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arXiv:cond-mat/0003018 · Physical review. B, Condensed matter

Quasiparticle localization transition in dirty superconductors

Smitha Vishveshwara, Matthew P. A. Fisher

Published Oct 15, 2001• 8 citations
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Abstract

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.

DOI
10.1103/physrevb.64.174511

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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.
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 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.
The thermal metal to thermal insulator transition due to random disorder is studied in the context of the symmetries of the Bogoliubov de Gennes Hamiltonian. We focus on a three dimensional system with gapless s-wave pairing that possesses time reversal and spin rotational symmetry. The quasiparticle excitations (bogolons) undergo a metal insulator transition as the disorder increases. We determine the critical disorder strength and correlation exponent first by the transfer matrix method (TMM). We then apply a multifractal finite sized scaling (MFSS) of the bogolon wavefunction obtained from large scale diagonalization of the Hamiltonian and obtain the critical disorder strength and exponent, in agreement with those found by TMM.
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.
We study the density of quasiparticle states of dirty d-wave superconductors. We show the existence of singular corrections to the density of states due to quantum interference effects. We then argue that the density of states actually vanishes in the localized phase as ∣E∣|E|∣E∣ or E2{E}^{2}E2 depending on whether time-reversal is a good symmetry or not. We verify this result for systems without time-reversal symmetry in one dimension using supersymmetry techniques. This simple, instructive calculation also provides the exact universal scaling function for the density of states for the crossover from ballistic to localized behavior in one dimension. Above two dimensions, we argue that in contrast to the conventional Anderson localization transition, the density of states has critical singularities which we calculate in a $2+\ensuremath{\epsilon}$ expansion. We discuss consequences of our results for various experiments on dirty high-Tc{T}_{c}Tc​ materials.
It is well known that the metal-insulator transition in two dimensions for non-interacting fermions takes place at infinitesimal disorder. In contrast, the superconductor-to-insulator transition takes place at a finite critical disorder (on the order of Vc ~ 2t), where V is the typical width of the distribution of random site energies and t is the hopping scale. In this article we compare the localization/delocalization properties of one and two particles. Whereas the metal-insulator transition is a consequence of single-particle Anderson localization, the superconductor-insulator transition (SIT) is due to pair localization – or, alternatively, fluctuations of the phase conjugate to pair density. The central question we address is how superconductivity emerges from localized single-particle states. We address this question using inhomogeneous mean field theory and quantum Monte Carlo techniques and make several testable predictions for local spectroscopic probes across the SIT. We show that with increasing disorder, the system forms superconducting blobs on the scale of the coherence length embedded in an insulating matrix. In the superconducting state, the phases on the different blobs are coherent across the system whereas in the insulator long-range phase coherence is disrupted by quantum fluctuations. As a consequence of this emergent granularity, we show that the single-particle energy gap in the density of states survives across the transition, but coherence peaks exist only in the superconductor. A characteristic pseudogap persists above the critical disorder and critical temperature, in contrast to conventional theories. Surprisingly, the insulator has a two-particle gap scale that vanishes at the SIT despite a robust single-particle gap.