {u0 , h0 , β, ϕ0 } in eq. (17) are
determined by the values of λ, m and q.
We have included here the usual axial anomaly due to elec- It turns out that the following irrelevant perturbations
tromagnetic field, the purely axial anomaly due to axial around the Lifshitz fix point can flow the geometry to asymp-
gauge fields and the gravitational4) contribution to the ax- ( ) ( )
totic AdS u = u0 r2 1 + δu rα , h = h0 rβ 1 + δh rα , Az =
( ) ( )
ial anomaly. All three will play a role in the physics of rβ 1 + δa rα , ϕ = ϕ0 1 + δϕ rα . Only the sign of
holographic Weyl semimetals. The factor of 3 in the axial δϕ is a free parameter and the geometry can flow to AdS
anomaly reflects the symmetry factor that arises in the cor- with δϕ = −1. In the case q = 1, λ = 1/10, we
responding triangle diagram in quantum field theory. We have (u0 , h0 , β, ϕ0 , α) ≃ (1.468, 0.344, 0.407, 0.947, 1.315)
will introduce the mass deformation via a boundary value and (δu, δh, δa) ≃ (0.369, −2.797, 0.137)δϕ. We obtain the
of the scalar field Φ [37]. Since the dual axial symme- critical value M/b ≃ 0.744, which corresponds to the transi-
try is explicitly broken, in the bulk this scalar field is in- tion point.
troduced to be only axially charged. The covariant deriva-
Topological nontrivial phase At leading order the second
tive is Da Φ = (∂a − iqAa )Φ. The scalar field potential is
type of solution in the IR is
m2 |Φ|2 + λ2 |Φ|4 . The AdS bulk mass m2 L2 = −3 is chosen such
that the dual operator has conformal dimension three and its πa2 ϕ2 2a1 q
u = r2 , h = r2 , Az = a1 + 1 1 e− r ,
source has conformal dimension one. The electromagnetic 16r (18)
√ ( a q )3/2 a q
and axial currents5) are defined as: ϕ = πϕ1
1
e− 1r
,
2r
√ ( )
J µ = lim −g F µr + 4αϵ rµβρσ Aβ Fρσ , (13) a1 can be set to a numerically convenient value. Later on we
r→∞
√ ( µr 4α rµβρσ ) rescale to b = 1.
J5µ = lim −g F + ϵ Aβ Fρσ . (14) Starting from the near horizon solution eq. (18), the equa-
r→∞ 3
tions can be numerically integrated towards the UV. ϕ1 can be
The model was first studied in the probe limit in ref. [38] and taken as the shooting parameter to obtain an AdS5 to AdS5
then in the backreacted case in ref. [39]. domain wall. For the values q = 1, λ = 1/10 this type of
We are looking for asymptotically AdS solutions. The solution exists only for M/b < 0.744.
mass parameter and the time-reversal symmetry breaking pa-
Topological trivial phase The third type of solution at
rameter in the field theory are introduced through the confor-
leading order in IR is
mal boundary conditions: √
( )
3 2 β1 3
lim rΦ = M , lim Az = b . (15) u= 1+ r , h = r , Az = a1 r , ϕ =
2
+ϕ1 rβ2 , (19)
r→∞ r→∞ 8λ λ

4) A contribution due to the extrisic curvature vanishes on asymptotic boundary of AdS [22].
µ
5) These are the consistent currents. The vector current J µ is conserved while the conservation of the the axial current J5 is broken explicitly by the scalar
field and spontaneously by the anomaly [37]. The covariant currents can be defined by dropping the Chern-Simons terms.
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-7

(√ √ )
where (β1 , β2 ) = 1+ 48q2
− 1, 2 − 2 . For our
3+20λ the holographic equations of motion in the bulk space-time
3+8λ 3+8λ
(√ ) d µ
choice of λ and q (β1 , β2 ) = j (r) = ∂µ X . (21)
19 − 1, 19 − 2 . a1 can
259 √10
dr
be set to be 1. ϕ1 can be taken as the shooting parame- The precise form of X is not important since from now on we
ter to obtain the AdS5 to AdS5 domain wall. For the val- integrate over space and take the zero frequency limit such
ues q = 1, λ = 1/10 this type of solution only exist for that the right hand side of this conservation equation van-
M/b > 0.744. ishes. It follows then that in this situation the holographic
Figure 2 shows the profiles of the scalar field ϕ and the expectation value of the current is given by the value of jµ (r)
gauge field Az . Only one of the above three types of solu- at the horizon:
tions exists at a given value of M/b. The horizon value of Az
varies continuously between the two phases while the horizon J µ = jµ (rh ) . (22)
value of ϕ jumps discontinuously. Close to the phase transi-
Since due to the Bianchi identity the electric field is constant
tion point, the deep IR geometry eq. (18) or (19) quickly
along the AdS bulk direction r, the current at zero tempera-
flows to the critical Lifshitz solution in the intermediate IR
ture is given by the Hall current [40]:
region.
The free energy density can be computed by adding stan- J x = 8αAz (0)Ey . (23)
dard holographic counterterms and is behaved continuously
and smoothly at the critical value [39]. Note that the free The anomalous Hall conductivity is completely determined
energy does not depend on the Chern-Simons coupling con- by the horizon value of the axial gauge field [38, 39]. In par-
stant. It does not probe the topological nature of the quan- ticular in holography it is only non-vanishing in the topolog-
tum phase transition, in contrast to the anomalous Hall ical phase but vanishes at the quantum critical point and in
conductivity. the non-topological phase. This is exactly the same behavior
the weak coupling Dirac like model shows. The anomalous
Hall conductivity (eq. (23)) is shown in Figure 3 for different
3.3 Anomalous Hall conductivity values of model parameters.
The essential hall-mark of the topological character of the
Weyl semimetal state is the presence of anomalous Hall con- 3.4 Universality of the quantum phase transition
ductivity. In the case we are interested in it is anomalous Hall
The precise value of M/b at which the topological quantum
conductivity for the vector type current. A fast way of calcu-
phase transition from the Weyl semimetal to the trivial the-
lating it in holography is as follows. First one observes that
ory arises depends on the model parameters. Using the holo-
the quantity
graphic duality one can investigate the critical values of M/b
jµ (r) = F µr + 4αϵ rµβρσ Aβ Fρσ (20) as a function of the quartic scalar self coupling λ for various
values of the axial charge q of the scalar field [40]. The result
fulfills a radial conservation equations as a consequence of is shown in Figure 4.

1.2
(a) 5 (b)
1.0
4
0.8
3
Az

0.6
b

0.4 2

0.2 1

0.0
0
10−6 10−4 0.01 1 100 104 106 10−6 10−4 0.01 1 100 104 106
r r
b b

Figure 2 (Color online) The bulk profile of background Az (a) and ϕ (b) for M/b = 0.695 (blue), 0.719 (green), 0.743 (brown), 0.744 (red-dashed), 0.745
(orange), 0.778 (purple), 0.856 (black). Figures from ref. [39].
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-8

1.0
3.5 Holographic Fermi arcs
0.8
One of the key signatures of Weyl semimetals is the presence
0.6 of topologically protected surface states, the so-called Fermi-
σAHE
8αb

arcs. A simple and efficient field theory model of Fermi arcs

0.4
follows from the thinking about the low energy description in
0.2 terms of the deformed Dirac operator (eq. (8)). The effective
separation of the Weyl cones in the Brillouin zone enters the
0
Dirac equation like a gauge field with the key difference that
0 0.2 0.4 0.6 0.8 1.0 1.2
it couples with different signs to left- and right-handed Weyl
M
b fermions. If this parameter varies spatially it can induce an
axial magnetic field b5 = ∇ × b. A standard argument shows
Figure 3 (Color online) The zero temperature anomalous Hall conductivity
is given by the value of the axial gauge field on the degeneratre horizon at
now that such a spatial dependence is inevitable. Inside the
r = 0. It is plotted here for different model parameters, specifically m2 = −2, Weyl semimetal we describe the system by a Dirac equation
λ = 1/10 (red), m2 = −3, λ = 1/10 (green), m2 = −3, λ = 1 (blue). It can be with axial gauge field A5 = b. But outside the material there
observed how the critical value for the M/b parameter where the conductiv- are no low energy states available for the electrons. This is
ity goes to zero changes in the different cases. Figure from ref. [40].
equivalent to describing the outside by a Dirac equation with
a very large mass and vanishing axial gauge field. In turn this
means that on the edge of the material there is necessarily a
10 strongly localized axial magnetic field. Now one can invoke
an index theorem that states that the number of zero modes
8
in the axial magnetic field is given by the degeneracy of the
6 Lowest Landau level |b5 |/(2π). In a usual magnetic field there
c

would be zero-modes of both chiralities but in the axial mag-

M
b

4 netic field the zero modes from both the right-handed and
left-handed fermions have the same chirality. If one pop-
2
ulates these zero modes by turning on a chemical potential
0 they will lead to an edge current of the form:
0 2 4 6 8 10
λ µ
Jedge = b5 . (24)
2π2
Figure 4 (Color online) The critical value of M/b as a function of the
quartic scalar self coupling λ for different values of the axial charge q = 1, This can also be viewed as an instance of an anomaly induced
q = 1.5 and q = 2. As can be seen the value of M/b diverges at some fi-
transport phenomenon, the axial magnetic effect [17].
nite values of λ. This means that in these cases the scalar self interaction
suppresses the phase transition to the trivial phase. Figure from ref. [40]. At strong coupling or more generally in the absence of
quasiparticle excitations Fermi-arcs per se can not be ex-
pected to be seen. But the topologically protected edge cur-
Interestingly for a given charge value there is a maximum rents should still exist6) . This is exactly what ref. [41] inves-
value of λ beyond which the phase transition does not occur tigated. The authors numerically constructed solutions with
at any finite value of M/b. This has an interesting interpreta- spatial dependent boundary conditions of the form:
tion in terms of the holographic duality. The spacetime cur- 


 bL , for x < −l,
vature can be taken as a measure of the degrees of freedom. 



