---
title: 2D Fermi-Hubbard Model Overview
url: https://www.emergentmind.com/topics/2d-fermi-hubbard-model
type: topic
---

# 2D Fermi-Hubbard Model Overview

The two-dimensional Fermi-Hubbard model is the single-band lattice model of spin-$\frac12$ fermions on a square lattice with nearest-neighbor hopping and on-site interaction, conventionally written
$$
H=-t\sum_{\langle ij\rangle,\sigma}\left(c^\dagger_{i\sigma}c_{j\sigma}+\text{h.c.}\right)+U\sum_i n_{i\uparrow}n_{i\downarrow}-\mu\sum_i n_i.
$$
In the repulsive case $U>0$, it organizes antiferromagnetic, Mott, pseudogap, and doped metallic or pairing regimes near half-filling; in the attractive case $U<0$, it provides a controlled setting for the BCS–BEC crossover and for correlation-driven changes in spectral topology [1706.07556][2301.04197]. Because the model is simple enough to define microscopically yet difficult enough to resist a complete solution in two dimensions, it has become a standard reference system for thermodynamics, transport, fluctuation diagnostics, quantum-gas-microscope measurements, tensor-network studies, diagrammatic approaches, and quantum-simulation algorithms [1305.6798][2008.02179].

## 1. Hamiltonian, lattice structure, and standard conventions

On the square lattice, $t$ sets the energy and temperature scale, and the nearest-neighbor tight-binding dispersion is
$$
\varepsilon_{\mathbf{k}}=-2t(\cos k_x+\cos k_y).
$$
For the repulsive model, the Brillouin zone is the square with components $q_x,q_y\in(-\pi,\pi]$, and the commensurate antiferromagnetic wavevector is $\mathbf{Q}=(\pi,\pi)$ [1706.07556]. The particle density per site is $n=\langle n_i\rangle$, while doping away from half-filling is commonly written as $x=1-n$ or $\delta=1-n$ [1706.07556][1305.6798].

Half-filling means $n=1$. In the particle-hole-symmetric convention often used for the square-lattice repulsive model, $\mu=0$ corresponds to half-filling when chemical potential is referenced to $U/2$ [1305.6798]. Many analyses adopt the idealized case $t'=0$, strictly local interactions, a single band, and no phonons. In that limit the square lattice has Fermi-surface nesting and particle-hole symmetry at $n=1$, while the Mermin–Wagner theorem forbids antiferromagnetic long-range order at any nonzero temperature in pure two dimensions [1706.07556].

Several extensions preserve the same basic Hubbard structure while modifying symmetry or control parameters. The attractive model changes only the sign of $U$ and is used to study pairing, pseudogap formation, and the BCS–BEC crossover [2301.04197][2308.06700]. A tilted version adds a linear potential $F\sum_{i,\sigma}x_i n_{i\sigma}$ and couples mass transport to local heating through energy conservation [1909.05848]. SU($N$) generalizations replace the two spin states by $N$ flavors and retain the same nearest-neighbor kinetic term and on-site interaction structure [2305.18967].

## 2. Ground-state organization near half-filling

For the square-lattice repulsive model, the zero-temperature phase structure near half-filling is highly structured but also model-specific. At $n=1$, the ground state is an antiferromagnetic insulator for any $U/t$ [1706.07556]. At moderate interactions, $U/t\lesssim4$, the ground state for $0.6<n\lesssim0.8$ is a $d_{x^2-y^2}$-wave superfluid, and for small doping $\delta=1-n\ll1$ the model is regarded as most likely unstable to phase separation [1706.07556]. The same source emphasizes that the phase-separation dome likely lies within the finite-temperature magnetic fluctuation regime rather than outside it.

These statements must be read together with the restrictions of the idealized model. The quoted phase diagram assumes a strictly two-dimensional, single-band, nearest-neighbor model with $t'=0$ and no phonons, so quantitative ranges depend on $U/t$ and are sensitive to finite-temperature and finite-cluster biases in numerical methods [1706.07556]. A recurrent issue is that large correlation lengths and phase-separation tendencies can bias finite-size calculations toward homogeneous states.

