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arXiv:1201.0244 · International Journal of Modern Physics Conference Series

CRITICAL EXPONENT FOR THE QUANTUM SPIN HALL TRANSITION IN ℤ2 NETWORK MODEL

K. KOBAYASHI, Tomi Ohtsuki, Keith Slevin

Published Jan 1, 2012• 6 citations
arXivHTMLPDF

Abstract

We have estimated the critical exponent describing the divergence of the localization length at the metal-quantum spin Hall insulator transition. The critical exponent for the metal-ordinary insulator transition in quantum spin Hall systems is known to be consistent with that of topologically trivial symplectic systems. However, the precise estimation of the critical exponent for the metal-quantum spin Hall insulator transition proved to be problematic because of the existence, in this case, of edge states in the localized phase. We have overcome this difficulty by analyzing the second smallest positive Lyapunov exponent instead of the smallest positive Lyapunov exponent. We find a value for the critical exponent ν = 2.73 ± 0.02 that is consistent with that for topologically trivial symplectic systems.

DOI
10.1142/s2010194512005995

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We have estimated the critical exponent describing the divergence of the localization length at the metal-quantum spin Hall insulator transition. The critical exponent for the metal-ordinary insulator transition in quantum spin Hall systems is known to be consistent with that of topologically trivial symplectic systems. However, the precise estimation of the critical exponent for the metal-quantum spin Hall insulator transition proved to be problematic because of the existence, in this case, of edge states in the localized phase. We have overcome this difficulty by analyzing the second smallest positive Lyapunov exponent instead of the smallest positive Lyapunov exponent. We find a value for the critical exponent ν = 2.73 ± 0.02 that is consistent with that for topologically trivial symplectic systems.
The localization problem of electronic states in a two-dimensional quantum spin Hall system (that is, a symplectic ensemble with topological term) is studied by the transfer matrix method. The phase diagram in the plane of energy and disorder strength is exposed, and demonstrates "levitation" and "pair annihilation" of the domains of extended states analogous to that of the integer quantum Hall system. The critical exponent nu for the divergence of the localization length is estimated as nu congruent with 1.6, which is distinct from both exponents pertaining to the conventional symplectic and the unitary quantum Hall systems. Our analysis strongly suggests a different universality class related to the topology of the pertinent system.
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.
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.
FINDING: Disordered 3D topological phase transition exhibits universal critical exponent ν ≈ 2.7, consistent with metal–ordinary insulator transitions, indicating a common universality class beyond topological specifics. MATH: - Critical exponent: ν ≈ 2.7 (from scaling analysis of localization length, arXiv:1211.5026v2) - Scaling relation: ξ ∝ |W − W_c|^(−ν), where ξ is localization length, W disorder strength, W_c critical disorder - Universality: ν(3D, symplectic class AII) ≈ 2.7 — matches known Anderson transition exponent for 3D symplectic class (ν ≈ 2.73 ± 0.02 in some numerical studies) - No new exponent for topological transition — the topological invariant (Z₂) does not alter the critical scaling CONNECTION: - ν ≈ 2.7 ≈ e (2.71828) within ~0.7% — a striking near-coincidence with Euler's number, though not proven exact. - 2.7 is close to 2.618 (φ²) — deviation ~3.1%, but not within numerical error bars (typically ±0.1). - No direct base-60 or crystallographic s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The quantum Hall effect is one of the most extensively studied topological effects in solid‐state physics. The transitions between different quantum Hall states exhibit critical phenomena described by universal critical exponents. Numerous numerical finite size scaling studies have focused on the critical exponent ν for the correlation length. In such studies, it is important to take proper account of irrelevant corrections to scaling. Recently, Dresselhaus et al. proposed a new scaling ansatz, which they applied to the two terminal conductance of the Chalker–Coddington model. Herein, their proposal is applied to the previously reported data for the Lyapunov exponents of that model using both polynomial fitting and Gaussian process fitting.
Machine learning has been successfully applied to identify phases and phase transitions in condensed matter systems. However, quantitative characterization of the critical fluctuations near phase transitions is lacking. In this paper, we propose a finite-size scaling approach based on a convolutional neural network and analyze the critical behavior of a quantum Hall plateau transition. The localization length critical exponent learned by the neural network is consistent with the value obtained by conventional approaches. We show that the general-purposed method can be used to extract critical exponents in models with drastically different physics and input data, such as the two-dimensional Ising model and four-state Potts model.