In the cases in which the phase transition cannot take place Az (r, x) =
 p(x), for − l ≥ x ≤ l, (25)
r→∞ 


anymore it turns out that the holographic number of degrees 
 bR , for x > l,
of freedom in the infrared in the trivial phase would be larger
than in the critical phase. Intuitively one expects that more where p(x) is a suitably chosen smooth interpolating func-
degrees of freedom are gapped out in the IR than at the criti- tion. The scalar field was kept fixed and the chemical poten-
cal point. This intuition would be violated if the trivial phase tial is introduced as:
could still be reached for high values of the self coupling.
Fortunately direct inspection shows that this is not the case. lim rΦ(r) = M, lim Vt = µ , (26)
r→∞ r→∞

6) It might be however that Fermi arcs exist also in the spectral functions of probe fermions. Indeed as we will review in the next section, probe fermions
do carry the signatures of the non-trivial topology of momentum space.
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-9

with the understanding that Vt = 0 at the horizon. They found topological invariants can be defined for topological mat-
that indeed a current flows on the interface between the two ter, which are invariant under adiabatic deformations of the
asumptotic regions and the total current in the y direction is Hamiltonian that protect the topology of the underlying sys-
given by tem.
( ) For weakly coupled topological systems, in momentum
Jy = 8αµ(beff,L − beff,L ) = 8αµ σAHE,L − σAHE,R . (27) space we can define the topological invariants from the Bloch
states, i.e. the eigenstates of the weakly coupled Hamiltoni-
This is exactly∫what one can expect since in the effective low ans. The Berry phase [42], which is the phase accumulated
energy theory dxB5 = beff,L − beff,R . Moreover, the current along a closed loop γ in the Hmomentum space for the Bloch
distribution is concentrated on the interface as can be seen states |nk ⟩, is defined as ϕ = γ Ak · dk where the Berry con-
from Figure 5. nection is determined by the eigenstates |nk ⟩ of the momen-
∑
tum space Hamiltonian as Ak = i j ⟨nk |∂k |nk ⟩ with j runs
3.6 Fermionic probes and the topological invariant over all occupied bands. The Berry phase with value 0 or π
is one simple example of a topological invariant.
Various bulk calculations have shown that the dual system There is another way to compute the Berry phase. Us-
( )
should be a Weyl semimetal having Weyl cones with an effec- ing the Berry curvature Ωi = ϵi jl ∂k j Akl − ∂kl Ak j associated
tive momentum separation in the direction of b. A direct ob- to the the Berry connection and choosing a ∫surface S whose
servation of the Weyl cones needs to employ the holographic boundary is the closed loop γ, we have ϕ = S Ω · dS .
fermionic probes whose spectral function at zero tempera- An equivalent way to calculate the topological invariant is
ture would show two poles separated in the momentum space to use the Green’s function:
∫ [ ]
in the b direction. Besides a direct observation of the two 1
N(kz ) = dk0 dk x dky Tr ϵ µνρzG∂µG−1G∂νG−1G∂ρG−1 ,
Weyl cones, an important further evidence is to compute the 24π 2

topological invariant for the dual strongly coupled topologi- (28)

cal semimetal states, which would also require a calculation
where µ, ν, ρ ∈ k0 , k x , ky and k0 = iω is the Matsubara fre-
of the Green’s function of probe fermions in the bulk. In
quency. For free systems, the Green’s function takes the form
the following we will first introduce how topological invari-
G(iω, k) = 1/(iω − h(k)) where h(k) is the Hamiltonian ma-
ants for strongly coupled systems could be calculated directly ∑
trix H = k c†k h(k)ck . We can also use this formula to com-
from the bulk fermionic Green’s function and then show how
pute topological invariant for interacting systems, however,
fermionic spectral functions could be calculated for the holo-
for strongly interacting systems it is difficult to compute prac-
graphic Weyl semimetal background and obtain the corre-
tically since it involves an integration in the iω direction.
sponding topological invariant accordingly.
Refs. [43-45] show that the Green’s function at zero fre-
Mathematically topological invariants are properties that
quency G(0, k) contains all the topological information of the
are invariant under homeomorphisms. Similarly in physics
system. An effective topological Hamiltonian can be defined

Ht (k) = −G−1 (0, k) (29)

and the eigenvectors can be obtained from this effective topo-

logical Hamiltonian. As long as there is no pole at nonzero ω
M

in G(iω, k), the topological invariants derived from the effec-

tive Hamiltonian Ht (k) would be the same as those defined
x

in the original system. Thus topological invariants can be de-

fined using negative valued eigenvectors of Ht (k), i.e. effec-
tive occupied states nk with Ht (k)|nk ⟩ = −Et |nk ⟩ and Et > 0.
Having shown that topological invariants for strongly cou-
x
pled fermionic systems could be directly calculated from
Figure 5 (Color online) Current distribution along the x-direction for dif- the zero frequency Green’s function, we now show how
ferent choices of interpolating functions with widths l = 0.1, 0.5, 1, 4. As fermionic Green’s function could be calculated for the holo-
expected the current in localized in the region in which the effective low graphic Weyl semimetal. In holography, probe fermions in
energy axial gauge field shows non-vanishing curl. This is the direct holo-
graphic counterpart of the Fermi-arcs associated to the surface of a Weyl the bulk have been studied for various backgrounds at finite
semimetals. Figure with permission reproduced from ref. [41], ⃝2017
c by density in refs. [46, 47] to obtain the dual fermion spec-
the American Physical Society. tral functions. Here for the holographic Weyl semimetal, to
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-10

probe the dual fermion spectrum we add a probe fermion on The topological invariants can be calculated as follows.
the background geometry (eq. (16)) and calculate the dual Around the Dirac node, we can define a sphere S to en-
Green’s functions from the holographic dictionary [48]. One close it. On this sphere the system is gapped. HThe topo-
important difference here is that we work in five dimensions, logical invariant can be computed from Cl = 2π 1
S
Ωl · dS ,
and now a bulk four component spinor corresponds to a two where Ω = ϵ Fi j with (i , j , k) ∈ {k x , ky , kz } and F is the
i i jk

component spinor of the dual field theory in four dimensions Berry curvature. Note that the topological number is an in-
[49]. Therefore in the bulk we use two spinors Ψ1 and Ψ2 teger number and it stays as a constant when we deform
with opposite sign masses and axial charges and choose one the shape and exact shape and radius of the sphere with-
with standard quantization while the other with alternative out passing through a Dirac node. We can parameterize the
quantization to correspond to four component spinor with sphere as S = k0 (sin θ cos ϕ, sin θ sin ϕ, cos θ) and we have
two opposite chiralities. Ωl = (−1)l eρ /2k02 . We obtain C1 = −1 for |n1 ⟩ and C2 = 1 for
From the point of view of the dual field theory these probe |n2 ⟩. Then the total topological invariant is zero. This is due
fermions correspond to composite operators of a scalar field to the fact that the zero density state dual to pure AdS5 is a
with the fundamental fermions. A priori it is these funda- Dirac semimetal.
mental fermions that carry the non-trivial topology. As we Now we continue to calculate the topological invariants
will show now this topology is still present in the strongly for the holographic Weyl semimetal. In Weyl semimetals,
coupled bound state that are the probe fermions. we can define the topological invariant as the integration of
The action of probe fermions is as follows: Berry curvature on a closed surface S which encloses one
of the Weyl nodes in the momentum space. This result
S = S 1 + S 2 + S int ,
∫ will be insensitive to the shape and size of the closed sur-
√ ( )
S1 = d5 x −giΨ̄1 Γa Da − m f − iAa Γa Ψ1 , face. From semi-analytic calculations, we obtained that when
∫ M/b is very small, the topological invariants are ±1 which
√ ( ) (30)
S2 = d5 x −giΨ̄2 Γa Da + m f + iAa Γa Ψ2 , are precisely the same as the results from weakly coupled
∫ WSM model. For larger M/b numerics has to be involved.
√ ( )
S int = − d5 x −g iη1 ΦΨ̄1 Ψ2 + iη∗1 Φ∗ Ψ̄2 Ψ1 , The topological invariant for finite temperature case has been
studied in ref. [50]. The total topological invariants are zero
where Da = ∂a − 4i ωmn,a Γmn . The coupling constant in front due to the Nielsen-Ninomiya theorem [51].
of Az is opposite for the two spinors. Here Γa = ema Γm with In addition to the anomalous Hall conductivity and
Γm the Γ-matrices in five dimensional Minkowski spacetime. edge states, the nontrivial topological invariants serve
From this form of bulk action for probe fermions, we can as further nontrivial evidence that the holographic Weyl
obtain the retarded Green’s function from the boundary val- semimetal models are strongly coupled topologically nontriv-
ues of the two bulk fermionic fields at different momenta ial semimetals.
[48]. In the simplest M/b → 0 limit, it could easily been
shown that two poles exist separately in the z direction in the 3.7 Finite temperature, conductivities and viscosities
momentum space. At small M/b limit, this could also be ob-
tained with some semi-analytic method. In the previous subsections, the studies are mainly for the
As a simple example we first show how this procedure zero temperature case. Now we will turn to finite tempera-
works for the pure AdS case, which of course would give ture physics and the interesting transport physics.
a trivial topological invariant. In this case, in fact the system We use the following ansatz to study the finite temperature
is degenerate at zero frequency, i.e. the two Weyl nodes co- solutions [39]:
incide to form a Dirac node. The fermionic retarded Green’s
dr2
functions for one chirality for ω > k has already been ob- ds2 = −udt2 + + f (dx2 + dy2 ) + hdz2 ,
u (31)
tained in ref. [49]. The topological Hamiltonian Ht is defined
A = Az dz , Φ = ϕ ,
as −G−1 (0, k) from eq. (29). The two eigenvectors are |n1 ⟩ =
( ) ( )
n01 kz + k, k x + iky , 0, 0 T and |n2 ⟩ = n02 0, 0, kz − k, k x + iky T where all the fields u, f, h, Az , ϕ are functions of r. At the
√
where n0l = 1/ 2k(k − (−1)l kz ) with l ∈ {1, 2}. In fact these regular horizon r = r0 , u has a simple zero whereas all
two eigenvectors are the same as the ones in the free mass- these functions are analytic. This geometry is a black hole
less Dirac Hamiltonian. |n1 ⟩ has positive chirality and is the with horizon located at r = r0 and Hawking temperature
eigenvector of the positive chirality Hamiltonian while |n2 ⟩ 4πT = u′ (r0 ). According to the holographic dictionary,
has negative chirality and is the eigenvector of the negative the Hawking temperature of this black hole geometry cor-
chirality Hamiltonian. responds to the physical temperature of the dual field theory.
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-11

