Relevant perturbations at the spin quantum Hall transition

Abstract

We study relevant perturbations at the spin quantum Hall critical point using a network model formulation. The model has been previously mapped to classical percolation on a square lattice, and we use the mapping to extract exact analytical values of the scaling dimensions of the relevant perturbations. We find that several perturbations that are distinct in the network model formulation correspond to the same operator in the percolation picture. We confirm our analytical results by comparing them with numerical simulations of the network model. PACS numbers: 72.15.Rn, 73.20.Fz, INTRODUCTIONAnderson localization of a quantum particle [1] or a classical wave in a random environment is a vibrant research field [2]. One of its central research directions is the physics of Anderson transitions [3], quantum critical points tuned by disorder. These include metal-insulator transitions and transitions of quantum Hall type separating distinct phases of topological insulators. While such transitions are conventionally observed in electronic (metallic and semiconductor) structures, there is also a considerable number of other experimental realizations actively studied in recent and current works. These include localization of light [4] and microwaves [5], cold atoms [6] (see a recent review [7]), ultrasound [8], and optically driven atomic systems [9].From the theoretical point of view, symmetries play a central role in determination of universality classes of critical phenomena. This idea was applied to Anderson localization by Altland and Zirnbaueer (AZ) [10] who identified ten distinct symmetry classes. In three of these classes, classes A, C, and D in AZ classification, the timereversal invariance is broken, and there is a possibility for a quantum Hall transition in two dimensions.The transition in class A is the usual integer quantum Hall (IQH) transition in a two-dimensional (2D) electronic system in a strong perpendicular magnetic field (see Ref.[11] for a review). Class A also includes the model of electrons in a random magnetic field, where all states are believed to be localized [12].Class C is one of the four Bogolyubov-de Gennes classes which describe transport of quasiparticles in disordered superconductors at a mean field level, and possess the particle-hole symmetry. In this class the spinrotation invariance is preserved, the quasiparticles have conserved spin, and one can study spin transport. The corresponding Hall transition is known as the spin quantum Hall (SQH) transition [13,14], at which the system exhibits a jump in the spin Hall conductance from 0 to 2 in appropriate units.In spite of tremendous efforts, most models of Anderson transitions have resisted analytical treatment. The IQH transition is one prominent example where only recently some analytical progress has been achieved [15]. On the other hand, the SQH transition enjoys a special status, since a network model of this transition was mapped exactly to classical percolation on a square lattice [16]. The original mapping used the supersymme...

Phase diagram of the spin quantum Hall transition

“…A numerical result found in [7] for a second critical exponent 1.45 µ ≈ was in a good agreement with analytical prediction 3/2 [12]. Both results were recently significantly corrected [13]. Percolation mapping of [12] was used to extract analytical value μ = 8/7.…”

“…It turned out that the deviation from Kramers degeneracy is a superior way to extract criti-cal exponents in this case, since we know its exact zero value at the critical point. It has been shown in [13] that both perturbations breaking spin-rotation invariance act as a random Zeeman field and must have the same critical exponent µ. The numerical results using deviations from Kramers degeneracy produced 1.15 µ ≈ in excellent agrement with analytical prediction μ = 8/7 1.14 ≈ .…”

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

“…The mapping has lead to a host of exact critical properties at the spin QH transition 90,[92][93][94][95][96][97] and was extended to arbitrary graphs.…”

“…(6) can be applied to all critical exponents obtained in Refs. [90,92,[94][95][96][97]. This includes, in particular, the dimension of the "two-leg" operator that determines the localization length exponent ν, as well as a few multifractal exponents.…”

“…Many exact results are known for classical percolation. Thus, the mapping has lead to a host of exact critical properties at the SQH transition [78,[80][81][82][83][84][85]. The mapping was extended to network models in class C on arbitrary graphs [63].…”