Metal-insulator transition in a 2D system of chiral unitary class Jonas F. Karcher1 2 3Ilya A. Gruzberg4and Alexander D. Mirlin2 3 1Pennsylvania State University Department of Physics University Park Pennsylvania 16802 USA

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Metal-insulator transition in a 2D system of chiral unitary class

Jonas F. Karcher,1, 2, 3 Ilya A. Gruzberg,4and Alexander D. Mirlin2, 3

1Pennsylvania State University, Department of Physics, University Park, Pennsylvania 16802, USA

2Institute for Quantum Materials and Technologies,

Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany

3Institut für Theorie der Kondensierten Materie,

Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

4Ohio State University, Department of Physics, 191 West Woodruﬀ Ave, Columbus OH, 43210, USA

(Dated: September 23, 2022)

We perform a numerical investigation of Anderson metal-insulator transition (MIT) in a two-

dimensional system of chiral symmetry class AIII by combining ﬁnite-size scaling, transport, den-

sity of states, and multifractality studies. The results are in agreement with the sigma-model

renormalization-group theory, where MIT is driven by proliferation of vortices. We determine the

phase diagram and ﬁnd an apparent non-universality of several parameters on the critical line of

MIT, which is consistent with the analytically predicted slow renormalization towards the ultimate

ﬁxed point of the MIT. The localization-length exponent νis estimated as ν= 1.55 ±0.1.

Introduction. Anderson transitions (ATs) in disor-

dered systems—which include metal-insulator transitions

(MITs) as well as transitions between topologically dis-

tinct insulating phases—remain a dynamic ﬁeld of re-

search [1]. In this context, two-dimensional (2D) systems

attract particular attention. On the experimental side,

there is a variety of realizations of 2D electronic disor-

dered systems, including semiconductor heterostructures

and MOSFETs, graphene and other 2D materials, oxide

heterostructures, as well as surfaces of topological insu-

lators and superconductors. Furthermore, investigation

of 2D disordered systems in photonic structures is an

emerging research area [2].

For the most conventional setting of a quantum par-

ticle in a random potential (Wigner-Dyson orthogonal

symmetry class AI), d= 2 is a lower critical dimension-

ality, as for conventional second-order phase transitions

with continuous symmetry. This implies that there is no

AT in 2D systems of this symmetry class and all states

are localized (although the localization length is exponen-

tially large for weak disorder). At the same time, it was

realized that there is a number of mechanisms generating

ATs in 2D disordered systems of other symmetry classes.

While ﬁeld theories of ATs are non-linear sigma models

with a continuous non-abelian symmetry, the existence of

metallic (symmetry-broken) phases in 2D geometry is not

in conﬂict with the Mermin-Wagner theorem, in view of

an unconventional character of the symmetry groups (in-

volving supersymmetry and non-compactness or replica

limit, depending on the formulation).

The 2D ATs include, in particular, MITs in classes AII,

D, and DIII with broken spin-rotation invariance play-

ing a crucial role, as well as quantum-Hall transitions

in classes A, C, and D that are governed by topology.

Whereas ATs of these types have been studied in a rather

detailed fashion, there is one more type of 2D ATs that

has received much less attention: MITs in chiral classes

AIII, BDI, and CII. In fact, early studies demonstrated a

resilience of chiral systems to Anderson localization, lead-

ing to a suggestion that 2D and 3D systems of chiral sym-

metry classes do not exhibit AT at all, remaining always

in a delocalized phase [3]. This has received an apparent

support from the renormalization-group (RG) analysis

of the corresponding sigma models performed in pioneer-

ing works of Gade and Wegner [4, 5], which yielded no

quantum corrections to conductivity (and thus no local-

ization) to all orders in perturbation theory. The Gade-

Wegner RG implies that 2D systems of chiral classes pos-

sess a metallic phase with a line of infrared-stable ﬁxed

points with diﬀerent values of conductivity. The special

character of RG in chiral classes is related to the fact

that the corresponding sigma-model manifolds contain

an additional U(1) degree of freedom.