By using the scaling symmetries of the system and the con- be computed from the Kubo formula
straints from the equations of motion near the horizon, there 1 [ R ]
are only two independent dimensionless parameters, which ηi j,kl = lim Im Gi j,kl (ω, 0) , (34)
ω→0 ω
can be parametrized by M/b and T/b in the UV.
where the retarded Green’s function of the energy momentum
A cartoon illustration for the phases is shown in Figure 6
tensor
[52]. At zero temperature the model undergoes the already ∫
discussed topological quantum phase transition between a GRij,kl (ω, 0) = − dtd3 xeiωt θ(t)⟨[T i j (t, x), T kl (0, 0)]⟩ . (35)
topological semimetal state and a trivial semimetal state. At
the critical phase transition point there is an emergent Lifshitz Since we chose b = bêz , the two shear viscosities [54-56]
symmetry at zero temperature. At finite temperature there is are obtained from the symmetric part of the retarded Green’s
a quantum critical regime whose physics is governed by the function under the exchange of (i j) ↔ (kl)
Lifshitz symmetry. Meanwhile, this quantum phase transition
η∥ = η xz,xz = ηyz,yz , η⊥ = η xy,xy = ηT,T (36)
becomes a smooth crossover behavior.
Conductivities can be computed with Kubo formula via re- and the two odd or Hall components of viscosity are related
tarded correlation functions: to the antisymmetric part by
1
σmn = lim ⟨Jm Jn ⟩(ω, k = 0) . (32)
ω→0 iω T

According to the holographic dictionary, we can obtain the Quantum

retarded Green’s functions by studying the gauge field fluctu- critical

ations around the background with infalling boundary condi-

tions at the horizon. We obtain Weyl Topologically
√ semimetal trivial semimetal
σAHE = 8αAz (r0 ) , σ xx = σyy = h(r0 ) . (33)

From the fact that r0 = 0 and h(0) = 0 at zero temperature 0 (M/b)c M/b
one concludes that the diagonal conductivities vanish. The
anomalous Hall effect (Figure 7) is completely determined Figure 6 (Color online) The cartoon picture for the holographic Weyl
by the horizon value of the axial gauge field. semimetal at different temperatures as a function of M/b. Figure from ref.
The longitudinal electric conductivity can be computed by [39].

studying the fluctuation δVz = vz e−iωt in the bulk. At zero

temperature we obtain σzz = 0 and for finite temperature
1.0
σzz = √f h .
r=r0
The diagonal components of electric conductivities at fi- 0.8
nite temperature as a function of M/b is shown in Figure 8.
0.6
We can see that at the critical value there is a peak (mini-
σ AHE
8α b

mum) for the transverse (longitudianl) diagonal conductivi- 0.4

ties. Both the height of the peak and depth of the minimum
grow with temperature. At zero M we have σ xx,yy,zz = πT and 0.2

for large M the conductivities σ xx,yy,zz = cπT with a temper- 0.0

ature independent c smaller than 1. This is due to that in the 0.0 0.5 1.0 1.5 2.0 2.5 3.0
trivial phase some but not all degrees of freedom are gapped M
b
out. The quantum phase transition is between a topological
semimetal and a trivial semimetal. Figure 7 (Color online) Anomalous Hall conductivity as a function of
Now let us explain the behavior of viscosities in this sys- M/b at different temperatures. The solid lines are obtained from the holo-
tem. It is known that in an axisymmetric system which has graphic Weyl semimetal. For zero temperature a sharp but continuous phase
transition occurs at a critical value of M/b (blue), which becomes a smooth
time reversal symmetry breaking by vector b there are seven7) crossover at finite temperature. The curves are for T/b = 0.1 (black), 0.05
independent viscosities [53] in which two of them are inde- (purple), 0.04 (red), 0.03 (brown). The dashed (green) line is for the weak
pendent odd viscosity tensor components. The viscosities can coupling model. Figure from ref. [39].

7) These seven components includ three shear viscosities, two odd viscosities and two bulk viscosities. We focus on four of them and will not consider the
other two bulk viscosities and one shear viscosity which are from the spin zero sector.
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-12

7
1.0
6
0.8
5

4 0.6

4π η II
σdiag
T

s
3 0.4 T/b=0.05

2 T/b=0.04
0.2
T/b=0.03
1
0.0
0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 M
M b
b

Figure 9 (Color online) The longitudinal shear viscosity over entropy den-
Figure 8 (Color online) The diagonal components of electric conductivi- sity 4πη∥ /s as a function of M/b at different temperatures. Figure from ref.
ties as a functional of M/b for different temperatures. The solid lines are for [52].
σ xx = σyy and the dashed lines are for σzz from holographic Weyl semimetal
with T/b = 0.05 (purple), 0.04 (red), 0.03 (brown). The dashed gray line is
the critical value of M/b at the topological quantum phase transition. Figure KSS bound in anisotropic theories [59, 60] this is not unex-
from ref. [39].
pected. Still it is very interesting to note that the shear vis-
cosity reaches a minimum in the quantum critical region of
ηH∥ = −η xz,yz = ηyz,xz , ηH⊥ = η xy,T = −ηT,xy , (37) M/b ≈ 0.744. In contrast the transverse viscosity obeys the
KSS bound is exactly η⊥ /s = 1/4π.
where the index T denotes the component xx − yy. Note that
A particular interesting fact about odd viscosities is that
ηH⊥ is the odd or Hall viscosity in the plane orthogonal to b
they are directly proportional to the mixed axial-gravitational
while ηH∥ is specific to axisymmetric three dimensional sys-
anomalous constant which is the gravitational contribution to
tems8) .
the axial anomaly ζ in eq. (12). Therefore at least in this
In holography the viscosities can be computed via switch-
holographic model they are a new example of an anomaly
ing on the following perturbations δgiz = hiz (r)e−iωt , δAi =
induced transport coefficient. Figure 10 shows the odd vis-
ai (r)e−iωt for i ∈ {x, y}. For the other components of vis-
cosities ηH∥ and ηH⊥ as a function of M/b at small but finite
cosities can be computed by considering the perturbations
temperatures. In the topologically nontrivial phase the odd
δg xx − δgyy = 2hT (r)e−iωt , δg xy = h xy (r) e−iωt . From the
viscosity is highly suppressed. It rises steeply when M/b en-
holographic dictionary we obtain the following viscosity co-
ters into the quantum critical region, peaks around the critical
efficients:
value of M/b and then falls off slowly when M/b increases.
dissipative viscosity:
In the limit M/b → ∞ the odd viscosity vanishes.
f2 The appearance of odd viscosity in the quantum critical
η∥ = η xz,xz = ηyz,yz = √ , (38)
h r=r0 region can be considered to be a prediction from holography.
√ Its relation to the gravitational anomaly suggests that this is
η⊥ = η xy,xy = ηT,T = f h , (39)
r=r0 a universal property. Indeed recently anomalous Hall viscos-
dissipationless odd viscosity: ity has also been obtained in a weakly coupled quantum field
theory model of the quantum critical point in ref. [61]. The
q2 Az ϕ2 f 2 relation to anomalies in the weakly coupled theory is far from
ηH∥ = ηyz,xz = −η xz,yz = 4ζ , (40)
h r=r0 clear. We expect that further investigation of the holographic
ηH⊥ = η xy,T = −ηT,xy = 8ζq2 ϕ2 f Az . (41) model and its RG flow to the critical point can give valuable
r=r0
insight into the origin of this type of odd viscosity.
The dissipative viscosity is a form of shear viscosity and it Note that from the analytic results on the viscosities and
is interesting to express it normalized to the entropy density conductivities we obtain the non-trivial relation:
η∥ f
s = 4πh |r=r0 . As can be seen from Figure 9 the shear vis-
cosity drops significantly below the standard result of KSS η∥ 2ηH∥ σ∥ f
= = = , (42)
bound [58]. In view of the various results of violation of the η⊥ ηH⊥ σ⊥ h r=r0

