--- title: Square Lattice Hubbard Model url: https://www.emergentmind.com/topics/square-lattice-hubbard-model type: topic --- # Square Lattice Hubbard Model to investigate Mott transitions, antiferromagnetism, superconductivity, and competing inhomogeneous states." That appears in various data. Good. Need maybe include a small table for variants with Hamiltonians? But tables can be brief. Yet detailed explanations should be paragraphs. We can maybe avoid table. Let's search arxiv. The square lattice Hubbard model is a class of lattice fermion models defined on a two-dimensional square lattice, most commonly in its one-band form with nearest-neighbor hopping \(t\) and on-site interaction \(U\). In the data considered here, it appears in several closely related realizations: the standard SU(2) model, extended models with nearest-neighbor interaction \(V\), frustrated \(t\text{-}t'\text{-}U\) variants, strong-coupling spin reductions with ring exchange, and multicomponent generalizations such as SU(3). Across these formulations, the square lattice geometry provides the common setting in which Mott physics, antiferromagnetism, charge order, superconductivity, phase separation, and inhomogeneous states are analyzed with exact diagonalization, determinant quantum Monte Carlo, variational cluster methods, density matrix embedding theory, variational Monte Carlo, and symmetry-based operator constructions [1111.0076]. ## 1. Standard formulation and principal generalizations In its standard one-band form on the square lattice, the Hubbard Hamiltonian is written as \[ H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow}, \] with nearest-neighbor hopping \(t\), on-site repulsion \(U\), and fermion operators \(c_{i\sigma}^{\dagger},c_{i\sigma}\). A closely related notation used in the strong-coupling spin-expansion literature writes \[ H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow}, \] at half filling with \(\mu=0\). The model is also written in a frustrated form with next-nearest-neighbor hopping, \[ H = -t\sum_{\langle ij\rangle \sigma}a_{i\sigma}^{\dagger}a_{j\sigma} -t^\prime\sum_{\langle\langle ij\rangle \rangle \sigma}a_{i\sigma}^{\dagger}a_{j\sigma} +U\sum_{i}n_{i\uparrow}n_{i\downarrow}, \] and in an anisotropic frustrated form with inequivalent diagonal hoppings \(t_2,t_2'\). These extensions retain the square lattice while altering magnetic competition and the low-energy phase structure [1504.01784]. A second major generalization is the extended Hubbard model, where a nearest-neighbor density interaction \(V\) supplements the on-site term: \[ H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_i n_{i\uparrow}n_{i\downarrow} + V \sum_{\langle i,j\rangle,\sigma,\sigma'} n_{i\sigma}n_{j\sigma'}. \] At half filling, another equivalent form writes the interaction as \(V\sum_{\langle i,j\rangle}n_in_j\), or as \(\frac12\sum_{\mathbf l\mathbf l'}v_{\mathbf l\mathbf l'}(n_{\mathbf l}-\bar n)(n_{\mathbf l'}-\bar n)\) with \(\bar n=1\). Setting \(V=0\) reduces the model exactly to the usual square-lattice Hubbard model. The data also include an SU(3) square-lattice Fermi-Hubbard model, \[ \hat H = \hat H_t + \hat H_U - \mu \sum_{i,\alpha}\hat n_{i\alpha}, \] with three fermionic flavors, nearest-neighbor hopping, and onsite interactions \(U \sum_i \sum_{\alpha<\beta}\hat n_{i\alpha}\hat n_{i\beta}\), as well as depleted, \(\pi\)-flux, and moiré-derived square-lattice descendants. This suggests that “square lattice Hubbard model” denotes not a single Hamiltonian, but a family of models sharing the same lattice backbone while differing in orbital content, symmetry, and range of couplings. ## 2. Half filling, strong coupling, and antiferromagnetic order At half filling, the square-lattice Hubbard model is repeatedly treated as a Mott and antiferromagnetic problem. In the symmetry-based operator formulation, the repulsive model on a bipartite square lattice has exact global symmetry \[ SO(3)\times SO(3)\times U(1) \equiv [SU(2)\times SU(2)\times U(1)]/Z_2, \] with spin, \(\eta\)-spin, and a hidden \(U(1)\) generated by the number of rotated-electron singly occupied sites. For \(x=0\), \(m=0\), and \(N_a^2\to\infty\), long-range Néel antiferromagnetic order is