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

# Periodic Spinful 2D Fermi-Hubbard Model

to=arxiv_search  天天中彩票官方  北京pk赛车query":"Periodic spinful 2D Fermi-Hubbard model square lattice finite temperature doping XTRG DQMC 2008.02179","max_results":5}
The periodic spinful 2D Fermi-Hubbard model is the standard lattice model of interacting spin-$\tfrac12$ fermions in two spatial dimensions, usually formulated on a translationally invariant square lattice with nearest-neighbor hopping and on-site Hubbard repulsion. In its canonical form it is controlled by the hopping amplitude $t$, the on-site interaction $U$, and either a chemical potential $\mu$ or a fixed filling, while closely related periodic generalizations add next-nearest-neighbor hopping, spin-dependent anisotropy, artificial gauge flux, or decorated-lattice unit cells [2008.02179], [2507.01856], [2412.20843], [1603.03439]. Across these formulations, the model serves as a common framework for half-filled antiferromagnetism, doped spin and charge correlations, stripe formation, noncollinear magnetism, symmetry-protected topology, and quantum Hall ferromagnetism [2510.05226], [2509.12184].

## 1. Canonical formulation and periodic lattice structure

The standard square-lattice Hamiltonian is
\[
H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},
\]
where $\hat c^\dagger_{i\sigma}$ and $\hat c_{i\sigma}$ create and annihilate a spinful fermion with $\sigma\in\{\uparrow,\downarrow\}$, $t$ is nearest-neighbor hopping, $U>0$ is the on-site repulsion, and $\mu$ controls filling [2008.02179]. A closely related ground-state form replaces the chemical-potential term by fixed filling and may include next-nearest-neighbor hopping $t'$,
\[
\hat{H}= -t\sum_{\langle ij\rangle,\sigma}\left(\hat{c}_{i\sigma}^\dagger \hat{c}_{j\sigma}+\mathrm{H.c.}\right) -t'\sum_{\langle\langle ij\rangle\rangle,\sigma}\left(\hat{c}_{i\sigma}^\dagger \hat{c}_{j\sigma}+\mathrm{H.c.}\right) +U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow},
\]
which is the form used in periodic variational studies of stripe formation [2507.01856].

In the periodic setting, the defining feature is translationally invariant hopping on a two-dimensional lattice with explicit spin-$\tfrac12$ degrees of freedom. Periodic square-lattice realizations appear directly in studies with periodic boundary conditions and translationally invariant Hamiltonians, including the spin-dependent anisotropic Hubbard model on finite $L\times L$ tori [2412.20843], the periodic square-lattice variational study of stripe order under PBC [2507.01856], and the hard-core one-hole problem in the thermodynamic limit on lattices interpolating between square and triangular geometry [2510.05226].

The periodic model also admits several structurally distinct extensions. Decorated triangular and decorated honeycomb lattices replace each site of an underlying Bravais lattice by a six-site hexagon and remain periodic at half filling [1603.03439]. Hofstadter-Hubbard variants retain the square lattice but introduce Peierls phases corresponding to a periodic magnetic flux pattern [1812.02001], while an extended Hofstadter-Fermi-Hubbard model supplements the on-site Hubbard repulsion by nearest-neighbor density repulsion $V$ [2509.12184]. Rashba-type synthetic gauge couplings provide another periodic square-lattice generalization in which hopping matrices mix spin states [1306.3124].

A basic but important distinction is between the canonical periodic Hamiltonian and the boundary conditions chosen in a specific numerical implementation. The finite-temperature XTRG study of the standard square-lattice model is directly about the canonical spinful 2D Fermi-Hubbard model, but its main simulations were performed on open finite clusters rather than periodic tori in order to match finite experimental systems [2008.02179]. This distinction matters when comparing periodic-field-theoretic intuition with finite-cluster numerics.

