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arXiv:2608.00849

Spin Quantum Hall Effect: the Critical Exponents

Hrant Topchyan, W. Nuding, Andreas Klümper, A. Sedrakyan

Published Aug 1, 2026• 0 citations
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Abstract

The spin quantum Hall effect (SQHE) provides one of the few examples of an Anderson localization transition for which exact critical exponents are known, making it an important testing ground for theories of disordered topological systems and conformal field theory. The corresponding network model, obtained by replacing the random $U(1)$ phases of the Chalker--Coddington model with random $SU(2)$ matrices, belongs to symmetry class C of the Altland--Zirnbauer classification and is believed to describe quasiparticle transport in two-dimensional disordered superconductors with broken time-reversal symmetry. In this work, we perform high-precision numerical calculations of the localization-length exponent ($ν$) and the boundary critical exponent ($μ$) for the SQHE network model using the recently developed $S$-matrix approach to random networks.

DOI
10.48550/arxiv.2608.00849

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The spin quantum Hall effect (SQHE) provides one of the few examples of an Anderson localization transition for which exact critical exponents are known, making it an important testing ground for theories of disordered topological systems and conformal field theory. The corresponding network model, obtained by replacing the random U(1)U(1)U(1) phases of the Chalker--Coddington model with random SU(2)SU(2)SU(2) matrices, belongs to symmetry class C of the Altland--Zirnbauer classification and is believed to describe quasiparticle transport in two-dimensional disordered superconductors with broken time-reversal symmetry. In this work, we perform high-precision numerical calculations of the localization-length exponent (ννν) and the boundary critical exponent (μμμ) for the SQHE network model using the recently developed SSS-matrix approach to random networks.
Level statistics for two classes of disordered systems at criticality are analyzed in terms of different realizations of the Chalker-Coddington network model. These include: 1) Re-examination of the standard U(1) model describing dynamics of electrons on the lowest Landau level in the quantum Hall effect, where it is shown that after proper local unfolding the nearest-neighbor spacing distribution (NNSD) at the critical energy follows the Wigner surmise for Gaussian unitary ensembles (GUE). 2) Quasi-particles in disordered superconductors with broken time reversal and spin rotation invariance (in the language of random matrix theory this system is a representative of symmetry class D in the classification scheme of Altland and Zirnbauer). Here again the NNSD obeys the Wigner surmise for GUE, reflecting therefore only “basic” discrete symmetries of the system (time reversal violation) and ignoring particle-hole symmetries and other finer details (criticality). In the localized regime level repulsion is suppressed.
We introduce and analyze a class of model systems to study transitions in the integer quantum Hall effect (IQHE). Even without disorder our model exhibits an IQHE transition as a control parameter is varied. We find that the transition is in the two-dimensional Ising universality class and compute all associated exponents and critical transport properties. The fixed point has time-reversal, particle-hole, and parity invariance. We then consider the effect of quenched disorder on the IQHE transition and find the following. (i) Randomness in the control parameter (which breaks all the above symmetries) translates into bond randomness in the Ising model and is hence marginally irrelevant. The transition may equally well be viewed as a quantum percolation of edge states localized on equipotentials. The absence of random-phase factors for the edge states is responsible for the nongeneric (Ising) critical properties. (ii) For a random magnetic field (which preserves particle-hole symmetry in every realization) the model exhibits an exactly solvable fixed line, described in terms of a product of a Luttinger liquid and an SU(n) spin chain. While exponents vary continuously along the fixed line, the longitudinal conductivity is constant due to a general conformal sum rule for Kac-Moody algebras (derived here), and is computed exactly. We also obtain a closed expression for the extended zero-energy wave function for every realization of disorder and compute its exact multifractal spectrum f(\ensuremath{\alpha}) and the exponents of all participation ratios. One point on the fixed line corresponds to a recently proposed model by Gade and Wegner. (iii) The model in the presence of a random on-site potential scales to a strong disorder regime, which is argued to be described by a symplectic nonlinear-sigma-model fixed point. (iv) We find a plausible global phase diagram in which all forms of disorder are simultaneously considered. In this generic case, the presence of random-phase factors in the edge-state description indicates that the transition is described by a Chalker-Coddington model, with a so far analytically inaccessible fixed point.
We study a physical system consisting of noninteracting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotation invariance. This system belongs to class D within the recent classification scheme of random matrix ensembles, and its phase diagram contains three different phases: metallic and two distinct localized phases with different quantized thermal Hall conductances. We find that critical exponents describing different transitions (insulator-to-insulator and insulator-to-metal) are identical within the error of numerical calculations and also find that critical disorder of the insulator-to-metal transition is energy-independent.
Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\ensuremath{\nu}$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a Z2{\mathbb{Z}}_{2}Z2​ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents, which we show (numerically) to depend on whether the insulating state is topologically trivial or not.
A fundamental property of the quantum Hall effect is the presence of edge states at the boundary of the sample, robust against localization and responsible for the perfect quantization of the Hall conductance. The transition between integer quantum Hall plateaus is a delocalization transition, which can be identified as a strong-coupling fixed point of a 1 + 1-dimensional supersymmetric sigma model with topological theta-term. The conformal field theory describing this transition displays unusual features such as non-unitarity, and resisted any attempt of solution so far. In this thesis we investigate the role of edge states at quantum Hall transitions using lattice discretizations of super sigma models. Edge states correspond to twisted boundary conditions for the fields, which can be discretized as quantum spin chains or geometrical (loop) models. For the spin quantum Hall effect, a counterpart of the integer quantum Hall effect for spin transport (class C), our techniques allow the exact computation of critical exponents of the boundary conformal field theories describing higher plateaus transitions. Our predictions for the mean spin conductance are validated by extensive numerical simulations of the related localization problems. Envisaging applications to transport in network models of 2 + 1-dimensional disordered electrons, and to quenches in one-dimensional gapless quantum systems, in this thesis we have also developed a new formalism for dealing with partition functions of critical systems in rectangular geometries. As an application, we derive formulas for probabilities of self-avoiding walks.
We study the spin quantum Hall effect and transitions between Hall plateaus in quasi-two-dimensional network models consisting of several coupled layers. Systems exhibiting the spin quantum Hall effect belong to class C in the symmetry classification for Anderson localization, and for network models in this class there is an established mapping between the quantum problem and a classical one involving random walks. This mapping permits numerical studies of plateau transitions in much larger samples than for other symmetry classes, and we use it to examine localization in systems consisting of nnn weakly coupled layers. Standard scaling ideas lead one to expect nnn distinct plateau transitions, but in the case of the unitary symmetry class this conclusion has been questioned. Focusing on a two-layer model, we demonstrate that there are two separate plateau transitions, with the same critical properties as in a single-layer model, even for very weak interlayer coupling.