More recently, numerical studies of suitably designed

2D models of chiral classes have provided evidence of An-

derson MITs [6, 7]. An analytical theory of 2D ATs in

chiral classes was developed in Ref. [8]. It was pointed out

in Ref. [8] that, since the sigma-model manifolds for chi-

ral classes are not simply connected [due to the U(1) de-

gree of freedom], they allow for topological excitations—

vortices. Inclusion of the vortices in the RG analysis leads

to a metal-insulator phase transition [8], in an analogy

with the famous Berezinskii-Kosterlitz-Thouless (BKT)

transition in the XY model. The analysis of the resulting

RG ﬂow showed, however, that there is an essential dif-

ference: the transition happens at a ﬁnite fugacity y > 0,

at variance with the ﬁxed point value y= 0 for the BKT

transition. This hinders a fully controllable analytical

calculation of critical exponents at MITs in chiral classes,

thus making numerical studies of these transitions even

more important. The central goal of this paper is a nu-

merical study of the 2D MIT in the chiral unitary class

AIII.

Chiral classes. The special character of disordered

systems of chiral symmetry classes has been understood

since the pioneering work of Dyson who found a singu-

arXiv:2210.03131v1 [cond-mat.dis-nn] 6 Oct 2022

larity of the density of states in 1D harmonic chains at

zero energy (chiral symmetry point) [9]. Further works

extended the analysis to localization properties and to

quasi-1D systems. It was found that an N-channel quasi-

1D system of chiral class has Ntopological phases. At

transitions between these phases, the density of states ex-

hibits the Dyson singularity [10, 11], and the localization

length diverges [12–20]. Critical points of these tran-

sitions have inﬁnite-randomness character, with critical

wave functions showing very strong ﬂuctuations [15, 21].

For 2D chiral-class systems, most of the past research

focussed on properties of the metallic phase. The Gade-

Wegner sigma model was re-derived and analyzed in

many works [22–25]. A particular attention was paid

to the asymptotic infrared behavior, with is of inﬁnite-

randomness character, exhibiting a very strong diver-

gence of the density of states and a “freezing” of the mul-

tifractality spectrum [6, 26–28]. On the numerical side,

most papers showed critical properties of the metallic

phase that are characterized by non-universal exponents

for various observables (such as multifractality, density

of states, localization length at ﬁnite energy) [29–34]

and are essentially diﬀerent from those expected in the

inﬁnite-randomness infrared limit. This is not surprising:

the Gade-Wegner ﬂow towards the line of infrared ﬁxed

points is logarithmically slow, so that in a typical situ-

ation the infrared limiting behavior can likely be out of

reach on any realistic length scale. In several works [35–

37], evidence of the asymptotic behavior of the lowest

Lyapunov exponents in the quasi-1D geometry has been

reported.

Apart from realizations in disordered electronic sys-

tems, the interest to models in the chiral classes is due

to their relation to models of Dirac fermions coupled to

ﬂuctuating gauge ﬁleds that are discussed in the con-

text of quantum chromodynamics (QCD) [38]. It was

proposed that ATs in such models may be connected to

QCD phase transitions; see Ref. [39] for a recent review.

It is also worth mentioning that chiral-class models can

be experimentally realized in microwave setups based on

coupled resonators [40]. Recently, MITs in 3D chiral-class

systems were studied in Refs. [41, 42].

Field theory of 2D chiral AT. In the fermionic replica

formalism, the sigma-model manifolds for classes AIII,

BDI, and CII are U(n), U(2n)/Sp(2n), and U(n)/O(n),

respectively. In the analytical and numerical analysis be-

low, we focus on the class AIII. The Gade-Wegner sigma-

model action has the form [4, 5]

S[Q] = −Zd2rhσ

8πTr(U−1∇U)2+κ

8π(TrU−1∇U)2i.