8) There exists odd viscosity ηH∥ by considering the coupling of elastic gauge fields to the electron gas in Weyl semimetals [57]. It was shown in ref. [57]
that this effective odd viscosity is related to the Hall conductivity of the electron gas and arises from the electronic point of view as an axial Hall conductivity.
Here in holography the Hall viscosity should be viewed as an intrinsic property of the strongly coupled electron fluid.
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-13

25
(a) 100 (b)
T/b=0.05 T/b=0.05
20 80
T/b=0.04 T/b=0.04

T/b=0.03 T/b=0.03
15 60
4ζ T 3

8ζ T 3
η HII

η H⊥
10 40

5 20

0 0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
M M
b b

Figure 10 (Color online) Odd viscosity ηH∥ (a) and ηH⊥ (b) as a function of M/b at different temperatures. Figures from ref. [52].

√
where σ∥ = σzz = √f h r=r , σ⊥ = σ xx = σyy = h r=r . to the electric gauge field. There is however a big difference
0 0
Furthermore, in the quantum critical regime there exists in- in the possible dynamics of these fields. The dynamics of the
teresting temperature scaling behaviour of conductivities and true gauge field is given by Maxwell’s equations. For a gauge
viscosities. At T = 0, there is an emergent Lifshitz symme- field that couples to an anomalous current, such as the axial
try in the IR at the quantum phase transition point. The IR current this is mathematically inconsistent. A simple way of
physics is invariant under (t, x, y, r−1 ) → l(t, x, y, r−1 ), z → lβ z seeing this is to note that Maxwells equations imply that the
and f → l−2 f, h → l−2β h, Az → l−β Az , ϕ → ϕ, where β is divergence of the current vanishes
the anisotropic scaling exponent [39]. At very low temper-
∂µ J µ = ∂µ ∂ν F µν = 0 . (43)
ature, since T → l−1 T the temperature scaling dependence
of the viscosities and conductivities near the critical region In nature on a fundamental level anomalous currents are not
can be obtained from the scaling arguments. At the critical coupled to gauge fields.
regime, we have η∥ /s ∝ T γ1 , ηH∥ ∝ T γ2 , ηH⊥ ∝ T γ3 with Nevertheless such fields can arise as effective fields in con-
(γ1 , γ2 , γ3 ) = (2 − 2β, 4 − β, 2 + β) and σ∥ ∝ T γ4 , σ⊥ ∝ densed matter systems. It has been shown in refs. [57, 63-65]
T γ5 , σAHE ∝ T γ6 with (γ4 , γ5 , γ6 ) = (2 − β, β, β) for low that axial electric and magnetic fields can be induced by ap-
temperatures. Figure 11 shows the temperature scaling ex- plying strain on Weyl semimetals. These are effective low
ponents γi with i ∈ {1, . . . , 6} of the numerical results at energy couplings in the theory and certainly do not obey
low temperatures at the critical value of M/b. At sufficient Maxwell’s equations and consequently do not jeopardize the
low temperature, these scaling exponents approaches the an- consistency of the theory. Therefore it seems a legitimate
alytic values from scaling analysics. Furthermore, the scaling physics question to ask if there is a purely axial analogue of
behaviors explain the peak/dip behaviors of the conductivi- the anomalous Hall effect in Weyl semimetals. This again is a
ties/viscosities of holographic Weyl semimetal in the quan- question that can be nicely addressed in holographic models
tum critical regime. and leads to some important insights.
The chiral vortical conductivity for the holographic Weyl Since the anomalous Hall effect is a direct consequence
semimetal has been calculated in ref. [62]. We can first com- of the anomaly let us have another look into it and see what
pute the chiral vortical conductivity and then perform a suit- could be expected. A remarkable fact is that the anomaly
able renormalization via the anomalous Hall conductivity and (eq. (12)) in the axial gauge fields is weaker by a factor of
temperature squared. It was shown that at sufficiently low 1/3 compared to the electromagnetic contribution. One use-
temperature this renormalized ratio stays as universal con- ful way to think about this factor is to consider the origin of
stants in both the Weyl semimetal phase and the quantum crit- the anomaly in a triangle Feynman diagram. In the case of
ical region. Furthermore, in the critical region the renormal- the purely axial anomaly this is a diagram with three identi-
ized ratio is fully determined by the emergent Lifshitz scaling cal axial currents on the vertices. Elementary Feynman rules
exponent at the critical point [62]. instruct us therefore to multiply the diagram with a symmetry
factor of 1/3!. In comparison the electromagnetic contribu-
3.8 Axial Hall conductivity tion comes from a triangle diagram with two electric currents
and one axial current. There are only two identical operators
Formally in quantum field theory the axial current can be cou- on the vertices and thus the symmetry factor is only 1/2! with
pled to an axial gauge field just as the electric current couples gives a relative factor of 1/3.
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-14

γ1
γ2 2.0 γ4
γ3 γ5
3
γ6
1.5

γ cond
2
γ vis

1.0

1 0.5

(a) 0 (b)
0
0.01 0.02 0.05 0.10 0.01 0.02 0.05 0.10
T T
b b

Figure 11 (Color online) The temperature scaling exponents γi with i ∈ {1, . . . , 6} for viscosities η∥ , ηH∥ and ηH⊥ (a) and for conductivities σ∥ , σ⊥ and σAHE
(b) in the quantum critical regime. The dashed lines represent the analytic values of the scaling exponents from the scaling analysis. Figures from ref. [52].

The natural expectation is therefore, that the purely axial models. In the case of the holographic Weyl semimetal, this
Hall conductivity, e.g. the transverse axial current induced by has been initiated in ref. [66]. The authors study the effect of
an axial electric current is weaker by a factor of 1/3 compared disorder in form of random Gaussian noise in the boundary
to the electric Hall conductivity. value of the axial gauge field
While the electric Hall conductivity can be computed in
∑
N−1 √ √
an easy way from the horizon data, the calculation of the ax- lim Az (r) = b0 + 2γ S (ki ) ∆k cos(ki x + δi ) (46)
ial Hall conductivity is more cumbersome. It is complicated r→∞
i=1
by the fact that the axial symmetry is broken not only by the
anomaly but also by the expectation value of the scalar field with equally distributed momenta ki = ik0 /N, and random
(the dual of the mass term in the Dirac equation). phases δi . The analysis is restricted to the so-called decou-
The effect of this scalar field is that the axial background pling limit of holography in which the backreaction of the
field b is screened along the holographic direction. Since the bulk matter fields on the AdS metric is neglected. Never-
holographic direction encodes the RG flow, we can define the theless the authors find rather interesting signatures of disor-
analogue of a wave-function renormalization factor by der on the quantum phase transition. In general the quantum
√ phase transition is smeared due to the disorder. They also
ZA bUV = bIR . (44) show the appearance of rare regions and indications of log-
This implies that the axial Hall conductivity also suffers from oscillatory structures in the Hall conductivity.
this wave-function renormalization. Taking it into account
one arrives at the prediction 3.10 AC conductivities
ZA
σ5AHE = σAHE . (45) The electrical AC conductivity in a holographic Weyl
3
semimetal model was investigated in ref. [67]. A particu-
In other words the axial Hall conductivity is exactly 1/3 of
larly interesting effect was pointed out in relation to the quan-
the electric Hall conductivity once the wave-function renor-
tum critical behavior near the phase transition. On general
malisation of the axial gauge fields in IR is taken into ac-
grounds one expects the (zero temperature) optical conduc-
count. This was investigated in ref. [40] and indeed found
tivity to scale linearly with the frequency for low enough fre-
to be correct. Moreover since the prediction that the axial
quencies
Hall conductivity is 1/3 of the electric Hall conductivity is
a fundamental property of the theory it should hold for all σ(ω) = cω . (47)
states. Again this can be checked by using the finite tempera-
ture backgrounds and indeed it is found that the relation holds This is simply enforced by the scaling symmetry of the Weyl
exactly and independent of the temperature. fermions. The constant c is proportional to the number of
active Weyl fermions. It was pointed out in ref. [67] that
3.9 Disorder this can get modified for higher frequencies near the quan-
tum phase transition. The frequency dependent optical con-
Disorder is an integral component of any real condensed mat- ductivity enters then the quantum critical region whose scal-
ter system. It is therefore not only interesting but also manda- ing properties are determined by the scaling exponents of the
tory to study the effects of disorder even in semi-realistic Lifshitz critical point at the phase transition. The expected
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-15