For the attractive model, the ground-state language shifts from antiferromagnetism and Mottness to pairing and crossover physics. In two dimensions with $U<0$, pairing evolves continuously from a BCS regime of large, overlapping Cooper pairs to a BEC regime of tightly bound on-site pairs, and the Gaussian pairing state is reported to be a good approximation to the ground states of the attractive Hubbard model, in particular in the strong and weak coupling limits [2308.06700]. This suggests that the sign of $U$ does not merely reverse an interaction term; it reorganizes the relevant low-energy degrees of freedom.

## 3. Finite-temperature magnetism, pseudogap behavior, and real-space correlations

A central finite-temperature object is the static spin susceptibility
$$
\chi(\mathbf{q},0)=\int_0^\beta d\tau\,\langle S_z(\tau,\mathbf{q})S_z(0,-\mathbf{q})\rangle\big|_{\omega=0},
$$
with $S_z=(n_\uparrow-n_\downarrow)/2$ [1706.07556]. In the repulsive model, a pronounced and narrow peak in $\chi(\mathbf{q},0)$ indicates a large magnetic correlation length. An operational magnetic crossover temperature $T_M(n)$ was defined by the condition $\chi_{\max}/\chi_{\min}=10$, marking the onset of a regime with $\xi\gg a$ [1706.07556]. Within the self-consistent skeleton-diagrammatic study of that regime, $T_M$ can be as high as $T\sim0.25\,t$ at half-filling, vanishes at both small and large $U$, and is maximal around $U/t\sim4$, with the strongest finite-temperature magnetic fluctuations for $U/t\approx4$–6 [1706.07556]. Away from half-filling, the dominant magnetic peak becomes incommensurate and can split around $(\pi,\pi)$, producing a diagonally oriented “carpet” of antiferromagnetic domains separated by diagonal domain walls; whether that texture survives to $T\to0$ remains open [1706.07556].

The pseudogap-like and non-Fermi-liquid regime has been resolved more microscopically through fluctuation diagnostics. In the ladder dual-fermion analysis, the low-frequency self-energy crossover is governed by relatively sharp spin fluctuations at $\mathbf{q}=(\pi,\pi)$, even at weak coupling [1905.07462]. At $U/t=5.6$, $t'/t=-0.3$, $\mu=0$, and $\beta t=5$, the antinodal spectral weight at $\omega=0$ is about $15\%$ of the nodal value, while the sign change of the standard Matsubara metric $\Delta\Sigma_k=\operatorname{Im}\Sigma_k(i\omega_0)-\operatorname{Im}\Sigma_k(i\omega_1)$ tracks the crossover from Fermi-liquid to non-Fermi-liquid behavior [1905.07462]. In the particle-hole-symmetric weak-coupling case $U/t=2$, the crossover temperatures were found near $T/t\approx0.06$ and $0.065$ for nodal and antinodal momenta, with correlation length $\xi\approx15$ near the crossover [1905.07462].

Quantum-gas-microscope experiments and finite-temperature simulations have made the same regime visible in real space. At $U/t=7.2(1)$ and temperatures $T/t\approx0.89$–$1.22$, nearest-neighbor spin correlations are maximal at half-filling and weaken monotonically upon doping; at the lowest reported temperature, the nearest-neighbor spin correlator reaches about $-0.09$ [1606.04089]. The nearest-neighbor moment correlator changes sign near a doping $x\approx0.21$, reflecting a crossover from anti-bunching of singly occupied sites to strong doublon-hole bunching, and the corresponding nonlocal moment correlations imply a nearest-neighbor contribution to potential-energy fluctuations of about $0.8\,t^2$ at half-filling [1606.04089]. XTRG and DQMC calculations on open clusters at $U/t=7.2$ further show that, upon doping, the diagonal spin correlation $C_S(\sqrt2)$ changes sign around $\delta\approx0.2$ and the next-nearest correlation $C_S(2)$ around $\delta\approx0.25$, a pattern interpreted in terms of magnetic polarons and geometric-string distortions of the antiferromagnetic background [2008.02179].