Here U(r)∈U(n)(with the replica limit n→0to be

taken at the end of the calculation), σis the conductivity

in units of e2/πh; the second term (known as “the Gade

term”) couples only to the U(1) degree of freedom and is

speciﬁc for chiral classes. To describe the transition, one

has to include also vortices, with a fugacity y[8]. The

RG equations for three couplings σ,κ, and yread

∂K/∂ ln L= 1/4−2Ky2,(1)

∂y/∂ ln L= (2 −K)y , (2)

∂σ/∂ ln L=−σy2,(3)

where K= (σ+κ)/4. Equations (1) and (2) form a

closed system, with a ﬁxed point at K= 2 and y=1

In the three-dimensional parameter space (σ,κ,y), this

corresponds to a critical line of MITs, σ+κ= 8,y=1

Along this line, there is a ﬂow according to Eq. (3) to-

wards the ultimate ﬁxed point σ= 0,κ= 8, and y=1

This ﬂow is, however, very slow: σ(L)=σ0L−1/16. There-

fore, while in the strict infrared limit the transition is

described by the ultimate ﬁxed point, on realistic scales

one expects to see a transition described by some point

on the critical line. This is expected to lead to an ap-

parent non-universality of some of critical properties, as

discussed below.

The RG ﬂow that follows from Eqs. (1)–(3) is illus-

trated in Fig. 1. The overall ﬂow is three-dimensional

and is thus diﬃcult to display. What is shown is the pro-

jection of the ﬂow on the σ–κplane, with all RG trajec-

tories having an initial value of the fugacity y0=1

4. The

ﬁxed points of the ﬂow are as follows. First, there is an

infrared-stable line of ﬁxed points describing the metal-

lic phase, with σbeing an arbitrary constant, κ→ ∞,

y→0. Second, there is an infrared-stable ﬁxed point

describing the insulating phase: σ, κ →0and y→ ∞.

Finally, there is a ﬁxed point σ= 0,κ= 8,y=1

4, de-

scribing the MIT. It has one unstable direction, so that

there is a two-dimensional critical surface with a ﬂow to-

wards this point. A cross-section of this surface with the

plane y=1

4is the critical line σ+κ= 8 shown in Fig. 1.

Linearizing the RG equations (1) and (2) near the

transition point, we get the critical exponent of the lo-

calization length ν= 1.54 and the irrelevant exponent

yirr = 0.77. In addition, there is a very slow ﬂow towards

the ﬁxed point along the critical line described by Eq. (3);

it yields an exponent y0

irr = 1/16 '0.06. The fact that

the ultimate ﬁxed point of the transition is at σ= 0

implies very strong ﬂuctuations of critical eigenfunctions

in the infrared limit (with freezing of the multifractal

spectrum). This is expected on physical grounds: we

know that eigenstates in the metallic phase possess this

property, and it would be surprising if eigenstates at the

transition would be “less localized” than in the metal.

Let us reiterate that the RG equations are only con-

trollable at y1. Since the ﬁxed point of the tran-

sition is at y=1

4, that is not parametrically small, all

quantitative conclusions about the transition should be

taken with caution. A plausible assumption is that the

obtained ﬂow is qualitatively correct but numbers de-

scribing the transition may diﬀer substantially. It is thus

crucially important to explore the transition numerically,

2 4 6 8 σ0

...

insulator

MIT

...

∞

metal

FIG. 1. Schematic representation of the RG ﬂow implied by

Eqs. (1)-(3). The starting value y0of the fugacity is taken to

be the critical one, y0=1

4, and the resulting ﬂow is projected

to the σ–κplane.

which is done below.

Model. We study the bipartite Hamiltonian deﬁned on

a square lattice,

H=X

i,j hc†

i,j t(x)

i,j ci+1,j +c†

i.j t(y)

i,j ci,j+1 +h.c.i,(4)

with hoppings

t(x)

i,j =1 + 1

2(e−δ−1)[(−1)i+ 1](1 + vi,j ),

t(y)

i,j =1 + 1

2(e−δ−1)[(−1)j+ 1](1 + wi,j ).(5)

Disorder is introduced via random vi,j and wi,j , whose

real and imaginary parts are drawn independently from

box distributions on [−W/2, W/2]. Since the matrix el-

ements are complex, the time-reversal symmetry is bro-

ken, which puts Hin the chiral unitary class AIII. The

real parameter δcontrols the degree of staggering, which

is absent for δ= 0 and maximal for δ→ ±∞, when the

system decouples into 2×2plaquettes.

