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arXiv:2512.11073 · Physical Review A

Self-consistent inclusion of disorder in the BCS-BEC crossover near the critical temperature

M. Iskin

Published Mar 10, 2026• 0 citations
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Abstract

We develop a systematic theoretical approach to incorporate the effects of a static white-noise disorder into the BCS-BEC crossover near the critical temperature ($T_c$) of the superfluid transition. Starting from a functional-integral formulation in momentum-frequency space, we derive an effective thermodynamic potential that fully accounts for Gaussian fluctuations of the order-parameter field and its coupling to the disorder potential. The effective action, expanded to second order in both the disorder potential and the bosonic field, naturally involves third- and fourth-order terms arising from the logarithmic expansion near $T_c$. By providing a controlled description of pairing fluctuations and disorder effects, this formalism correctly recovers the well-established BCS and BEC limits. This ensures a consistent physical foundation for analyzing the entire BCS-BEC crossover, effectively anchoring the intermediate regime between these two analytically robust endpoints. The approach applies equally to continuum and lattice systems and provides a natural framework for generalizations to multiband models.

DOI
10.1103/91hc-kxh6

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We develop a systematic theoretical approach to incorporate the effects of a static white-noise disorder into the BCS-BEC crossover near the critical temperature (TcT_cTc​) of the superfluid transition. Starting from a functional-integral formulation in momentum-frequency space, we derive an effective thermodynamic potential that fully accounts for Gaussian fluctuations of the order-parameter field and its coupling to the disorder potential. The effective action, expanded to second order in both the disorder potential and the bosonic field, naturally involves third- and fourth-order terms arising from the logarithmic expansion near TcT_cTc​. By providing a controlled description of pairing fluctuations and disorder effects, this formalism correctly recovers the well-established BCS and BEC limits. This ensures a consistent physical foundation for analyzing the entire BCS-BEC crossover, effectively anchoring the intermediate regime between these two analytically robust endpoints. The approach applies equally to continuum and lattice systems and provides a natural framework for generalizations to multiband models.
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A systematic investigation of the effects of disorder on the BCS-BEC crossover at the lowest order in the impurity potential is presented for the normal phase above the critical temperature Tc{T}_{c}Tc​. Starting with the ttt-matrix approach for the clean system, by which pairing correlations between opposite-spin fermions evolve from the weak-coupling (BCS) to the strong-coupling (BEC) limits by increasing the strength of the attractive interparticle interaction, all possible diagrammatic processes are considered where the effects of a disordered potential are retained in the self-energy at the lowest order. An accurate numerical investigation is carried out for all these diagrammatic terms, to determine which of them are mostly important throughout the BCS-BEC crossover. Explicit calculations for the values of Tc{T}_{c}Tc​, the chemical potential, and the Tan's contact are carried out. In addition, the effect of disorder on the single-particle spectral function is analyzed, and a correlation is found between an increase of Tc{T}_{c}Tc​ and a widening of the pseudogap energy at Tc{T}_{c}Tc​ on the BCS side of unitarity in the presence of disorder, while on the BEC side of unitarity the presence of disorder favors the collapse of the underlying Fermi surface. The present investigation is meant to orient future studies when the effects of disorder will be considered at higher orders, with the purpose of limiting the proliferation of diagrammatic terms in which interaction and disorder are considered simultaneously.
We address the thermodynamics, density profiles, and superfluid density of trapped fermions undergoing BCS-BEC crossover. We consider the case of zero and finite population imbalance and apply the local density approximation (LDA) to include the trap potential. Our approach represents a fully self-consistent treatment of ``pseudogap effects.'' These effects reflect the distinction between the pair formation temperature T∗{T}^{*}T∗ and the pair condensation temperature Tc{T}_{c}Tc​. As a natural corollary, this temperature difference must be accommodated in the fermionic excitation spectrum Ek{E}_{\mathbf{k}}Ek​ to reflect the fact that fermions are paired at and above Tc{T}_{c}Tc​. It is precisely this natural corollary which has been omitted from all other many body approaches in the literature. At a formal level, we show how enforcing this corollary implies that pairing fluctuation or self-energy contributions enter into both the gap and the number equations; this is necessary in order to be consistent with a generalized Ward identity. At a less formal level, we demonstrate that we obtain physical results for the superfluid density ns(T){n}_{s}(T)ns​(T) at all temperatures. In contrast, previous work in the literature has led to nonmonotonic, or multivalued or discontinuous behavior for ns(T){n}_{s}(T)ns​(T). Because it reflects the essence of the superfluid state, we view the superfluid density as a critical measure of the physicality of a given crossover theory. In a similarly unique fashion, we emphasize that in order to properly address thermodynamic properties of a trapped Fermi gas, a necessary first step is to demonstrate that the particle density profiles are consistent with experiment. Without a careful incorporation of the distinction between the pairing gap and the order parameter, the LDA-derived density profiles (of the unpolarized gas) tend to exhibit sharp features at the condensate edge, which are never seen experimentally in the crossover regime. The lack of demonstrable consistency between theoretical and experimental density profiles, along with problematic behavior found for the superfluid density, casts doubt on previous claims in the literature concerning quantitative agreement between thermodynamical calculations and experiment.
In this paper, we have investigated the effect of weak random disorder in the Bardeen–Cooper–Schrieffer to Bose–Einstein condensation (BCS–BEC) crossover region. The disorder is included in the mean field formalism through Nozières–Smith-Rink (NSR) theory of superconducting fluctuations. A self-consistent numerical solution of the coupled equations involving the superfluid gap parameter and density as a function of the disorder strength, albeit unaffected in the BCS phase, yields a depleted order parameter in the BEC regime and an interesting nonmonotonic behaviour of the condensate fraction in the vicinity of the unitary region, and a gradual depletion thereafter, as the pairing interaction is continuously tuned across the BCS–BEC crossover. The unitary regime thus demonstrates a robust paradigm of superfluidity even when the disorder is introduced. To support the above feature and shed light on a lingering controversial issue, we have computed the behaviour of the sound mode across the crossover that distinctly reveals a suppression of the sound velocity. We also find the Landau critical velocity that shows similar nonmonotonicity as that of the condensate fraction data, thereby supporting a stable superfluid scenario in the unitary limit.