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Thermal quantum Hall effect in disordered superconductors

V. Kagalovsky, B. Horovitz, Y. Avishai

Published Jan 1, 2004• 0 citations
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Abstract

We study the symmetry class for localization which arises from models of noninteracting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotation invariance. Twodimensional systems in this category, which is known as class D, can display phases with three different types of quasiparticle dynamics: metallic, localized, or with a quantized (thermal) Hall conductance. Correspondingly, they can show a variety of delocalization transitions. We illustrate this behavior by investigating numerically the phase diagrams of network models with the appropriate symmetry and show the appearance of the metallic phase. We also study level statistics for this symmetry class and find that the nearest neighbor spacing distribution (NNSD) at the critical energy follows the Wigner’s surmise for Gaussian Unitary Ensembles (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.

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We study the symmetry class for localization which arises from models of noninteracting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotation invariance. Twodimensional systems in this category, which is known as class D, can display phases with three different types of quasiparticle dynamics: metallic, localized, or with a quantized (thermal) Hall conductance. Correspondingly, they can show a variety of delocalization transitions. We illustrate this behavior by investigating numerically the phase diagrams of network models with the appropriate symmetry and show the appearance of the metallic phase. We also study level statistics for this symmetry class and find that the nearest neighbor spacing distribution (NNSD) at the critical energy follows the Wigner’s surmise for Gaussian Unitary Ensembles (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 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.
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 study in detail a critical line on the phase diagram of the Cho-Fisher network model separating three different phases: metallic and two distinct localized phases with different quantized thermal Hall conductances. This system describes noninteracting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotational invariance. We find that in addition to a tricritical fixed point WT{W}_{T}WT​ on the critical line, separating two localized phases, there exists an additional repulsive fixed point WN{W}_{N}WN​ (where the vortex disorder concentration WN<WT{W}_{N}<{W}_{T}WN​<WT​), which splits RG flow into opposite directions: toward a clean Ising model at W=0W=0W=0 and toward WT{W}_{T}WT​.
In this paper, a delocalization-localization transition within a superconducting state is explored. The symmetries of the Bogoliubov deGennes Hamiltonian endow the two associated superconducting phases---the thermal metal and the thermal insulator---and the critical point between them, with properties that are in stark contrast with analogous phases in normal systems. Here, for systems preserving spin-rotational invariance and time-reversal symmetry in three dimensions, the transition is shown to be of a different universality class from its normal partner by extracting and comparing the localization length exponent. The density of states, which may be regarded as the ``order parameter'' for the field theory describing the superconducting system, is studied for its unusual properties about criticality.
A generic two-dimensional disordered topological superconductor in symmetry class D exhibits rich phenomenology and multiple phases: diffusive thermal metal (DTM), Anderson insulator (AI), and thermal quantum Hall (TQH) phase (a topological superconductor). We numerically investigate the phase diagram of a lattice model of such class-D superconductor, specifically focusing on transitions between the phases and the associated universal critical behaviors. We confirm the existence of a tricritical point and its repulsive nature at the point on the phase diagram where the three phases meet. We characterize the critical behaviors at various critical points and the tricritical point using numerical evaluation of the localization length, the conductance (or conductivity), and the density of states. We conclude that the two metal-insulator transitions (DTM-TQH and DTM-AI) belong to the same universality class, whereas the tricritical point represents a distinct universality class.
We study how quasiparticle interactions affect their localization properties in dirty superconductors with broken time reversal symmetry -- for example in a magnetic field. For SU(2) spin-rotation invariant (class C) systems, the only important coupling is the spin-spin triplet interaction, which we study within a renormalization group approach. Either an additional Zeeman coupling or a complete breaking of spin rotation symmetry renders all interactions irrelevant. These two situations realize, respectively, the non-interacting unitary Anderson and the ``thermal'' (class D) universality class. Our results imply a stable metallic phase in 2D for class D. Experimental implications are discussed.