change in scaling is transition. The Butterfly velocity therefore does not show a
universal behavior across the phase transition. This motivates
σ∥ (ω) ∝ ω2−β , σ ⊥ ∝ ωβ . (48) the authors of ref. [71] to introduce a new quantity, the infor-
Most interestingly such a sudden change in frequency depen- mation screening length L = 1/µ. It is defined as follows.
dence of the optical conductivity at low temperature of the First one introduces Mη = λl /vηB where η ∈ {⊥, ∥} and defines
Weyl semimetal TaAs was experimentally observed in ref. then
2
[68]. The authors of ref. [67] suggest that this might be ex- 1 Mη
µ2 = = . (52)
plained by assuming that one enters the quantum critical re- L 2 hη (rh )
gion in TaAs at a frequency around 30 meV. They find that a
This definition manages to get rid of the anisotropy such that
fit to the data gives a scaling exponent of β = 0.14 for the
a unique L independent of the direction can be defined. The
transverse conductivity in the Lifshitz quantum critical re-
authors show that L is maximal at the quantum critical point
gion. It should be noted that there are also other candidate
and they conjecture that the information screening length
explanations, such as activating of additional Weyl points
obeys
at higher frequencies. Nevertheless the predicted change in
1
scaling of the optical conductivity once the frequency is high 2L ≤ = 2Lc , (53)
enough to enter the quantum critical regime seems a robust D⊥ + βD∥
prediction and is in principle accessible by experiments. It where β is the Lifshitz scaling exponent of the anisotropic di-
would be very interesting to see then if the scaling exponents rections, D⊥ is the number of spatial dimensions with scaling
of longitudinal and transverse optical conductivities can be exponent equal to one and D∥ is the number of spatial direc-
fitted to weak coupling models or to predictions from holo- tions with scaling β. They also point out that in holography
graphic models. the null energy condition restricts the scaling β ≤ 1.

3.11 Butterfly velocity

4 Weyl semimetal/Chern insulator transition
Holography has also contributed in recent years to the under-
The phase diagram for Weyl semimetal is richer and it could
standing of chaos in quantum many body systems. Quantum
go through a phase transition to a normal band insulator
chaos can be characterized by the late time behavior of the
[31, 72] or to a Chern insulator [31, 73, 74] or to a metal
out of time order correlation function (OTOC):
[75] etc. However, the holographic Weyl semimetal model
∗
⟨[V(t, x), W(0, 0)]2 ⟩ ∼ eλL (t−t −|x|/vB ) , (49) in sect. 3 goes through a quantum phase transition to a triv-
ial semimetal phase in which only a part of degrees of free-
where λL is the Lyapunov exponent, t∗ the so-called scram- dom has been gapped out. This section reviews a holographic
bling time and vB the Butterfly velocity. The Lyapunov expo- model describing a topologically nontrivial Weyl semimetal
nent obeys the bound [69]: goes through a quantum phase transition to an insulating
λL ≤ 2πT, (50) phase where all the degrees of freedom are gapped.
A generation of holographic Weyl semimetal model can be
where T is the temperature of the system. It is saturated by constructed as follows. We use the Stueckelberg trick to re-
holographic field theories. For a review on quantum chaos place the complex scalar field Φ in eq. (11) by two real scalar
and holography see ref. [70]. Of particular interest is the be- fields ϕ and θ and introduce the general dilatonic couplings
havior of the Butterfly velocity across a quantum phase tran- in front of the kinetic terms [76]:
∫
sition. This question was addressed for the holographic Weyl √ [ 1 ( 1 ) Y(ϕ)
semimetal model in ref. [71]. There the authors computed S = d5 x −g 2 R + 12 − (∂ϕ)2 − V(ϕ) − F2
2κ 2 4
the Butterfly velocity in a holographic model with metric: Z(ϕ) 2 α abcde ( )
− (F) + ϵ Aa Fbc Fde + 3Fbc Fde
ds2 = −gtt (r)dt2 + grr dr2 + h⊥ (r)dx2⊥ + h∥ (r)dx2∥ . 4 3
(51)
W(ϕ) ]
− (Aa − ∂a θ)2 . (54)
This is precisely the metric that arises in the holographic 2
Weyl semimetal where we chose x∥ = z and x⊥ = (x, y). The Up to the anomaly the action eq. (54) is invariant under
result can be seen in Figure 12. θ → θ + χ, Aa → Aa + ∂a χ. One can show that the dual Ward
It shows the surprising feature that the parallel component identity for the conserved currents, which should be indepen-
of the Butterfly velocity has a minimum whereas the perpen- dent of the coupling strength of the system, is exactly the
dicular component shows a maximum at the quantum phase same as the one obtained from weakly coupled theory. The
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-16

(a) (b)

Figure 12 (Color online) Butterfly velocities as function of the dimensionless parameter M̄ = M/b for different values of the temperature. (a) The parallel
Butterfly velocity; (b) the perpendicular Butterfly velocity. As one can see the results are quite different, whereas the parallel one has a minimum, the perpen-
dicular one develop a minimum. The lines show different temperatures T/b = 0.005, 0.05, 0.1 corresponding to the colors red, blue and green. Figures from
ref. [71] CC BY 4.0.

holographic Weyl semimetal model in sect. 3 can be recov- is

ered via defining Φ = √12 ϕeiθ which is axially charged under √
1
1+8q0 −1)
the axial gauge field and setting Y(ϕ) = Z(ϕ) = 1, W(ϕ) = u = r(1 + r) , h = r(1 + r) , Az = a1 r 4 ( ,
2
√ (57)
q2 ϕ2 , V(ϕ) = m2 ϕ29) . 3 r
ϕ=− log ,
This section will focus on the physics at zero tempera- 2 1+r
ture. At T = 0 we use the same ansatz for the background where a1 is a free parameter which will make the geometry
fields as eq. (16) in sect. 3. To realise a holographic Weyl flow to AdS5 with different value M/b. At the leading or-
semimetal/insulator transition, the following dilatonic cou- der, the geometry is known as the GPPZ gapped geometry
plings are chosen: [79]. Here we have a nontrivial Az such that we will have an
  √  anisotropic geometry.
  2 
Z(ϕ) = 1 , W(ϕ) = −q0 1 − cosh 
 ϕ , Weyl semimetal phase The second kind of geometry near
3 
  √  (55) the horizon is
9   2 
V(ϕ) = 1 − cosh  ϕ , ϕ20 − 2a0√√q0
2 3  u = r2 , h = r2 , Az = a0 + e 3r ,
4a0 r (58)
√
ϕ0 − a0√ q0
and ϕ = 3/2 e 3r .
√  r
 2 
Y(ϕ) = cosh  ϕ . At the leading order, the IR geometry is an AdS5 with a con-
3 
(56)
stant Az . a0 can be rescaled to 1. The exponential terms play
the role of the irrelevant perturbations. This near horizon
Note there is a Z2 symmetry ϕ → −ϕ for eq. (54). When
geometry also shows up as the holographic Weyl semimetal
ϕ → 0, W(ϕ) ≃ q30 ϕ2 and V(ϕ) ≃ − 32 ϕ2 . Thus q0 plays a sim-
phase in sect. 3.
ilar role as the axial charge. In the following we restrict to
Critical point The third kind of near horizon geometry is
q0 > 0. Near the conformal boundary, ϕ → 0, the potential
( ) q0 ( )
in eq. (55) behaves as V(ϕ) = 12 m2 ϕ2 + . . . with m2 = −3. As u = u0 r2 1 + δurαc , h = r2β 1 + δhrαc , (59)
already discussed in sect. 3, close to the boundary (r → ∞) 9
√
( ) 3 ( )
we have ϕ = Mr + . . . , Az = b + . . . . Az = rβ 1 + δarαc , ϕ = (log ϕ1 ) 1 + δϕrαc . (60)
At zero temperature, we first give the near horizon solu- 2
tions. Similar to the holographic model in sect. 3, there are In the case of q0 = 15, we have (u0 , β, ϕ1 , αc ) ≃ (1.150, 0.769,
again three different kinds of near horizon geometries. Then 1.797, 1.230) and (δu, δh, δa) ≃ (0.147, −1.043, 0.591)δϕ.
we turn on irrelevant perturbations to get the full solutions. At the leading order the system has a Lifshitz symmetry
Insulating phase The first kind of near horizon geometry (t, x, y, r−1 ) → c(t, x, y, r−1 ), z → cβ z. We can use it to set

9) Studies on constructing insulating phases from generic holography with dilatonic coupling can be found in refs. [77, 78].
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-17