argued to occur in the half-filled ground state, lowering the ground-state symmetry from \(SO(3)\times SO(3)\times U(1)\) for large finite \(N_a^2\) to \[ [U(2)\times U(1)]/Z_2^2 \equiv [SO(3)\times U(1)\times U(1)]/Z_2 \] in the thermodynamic limit. In that construction, the equality of the spin effective lattice and the original square lattice is identified as a necessary condition for long-range ground-state antiferromagnetic order [1003.4619]. In the strong-coupling regime \(U\gg t\), the half-filled square-lattice model reduces to an effective spin problem. To second order in \(t/U\), one recovers the nearest-neighbor Heisenberg antiferromagnet; to fourth order, the effective Hamiltonian contains second- and third-neighbor exchanges and a four-spin ring-exchange term, \[ \hat{H}^{(4)} = J \sum_{\langle i,j \rangle} \hat{\mathbf S}_i \cdot \hat{\mathbf S}_j + J' \sum_{\langle i,j'\rangle} \hat{\mathbf S}_i \cdot \hat{\mathbf S}_{j'} + J'' \sum_{\langle i,j''\rangle} \hat{\mathbf S}_i \cdot \hat{\mathbf S}_{j''} \] \[ + J_r \sum_{\langle i,j,k,l \rangle} \left[ (\hat{\mathbf S}_i \cdot \hat{\mathbf S}_j)(\hat{\mathbf S}_k\cdot \hat{\mathbf S}_l) + (\hat{\mathbf S}_i \cdot \hat{\mathbf S}_l)(\hat{\mathbf S}_k \cdot \hat{\mathbf S}_j) - (\hat{\mathbf S}_i \cdot \hat{\mathbf S}_k)(\hat{\mathbf S}_j \cdot \hat{\mathbf S}_l) \right], \] with \[ J = 4 t^2/U - 24 t^4/U^3,\quad J' = J'' = 4 t^4/U^3,\quad J_r = 80 t^4/U^3. \] Exact diagonalization of this Hubbard-parametrized spin model finds a quantum phase transition from \((\pi,\pi)\) Néel order to a \((\pi/2,\pi/2)\) ordered state, with the critical ratio reduced by quantum fluctuations from the mean-field value \(J_r/J=2\) to approximately \(J_r/J\approx 1.1\). Near that transition, the dynamical spin correlation function develops a pseudo-continuum at wavevectors between the competing ordering vectors [1812.04277]. A complementary cluster exact-diagonalization study of the extended model on a \(4\times4\) square lattice with four electrons and periodic boundary conditions makes the low-filling many-body spectrum fully explicit. In the \(V=0\) limit, the first excited state energy is \[ E_1(U,0)=-8t \] for all \(U\), implying a purely kinetic state with no double occupancy, whereas the ground state retains double occupancy for all \(U\). This finite-cluster result is not a thermodynamic statement, but it provides an exact benchmark for how on-site repulsion alone organizes low-lying states on the square lattice [1111.0076]. ## 3. Charge order, superconductivity, and phase separation in the extended model Adding nearest-neighbor interaction \(V\) qualitatively restructures the half-filled square-lattice phase diagram. Large-scale determinant quantum Monte Carlo for the half-filled extended Hubbard model in the \(U\)–\(V\) plane finds quantitatively resolved boundaries among antiferromagnetic, charge-ordered, \(s\)-wave superconducting, \(d\)-wave superconducting, and phase-separated ground states. In the repulsive first quadrant, the antiferromagnetic–charge-density-wave boundary lies slightly above the strong-coupling line \(V=U/4\); finite-size scaling gives representative critical values \(V_c\approx 0.3\pm0.1\) at \(U=1\), \(V_c\approx 0.65\pm0.05\) at \(U=2\), and \(V_c\approx 1.1\pm0.1\) at \(U=4\). In the attractive quadrants, \(U<0,V<0\) supports \(s\)-wave superconductivity above the line \(V=U/2\), \(d\)-wave superconductivity below that line, and phase separation at sufficiently large negative \(V\). In the repulsive sector with \(U>0,V<0\), an intermediate \(d\)-wave region appears for \(U\lesssim2\), flanked by antiferromagnetism and phase separation [2304.08683]. Exact diagonalization on a \(4\times4\) periodic cluster yields a closely related but cluster-limited picture. At half filling and in the strong-coupling regime, the phase diagram contains spin density wave and \(d_{x^2-y^2}\)-wave superconductivity at large positive \(U\), charge density wave and extended \(s^*\)-wave superconductivity at large positive and negative \(V\), and \(s\)-wave superconductivity at large negative \(U\) with vanishing \(V\). The energies of different particle sectors delineate the phase separation region. With hole