## 2. Half filling, symmetries, and strong-coupling limits

At half filling on the bipartite square lattice, the canonical model is obtained by setting
\[
\mu = U/2,
\]
which yields particle-hole symmetry [2008.02179]. In this limit the model has enhanced symmetry,
\[
SU(2)_{\text{charge}}\otimes SU(2)_{\text{spin}},
\]
and, in the large-$U$ regime, it maps to a Heisenberg antiferromagnet with exchange
\[
J=\frac{4t^2}{U}.
\]
Accordingly, low-temperature half-filled states develop strong antiferromagnetic correlations [2008.02179].

For the isotropic square-lattice model, half filling is the reference point for both numerics and experiment. In the site-resolved cold-atom realization, repulsion suppresses double occupancy and favors Mott physics near half filling, while short-range antiferromagnetic correlations strengthen as the filling approaches one fermion per site [1606.04089]. In finite-temperature tensor-network calculations, the antiferromagnetic pattern is well developed for $T\lesssim 0.12$ and fades above $T\sim 0.49$ on the studied clusters, with $L=8$ data agreeing well with the 2D optical-lattice experiment of Mazurenko et al. [2008.02179].

A common misconception is that all half-filled periodic 2D Hubbard models share the same finite-temperature magnetic structure. The anisotropic spin-dependent square-lattice model provides a direct counterexample. There the hopping amplitudes are
\[
t_{x\uparrow}=t_{y\downarrow}=t,\qquad t_{x\downarrow}=t_{y\uparrow}=\alpha t,\qquad \alpha\in[0,1],
\]
so spin rotation symmetry is reduced from the isotropic case to a residual symmetry preserving $S^z$ [1702.00035], [2412.20843]. Because the symmetry is reduced to a discrete $\mathbb{Z}_2$ structure, the half-filled system can exhibit a genuine finite-temperature Ising transition in 2D: for repulsive interactions the ordered phase is an Ising antiferromagnet, and for attractive interactions a partial particle-hole transformation maps it to charge-density-wave order [2412.20843].

The same anisotropic model has a controlled strong-coupling reduction to an XXZ Hamiltonian,
\[
\hat{H}_{\rm XXZ} =\sum_{\langle \mathbf{i}\mathbf{j}\rangle} \Big[ J_{xx}(\hat{S}^x_{\mathbf{i}}\hat{S}^x_{\mathbf{j}}+\hat{S}^y_{\mathbf{i}}\hat{S}^y_{\mathbf{j}}) +J_z \hat{S}^z_{\mathbf{i}}\hat{S}^z_{\mathbf{j}} \Big],
\]
with
\[
J_{xx}=\frac{4\alpha t^2}{U},\qquad J_z=\frac{2(1+\alpha^2)t^2}{U}, \qquad \frac{J_z}{J_{xx}}=\frac{1+\alpha^2}{2\alpha}\ge 1,
\]
which explicitly favors Ising antiferromagnetism [2412.20843]. Numerically, the critical temperature is nonmonotonic in $U$, rising from weak coupling, reaching a maximum around $U/t\approx 5$, and then decreasing in the strong-coupling regime. For $\alpha=0.5$, the largest reported critical temperature is $T_N/t \approx 0.1847(2)$ with critical entropy $\boldsymbol{s}_N/k_B \approx 0.219(2)$ [2412.20843].

In weak coupling, the same model shows why the anisotropic case differs qualitatively from the isotropic one. Each spin species retains perfect nesting at $\mathbf Q=(\pi,\pi)$, the longitudinal susceptibility diverges logarithmically as $T\to0$, and the dominant instability is in the longitudinal spin channel. The result is an Ising antiferromagnetic ground state for any infinitesimal repulsive interaction $U>0$, together with a finite-temperature Ising transition for all $\alpha<1$; at $\alpha=1$, by contrast, spin $SU(2)$ symmetry is restored and there is no finite-temperature magnetic order in purely 2D [1702.00035].

