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arXiv:2504.20695 · Physical review. B./Physical review. B

Inhomogeneous phase stiffness in two-dimensional s -wave disordered superconductors

Sudipta Biswas, A. Taraphder, Sudhansu S. Mandal

Published Sep 25, 2025• 2 citations
arXivHTMLPDF

Abstract

We investigate the effect of white-noise disorder on the local phase stiffness and thermodynamic properties of a two-dimensional $s$-wave superconductor. Starting from a local attractive model and using path-integral formalism, we derive an effective action by decoupling the superconducting order parameter into amplitude and phase components in a gauge-invariant manner. Perturbative techniques are applied to the phase fluctuation sector to derive an effective phase-only XY model for disordered superconducting systems. Solving the saddle-point Green's function using Bogoliubov-de Gennes theory, we calculate the distributions of nearest-neighbor couplings for various disorder strengths. A single-peak distribution is observed for low disorder strength, which becomes bimodal with one peak at negative couplings as the disorder strength increases. The local phase stiffness remains randomly distributed throughout the lattice and shows no correlation with pairing amplitudes. The temperature dependence of the superfluid stiffness (${J}_{s}$) is studied using Monte Carlo simulations. At strong disorder and low temperatures, ${J}_{s}$ increases with increasing temperature, exhibiting anomalous behavior that may indicate the onset of a glassy transition. Additionally, calculations of the Edwards-Anderson order parameter in this disorder regime suggest the emergence of a phase-glass state at very low temperatures.