doping, charge fluctuation produces strong competition between different orders, and at \(37.5\%\)–\(50\%\) hole doping the \(p\)-wave pairing channel is found to be enhanced, especially in an interaction range described as relevant to the cuprate superconductors [2206.01119]. A strong-coupling diagram-technique treatment of the same half-filled extended model emphasizes the charge sector. Summing infinite ladder series for spin and charge fluctuations, it finds a first-order charge transition at \(v=v_c\gtrsim U/4\), visible as an abrupt sign change of a sharp maximum in the zero-frequency charge susceptibility at the Brillouin-zone corner \((\pi/a,\pi/a)\). The associated states have alternating deviations of electron occupations from the mean value on neighboring sites; due to fluctuations these deviations possess short-range order. For the parameters considered, this behavior appears for \(U\lesssim5t\), while for insulating \(U\gtrsim6t\) the transition is not observed and the Mott gap grows continuously with \(v\) [2302.01062]. The finite-temperature sector is partly resolved as well. Determinant quantum Monte Carlo determines a partial \(T_c(U,V)\) diagram in which CDW critical temperatures in the repulsive \(V>0\) sector rise rapidly with \(V\), and \(s\)-wave superconducting critical temperatures in the attractive \(U<0,V<0\) sector are enhanced by nearest-neighbor attraction relative to the pure attractive Hubbard limit. By contrast, at half filling in the repulsive \(U>0,V<0\) sector, \(d\)-wave superconductivity is found to have a much smaller \(T_c\) than the \(s\)-wave state in the attractive sector. This suggests that nearest-neighbor interactions enrich square-lattice Hubbard physics not only by introducing new phases, but also by splitting the energy scales of ordering channels that are degenerate or absent in the on-site model. ## 4. Doping, frustration, and competing inhomogeneous states Away from half filling, the square-lattice Hubbard model supports metallic, antiferromagnetic, superconducting, and inhomogeneous states whose balance depends strongly on \(U\), \(t'\), and density. Density matrix embedding theory on the thermodynamic-limit square lattice, with impurity clusters up to 16 sites and an explicit error model, finds antiferromagnetism at half filling, a d-wave superconducting region at intermediate doping, a metallic regime at large doping and small \(U\), and a low-doping sector in which inhomogeneous charge, spin, and pairing states compete very closely in energy. For \(t'=0\) and \(U=4\), the d-wave order parameter has a dome-like dependence on filling, peaking near \(\langle n\rangle\approx0.9\) and extending roughly over \(0.8\lesssim n\lesssim0.95\). Positive \(t'=+0.2\) stabilizes antiferromagnetism and suppresses d-wave superconductivity, while negative \(t'=-0.2\) enhances d-wave superconductivity and strengthens low-doping inhomogeneity, mirroring the contrast between electron-doped and hole-doped cuprates [1504.01784]. In the underdoped strong-coupling regime, the inhomogeneous solutions found by embedding include superconducting ribbons, classic stripes with period-4 charge modulation and period-8 spin modulation, and states combining spin-density wave, charge-density wave, and pair-density wave structure. The energies of these states are nearly degenerate with more homogeneous d-wave solutions, and the dominant uncertainty comes from cluster-size extrapolation rather than from the impurity solver. This suggests that the low-doping square-lattice Hubbard model is intrinsically a near-degenerate problem in which stripe, pair-density-wave, and uniform superconducting tendencies remain in close competition. Frustration introduced through anisotropic next-nearest-neighbor hoppings \(t_2,t_2'\) produces a distinct but related landscape. Variational cluster calculations at half filling on the square, crossed-square, and triangular limits of the same two-parameter lattice family find Néel, collinear, and \(120^\circ\) spiral magnetic phases, together with a nonmagnetic insulating phase separating them in broad parameter windows at strong coupling. For \(U/t_1=60\), along the line \(t_2=t_2'\), the nonmagnetic insulator occupies \(0.73