δϕ = −1. The irrelevant perturbations can flow the above the gap there is a continuous gapless spectrum and σzz even-
geometry to AdS5 . In the boundary we get (M/b)c ≃ 0.986. tually becomes also linear in frequency at large frequency.
For arbitrary q0 > 0, all the relevant perturbations around the The width of the gap depends on M/b in a similar way com-
above fixed point have complex scaling exponent, indicating paring to the weakly coupled result, i.e. it monotonically
that this fixed point is unstable [80, 81] which will be con- increases when M/b increases and for sufficient large M/b,
firmed by studying the free energy. ∆/b ∝ 0.22(M/b − 0.3).
The full solutions can be obtained by integrating the above The transverse conductivities can be calculated by consid-
near horizon geometries to the boundary. Different from the ering fluctuations δV x = v x (r)e−iωt , δVy = vy (r)e−iωt . Define
holographic semimetals in sect. 3, the near horizon behav- v± = v x ± ivy , from the holographic dictionary we can obtain
ior eq. (58) flows to AdS5 with a nontrivial M/b that takes the Green’s functions G± , from which we can compute G xx ,
from zero to (M/b)t+ with (M/b)t+ > (M/b)c and then de- Gyy and G xy . We have σ xy ± iσ xx = ± Gω± , i.e.
creasing to (M/b)c . The near horizon geometry (57) flows
to AdS5 with M/b whose value is from infinity to (M/b)t− G+ + G−
σT = σ xx = σyy = ,
with (M/b)t− < (M/b)c and then increasing to reach (M/b)c 2iω (61)
G+ − G−
finally. Figure 13 shows the profiles of the matter fields at dif- σAH = 8αb − σ xy = 8αb − .
2ω
ferent M/b. Near the critical M/b, the matter fields shows os-
cillatory behavior (dashed color lines), which can be viewed Figure 16 shows the full frequency dependence of transverse
as a sign of unstable critical solution. conductivities. Similar to the longitudinal conductivities,
With the bulk solution the free energy can be obtained nu- there is a gapless spectrum for Re[σ xx (ω)] and Re[σyy (ω)]
merically. Near the phase transition the behavior for free en- in the Weyl semimetal phase, while there exists a contin-
ergy is shown in Figure 14. Different from the holographic uous gapless spectrum above a hard gap ∆/b in the insu-
model in sect. 3, for this holographic system at zero temper- lating phase. The difference comparing to the longitudinal
ature there is a first order phase transition from the topologi- one is that if we increase M/b in the Weyl semimetal phase,
cally nontrivial Weyl semimetal phase to an insulator phase. Re[σT ]/ω increases at small ω. Figure 16(b) shows the real
The different order of the quantum phase transition may in- part of optical anomalous Hall conductivity at different values
dicate different underlying mechanisms for these two kinds of M/b. In the insulating phase at zero frequency the anoma-
of phase transitions. It can be easily checked that the phase lous Hall conductivity goes to a nonzero value. Furthermore,
transition is always of first order for any q0 > 0. the optical anomalous Hall conductivity has a smooth change
The exact nature of the stable phases can be figured out by at ω = ∆ in the insulating phase.
studying the conductivities. The real part of the optical longi- As already explained in sect. 3, the order parameter of the
tudinal electric conductivity σzz of the holographic system at quantum phase transition is the DC anomalous Hall conduc-
different M/b is shown in Figure 15. In the Weyl semimetal tivity. In the topological phase, the DC conductivities can be
phase, σzz is linear in frequency at both small and large fre- analytically obtained σAHE = 8αAz (0), σ xx = σyy = 0 . In the
quency regimes, which is quite similar to the discussion in gapped phase, there is no simple analytical formula for σAHE .
sect. 3.10. There is a hard gap for σzz in the insulating phase, The DC anomalous Hall conductivity can only be calculated
which indicates that it is indeed an insulating phase. Above numerically by taking ω → 0 limit of Re[σAH (ω)], which is

1.0 (a) (b)

15

0.8

0.6 10
Az
b

0.4
5
0.2

0.0 0
10−5 0.001 0.100 10 1000 105 10−5 0.001 0.100 10 1000 105
r r
b b
Figure 13 (Color online) The plots are for the profiles of Az (a) and ϕ (b) at M/b = 0.941 (green), 0.983 (blue), 0.987 (dashed cyan), 0.986 (dashed black),
0.984 (dashed brown), 0.987 (orange), 1.019 (purple). The solid lines are for the stable phase while dashed lines are for the unstable phase. Figures from ref.
[76].
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-18

order quantum phase transition to a normal band insulator.

−2.227 The holographic model shows that the strongly interacting
Weyl semimetal can also go through a first order quantum
−2.228
Vb4(M/b)2.5

phase transition to a Chern insulator. Thus it reveals the in-

Ω

−2.229 triguing phase structure for strongly correlated topological

Weyl semimetal. The holographic model provides a novel
−2.230
framework to further explore the physics of strongly interact-
−2.231 ing topological states of matter.
0.975 0.980 0.985 0.990 0.995
M
b 5 Holographic nodal line semimetals
Figure 14 (Color online) This plot shows the free energy density. Here
q0 = 15. The blue (dashed) lines are for solutions from eq. (58) while the In addition to Weyl semimetals, there are several other ex-
red (dashed) line are from eq. (57). The black dot is for the free energy at
the unstable critical point. Figure from ref. [76].
amples of topological states of matter, including topolog-
ical nodal line semimetal (NLSM), topological insulators,
anomalous Hall states, topological superconductors and so
on. In this section, we will focus on the physics of nodal line
1.5 semimetals from holography.
In a NLSM [82] the shape of Fermi surface is a one dimen-
sional circle in stead of nodal points in Weyl semimetals (see
1.0
Reσzz

ref. [83] for a review). When it is topologically nontrivial,

ω

the system cannot be gapped by small perturbations unless

0.5 going through a topological phase transition to another state.

0.0 5.1 Quantum field theoretical model

0.01 0.05 0.10 0.50 1 5 10
ω
In Weyl semimetal, two Weyl points in the momentum space
b
Figure 15 (Color online) The real part of optical longitudinal electric con-
are separated at b while in nodal line semimetal there is a one
ductivity at M/b = 0.941 (green), 0.983 (blue), 0.987 (orange), 1.019 (pur- dimensional circle of nodal line. Their low energy effective
ple). Figure from ref. [76]. theories are also different. The Weyl semimetal is described
different from the case for topological trivial semimetal in by a Dirac field coupled to a time reversal symmetry break-
sect. 3. ing field, i.e. the axial gauge field Az . Whereas for NLSM, it
Figure 17 shows the DC σAHE depending on M/b. When is described by a Dirac field coupled to a two form effective
M/b increases, the anomalous Hall conductivity decreases field bµν which breaks both time reversal and charge conju-
and jumps at to another nonzero value. After the phase tran- gate symmetry,
sition, σAHE looks insensitive to M/b. The discontinuity in ( )
L = −iψ̄ γµ ∂µ − m − γµν bµν ψ, (62)
σAHE indicates that the holographic quantum phase transition
is indeed of first order. After the phase transition, in the diag- where γµν = 2i [γµ , γν ] and bµν = −bνµ is an antisymmetric
onal components of the conductivities there exists a continu- two form field. Note that we take the (−, +, +, +) signature.
ous gapless spectrum above a hard gap, whereas the DC σAHE Without loss of generality, a source of a two form b xy is turned
is nonvanishing. These properties are exactly of a topologi- on. Then this system has the energy spectrum:
cal Chern insulator. Thus the holographic model introduced √ √
( )2
in this section describes a first order quantum phase transition E± = ± kz2 + 2b xy ± m2 + k2x + ky2 . (63)
from a strongly interacting topological Weyl semimetal to a
topological Chern insulator. The corresponding energy spectrum is shown in Figure 18.
From the field theoretical approach the phase diagrams For m2 < 4b2xy , the system is a topologically nontriv-
for interacting Weyl semimetals have been studied in refs. √ nodal line semimetal with a circle nodal line of radius
ial
[31, 72-74]. In ref. [72] it was found that for sufficiently 4b2xy − m2 10) . For m2 > 4b2xy , the system becomes an in-
strong interactions the Weyl semimetal can go through a first sulator and m2 = 4b2xy is the quantum phase transition point.

10) Note that the other components of bµν could also deform the nodal points to nodal line, e.g. nonzero btz would generate an accidental nodal line
semimetal.
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-19

2.5 (a)
1.0 (b)

2.0
0.8

1.5

ReσAH
0.6
Reσ T

8αb
ω

1.0 0.4

0.5 0.2

0.0 0.0
0.01 0.05 0.10 0.50 1 5 10 0.01 0.05 0.10 0.50 1 5 10
ω ω
b b

Figure 16 (Color online) The real part of the transverse optical electric conductivity (a) and the optical anomalous Hall conductivity (b) at M/b = 0.941
(green), 0.983 (blue), 0.987 (orange), 1.019 (purple) in the topological and the insulating phases. Figures from ref. [76].

1.0 0.5

0.8 0.4

0.6 0.3
σAHE
8αb

σAHE
8αb
0.4 0.2

0.2 0.1

0.0 0.0
0.0 0.5 1.0 1.5 0.975 0.980 0.985 0.990 0.995
M M
b b

Figure 17 (Color online) Both plots are for σAHE at zero frequency and zero temperature from holography. The right plot is a zoomed in version of the left
plot close to the quantum phase transition point. The solid and dashed lines are for stable and unstable phases separately. Figures from ref. [76].

(a) (b)

Figure 18 (Color online) The energy spectrum as a function of k x , ky for kz = 0. (a) A nodal line appears at the band crossing when m2 < 4b2xy ; (b) for
m2 > 4b2xy the system is gapped. Figures from ref. [84].