Finite-size scaling. To locate the MIT, we use the

transfer-matrix method for a quasi-1D strip of width

M= 12,...,256 and large length L= 105, with peri-

odic boundary conditions in the transverse (M) direc-

tion. The extracted Lyapunov exponents λk,M become

self-averaging at large L. The inverse of the smallest Lya-

punov exponent yields the quasi-1D localization length

ξM=λ−1

0,M . In the localized phase, ξMis determined,

for large M, by the 2D localization length ξ2D, so that

ξM/M →0at M→ ∞. In contrast, in the metallic

phase, the large-Mlimit of ξM/M is nonzero. Note that

this limit is ﬁnite (at variance with conventional MITs),

which reﬂects a peculiar critical nature of the metallic

phase in 2D chiral-class systems.

In Fig. 2, we show the ratio ξM/M for various Mas a

function of δfor W= 0.5. The plot clearly shows an MIT

at δc'1.2. The same analysis is carried out for W=

0.3,1.0,2.0,3.0, see Supplementary Material (SM) [43].

The resulting phase diagram is shown in Fig. 3. Applying

a scaling ﬁt (see inset of Fig. 2), ξM/M =F(dνM)with

d=δ−δc, we ﬁnd the exponent of the localization length,

0.0 0.5 1.0 1.5 2.0

0.01

0.05

0.10

0.50

ξM/M

W=0.50

ν=1.55, δc=1.22

F[νM]

0.010 0.100 1 10 100 1000

νM

0.001

0.005

0.010

0.050

0.100

0.500

ξM/M

FIG. 2. Finite-size scaling analysis. Ratio ξM/M as a function

of staggering δfor disorder W= 0.5and M= 12,...,256.

Inset: data collapse ξM/M =F(dνM), with d=δ−δc,

critical staggering δc= 1.22, and the exponent ν= 1.55.

0.5 1.0 1.5 2.0 2.5 3.0

0.5

1.0

1.5

0.25

0.50

0.75

1.00

1.25

1.50

1.75

FIG. 3. Phase diagram of MIT in the (W,δ) plane. Red

symbols: MIT critical values δc(W)obtained by transfer-

matrix analysis, see Fig. 2. Color code: IPR exponent,

τ2(L) = −∂ln P2(L)/∂ ln Lfor largest available L.

ν= 1.55 ±0.1, in a remarkable agreement with the value

ν= 1.54 obtained from the RG equations (1) and (2). A

very close result for νwas obtained very recently for a

related non-Hermitian model [44].

Let us emphasize an apparent non-universality of the

ratio ξM/M at criticality, see Table I. This is consistent

with a very slow RG ﬂow along the critical line σ+κ= 8

predicted analytically.

Inverse participation ratio (IPR). A complementary

approach is to study directly properties of eigenstates

ψ(r)of a 2D system. We performed the exact diago-

nalization of L×Lsystems with L= 24,...,768 and

periodic boundary conditions, averaging over N= 500

W δcξM/M ανσ1/[2π(α0−2 + xν)]

0.3 1.64 0.72 0.015 3.3 0.70

0.5 1.22 0.73 −0.004 3.6 0.71

1.0 0.73 0.41 −0.025 2.9 0.44

2.0 0.33 0.45 −0.09 2.7 0.45

3.0 0.22 0.42 −0.11 2.6 0.40

TABLE I. Critical parameters on the MIT line.

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Metal-insulatortransitionina2DsystemofchiralunitaryclassJonasF.Karcher,1,2,3IlyaA.Gruzberg,4andAlexanderD.Mirlin2,31PennsylvaniaStateUniversity,DepartmentofPhysics,UniversityPark,Pennsylvania16802,USA2InstituteforQuantumMaterialsandTechnologies,KarlsruheInstituteofTechnology,76021Karlsruhe,Germany3I...

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Metal-insulator transition in a 2D system of chiral unitary class Jonas F. Karcher1 2 3Ilya A. Gruzberg4and Alexander D. Mirlin2 3 1Pennsylvania State University Department of Physics University Park Pennsylvania 16802 USA

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