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arXiv:1707.04597

Multifractal Study of Quasiparticle Localization in Disordered Superconductors

Conrad Moore, Ka-Ming Tam, Zhang, Yi, Mark Jarrell

Published Jul 14, 2017• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.1707.04597

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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.
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 analyze the disorder-driven localization of the two-dimensional Bose-Hubbard model by evaluating the full low-energy quasiparticle spectrum via a recently developed fluctuation operator expansion method.For any strength of the local interaction we find a mobility edge that displays an approximately exponential decay with increasing disorder strength.We determine the finite-size scaling collapse and exponents at this critical line finding that the localization of excitations is characterized by weak multifractality and a thermal-like critical gap ratio.A direct comparison to a recent experiment yields an excellent match of the predicted finite-size transition point and scaling of single particle correlations.
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.
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.
We describe a new multifractal finite-size scaling (MFSS) procedure and its application to the Anderson localization-delocalization transition. MFSS permits the simultaneous estimation of the critical parameters and the multifractal exponents. Simulations of system sizes up to L3=1203{L}^{3}={120}^{3}L3=1203 and involving nearly 106{10}^{6}106 independent wave functions have yielded unprecedented precision for the critical disorder Wc=16.530(16.524,16.536){W}_{c}=16.530(16.524,16.536)Wc​=16.530(16.524,16.536) and the critical exponent $\ensuremath{\nu}=1.590(1.579,1.602)$. We find that the multifractal exponents ${\ensuremath{\Delta}}_{q}$ exhibit a previously predicted symmetry relation and we confirm the nonparabolic nature of their spectrum. We explain in detail the MFSS procedure first introduced in our Letter [Phys. Rev. Lett. 105, 046403 (2010)] and, in addition, we show how to take account of correlations in the simulation data. The MFSS procedure is applicable to any continuous phase transition exhibiting multifractal fluctuations in the vicinity of the critical point.