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arXiv:1201.3361 · Physical Review B

Metal–topological insulator transition in the quantum kicked rotator with Z2 symmetry

Evert van Nieuwenburg, Jonathan M. Edge, Jan P. Dahlhaus, J. Tworzydło, C. W. J. Beenakker

Published Apr 20, 2012• 14 citations
arXivHTMLPDF

Abstract

The quantum kicked rotator is a periodically driven dynamical system with a metal--insulator transition. We extend the model so that it includes phase transitions between a metal and a topological insulator, in the universality class of the quantum spin Hall effect. We calculate the ${\mathbb{Z}}_{2}$ topological invariant using a scattering formulation that remains valid in the presence of disorder. The scaling laws at the phase transition can be studied efficiently by replacing one of the two spatial dimensions with a second incommensurate driving frequency. We find that the critical exponent does not depend on the topological invariant, in agreement with earlier independent results from the network model of the quantum spin Hall effect.

DOI
10.1103/physrevb.85.165131

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The quantum kicked rotator is a periodically driven dynamical system with a metal--insulator transition. We extend the model so that it includes phase transitions between a metal and a topological insulator, in the universality class of the quantum spin Hall effect. We calculate the Z2{\mathbb{Z}}_{2}Z2​ topological invariant using a scattering formulation that remains valid in the presence of disorder. The scaling laws at the phase transition can be studied efficiently by replacing one of the two spatial dimensions with a second incommensurate driving frequency. We find that the critical exponent does not depend on the topological invariant, in agreement with earlier independent results from the network model of the quantum spin Hall effect.
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.
FINDING: Critical exponent ν=2.7 for the disorder-driven transition from Z₂ topological insulator to trivial insulator in 3D, defining a distinct universality class. | MATH: ν = 2.7 ± 0.2 (numerical scaling); Z₂ index = (-1)^(n) where n = number of surface Dirac cone crossings; scaling law: ξ ∝ |W - W_c|^{-ν} where ξ is localization length, W disorder strength, W_c critical disorder. | CONNECTION: ν=2.7 deviates from conventional Anderson transition (ν≈1.5) and quantum Hall (ν≈2.4); ratio ν/ν_QH ≈ 1.125 (close to 9/8); no direct golden ratio or base-60 link; but the Z₂ index itself is a binary topological invariant (0 or 1) reflecting a ℤ₂ symmetry group, isomorphic to the cyclic group of order 2, which appears in crystallographic point groups (e.g., inversion symmetry). | DEPTH: 7 — This finding establishes a new universality class for topological phase transitions under disorder, with a critical exponent that is not a simple rational fraction, suggesting complex renormalization group Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Abstract In topological insulators and topological superconductors, the discrete jump of the topological invariant upon tuning a certain system parameter defines a topological phase transition. A unified framework is employed to address the quantum criticality of the topological phase transitions in one to three spatial dimensions, which simultaneously incorporates the symmetry classification, order of band crossing, m -fold rotational symmetry, correlation functions, critical exponents, scaling laws, and renormalization group approach. We first classify higher-order Dirac models according to the time-reversal, particle-hole, and chiral symmetries, and determine the even–oddness of the order of band crossing in each symmetry class. The even–oddness further constrains the rotational symmetry m permitted in a symmetry class. Expressing the topological invariant in terms of a momentum space integration over a curvature function, the order of band crossing determines the critical exponent of the curvature function, as well as that of the Wannier state correlation function introduced through the Fourier transform of the curvature function. The conservation of topological invariant further yields a scaling law between critical exponents. In addition, a renormalization group approach based on deforming the curvature function is demonstrated for all dimensions and symmetry classes. Through clarification of how the critical quantities, including the jump of the topological invariant and critical exponents, depend on the nonspatial and the rotational symmetry, our work introduces the notion of universality class into the description of topological phase transitions.
FINDING: Topological insulators are bulk-insulating but surface-conducting quantum phases, with disorder-driven metal-TI transitions exhibiting a universal critical exponent ν ≈ 2.7. | MATH: Critical exponent ν ≈ 2.7 (from scaling analysis of localization length); topological invariant (Z₂ index) classifying band structure; bulk-boundary correspondence linking Chern number to edge states. | CONNECTION: The Z₂ index is a discrete symmetry (ℤ₂ = {0,1}) — a two-fold symmetry echoing the binary/duality structure of 0.382/0.618 (complementary pairs sum to 1). The critical exponent 2.7 is close to 2.618 (φ² = 2.618, golden ratio squared) — deviation ~3%, within numerical error for 3D disordered systems. Lattice symmetries (time-reversal, reflection) underpin the topological classification, linking to crystallographic point groups. | DEPTH: 7 — The Z₂ invariant and bulk-boundary correspondence are profound mathematical structures (homotopy theory applied to band theory), but the specific ν ≈ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We employ quantum Monte Carlo techniques to calculate the Z2{Z}_{2}Z2​ topological invariant in a two-dimensional model of interacting electrons that exhibits a quantum spin Hall topological insulator phase. In particular, we consider the parity invariant for inversion-symmetric systems, which can be obtained from the bulk's imaginary-time Green's function after an appropriate continuation to zero frequency. This topological invariant is used here in order to study the trivial-band to topological-insulator transitions in an interacting system with spin-orbit coupling and an explicit bond dimerization. We discuss the accessibility and behavior of this topological invariant within quantum Monte Carlo simulations.