## 3. Doping, spin correlations, charge correlations, and stripe order

Doping away from half filling is introduced by tuning the chemical potential below the particle-hole-symmetric value,
\[
\mu < U/2,
\]
thereby introducing holes with density denoted by $\delta$ [2008.02179]. In the canonical square-lattice model, the spin sector is commonly characterized by the normalized correlator
\[
C_S(d) = \frac{1}{N_d}\sum_{|i-j|=d} \frac{\langle \hat S_i\cdot \hat S_j\rangle}{S(S+1)},
\]
together with the structure factor
\[
S(q)= \frac{1}{N(N-1)}\sum_{i\neq j} \frac{\langle \hat S_i\cdot \hat S_j\rangle}{S(S+1)} e^{-iq\cdot(i-j)},
\]
and the finite-size staggered magnetization
\[
m_s = \sqrt{S(\pi,\pi)}.
\]
These observables resolve nearest-neighbor, diagonal, and next-nearest-neighbor antiferromagnetic correlations as well as their finite-size melting with temperature [2008.02179].

The basic doped trend is a rapid suppression of antiferromagnetism. In XTRG calculations, AF correlations are strong near half filling, weaken with hole doping, and for $\delta \gtrsim 0.15$ finite-size AF order is essentially lost in the $6\times6$ data [2008.02179]. The site-resolved optical-lattice experiment reported the same qualitative evolution: antiferromagnetic spin correlations are strongest at half filling and weaken monotonically upon doping [1606.04089].

A more specific doped signature is the sign reversal of longer-range spin correlations. In the finite-temperature square-lattice study, the diagonal correlator $C_S(\sqrt{2})$ changes sign around $\delta \simeq 0.2$ and the next-nearest-neighbor correlator $C_S(2)$ reverses sign around $\delta \simeq 0.25$ [2008.02179]. The interpretation given there is magnetic polaron formation: a doped hole moving through the antiferromagnetic background creates a string of misaligned spins, distorts the magnetic environment, and generates nontrivial sign changes in longer-range correlations, consistent with the geometric-string picture and with doped-Hubbard experiments [2008.02179].

The charge sector is naturally expressed in terms of the hole projector
\[
\hat h_i = |0\rangle\langle 0|_i,
\]
the doublon projector
\[
\hat d_i = |\uparrow\downarrow\rangle\langle\uparrow\downarrow|_i
= \hat n_{i\uparrow}\hat n_{i\downarrow},
\]
and the antimoment operator
\[
\hat\alpha_i = \hat h_i + \hat d_i.
\]
In the finite-temperature square-lattice calculations, hole-hole correlations are repulsive or anticorrelated, $g^{hh}_2(d)<1$, while hole-doublon correlations are attractive or bunched, $g^{hd}_2(d)>1$. Near half filling, hole-doublon attraction dominates and gives antimoment bunching; at larger doping, hole-hole repulsion dominates and antimoments become antibunched [2008.02179]. The experimental moment correlator shows the same sign structure in a directly measured observable: at low fillings the nearest-neighbor moment correlator is negative because of the combined Pauli hole and correlation hole, while near half filling it becomes positive, with the crossover occurring around a moment of about $0.75$, corresponding to a doping $x\approx 0.21$, and the positive signal is attributed primarily to neighboring doublon-hole pairs [1606.04089].

Periodic-boundary-condition ground-state studies add a distinct phenomenon: spontaneous stripe formation. In the Tensor-Backflow variational study, the optimized state under PBC at $n=0.875$ and $U=8$ on the $16\times16$ lattice develops a linear stripe ground state without any imposed symmetry constraint. The spin-density-wave period is 16, the charge-density-wave period is 8, the charge structure factor peaks at $\pi/4$, and the spin structure factor peaks at $7\pi/8$ [2507.01856]. The same work reports that results at $n=0.8$ and $n=0.9375$ are consistent with the AFQMC phase diagram, and it emphasizes that translational symmetry need not be enforced for stripe order to emerge in periodic variational states [2507.01856].

