Attractive Hubbard Model

• Attractive Hubbard model is a paradigmatic system with onsite fermion attraction that leads to s-wave superconductivity and charge-density wave order.
• Competing kinetic energy and local attraction drive rich quantum phase transitions and enable mapping to pseudospin models at strong coupling.
• Advanced methods like DQMC, DMFT, and Bethe ansatz yield quantitative insights into pairing, pseudogap formation, and topological transitions.

The attractive Hubbard model is a paradigmatic strongly correlated electron system defined by local onsite attraction between fermions on a lattice. In its canonical form, the Hamiltonian is

$$H = -t\, \sum_{\langle ij \rangle, \sigma} (c_{i\sigma}^\dagger c_{j\sigma} + \text{h.c.}) - U \sum_i n_{i \uparrow} n_{i \downarrow} - \mu \sum_{i,\sigma} n_{i \sigma}$$

with $t>0$ the nearest-neighbor hopping, $U > 0$ the onsite attraction, $\mu$ the chemical potential, $c_{i\sigma}$ the fermion annihilation operator at site $i$ with spin $\sigma\in\{\uparrow, \downarrow\}$, and $n_{i\sigma}$ the number operator. The model exhibits rich phenomena including s-wave superconductivity (pairing), charge-density-wave (CDW) order, quantum phase transitions, and a nontrivial interplay between spin, charge, and lattice degrees of freedom, especially under the effects of disorder, dimensionality, interaction anisotropy, external gauge fields, population (spin) imbalance, and extensions beyond onsite attraction.

1. Ground-State and Finite-Temperature Properties

The system’s ground state and low-temperature physics are governed by the competition between kinetic energy and local attractive interaction, favoring pairing of opposite-spin fermions on a site. At half filling, the model exhibits a coexistence of s-wave superconductivity and CDW order—linked via a hidden SU(2) pseudo-spin symmetry—while away from half filling, CDW order weakens and pure onsite s-wave superfluidity dominates (Mitra et al., 2017, Qin et al., 2021). In strong-coupling $U \gg t$, electrons form tightly bound onsite pairs that behave as hard-core bosons, and the system maps onto low-energy pseudospin models with $SO(3)$ symmetry, where the in-plane pseudospin components represent superconductivity and the out-of-plane component encodes CDW order (Karmakar et al., 2020).

The superfluid (pairing) order parameter is

$\Delta = \langle c_\uparrow c_\downarrow \rangle$

and at low temperature is nonzero, indicating long-range order, provided the system is not at exactly half filling in two dimensions (where Mermin–Wagner constraints preclude true LRO at $T > 0$ but allow quasi-long-range order via a Berezinskii–Kosterlitz–Thouless (BKT) transition). In three dimensions, true superconducting order exists even at half filling (Xiong et al., 17 Feb 2025).

Population imbalance introduced by a Zeeman field ($h$) leads to a polarized superfluid (PSF) phase characterized by finite $\Delta$ with finite magnetization $$m = \sum_\sigma \sigma \langle c^\dagger_\sigma c_\sigma \rangle$$. In this state, the density of states (DOS) becomes asymmetric for up and down spins, leading to spin-split spectral peaks and, at low temperatures, an eventual first-order transition to the normal metallic phase as pairing correlations are suppressed (Koga et al., 2010). At strong couplings and low density, single-fermion excitations in the even-number sector can be fully gapped, while pairing excitations (Goldstone modes) are gapless when deviating from half-filling by an extensive number of particles (Goto et al., 24 Sep 2025).