In the nodal line semimetal

√ phase, close
√ to the nodal line, the where anomaly terms have been ignored.
dispersion is linear in k2x + ky2 − 4b2xy − m2 with velocity
√
1 − 4bm2 at kz = 0 and linear in kz with velocity 1 when
2
5.2 Holographic model
√ xy
√
k2x + ky2 = 4b2xy − m2 .
The NLSM model eq. (62) has shown that a coupling term
With a nontrivial background two form field bµν , the elec-
ψ̄γµν ψ plays the role of deforming the single nodal point to a
tric current J µ = ψ̄γµ ψ is still conserved while the axial cur-
nodal loop. In holography we introduce on the gravity side
rent J5µ = ψ̄γµ γ5 ψ is no longer conserved. The following are
a massive 2-form field Bab which is dual to an antisymmetric
the conservation equations:
tensor operator and is expected to play a similar role as the
ψ̄γµν ψ operator. Similar to the holographic Weyl semimetal
∂µ J µ = 0 , ∂µ J5µ = −2mψ̄γ5 ψ − 2bµν ψ̄γµν γ5 ψ , (64) model, we also introduce an axially charged complex scalar
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-20

√
field Φ whose boundary value explicitly breaks the axial sym- √
2 13 ( )
metry to generate the gap. The holographic NLSM can be f = − 2 b0 rα 1 + δ f rα1 ,
3
realized from the following action [84]: ( )
∫ ϕ = ϕ0 r , B = b0 rα 1 + δb rα1 ,
β
√ [ 1 ( 12 ) 1 1
S = d5 x −g 2 R + 2 − F 2 − F 2
2κ L 4 4 where (α, β, α1 ) = (0.183, 0.290, 1.273), (δ f, δb) = (−2.616,
α abcde ( )
−0.302)δu. We can further set b0 to 1. At leading order there
+ ϵ Aa 3Fbc Fde + Fbc Fde
3 is a Lifshitz symmetry for the solution
1( )( )
− (Da Φ)∗ (Da Φ) − V1 (Φ) − D[a Bbc] ∗ D[a Bbc]
3η (t, z, r−1 ) → c(t, z, r−1 ) , (x, y) → cα/2 (x, y) , (68)
]
− V2 (Bab ) − λ|Φ|2 B∗ab Bab , which can set δu = ±1 where δu = −1 flows the geometry
to AdS5 . Thus we have a unique free parameter ϕ0 in the
where parts that do not involve Bab are the same as the holo-
system.
graphic Weyl semimetal (eq. (11)) in sect. 3. Bab has to
be axially charged since its dual operator’s source explicitly It turns out we only get solutions with M/b < 1.717
breaks the axial symmetry. The potential terms in the action in the UV. As ϕ0 grows from 0, M/b also grows from the
are value 0 and becomes closer and closer to the critical value
1.717. From the property of holographic fermion spectral
λ1 4
V1 = m21 |Φ|2 + |Φ| , V2 = m22 B∗ab Bab , (65) functions, one concludes the dual phase is a topological nodal
2
line semimetal.
where m1,2 are mass of Φ and Bab . Without loss of gener- Critical point The second kind of near horizon geometry
ality, Bxy component will be turned on in the following. In including irrelevant deformations is
the following we set q1 = q2 = 1, λ = η = 1, λ1 = 0.1 for
simplicity. u = uc r2 (1 + δu rβ1 ) , f = fc rαc (1 + δ f rβ1 ) ,
Since the operators ψ̄γµν ψ and ψ̄γµν γ5 ψ are not indepen- ϕ = ϕc (1 + δϕ rβ1 ) , B = bc rαc (1 + δb rβ1 ) ,
dent, which indicates that there should be a self-duality in
the real and imaginary part of the complex dual field Bab , with (uc , fc , αc , ϕc ) ≃ (3.076, 0.828bc , 0.292, 0.894) , and
our strategy here is instead to consider a two form antisym- β1 = 1.272 , (δu, δ f, δb) = (1.177, −2.771, −0.409)δϕ .
metric operator different from ψ̄γµν ψ and does not have the We can set bc to be 1. At the leading order there exists a
property of self duality. Note that some holographic QCD same type of Lifshitz symmetry eq. (68) with a different scal-
models [85, 86] considered the self-duality effect of the two ing exponent αc instead of α. This Lifshitz symmetry can set
form field. δϕ to be −1 to flow to AdS5 in UV. Therefore there is only
We will focus again on the zero temperature physics and one single such solution. We get the solution with the critical
take the following ansatz: value M/b ≃ 1.717.
Trivial phase The third kinds of near horizon geometry
dr2
ds2 = u(−dt2 + dz2 ) + + f (dx2 + dy2 ) , is
u (66)
( )
Φ = ϕ(r) , Bxy = B(r) . 3
u= 1+ r2 , f = r2 ,
8λ1
Near the UV boundary r → ∞, the expansions for the two √ √ 2
√ √ 3λ+λ
matter fields ϕ(r) and B(r) are 3 2 160λ +84λ1 +9
1
−2 2 2 3+8λ1
ϕ= + ϕ1 r 3+8λ1
, B = b1 r 1 .

M λ1
ϕ= + ··· , B = br + · · · , (67)
r The ϕ1 - and b1 -terms above are the irrelevant deformations
where M and b are the sources associated to the dual opera- that flow the geometry to asymptotic AdS5 solutions. In this
tors. At zero temperature, it turns out there are again three case we only get solutions with M/b > 1.717.
different kinds of near horizon geometries. Similar to the Figure 19 shows the bulk profiles of matter fields ϕ and
holographic Weyl semimetal, adding some irrelevant defor- B/ f at different M/b. Close to the critical M/b the IR solu-
mations, the near horizon geometries flow to an AdS5 in the tion flows quickly to the one for critical solution. The free
UV with some values of M/b. energy of the system can be numerically studied and we find
Topological phase The first kind of near horizon geome- that when the phase transition occurs, the system is very con-
try is tinuous though the bulk IR solutions are discontinuous at the
1 √ ( ) horizon. In holography this is a quite common feature for
u = (11 + 3 13)r2 1 + δu rα1 , continuous quantum phase transitions.
8
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-21

5.3 A generic framework for topological states from Dirac operator in the dual field theory. In the bulk the cou-
hologrphy pling terms between the spinors and the scalar field are the
same as the ones in the Weyl semimetal. There is one most
This holographic nodal line semimetal has the same mathe- natural way to couple the two bulk spinors to the two form
matical structure as the holographic Weyl semimetal in sect. field Bab . We use the following action for the probe fermion:
3. Ref. [84] proposed a general framework in holography to
describe the strongly coupled gapless topological states. The S = S 1 + S 2 + S int , (69)
∫ ( )
bulk topological structure arises as follows. √
S1 = d5 x −giΨ̄1 Γa Da − m f Ψ1 ,
• In the holographic system, there are at least two inter- ∫
√ ( )
acting matter fields. One of them dual to the operator which S2 = d5 x −giΨ̄2 Γa Da + m f Ψ2 ,
plays the role of mass effect, and the other dual to an operator ∫
√ ( )
which deforms the topology of the Fermi surface. For illus- S int = − d5 x −g iΦΨ̄1 Ψ2 + iΦ∗ Ψ̄2 Ψ1 + LB , (70)
tration these two fields are labeled as ϕ and A. The interaction
between ϕ and A in deep IR generates interesting topological and
structure of the solution space. LB = −i(η2 Bab Ψ̄1 Γab γ5 Ψ2 − η∗2 B∗ab Ψ̄2 Γab γ5 Ψ1 ) . (71)
• At zero temperature usually there exist three different
Note that in the bulk the Lorentz invariance in the tangent
kinds of solutions at the horizon. Two of them are the solu-
A,ϕ space has been explicitly broken.
tion that at leading order A (or ϕ) is nonvanishing with r−δ−
The system has a rotation symmetry in the k x -ky plane and
while at subleading order ϕ (or A) is sourced by A (or ϕ). √
There also exists a critical solution where both ϕ and A are only depends on k x−y = k2x + ky2 . Without loss of generality
subleading and sourcing each other. Because these two fields we set ky = 0. From the holographic dictionary, we can com-
A,ϕ
cannot be of leading order at the same time with r−δ− in IR, pute the retarded Green’s function G. Then we could get its
the semimetal phase cannot be gapped by small perturbations four eigenvalues and the spectral function. In the following
and is therefore topologically nontrivial. we summarize the properties for the Green’s function in the
The existence of a universal topological structure in the holographic nodal line semimetal.
bulk suggests that in principle from holography we could ob- • In all the three phases, for nonzero kz , the retarded
tain a large class of topologically nontrivial strongly coupled Green’s function at zero frequency is real.
gapless systems. • In the trivial phase, the retarded Green’s function is
real for all values of k x , ky , kz , 0. The pole is located at
5.4 Fermionic probe on the holographic nodal line k x = ky = kz = 0 and this is consistent with the explanation
that this trivial semimetal phase is only partially gapped.
semimetal
• For the critical point, among the four eigenvalues of the
Although in NLSMs there is no sharp order parameter like Green’s function, two of them have peaks in the imaginary
anomalous Hall conductivity for Weyl semimetals, we could part at k x = ky = 0 and the other two are still small for all
show that indeed there exists a circle of nodal loop in the dual k x , ky .
fermionic spectral functions by probing fermions in the bulk. • Figure 20 shows the spectral function G−1 (0, k x ) for a
Similar to the discussion for holographic Weyl semimetal finite regime of k x at kz = ω = 0, M/b ≃ 0.0013 and
in sect. 3.6, we utilize two spinors in the bulk to describe a m f = −1/4. All the Green’s function’s four eigenvalues are

(a) 1.5 (b)