## 4. Ordered, noncollinear, and topological phases in periodic variants

The periodic spinful 2D Fermi-Hubbard framework supports a wide family of ordered and topological states once lattice geometry, gauge structure, or interaction content is modified. The following representative variants illustrate the range already established in explicit periodic models.

| Variant | Periodic structure | Representative result |
|---|---|---|
| Decorated triangular Hubbard model | Hexagon-decorated triangular lattice | Stripe pseudospin order maps to $d$-density-wave charge order |
| Decorated honeycomb Hubbard model | Hexagon-decorated honeycomb lattice | Gapped fermionic SPT protected by time reversal and reflection |
| Square-to-triangular hard-core model | Square lattice plus one diagonal hopping | Nagaoka ferromagnet destabilized by a spin spiral |
| Hofstadter-Hubbard with imbalance | Square lattice with periodic flux | Transverse net magnetization phase |
| Extended Hofstadter-Fermi-Hubbard | Square lattice with flux and nearest-neighbor repulsion | $\nu=\tfrac13$ Laughlin-like FCI; skyrmions near $\nu=1$ |

On decorated lattices, the half-filled periodic Hubbard problem can reduce to a quantum compass model. For the decorated triangular case, exact diagonalization finds collinear stripe antiferromagnetism in the pseudospins, and the corresponding fermionic interpretation is a time-reversal-invariant $d$-density-wave charge order that doubles the unit cell [1603.03439]. On the decorated honeycomb lattice, the effective compass model instead has a unique ground state on the torus, a finite many-body gap in the thermodynamic limit, no long-range Néel order, no long-range plaquette order, and a very short correlation length. Because the many-body ground state transforms as a nontrivial irreducible representation of the point group for some clusters, it cannot be adiabatically connected to a free-fermion band insulator while preserving time reversal and reflection symmetry; the resulting phase is therefore identified as a 2D interacting fermionic SPT [1603.03439].

In the extreme strong-coupling and low-doping limit, periodic geometry alone can change the magnetic ground state. In the hard-core one-hole model interpolating between square and triangular lattices, the square-lattice endpoint $t'=0$ realizes the Nagaoka ferromagnetic ground state with maximal total spin, while the triangular-lattice endpoint $t'=t$ yields a spin-singlet state associated with a $120^\circ$ spiral [2510.05226]. The ferromagnet becomes unstable not to a conventional spin wave but to a spin spiral with wavevector $\mathbf Q=(q,q)$, and the exact critical point is
\[
(t_c')_{\rm exact}=0.24\,t.
\]
At the triangular limit the spiral reaches the $120^\circ$ noncollinear state with $\mathbf Q=(2\pi/3,\,2\pi/3)$ [2510.05226].

Artificial gauge fields and spin imbalance generate additional ordered states. In the spin-imbalanced Hofstadter-Hubbard model with $\alpha=1/4$, the strong-coupling spin description contains frustrated exchange along the $y$ direction, and the interplay of this frustration with longitudinal spin imbalance produces a transverse magnetization phase characterized by a nonzero in-plane net moment
\[
S^r = \frac{1}{N_xN_y} \sqrt{ \left(\sum_{\mathbf{j}} S^x_{\mathbf{j}}\right)^2 + \left(\sum_{\mathbf{j}} S^y_{\mathbf{j}}\right)^2 }.
\]
This phase is distinct from canted antiferromagnetism because the transverse components do not cancel globally [1812.02001].