2. Thermodynamics and Quantum Criticality

In one dimension, Bethe ansatz and thermodynamic Bethe ansatz (TBA) yield exact thermodynamics: the FFLO-like partially polarized phase (finite magnetization, finite $\Delta$) is described as a “two-free-quantum-fluids” system, with compressibility and susceptibility additivity rules arising from decoupled bound pairs and excess fermions. The additivity rules are, for example,

$$\kappa = \kappa_1 + \frac{2}{\alpha_1} \kappa_2, \quad \frac{1}{\chi} = \frac{1}{\chi_1} + \frac{\alpha_1}{2} \frac{1}{\chi_2}$$

where $\kappa_{1,2}$ and $\chi_{1,2}$ are compressibilities and susceptibilities for unpaired and paired degrees of freedom, and $\alpha_1$ is a lattice-dependent constant (Cheng et al., 2017). Quantum criticality near phase transitions is described by universal exponents $z=2$ (dynamical) and $\nu=1/2$ (correlation length).

In infinite dimensions ($d\to\infty$) and three-dimensional systems, the normal state and superfluid transition are described using dynamical mean-field theory (DMFT) and advanced quantum Monte Carlo (CTQMC, DQMC). The model interpolates between BCS and BEC regimes: weak coupling features a small gap $\Delta$ and $T_c \sim e^{-1/U}$, strong coupling features tightly-bound pairs with $T_c\sim 1/U$, superfluid order is robust, and a pseudogap regime with suppressed spectral weight above $T_c$ emerges, especially at low densities (Koga et al., 2011, Xiong et al., 17 Feb 2025).

3. Extensions and Exotic Superconductivity

Non-local attractive interactions, e.g., extended Hubbard models incorporating nearest-neighbor attraction, support a broader set of pairing symmetries, including $d$-wave, mixed $s+d$-wave, $d_{x^2-y^2}+p_x$, and topologically nontrivial $p_x + i p_y$ phases (Nayak et al., 2017, Calegari et al., 2011). Mean-field Bogoliubov–de Gennes (BdG) calculations with no a priori symmetry restrictions find energetically stable unconventional phases with mixed or chiral order parameters (e.g., $d_{x^2-y^2}+i[s+d_{x^2+y^2}]$). These can lead to full energy gaps, multiple coherence peaks, or nodal suppression in the excitation spectrum—features potentially relevant to high-$T_c$ cuprates and ruthenates.

Further, in lattices with artificial gauge fields or in the presence of Peierls phase factors, as in the attractive Hofstadter–Hubbard model, the interplay between the (fractal) Hofstadter spectrum, strong interaction, and Zeeman field leads to spatially modulated superfluid phases, including stripe-ordered (FFLO-like) superfluids and supersolids with coexisting pair-density (PDW), charge-density (CDW), and spin-density (SDW) wave orders. The spatial period of the modulations is set by the effective flux seen by tightly bound pairs ($\phi_d = \phi_\uparrow + \phi_\downarrow$), and imbalanced vector potentials generate additional SDW ordering with global zero polarization (Iskin, 2015).

4. Disorder and Universality

Disorder enters via random on-site energies, broadening the density of states. Within a generalized Anderson theorem framework, the superconducting transition temperature $T_c$ and Ginzburg–Landau (GL) expansion coefficients ($A$, $B$) for the homogeneous order parameter depend on disorder only through the effective bandwidth $D_\mathrm{eff}$, i.e., $$D \rightarrow D_\mathrm{eff} = D\sqrt{1 + 4\Delta^2/D^2}$$ for a semi-elliptic DOS. This “universality” holds across the BCS–BEC crossover: $T_c$ and the specific heat jump at the phase transition display scaling governed by $D_\mathrm{eff}$, validating the Anderson theorem even at strong coupling, as long as the gradient (GL) term is neglected (Kuchinskii et al., 2015).

5. Many-Body, Numerical, and Functional Methods

A suite of nonperturbative computational techniques underpin quantitative understanding, including minus-sign–free DQMC, CTQMC (auxiliary-field and strong-coupling expansions), and recent many-body diagrammatic approaches such as GW, fluctuation exchange, and dynamical vertex approximation (D$\Gamma$A). GW-type approximations and their variants (HGW, post-GW with covariant corrections) provide accurate results for Green’s functions, pairing, and transition temperatures in the weak to intermediate coupling regime, recovering critical exponents close to the 3D XY universality class, especially for superconducting criticality (Xiong et al., 17 Feb 2025, Re et al., 2018).