DOI
10.1103/p64s-p7t3

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We investigate the effect of white-noise disorder on the local phase stiffness and thermodynamic properties of a two-dimensional sss-wave superconductor. Starting from a local attractive model and using path-integral formalism, we derive an effective action by decoupling the superconducting order parameter into amplitude and phase components in a gauge-invariant manner. Perturbative techniques are applied to the phase fluctuation sector to derive an effective phase-only XY model for disordered superconducting systems. Solving the saddle-point Green's function using Bogoliubov-de Gennes theory, we calculate the distributions of nearest-neighbor couplings for various disorder strengths. A single-peak distribution is observed for low disorder strength, which becomes bimodal with one peak at negative couplings as the disorder strength increases. The local phase stiffness remains randomly distributed throughout the lattice and shows no correlation with pairing amplitudes. The temperature dependence of the superfluid stiffness (Js{J}_{s}Js​) is studied using Monte Carlo simulations. At strong disorder and low temperatures, Js{J}_{s}Js​ increases with increasing temperature, exhibiting anomalous behavior that may indicate the onset of a glassy transition. Additionally, calculations of the Edwards-Anderson order parameter in this disorder regime suggest the emergence of a phase-glass state at very low temperatures.
We study a simple model of a two-dimensional s-wave superconductor in the presence of a random potential as a function of disorder strength. We first use the Bogoliubov--de Gennes (BdG) approach to show that, with increasing disorder, the pairing amplitude becomes spatially inhomogeneous, and the system cannot be described within conventional approaches for studying disordered superconductors that assume a uniform order parameter. In the high-disorder regime, we find that the system breaks up into superconducting islands, with large pairing amplitude, separated by an insulating sea. We show that this inhomogeneity has important implications for the physical properties of this system, such as superfluid density and the density of states. We find that a finite spectral gap persists in the density of states, even in the weak-coupling regime, for all values of disorder, and we provide a detailed understanding of this remarkable result. We next generalize Anderson's idea of the pairing of exact eigenstates to include an inhomogeneous pairing amplitude, and show that it is able to qualitatively capture many of the nontrivial features of the full BdG analysis. Finally, we study the transition to a gapped insulating state driven by quantum phase fluctuations about the inhomogeneous superconducting state.
We show, using a Bogoliubov-de Gennes (BdG) mean field theory, that the local pairing amplitude Δ(r) becomes highly inhomogeneous with increasing disorder in an s-wave superconductor. The probability distribution P(Δ) is peaked about the BCS value at low disorder, but with increasing disorder, progressively develops into a broad distribution with significant build up of weight near Δ≈0. At high disorder, the system is found to form superconducting "islands" separated by a non-superconducting sea. Surprisingly, a finite energy gap persists into the highly disordered state in spite of many sites having negligible pairing amplitude and is understood in detail within the BdG framework. Once the pairing amplitude becomes inhomogeneous, the role of quantum phase fluctuations becomes crucial in driving a superconductor-insulator transition at a critical disorder. The insulator is unusual as it has a finite gap for all disorder strengths in marked contrast to the Anderson insulator in non-interacting systems. We treat the phase fluctuations within a self consistent harmonic approximation and obtain the superfluid stiffness as a function of disorder, which agrees well with our earlier quantum Monte Carlo studies.
We study the effect of disorder on the superconductor-insulator transition in an inhomogeneous d-wave superconductor using the kernel polynomial method. As the Bogoliubov-de Gennes equations of the two-dimensional square lattice are solved self-consistently for the cases with more than 100000 unit cells, it is possible to observe the spatial fluctuations of the superconducting order parameters at the nanoscale. We find that strong spatial fluctuation of the superconducting order parameters can be introduced by disorder, and some superconducting specific order parameters are even enhanced. Moreover, we find that some isolated superconducting “islands” can survive the strong disorder, giving a boson insulator with some localized Cooper pairs. Our numerical calculations predict the existence of two sequential transitions with the increasing disorder strength: a d-wave to s-wave superconductor transition, and then an s-wave superconductor to insulator transition. The possibility of the appearance of a metallic phase between the superconducting and insulating phases is excluded by performing the lattice-size scaling of the generalized inverse participation ratio. In addition, we also discuss the effect of disorder on the optical conductivity of the d-wave superconductors.
We consider the attractive Hubbard model with on-site disorder as a prototype of a disordered superconductor. We solve the Bogoliubov-de Gennes equations on two-dimensional finite clusters at zero temperature and evaluate the electromagnetic response to a vector potential. We find that the standard decoupling between transverse and longitudinal response does not apply in the presence of disorder. Moreover, the superfluid density is strongly reduced by the relaxation of the phase of the order parameter already at mean-field level when disorder is large. We also find that the anharmonicity of the phase fluctuations is strongly enhanced by disorder. Beyond mean field, this provides an enhancement of quantum fluctuations inducing a zero-temperature transition to a nonsuperconducting phase of disordered preformed pairs. Finally, the connection of our findings with the glassy physics for extreme dirty superconductors is discussed.
The effect of nonmagnetic impurities on 2D sss-wave superconductors is studied beyond the weak disorder regime. Within the Bogoliubov--de Gennes (BdG) framework, the local pairing amplitude develops a broad distribution with significant weight near zero with increasing disorder. Surprisingly, the density of states continues to show a finite spectral gap. The persistence of the spectral gap at large disorder is shown to arise from the breakup of the system into superconducting ``islands.'' Superfluid density and off-diagonal correlations show a substantial reduction at high disorder. A simple analysis of phase fluctuations about the highly inhomogeneous BdG state is shown to lead to a transition to a nonsuperconducting state.
In this article we study superconductor-insulator transitions within the general framework of an attractive Hubbard model. This is a well-defined model of s-wave superconductivity which permits different tuning parameters (disorder and field). Furthermore, it allows a comparison of various analytical and computational approaches in order to gain a complete understanding of the various effects of amplitude and phase fluctuations. We present a systematic pedagogical approach, aiming to equip the lay reader with enough apparatus to be able to understand the numerical calculations, reproduce some of the simpler results, and be able to tackle future problems related to inhomogeneous phases. We go into considerable detail on mean-field theory (MFT) and the Bogoliubov-de Gennes (BdG) approach, as these are a first line of attack which can capture much of the physics, but we also outline cases where this fails to capture phase fluctuations and more sophisticated Quantum Monte Carlo (QMC) calculations are necessary. We discuss the behavior of many observables, including densities of states, superfluid stiffness, and dynamical conductivity, for the disorder-tuned superconductor-insulator transition. We also discuss SITs tuned by parallel magnetic field, which are quite different due to pairbreaking.