BEC-BCS Crossover in Fermionic Systems

• BEC-BCS crossover is a continuous transition where fermionic systems evolve from weakly coupled Cooper pair superfluidity to tightly bound Bose-Einstein condensates.
• Advanced techniques including QMC, T-matrix, and diagrammatic methods accurately capture its thermodynamic, pseudogap, and transport properties.
• Practical realizations in ultracold atomic gases, layered superconductors, and lattice systems provide experimental access to study this unifying many-body phenomenon.

The BEC-BCS crossover describes the continuous evolution of the ground state of a fermionic many-body system with attractive interactions from the weak-coupling Bardeen-Cooper-Schrieffer (BCS) superfluid regime, characterized by large, overlapping Cooper pairs, to the strong-coupling Bose-Einstein condensate (BEC) regime, composed of tightly bound fermion pairs behaving as composite bosons. This crossover is realized by tuning the effective attraction, density, or other control parameters in systems ranging from ultracold atomic Fermi gases to layered and multiband superconductors. In recent years, the BEC-BCS crossover has provided a unifying framework for the study of strongly correlated fermionic systems, allowing experimental access to the full range of superfluidity and superconductivity in varying dimensions, band structures, and disorder environments.

1. Theoretical Foundations and Mean-Field Descriptions

The minimal model for the crossover consists of fermions with two "spin" components and an attractive interaction, most commonly parameterized by the Hamiltonian

$$\hat H = \sum_{\mathbf{k},\sigma} (\epsilon_{\mathbf{k}}-\mu) c_{\mathbf{k}\sigma}^\dagger c_{\mathbf{k}\sigma} - g \sum_{\mathbf{k},\mathbf{k}'} c_{\mathbf{k}\uparrow}^\dagger c_{-\mathbf{k}\downarrow}^\dagger c_{-\mathbf{k}'\downarrow} c_{\mathbf{k}'\uparrow},$$

where $g$ is the interaction, $\epsilon_{\mathbf{k}}$ is the dispersion, and $\mu$ is the chemical potential. The ground-state properties are determined by the self-consistent gap and number equations: $$1/g = \sum_{\mathbf{k}} \frac{1}{2E_{\mathbf{k}}}, \qquad n = 2 \sum_{\mathbf{k}}\left[1 - \frac{\epsilon_{\mathbf{k}}}{E_{\mathbf{k}}}\right], \quad E_{\mathbf{k}} = \sqrt{(\epsilon_{\mathbf{k}} - \mu)^2 + \Delta^2},$$ with the mean-field order parameter $$\Delta = g \sum_{\mathbf{k}} \langle c_{-\mathbf{k}\downarrow} c_{\mathbf{k}\uparrow}\rangle$$.

In the weak-coupling limit, $\mu \approx E_F$ and $\Delta \ll E_F$ (standard BCS pairing). In the strong-coupling limit, $\mu \to -E_b/2$ (with $E_b$ the two-body binding energy), and $\Delta \gg E_F$; fermions form tightly bound pairs that Bose-condense, realizing the BEC regime (Randeria et al., 2013).

The effective dimensionality and underlying band structure affect crossover physics: for example, in 2D, tuning the carrier density directly adjusts $\Delta/E_F$ due to the parabolic dispersion $\mu = E_F = \hbar^2 \pi n_{2D}/m^*$, and the crossover occurs for $\Delta/E_F \sim 0.2 - 0.5$ (Nakagawa et al., 2020, Zhou et al., 2022).

2. Thermodynamics, Critical Temperatures, and Universal Behavior

Thermodynamics and superfluid transition temperatures across the crossover are accurately characterized by Quantum Monte Carlo (QMC), T-matrix, and diagrammatic approaches. The critical temperature for superfluidity has a nonmonotonic dependence on interaction strength. In the three-dimensional uniform gas, $T_c/E_F$:

• Is exponentially small in the BCS limit, $T_c/E_F \sim \exp[-\pi/(2k_F|a|)]$
• Peaks at $T_c/E_F \simeq 0.26$ slightly on the BEC side of unitarity $(1/k_F a = 0.5)$
• Saturates to $T_c/E_F \to 0.218$ in the deep BEC limit, set by the condensation temperature of a gas of tightly bound dimers (0805.3047, Randeria et al., 2013).

At unitarity $(1/k_F a = 0)$, the system displays universal thermodynamics: $$\mathcal{E}_0 = \xi_s \frac{3}{5} n E_F, \quad \mu = \xi_s E_F,$$ with $\xi_s \simeq 0.37$ (the Bertsch parameter). Absence of a small parameter motivates the use of QMC, large-$N$, and renormalization group methods (Randeria et al., 2013, Ghosh, 2011).