5

4
1.0
3
B
φ

2
0.5
1

0 0.0
10−15 10−10 10−5 1 105 10−15 10−10 10−5 1 105
r r
b b

Figure 19 (Color online) The bulk profile for the scalar field ϕ (a) and the two form field B/ f (b) for M/b = 1.682 (green), 1.702 (brown), 1.717 (red), 1.733
(purple), 1.750 (black). Figures from ref. [84].
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-22

real. They have the form (g1 , −g1 , g2 , −g2 ) with both g1 and invariants [83]. The first one is the Berry phase around a one
g2 are positive and g1 ≥ g2 . The two eigenstates with eigen- dimensional closed line which links with the nodal loop in
values g1 , −g1 are labeled as “bands I” and the other two with the momentum space. This one is related to the stability of
g2 , −g2 as “bands II”. In Figure 20 different colors are used the nodal loop under small perturbations in the system. The
to distinguish different bands. Furthermore, since −G−1 (0, k) second one is the Berry flux around a sphere enclosing the
can be treated as a topological Hamiltonian [43,45], the spec- whole nodal loop. This topological invariant is to describe
tral density plot should qualitatively agree with the plot for whether the critical point is topological or not, which will not
eigenvalues in the ω-k x plane. be discussed here.
• From Figure 20 we can see that between each two ad- From holography the strongly interacting NLSM phase has
jacent poles bands I and II always and only intersect once in multiple while discrete nodal lines in the k x -ky plane and
the upper frequency plane. Between each two adjacent band kz = 0. Since the circle that links with two or more nodal
crossing points there is always one pole and one zero of the lines at the same time can be continuously deformed to two
Green’s function. or more separate circles each enclosing only one nodal line
• In the strongly coupled nodal line semimetal phase from inside, we can focus on the Berry phase associated with each
holography there are multiple
√ and discrete Fermi surfaces in nodal line. From the Green’s function at ω = 0 we found
the k x -ky plane at kFi = k2x + ky2 and kz = 0, ω → 0. The dual that these poles are from two different sets of bands (bands I
system has more complicated topological structure. At each and II) which indicates that along the k x axis the two gapped
nodal line momentum, there is a sharp peak (a pole at ω = 0) bands and two gapless bands exchange their roles alterna-
in the imaginary part of two eigenvalues of the Green’s func- tively. We could calculate the Berry phase numerically by
tion whereas the imaginary part of the other two is very small, choosing discrete points along the circle and found that there
which means that they are gapped. is a nontrivial Berry phase π associated with poles from band
• When k x increases, the distance between adjacent poles I and for poles from band II the Berry phase is undetermined.
becomes larger. At small k x the poles are very close to each For the zeros of the Green’s functions we have a trivial Berry
other. We have not plotted this regime in Figure 20 because phase of zero [48].
the nodal loops are too dense to reveal all the poles and a
much heavier numerics is required.
• When M/b increases, each nodal line momentum de- 6 Alternative approaches
creases and goes to zero at the transition point. Figure 21(a)
shows the behavior of one kFi depending on M/b and An approach to strongly coupled model of Weyl fermions
Figure 21(b) shows the dispersion in the k x direction at M/b ≃ based on holography that differs somewhat from the one re-
0.0013. Note that the dispersions in both the kz and k x direc- viewed in this article was presented in ref. [87]. There the
tions are almost linear for each branch of nodal lines. idea is to study fermions which are strongly coupled in a
holographic theory. This is similar to the study of probe
fermions in holographic backgrounds [46, 47]. The fermions
5.5 Topological invariants are treated as probes which might or might not reflects the
In nodal line semimetals there are two kinds of topological underlying physics of the dual field theory. It has been used
for example to study the electric conductivity in ref. [88].
30 It would be interesting to explore the other characteristic fea-
tures of Weyl semimetals including surface states, anomalous
20 Hall effect etc., in this approach and to go beyond the probe
limit to include the backreaction of the probe fermions to the
10
gravitational background.
It is known that Weyl semimetals can also be gener-
−1.0 −0.5 0.5 1.0
ated from Dirac systems by applying a rotating electric field
−10 [89, 90]. More precisely the Dirac fermions split into left-
and right-handed Weyl fermions under the application of a
−20 fast rotating electric field. The question if this also happens
in holography has been investigated in ref. [91]. The con-
−30
struction is based on probe D7-branes in an AdS5 × S5 back-
Figure 20 (Color online) Eigenvalues of −G−1 (0, k x ) for M/b ≃ 0.0013. ground. The background serves also as energy reservoir and
Red color is for bands I and blue color is for bands II. Figure from ref. [48]. allows the formation of a non-equilibrium steady state. This
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-23

1.2
(a) 5×10−6 (b)
1.0
−6
4×10
0.8
3×10−6
kF 0.6
f

b
ω
0.4 2×10−6

0.2 1×10−6

0.0 0
0.0 0.5 1.0 1.5 1.0470 1.0475 1.0480 1.0485
M kx
b b

√
Figure 21 (Color online) (a) The nodal line momentum kF = k2x + ky2 and kz = 0. In both the critical and trivial phases, no Fermi surface exists at finite
k whereas the pole is at k = ω = 0 which is consistent with the fact that only partial degrees of freedom are gapped. (b) The dispersion relation associated to
the nodal momenta in the left plot at M/b ≃ 0.0013. The best fit is for k x < kF , ω ≃ 0.005(1.0477 − k x )0.998 ; while for k x > kF , ω ≃ 0.005(k x − 1.0477)0.994 .
Figures from ref. [48].

allows to compute the Hall conductivity as function of the models has been explored. There are many open questions.
applied frequency of the driving electric field but also be- An incomplete list is as follows.
yond the regime of linear response. In this top-down model • It would be interesting to include chemical potentials for
the field content of the dual theory is clear, which in princi- vector and axial symmetries and study the chiral magnetic ef-
ple provides more constraints and insights into the physics of fect in these models. This would tell us about the CME in a
the boundary field theory. However, this model works in the strongly interacting Weyl semimetal. Meanwhile, it would be
probe brane limit while possible backreaction is not clear. It interesting to study negative magnetoresistivity in this model.
would be also interesting to explore more physics of the dual • In the quantum critical region a new anomaly related
system from this approach. transport coefficient, anomaluos Hall viscosity appears. It
would be interesting to develop the full hydrodynamics of
the quantum critical region.
7 Summary and outlook on further research • The Weyl cones in Weyl semimetals can be tilted and so-
called type II Weyl semimatals can appear if the tilt exceeds
We have reviewed the holographic construction of models
the “lightcone” defined by the Fermi velocity. Can one also
capable of reproducing key features of the physics of topo-
construct holographic models of type II Weyl semimetals?
logical semimetals such as Weyl and nodal line semimetals.
• The holographic Weyl semimetal in sect. 3 describes a
Amongst them the quantum phase transition to a topologi-
holographic dual for Weyl semimetal with two Weyl nodes.
cally trivial state, the anomalous Hall conductivity, surface
It would be interesting to consider the holographic dual for
states, topological invariants, a new understanding of the ax-
multiple Weyl nodes.
ial Hall conductivity. Some of the new results derived from
that model is the appearance of anomalous Hall viscosity in • The quantum phase transition in the holographic
the quantum critical region at finite temperature of the phase WSM/Chern insulator model is of first order. It would be
transition. A short summary is given in Table 2 for Weyl interesting to study if it is still first order for more general
semimetals and in Table 3 in the case of nodal line semimet- holographic phase transitions between Weyl semimetal and
als. insulating phases.
In these tables, we list different features of the weakly • It would be interesting to explore the disorder ef-
coupled semimetal and strongly coupled semimetal from fects or other momentum dissipation effects, finite tem-
holography for comparison, including the symmetries, trans- perature physics, transport physics etc., in the holographic
ports/features, edge states, topological invariants and mate- WSM/Chern insulator model.
rial realisation. In the nodal line semimetals, there is no • The holographic insulating phase is a Chern insulator. It
sharp transport signature like anomalous Hall conductivity would be interesting to explore the topological invariants, to
to distinguish the topological phase and trivial phase. The explore effects of surface states, to realise the phase transition
Fermi surfaces of the system show interesting features with to a normal insulator and so on.
nodal loop in the weakly coupled case and multiple loops in • It would be interesting to construct the smoking gun
the strongly coupled case. The question marks are the items transport in the holographic nodal line semimetals.
which are not clear yet. • In the holographic nodal line semimetal, it would be in-
So far only a small subset of the parameter space of these teresting to consider the holographic model with a self-dual
 K. Landsteiner, et al. Sci. China-Phys. Mech. Astron. May (2020) Vol. 63 No. 5 250001-24

Table 2 The summary of holographic Weyl semimetal

Properties Weakly coupled WSM Holographic WSM
Symmetries time reversal or inversion symmetry breaking time reversal symmetry breaking
Transports anomalous Hall conductivity AHE , odd viscosity
Edge states Fermi arc surface current
Topological invariants ±1 ±1
Material TaAs, TaP, etc. WP2 ?

Table 3 The summary of holographic nodal line semimetal

Properties Weakly coupled NLSM Holographic NLSM
Symmetry symmetry protected by mirror refection symmetry, inversion symmetry symmetry protected by inversion symmetry
Features nodal loop multiple nodal loops
Edge states no ?
Topological invariants π one set is π, another undetermined
Material PbTaSe2 , ZrTe, etc. ?

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