Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2106.11848 · Physical review. B./Physical review. B

Distribution of the order parameter in strongly disordered superconductors: An analytic theory

Anton V. Khvalyuk, M. V. Feigel’man

Published Dec 15, 2021• 3 citations
arXivHTMLPDF

Abstract

We developed an analytic theory of inhomogeneous superconducting pairing in strongly disordered materials, which are moderately close to superconducting-insulator transition. Single-electron eigenstates are assumed to be Anderson localized, with a large localization volume. Superconductivity develops due to coherent delocalization of originally localized preformed Cooper pairs. The key assumption of the theory is that each such pair is coupled to a large number $Z\ensuremath{\gg}1$ of similar neighboring pairs. We derived integral equations for the probability distribution $P(\mathrm{\ensuremath{\Delta}})$ of local superconducting order parameter $\mathrm{\ensuremath{\Delta}}(\mathbit{r})$ and analyzed their solutions in the limit of small dimensionless Cooper coupling constant $\ensuremath{\lambda}\ensuremath{\ll}1$. The shape of the order-parameter distribution is found to depend crucially upon the effective number of nearest neighbors ${Z}_{\text{eff}}=2{\ensuremath{\nu}}_{0}{\mathrm{\ensuremath{\Delta}}}_{0}Z$, with ${\ensuremath{\nu}}_{0}$ being the single-particle density of states at the Fermi level. The solution we provide is valid both at large and small ${Z}_{\text{eff}}$; the latter case is nontrivial as the function $P(\mathrm{\ensuremath{\Delta}})$ is heavily non-Gaussian. One of our key findings is the discovery of a broad range of parameters where the distribution function $P(\mathrm{\ensuremath{\Delta}})$ is non-Gaussian but also noncritical (in the sense of superconductor-insulator transition criticality). The analytic results are supplemented by numerical data and good agreement between them is observed.

DOI
10.1103/physrevb.104.224505

Related papers

Superconductor-insulator transition and energy localization

M. V. Feigel’man, L. B. Ioffe, Marc Mézard

We develop an analytical theory for generic disorder-driven quantum phase transitions. We apply this formalism to the superconductor-insulator transition and we briefly discuss the applications to the order-disorder transition in quantum magnets. The effective spin-$\frac{1}{2}$ models for these transitions are solved in the cavity approximation which becomes exact on a Bethe lattice with large branching number $K⪢1$ and weak dimensionless coupling $g⪡1$. The characteristic feature of the low-temperature phase is a large self-formed inhomogeneity of the order-parameter distribution near the critical point $K\ensuremath{\ge}{K}_{c}(g)$, where the critical temperature ${T}_{c}$ of the ordering transition vanishes. We find that the local probability distribution $P(B)$ of the order parameter $B$ has a long power-law tail in the region where $B$ is much larger than its typical value ${B}_{0}$. Near the quantum-critical point, at $K\ensuremath{\rightarrow}{K}_{c}(g)$, the typical value of the order parameter vanishes exponentially, ${B}_{0}\ensuremath{\propto}{e}^{\ensuremath{-}C/[K\ensuremath{-}{K}_{c}(g)]}$ while the spatial scale ${N}_{inh}$ of the order parameter inhomogeneities diverges as ${[K\ensuremath{-}{K}_{c}(g)]}^{\ensuremath{-}2}$. In the disordered regime, realized at $K<{K}_{c}(g)$ we find actually two distinct phases characterized by different behavior of relaxation rates. The first phase exists in an intermediate range of ${K}^{\ensuremath{\ast}}(g)<K<{K}_{c}(g)$. It has two regimes of energies: at low excitation energies, $\ensuremath{\omega}<{\ensuremath{\omega}}_{d}(K,g)$, the many-body spectrum of the model is discrete, with zero-level widths, while at $\ensuremath{\omega}>{\ensuremath{\omega}}_{d}$ the level acquire a nonzero width which is self-generated by the many-body interactions. In this phase the spin model provides by itself an intrinsic thermal bath. Another phase is obtained at smaller $K<{K}^{\ensuremath{\ast}}(g)$, where all the eigenstates are discrete, corresponding to full many-body localization. These results provide an explanation for the activated behavior of the resistivity in amorphous materials on the insulating side near the superconductor-insulator transition and a semiquantitative description of the scanning tunneling data on its superconductive side.

arXiv:1006.5767•Physical Review B•Nov 29, 2010•100 citations

RECENT PROGRESS ON MODELS OF HIGHLY DISORDERED SUPERCONDUCTORS

Nandini Trivedi, Amit Ghosal, Mohit Randeria

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.