3. Experimental Signatures and Physical Observables

Observable features of the crossover include:

• Pseudogap phenomenon: In the crossover and BEC regimes, pairing correlations persist above $T_c$, leading to a suppression of the single-particle density of states—termed the pseudogap. In 2D systems (e.g., gate-tuned ZrNCl), the window between the pseudogap onset $T^*$ and $T_c$ widens as the coupling increases, consistent with the separation of pair formation and superfluid coherence (Nakagawa et al., 2020).
• Cooper pair size and momentum distribution: The pair size shrinks from much larger than interparticle spacing in BCS to the molecular size in BEC. The Cooper pair momentum distribution exhibits a crossover from a Fermi-surface-peaked function to a broad molecular form. Large-momentum tails of the distribution encode the Tan contact parameter, $C \sim m^2 \Delta^2$, which increases through the crossover (Tajima et al., 2019).
• Transport and collective response: The unitary Fermi gas minimizes viscosity relative to entropy, $\eta/s \approx 0.2 \hbar/k_B$, closely approaching the conjectured AdS/CFT bound, and displays universal sound velocities and collective mode behavior (Randeria et al., 2013, Zhou et al., 2022). Nernst and flux-flow Hall effect measurements in superconductors (e.g., in $\kappa$-(BEDT-TTF)$_4$Hg$_{2.89}$Br$_8$ and Li$_x$ZrNCl) are used to distinguish BCS from BEC-like condensates (Suzuki et al., 2022, Heyl et al., 2022).

4. Multiband and Lattice Crossover Physics

Multiband superconductivity demonstrates enhanced complexity in the crossover. In two-band systems (deep and shallow bands), tuning the shallow band's coupling drives a BEC-BCS crossover while the deep band remains weakly paired. Interband pair exchange amplifies the effective pairing and critical temperature, and the two Tan contacts $C_1, C_2$ reflect the evolution in both bands (Tajima et al., 2019, Guidini et al., 2014, Wolf et al., 2016). Applications to Fe-based superconductors reveal coexistence of small- and large-pair condensates, with optimal high-$T_c$ realized when the gap is comparable to the chemical potential in a small band (Guidini et al., 2014).

On optical lattices, the presence of discrete band structure, Hartree shifts, and broken Galilean invariance modifies the crossover. The superfluid stiffness and compressibility deviate from continuum expectations, and in the BEC limit, the pairs hop with renormalized mass $\sim U/t^2$, altering $n_s$ and $T_c$ scaling from BCS ($\sim \Delta_0$) to BEC ($\sim t^2/U$) (Ghosh, 2011).

5. Dimensionality, Topological, and Inhomogeneous Effects

Dimensional reduction intensifies fluctuation effects. In quasi-2D, universal relations known from 3D BCS theory between $\Delta$, $\xi$, and $\mu$ fail in the BEC regime; Gaussian pair fluctuation theory becomes essential, as confirmed by layered superconductors and cold atomic gases (Zhou et al., 2022, Nakagawa et al., 2020). The critical Berezinskii-Kosterlitz-Thouless (BKT) temperature in the BEC–BCS crossover regime in 2D saturates to $T_{BKT}/T_F \simeq 0.12$, in accord with theoretical bounds (Nakagawa et al., 2020).

Topologically, the transition from BEC to BCS pairing is sharp under specific observables: the Z$_2$ Berry phase (defined via a local twist in the pairing field) jumps by $\pi$ at a critical $\Delta_c$ without bulk gap closure, delineating two topological classes. This criterion is strictly realized only under certain chiral symmetry and localized-twist conditions (Arikawa et al., 2010). In driven (nonequilibrium) systems, such as under rf radiation, the BCS side can be selected into a supersolid phase due to features in the fermionic excitation spectrum not present in the BEC regime (Lemonik et al., 2015).

Spatially inhomogeneous profiles, as in superconducting p-n junctions, naturally facilitate real-space BCS–BEC crossover, resulting in interfaces with confined BEC condensates adjacent to BCS regions. These are experimentally observable via tunneling, transport, and optical probes (Niroula et al., 2019).

6. Emergent and Disorder-Induced Phenomena

Disorder introduces nontrivial superfluid-insulator transitions. Across the BEC-BCS crossover in the presence of strong randomness, the critical disorder strength for superfluidity is nonmonotonic in detuning; reentrant superfluid phases can occur near the fermionic mobility edge, with coexisting Lifshitz (bosonic) and fractal (fermionic) insulators in the intermediate regime. This phenomenon is governed by the interplay of Anderson localization, bosonic puddling, and resonance-enhanced pairing (Gopalakrishnan, 2012).

Quantum-size effects in confined geometries (cigar-shaped traps, sub-bands in nanostructures) manifest as stepwise local BEC transitions in transverse subbands each time a subband bottom crosses the chemical potential, resulting in a condensate that is a coherent sum of BCS- and BEC-like components (Shanenko et al., 2012).

7. Broader Impact and Open Questions

The BEC-BCS crossover remains central to understanding strong-coupling superfluidity, high-temperature superconductivity, and quantum transport in both atomic and condensed-matter systems. Outstanding issues include the quantitative understanding of the pseudogap regime, the robustness of superfluidity in disordered and multiband systems, and the identification of topological distinctions beyond mean-field theory. The crossover framework connects with research into perfect fluids, gauge–gravity dualities, and nonperturbative quantum field phenomena (Randeria et al., 2013). Multiband superconductors, low-density 2D systems, and hybrid platforms such as exciton-polariton condensates extend the domain of applicability, offering new arenas for exploring the interplay of pairing, coherence, dimensionality, and topology (Byrnes et al., 2010).

Ongoing experimental progress in fine-tuning interactions and probing thermodynamics, transport, and dynamical response functions continues to refine theoretical models, with explicit emphasis on non-perturbative, diagrammatic, and real-space stochastic methodologies (0805.3047, Tarat et al., 2014, Dalton et al., 2022). The BEC-BCS crossover remains a rich and unifying paradigm for many-body quantum physics.