Sum Rule of Hall Conductance in a Random Quantum Phase Transition

Abstract

The Hall conductance σ xy of two-dimensional lattice electrons with random potential is investigated. The change of σ xy due to randomness is focused on. It is a quantum phase transition where the sum rule of σ xy plays an important role. By the string (anyon) gauge, numerical study becomes possible in sufficiently weak magnetic field regime which is essential to discuss the floating scenario in the continuum model. Topological objects in the Bloch wavefunctions, charged vortices, are obtained explicitly. The anomalous plateau transitions ( ∆σ xy = 2, 3, • • • > 1) and the trajectory of delocalized states are discussed.

Localization and mobility gap in the topological Anderson insulator

“…each other to avoid a band-touching which will trigger a transition to a trivial insulator [11]. This is a vivid manifestation of "robustness against weak disorder" for TI.…”

“…In two dimensions (2D), the TI (ν = 1) exhibits quantum spin Hall effect, whose edge currents are robust against weak non-magnetic disorder [4,8,9]. This dissipation-less transport can only be destroyed by extremely strong disorder, which drives the system into a traditional Anderson insulator [10,11]. The 2D TI has been experimentally realized in HgTe/CdTe quantum wells, where the thickness of the quantum well can be varied to tune the system between TI and normal insulator [12,13].…”

Topological quantum phase transitions in superconductivity on lattices

“…Typical examples include the Mott transition in strongly correlated systems and the Hall plateau transition in the quantized Hall effect. [1][2][3][4] The latter is special in the sense that the ground state is characterized by a quantized quantity ͑quantized Hall conductance xy ). As is known today, its quantization originates from a topological character of the Hall conductance.…”

“…Some of them are a special form of a selection rule, and the possible stability of the phase, which puts special importance on the topological transitions and discriminates from other quantum phase transitions. 4,10 Recently, there have been trials to extend the concept of this topological phase transition to unconventional singlet superconductivity. [11][12][13][14] In the discussions, mapping the system into the standard quantum Hall system is essential.…”

Levitation and percolation in quantum Hall systems with correlated disorder

“…7 In the quantum Hall system, the topological structure of the ground state wave function plays an essential role for quantization of the Hall conductance. Each quantum Hall state can be assigned a topological quantum number as the Chern number, 8 which is the Hall conductance in units of e 2 / h. Although there are several works [9][10][11][12][13] to calculate the Chern numbers and succeeded to explain experimentally observed direct transition, the random potential, which is spatially uncorrelated, is not compatible with the continuum limit. 14,15 Koschny et al proposed that unless the random potential has long range correlation, the floating theory in a lattice system cannot be consistent with that in a continuous system.…”