## 4. Equation of state, thermodynamics, and transport

Large-cluster DCA calculations provide a thermodynamic equation of state for the repulsive square-lattice model in the thermodynamic limit, including $E(T,n)$, $S(T,n)$, double occupancy $D$, nearest-neighbor spin correlations, and $n(\mu,T)$ for $U/t=4$, $8$, and $12$ near half-filling [1305.6798]. At $U/t=8$, a pronounced incompressible region with $\kappa_T\approx0$ appears near half-filling at low temperature; for $T/t=0.25$, $n$ is nearly flat as $\mu/t$ varies from roughly $-1$ to $0$ [1305.6798]. The same study identifies a behavioral shift in the energy below a pseudogap crossover temperature $T^*\approx0.3\,t$, accompanied by entropy suppression and a low-temperature increase of double occupancy in the thermodynamic-limit extrapolation [1305.6798]. The specific heat has a high-temperature “charge” peak near $T/t\approx2.0$ and a low-temperature “spin” feature tied to the same crossover [1305.6798].

Transport in the weak-coupling regime shows a distinct but equally structured set of regularities. In quantum kinetic theory for the 2D single-band model, the electrical resistivity is $T$-linear at high temperature,
$$
\rho(T)\approx0.076\,(T/t)\,(e^2/\hbar)\,(U/t)^2,
$$
and the thermal resistivity behaves as
$$
1/\kappa(T)\approx0.018\,\frac{U^2}{\hbar t^3}\,T^2
$$
for generic filling [2106.04479]. At half-filling, perfect nesting and a van Hove singularity produce nearly $T$-linear electrical resistivity down to the lowest temperatures in that framework, while the low-temperature Wiedemann–Franz ratio tends to $\frac{\pi^2}{3}(k_B/e)^2$ at half-filling and to $\frac{5\pi^2}{36}(k_B/e)^2$ away from half-filling [2106.04479]. By contrast, for $|\mu_F|>2t$ the zero-temperature Fermi surface is too small to allow Umklapp scattering, and resistivity is exponentially suppressed with an Umklapp gap $\Delta_U=2(|\mu_F|-2t)$ [2106.04479].

A distinct hydrodynamic regime emerges when the repulsive 2D model is placed under a uniform tilt. In quantum-gas-microscope experiments on a tilted square lattice with $U/t_h=3.9(1)$ and average filling $\bar n\approx0.6$, the decay time of prepared density waves crosses over from diffusive scaling $\tau\propto\lambda^2$ at weak tilt to subdiffusive scaling $\tau\propto\lambda^4$ at strong tilt [1909.05848]. In the strong-tilt regime, the decay rate is controlled by a thermal diffusivity through
$$
\frac{1}{\tau}=\frac{D_{\mathrm{th}}\,C}{F^2}\,k^4,
$$
reflecting the fact that particle currents generate local heating and are bottlenecked by heat diffusion [1909.05848]. This identifies a transport regime in which density relaxation is no longer set by ordinary charge diffusion alone.

## 5. Numerical and quantum-simulation methods

Methodologically, the 2D Fermi-Hubbard model has served as a benchmark for controlled but mutually complementary approaches. A self-consistent skeleton-diagrammatic GGGW scheme dresses the single-particle propagator, particle-particle and particle-hole pair propagators, and screened interaction simultaneously; at moderate $U$ it reproduces thermodynamic observables at the few-percent level and the amplitudes of $\chi$ and $\Sigma$ within $\lesssim30\%$ of DDMC benchmarks at half-filling [1706.07556]. Large-cluster DCA with CT-AUX provides thermodynamic-limit extrapolations of the equation of state and spin correlations without analytic continuation for thermodynamic observables [1305.6798]. Ladder dual fermions give momentum-resolved fluctuation diagnostics on fine grids and isolate the $(\pi,\pi)$ spin channel as the dominant source of non-Fermi-liquid self-energy structure [1905.07462].

Tensor-network and variational approaches cover complementary regions of parameter space. XTRG, combined with DQMC, reaches temperatures down to $T/t\approx0.06$ on open clusters up to $L=8$ at $U/t=7.2$, quantitatively reproducing finite-temperature spin and charge correlators measured in ultracold-atom experiments [2008.02179]. For the attractive model, a fermionic Gaussian variational method on 2D square lattices with periodic boundary conditions yields total energies with maximal systematic error $\le4\%$ in the intermediate regime around $U/t\sim-4$, with negligible dependence on lattice size up to $L=10$, and maps directly to repulsive-case energies and double occupancies through a partial particle-hole transformation [2308.06700]. A fermionic PEPS-plus-imaginary-time-TEBD construction has also been used on square lattices with periodic boundaries up to $L=10$, explicitly tracking Jordan–Wigner signs for vertical hopping by a doubling construction, although its small-bond-dimension energies remain visibly above exact results [1311.5300].

