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Quenched disorder and the BCS-BEC crossover in the Hubbard model

M. Iskin

TL;DR

This paper develops a controlled functional-integral framework to study weak quenched disorder on the BCS-BEC crossover in the attractive Hubbard model. By deriving the grand potential up to second order in both disorder and pairing fluctuations, the authors obtain self-consistent expressions for the number equation, condensate fraction, superfluid fraction, and sound speed at $T=0$, and they show that in the dilute BEC limit the lattice theory reproduces the known continuum results. The work demonstrates that disorder depletes the superfluid more than the condensate and can enhance the sound speed via overcompensation of the static compressibility, providing a unified description across the crossover and a basis for finite-temperature and multiband extensions. The methodology and results offer a robust tool for understanding disordered superconductors and ultracold lattice systems with potential applications to experiments and future theoretical generalizations.

Abstract

We study the impact of weak quenched disorder on the BCS-BEC crossover in the Hubbard model within a functional-integral approach. By deriving the thermodynamic potential up to second order in both the disorder potential and pairing fluctuations, we obtain self-consistent expressions for the number equation, condensate fraction, superfluid fraction and sound speed at zero temperature. In the dilute BEC limit, our results analytically reproduce the known continuum limits of weakly-interacting bosons, where weak disorder depletes the superfluid more strongly than the condensate due to broken translational symmetry, and enhances the sound speed through the overcompensation of the static compressibility. These findings establish a unified and controlled approach for describing the BCS-BEC crossover in disordered lattice models, and they provide a foundation for future extensions to finite temperatures and multiband Hubbard models.

Quenched disorder and the BCS-BEC crossover in the Hubbard model

TL;DR

This paper develops a controlled functional-integral framework to study weak quenched disorder on the BCS-BEC crossover in the attractive Hubbard model. By deriving the grand potential up to second order in both disorder and pairing fluctuations, the authors obtain self-consistent expressions for the number equation, condensate fraction, superfluid fraction, and sound speed at , and they show that in the dilute BEC limit the lattice theory reproduces the known continuum results. The work demonstrates that disorder depletes the superfluid more than the condensate and can enhance the sound speed via overcompensation of the static compressibility, providing a unified description across the crossover and a basis for finite-temperature and multiband extensions. The methodology and results offer a robust tool for understanding disordered superconductors and ultracold lattice systems with potential applications to experiments and future theoretical generalizations.

Abstract

We study the impact of weak quenched disorder on the BCS-BEC crossover in the Hubbard model within a functional-integral approach. By deriving the thermodynamic potential up to second order in both the disorder potential and pairing fluctuations, we obtain self-consistent expressions for the number equation, condensate fraction, superfluid fraction and sound speed at zero temperature. In the dilute BEC limit, our results analytically reproduce the known continuum limits of weakly-interacting bosons, where weak disorder depletes the superfluid more strongly than the condensate due to broken translational symmetry, and enhances the sound speed through the overcompensation of the static compressibility. These findings establish a unified and controlled approach for describing the BCS-BEC crossover in disordered lattice models, and they provide a foundation for future extensions to finite temperatures and multiband Hubbard models.

Paper Structure

This paper contains 14 sections, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Functional-Integral Approach
  3. Grand Partition Function
  4. Effective Gaussian Action
  5. Thermodynamic Potential
  6. BCS-BEC crossover
  7. Condensate Fraction
  8. Current-carrying Superfluid
  9. Superfluid Fraction
  10. Dilute BEC limit
  11. Speed of Sound
  12. Numerical Illustration
  13. Conclusion
  14. Derivation of $D_{\mathrm{B}, d}^{ij}$ given in Eq. (\ref{['eqn:DBd']})

Figures (2)

  • Figure 1: Self-consistent solutions for (a) the order parameter $\Delta$ and (b) the chemical potential $\mu$ as functions of the particle filling $n$ for $\kappa = 0.1t^2$. The results exhibit exact particle-hole symmetry about half filling ($n = 1$); therefore, only the particle side of the filling range is shown in Fig. \ref{['fig:fig2']}.
  • Figure 2: In (a) the condensate fraction $n_{c,0}$, and in (c) the superfluid fraction $D_{s,0}$ are shown as functions of the filling $0 \lesssim n \leq 1$ in the absence of disorder ($\kappa = 0$). For comparison, panels (b) and (d) display the corresponding disorder-induced reductions, $\delta n_c = n_{c,0} - n_c$ and $\delta D_s = D_{s,0} - D_s$, respectively, for weak disorder ($\kappa = 0.1t^2$).