International Journal of Modern Physics B•May 10, 2001•5 citations

Site-disorder driven superconductor–insulator transition: a dynamical mean field study

Naushad Ahmad Kamar, N S Vidhyadhiraja

We investigate the effect of site disorder on the superconducting state in the attractive Hubbard model within the framework of dynamical mean field theory. For a fixed interaction strength (U), the superconducting order parameter decreases monotonically with increasing disorder (x), while the single-particle spectral gap decreases for small x, reaches a minimum and keeps increasing for larger x. Thus, the system remains gapped beyond the destruction of the superconducting state, indicating a disorder-driven superconductor-insulator transition. We investigate this transition in depth considering the effects of weak and strong disorder for a range of interaction strengths. In the clean case, the order parameter is known to increase monotonically with increasing interaction, saturating at a finite value asymptotically for U→∞. The presence of disorder results in destruction of superconductivity at large U, thus drastically modifying the clean case behaviour. A physical understanding of our findings is obtained by invoking particle-hole asymmetry and the probability distributions of the order parameter and spectral gap.

arXiv:1309.7224•Journal of Physics Condensed Matter•Feb 13, 2014•2 citations

Disorder driven superconductor-insulator transition in inhomoge- nous d-wave superconductor

Jian Colin Sun, Yangyang Zhao, Yun Song

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.

•Jan 1, 2016•0 citations

Quantum circuits and the superconductor-insulator transition in a disordered superconductor

Thibault Charpentier

Strongly disordered superconductors undergo a transition to insulation upon increasingdisorder, driven by the localization of preformed Cooper pairs. A key and implicit question in this transition is the fate of the superfluid density as the critical disorder approaches.By measuring the kinetic inductance of superconducting resonators ofamorphous indium oxide, we show that the transition to superconductivity is bosonicwith a critical temperature ruled by superconducting phase fluctuations. Upon reachingthe critical disorder terminating superconductivity, the superfluid stiffness remainssurprisingly finite at the transition to insulation. This indicates an unexpected first-ordernature of the disorder-driven quantum phase transition to insulator, opening anew paradigm for superfluid disordered systems of quantum particles.An important by-product of the interplay between localization and superconductivity is the obtention of large kinetic inductance materials, allowing the design of the sought-after superinductances which play a major role in recent quantum circuit developments.

•Oct 10, 2023•0 citations

ASPECTS OF LOCALIZATION ACROSS THE 2D SUPERCONDUCTOR-INSULATOR TRANSITION

NANDINI TRIVEDI, YEN LEE LOH, Karim Bouadim +1 more

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.