Quantum simulation has become an additional methodological branch rather than a replacement for classical methods. On a 16-qubit superconducting processor, a low-depth symmetry-preserving variational algorithm for a $2\times4$ ladder observed antiferromagnetic order at half-filling and its suppression away from half-filling [2112.02025]. A QETU implementation for the $2\times2$ model on a 9-qubit grid-like architecture achieved ground-state fidelities above $0.99$ in noiseless simulations with a sufficiently high polynomial degree, though direct energy estimation remained shot-noise limited at about $10^{-2}$ with $10^7$ shots [2411.18535]. In a different direction, a hybrid quantum-classical analog simulation on neutral-atom Rydberg hardware reformulated the half-filled 2D model into free fermions coupled to auxiliary spins and reproduced a metal-to-Mott crossover on a $6\times6$ lattice through a self-consistent auxiliary-spin solver [2510.05897]. The common pattern is not methodological convergence but decomposition of the problem into regimes where different approximations are controlled.

## 6. Extensions, generalizations, and open questions

The attractive 2D Hubbard model has become a distinct arena for studying how strong correlations alter single-particle structure. At $U=-5|t|$, diagrammatic Monte Carlo showed that the spectrum can become disconnected or multiply connected in momentum space because the electron operator fractionalizes into singlon-like and doublon-like components whose spectral weights collapse in momentum-selective fashion [2301.04197]. In that framework, open Fermi-arc-like structures are not themselves computed in all regimes, because some cases are gapped, but the nontrivial spectral connectivity is identified as the prerequisite for arcs when $\omega=0$ intersects the excitation set [2301.04197]. This suggests a route to arc phenomenology that does not rely on explicit translation-symmetry breaking.

Generalizations in symmetry and lattice structure broaden the model’s scope without removing its Hubbard core. In a 2D single-layer square optical lattice, the SU($N$) Fermi-Hubbard model with $N=3,4,6$ has been characterized experimentally through its equation of state, compressibility, occupation probabilities, and density fluctuations for $U/t=2.3(1)$, $7.5(4)$, $10.4(6)$, and $33(2)$, with Mott-like incompressible regions near integer filling and model-independent fluctuation thermometry consistent with EoS-based fits [2305.18967]. On decorated honeycomb and triangular lattices, a half-filled Hubbard model can map to a 120-degree compass model; in that asymptotic regime, the triangular case yields collinear stripe order corresponding to a time-reversal-invariant $d$-form-factor charge-density wave, whereas the decorated honeycomb case yields a unique, gapped fermionic symmetry-protected topological phase protected by time-reversal and reflection symmetries [1603.03439].

Several open problems remain explicit in the current literature. For the repulsive square-lattice model, the precise low-temperature location of the phase-separation dome relative to magnetic and pairing regimes remains unsettled, and finite-size methods may bias toward homogeneous superfluid states or miss phase separation altogether when $\xi$ becomes large [1706.07556]. In the attractive model, mapping the precise arc locus as a function of $\mu$ and $T$ remains future work [2301.04197]. In an altermagnetic extension of the 2D Hubbard model with sublattice-odd next-nearest-neighbor hopping, Hartree–Fock analysis finds broad mixed-phase regions even at half-filling; for $U/t=3.5$, $t'/t=0.3$, and $\delta=0.9$, the coexistence window at the crossing chemical potential spans densities from about $0.935$ to $1.024$, so half-filling lies inside a mixed phase rather than a homogeneous one [2506.16503]. A plausible implication is that inhomogeneity, rather than a single uniform broken-symmetry state, may organize parts of the extended 2D Hubbard phase diagram that are experimentally accessible in cold atoms.

The 2D Fermi-Hubbard model therefore remains less a single solved phase diagram than a unifying framework in which antiferromagnetism, pseudogap behavior, doublon-hole correlations, transport anomalies, pairing tendencies, spectral-topology changes, symmetry-enriched descendants, and quantum-simulation strategies can all be formulated with a common microscopic language. Its importance derives not from analytical closure but from the unusually sharp way in which many different many-body phenomena can be traced back to the same square-lattice Hamiltonian.

Source: https://www.emergentmind.com/topics/2d-fermi-hubbard-model