D$\Gamma$A, extended to the attractive model in ladder approximation, allows systematic inclusion of nonlocal correlations, crucial for capturing the reduction of ordering temperatures and the modification of single-particle self-energies near phase transitions. The method is explicitly constructed to preserve the mapping between the attractive and repulsive cases at half filling, by channel-selective summing of ladder diagrams (Re et al., 2018).

Density-matrix functional theory (LDFT), formulated on the lattice, provides a statistical analogy between the interaction-energy functional $W[\eta]$ (for natural orbital occupations $\eta$) and the entropy $S[\eta]$ of a noninteracting Fermi gas with the same $\eta$: $$W[\eta] \approx W_\text{min} + (W_\text{max} - W_\text{min})(S[\eta]/S_\text{max})$$ (Müller et al., 2021). This yields a powerful variational framework that interpolates efficiently between weak and strong coupling, validated by exact diagonalizations across various geometries.

6. Application to Superfluidity and Experimental Realizations

The attractive Hubbard model underpins realizations of lattice superfluidity in cold atomic gases, where parameters ($U$, $t$, density, trapping, population imbalance, disorder, gauge fields) are highly tunable (Mitra et al., 2017, Chan et al., 2020). Quantum gas microscopy and site-resolved imaging enable the measurement of local and nonlocal observables, such as doublon (paired site) density, charge-density correlations, and direct visualization of CDW and pairing order. At half-filling, the hidden SU(2) pseudo-spin symmetry guarantees the equivalence (degeneracy) of CDW and superfluid correlations; the measurement of checkerboard CDW order provides a lower bound for superfluid pairing (since $$|C^d(a)| \le \langle \Delta^{x}_r \Delta^{x}_{r+a}\rangle_c$$).

Layering strategies (bilayers or three-dimensional stacking) have been shown—via minus-sign–free DQMC—to increase $T_c$ by up to $\sim$70% compared to a single plane, with optimal interlayer hopping $t_z$ and density; such findings are immediately relevant for engineering higher-$T_c$ cold-atom superfluids (Fontenele et al., 30 Aug 2024). The agreement between many-body theory and QMC provides quantitative guidance for exploring the BCS–BEC crossover and pseudogap regimes.

7. Topological, Thermal, and Quantum Phase Transitions

Attractive interactions can drive quantum phase transitions from trivial band insulators to quantum anomalous Hall (QAH) states in two-band magnetic Dirac models through band inversion and gap reopening as a function of $U$. The attractive interaction increases the topological bulk gap, enhances edge state localization, and protects quantized edge transport in the presence of disorder or finite size effects (Qi et al., 2019). In strong-coupling and orbital field regimes, the mapping to pseudospin models reveals thermal melting of supersolid (coexisting SC and CDW) order via a sharp Ising transition for charge order, and a more gradual (possibly vortex or crossover) melting of superconductivity (Karmakar et al., 2020).

In one dimension, quantum criticality and the crossover between vacuum, fully paired, FFLO, and fully polarized phases are governed by the vanishing of Fermi seas in different excitation branches, with scaling described by Bethe ansatz. Dimensional crossover, both in critical behavior and in the nature and robustness of pairing, has been extensively studied using scaling and finite-size analysis in QMC and many-body frameworks.

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In summary, the attractive Hubbard model and its extensions serve as a fundamental testbed for understanding strongly correlated pairing physics on a lattice. Its analytical and numerical tractability, broad range of accessible phases (superfluid, CDW, supersolid, topological), sensitivity to dimensionality and disorder, and relevance to modern cold-atom experiments make it central to contemporary theoretical and experimental studies of unconventional superconductivity, quantum criticality, and topological matter.