arXiv:1309.4769•International Journal of Modern Physics Conference Series•Jan 1, 2012•2 citations
We developed an analytic theory of inhomogeneous superconducting pairing in strongly disordered materials, which are moderately close to superconducting-insulator transition. Single-electron eigenstates are assumed to be Anderson localized, with a large localization volume. Superconductivity develops due to coherent delocalization of originally localized preformed Cooper pairs. The key assumption of the theory is that each such pair is coupled to a large number Z\ensuremath≫1Z\ensuremath{\gg}1Z\ensuremath≫1 of similar neighboring pairs. We derived integral equations for the probability distribution P(\ensuremathΔ)P(\mathrm{\ensuremath{\Delta}})P(\ensuremathΔ) of local superconducting order parameter \ensuremathΔ(\mathbitr)\mathrm{\ensuremath{\Delta}}(\mathbit{r})\ensuremathΔ(\mathbitr) and analyzed their solutions in the limit of small dimensionless Cooper coupling constant \ensuremathλ\ensuremath≪1\ensuremath{\lambda}\ensuremath{\ll}1\ensuremathλ\ensuremath≪1. The shape of the order-parameter distribution is found to depend crucially upon the effective number of nearest neighbors Zeff=2\ensuremathν0\ensuremathΔ0Z{Z}_{\text{eff}}=2{\ensuremath{\nu}}_{0}{\mathrm{\ensuremath{\Delta}}}_{0}ZZeff​=2\ensuremathν0​\ensuremathΔ0​Z, with \ensuremathν0{\ensuremath{\nu}}_{0}\ensuremathν0​ being the single-particle density of states at the Fermi level. The solution we provide is valid both at large and small Zeff{Z}_{\text{eff}}Zeff​; the latter case is nontrivial as the function P(\ensuremathΔ)P(\mathrm{\ensuremath{\Delta}})P(\ensuremathΔ) is heavily non-Gaussian. One of our key findings is the discovery of a broad range of parameters where the distribution function P(\ensuremathΔ)P(\mathrm{\ensuremath{\Delta}})P(\ensuremathΔ) is non-Gaussian but also noncritical (in the sense of superconductor-insulator transition criticality). The analytic results are supplemented by numerical data and good agreement between them is observed.
We develop an analytical theory for generic disorder-driven quantum phase transitions. We apply this formalism to the superconductor-insulator transition and we briefly discuss the applications to the order-disorder transition in quantum magnets. The effective spin-12\frac{1}{2}21​ models for these transitions are solved in the cavity approximation which becomes exact on a Bethe lattice with large branching number K⪢1K⪢1K⪢1 and weak dimensionless coupling g⪡1g⪡1g⪡1. The characteristic feature of the low-temperature phase is a large self-formed inhomogeneity of the order-parameter distribution near the critical point $K\ensuremath{\ge}{K}_{c}(g)$, where the critical temperature Tc{T}_{c}Tc​ of the ordering transition vanishes. We find that the local probability distribution P(B)P(B)P(B) of the order parameter BBB has a long power-law tail in the region where BBB is much larger than its typical value B0{B}_{0}B0​. Near the quantum-critical point, at $K\ensuremath{\rightarrow}{K}_{c}(g)$, the typical value of the order parameter vanishes exponentially, ${B}_{0}\ensuremath{\propto}{e}^{\ensuremath{-}C/[K\ensuremath{-}{K}_{c}(g)]}$ while the spatial scale Ninh{N}_{inh}Ninh​ of the order parameter inhomogeneities diverges as ${[K\ensuremath{-}{K}_{c}(g)]}^{\ensuremath{-}2}$. In the disordered regime, realized at K<Kc(g)K<{K}_{c}(g)K<Kc​(g) we find actually two distinct phases characterized by different behavior of relaxation rates. The first phase exists in an intermediate range of ${K}^{\ensuremath{\ast}}(g)<K<{K}_{c}(g)$. It has two regimes of energies: at low excitation energies, $\ensuremath{\omega}<{\ensuremath{\omega}}_{d}(K,g)$, the many-body spectrum of the model is discrete, with zero-level widths, while at $\ensuremath{\omega}>{\ensuremath{\omega}}_{d}$ the level acquire a nonzero width which is self-generated by the many-body interactions. In this phase the spin model provides by itself an intrinsic thermal bath. Another phase is obtained at smaller $K<{K}^{\ensuremath{\ast}}(g)$, where all the eigenstates are discrete, corresponding to full many-body localization. These results provide an explanation for the activated behavior of the resistivity in amorphous materials on the insulating side near the superconductor-insulator transition and a semiquantitative description of the scanning tunneling data on its superconductive side.
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 investigate the effect of site disorder on the superconducting state in the attractive Hubbard model within the framework of dynamical mean field theory. For a fixed interaction strength (U), the superconducting order parameter decreases monotonically with increasing disorder (x), while the single-particle spectral gap decreases for small x, reaches a minimum and keeps increasing for larger x. Thus, the system remains gapped beyond the destruction of the superconducting state, indicating a disorder-driven superconductor-insulator transition. We investigate this transition in depth considering the effects of weak and strong disorder for a range of interaction strengths. In the clean case, the order parameter is known to increase monotonically with increasing interaction, saturating at a finite value asymptotically for U→∞. The presence of disorder results in destruction of superconductivity at large U, thus drastically modifying the clean case behaviour. A physical understanding of our findings is obtained by invoking particle-hole asymmetry and the probability distributions of the order parameter and spectral gap.
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.
Strongly disordered superconductors undergo a transition to insulation upon increasingdisorder, driven by the localization of preformed Cooper pairs. A key and implicit question in this transition is the fate of the superfluid density as the critical disorder approaches.By measuring the kinetic inductance of superconducting resonators ofamorphous indium oxide, we show that the transition to superconductivity is bosonicwith a critical temperature ruled by superconducting phase fluctuations. Upon reachingthe critical disorder terminating superconductivity, the superfluid stiffness remainssurprisingly finite at the transition to insulation. This indicates an unexpected first-ordernature of the disorder-driven quantum phase transition to insulator, opening anew paradigm for superfluid disordered systems of quantum particles.An important by-product of the interplay between localization and superconductivity is the obtention of large kinetic inductance materials, allowing the design of the sought-after superinductances which play a major role in recent quantum circuit developments.
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.