The extended Hofstadter-Fermi-Hubbard model goes further by combining flux and nearest-neighbor repulsion. At flux $\alpha=\tfrac14$ and $U/J=8$, DMRG finds a spin-polarized $\nu=\tfrac13$ Laughlin-like fractional Chern insulator once the nearest-neighbor repulsion exceeds approximately $V_c/J\approx0.5$, with bulk Chern number $\mathcal{C}_{\rm bulk}=0.33\pm0.05$, a finite charge gap, short-distance suppression in density-density correlations, and hidden off-diagonal long-range order in the composite-boson basis [2509.12184]. At $\nu=1$, the same model realizes a quantum Hall ferromagnet or Chern insulator with $\mathcal{C}_{\rm bulk}=1.00\pm0.01$, and doping away from $\nu=1$ produces skyrmionic excitations; for $V/J=1$, both particle- and hole-skyrmions are stabilized, with fitted spin-flip numbers approximately $\mathcal{A}=3.1(4)$ and $\mathcal{S}=4.2(9)$ [2509.12184].

Synthetic spin-orbit coupling supplies a different periodic extension. In the half-filled square-lattice Rashba-Hubbard model, real-space mean-field theory finds antiferromagnetic order at weak repulsion because the susceptibility still diverges at $\mathbf q_\pi=(\pi,\pi)$, but increasing $U$ drives a first-order transition into noncollinear magnetic textures, including spiral phases and a skyrmion crystal, consistent with the effective strong-coupling spin model containing anisotropic exchange and Dzyaloshinskii-Moriya terms [1306.3124]. A plausible implication is that periodic 2D Hubbard physics is constrained less by the bare on-site interaction than by how lattice geometry and gauge structure reorganize the low-energy spin manifold.

## 5. Numerical methodologies and experimental benchmarks

The periodic spinful 2D Fermi-Hubbard model is also a testing ground for controlled many-body methods. At finite temperature, the square-lattice study combining XTRG and DQMC shows how complementary numerical approaches are used in practice: XTRG represents the thermal density matrix as an MPO on a one-dimensional snake mapping of the 2D lattice, cools exponentially by repeated squaring, exploits non-Abelian symmetries such as $SU(2)_{\rm spin}$ and, at half filling, $SU(2)_{\rm charge}$, and reaches temperatures down to about $T/t\simeq0.06$; DQMC decouples the interaction with a Hubbard-Stratonovich transformation and samples the finite-temperature path integral, but at finite doping it is limited by the minus-sign problem [2008.02179]. In their overlap regime the agreement is excellent [2008.02179].

At half filling in the anisotropic periodic model, finite-temperature AFQMC becomes numerically exact because an appropriate Hubbard-Stratonovich decomposition in the $S^x$ or $S^y$ channel makes the simulations sign-problem-free for both $U>0$ and $U<0$ [2412.20843]. This permits precision studies of the energy, double occupancy, specific heat, compressibility, entropy, and spin, singlon, and doublon correlations, as well as critical-temperature estimates across weak and strong coupling [2412.20843]. By contrast, away from half filling the sign problem reappears, and even in the anisotropic model the evidence for stripe spin-density-wave tendencies at $1/8$-hole doping remains suggestive rather than conclusive [2412.20843].

Other periodic regimes are accessed by different tools. Exact diagonalization establishes the pseudospin ordering and the interacting SPT diagnosis in decorated-lattice Hubbard models [1603.03439]. DMRG on cylinders and open squares, with bond dimension up to $\chi=8192$, resolves incompressibility, bulk Chern numbers, Tao-Thouless order, hidden composite-boson correlators, and skyrmionic textures in the extended Hofstadter-Fermi-Hubbard model [2509.12184]. Tensor-Backflow variational Monte Carlo targets large periodic lattices up to $16\times16$ and $8\times32$, discovers stripe order spontaneously under PBC, and reaches energies competitive with state-of-the-art tensor-network methods; for $n=0.875$ and $U=8$ on the $16\times16$ lattice under open boundary condition, the nearest-neighbor-backflow energy is only $4.5\times10^{-4}$ above the fPEPS result with bond dimension $D=20$ [2507.01856].

Cross-validation against experiment is unusually direct in this field. Site-resolved quantum gas microscopy has measured real-space charge and spin correlators in the 2D Fermi-Hubbard model, including the nearest-neighbor spin correlator, the moment correlator, and the normalized moment pair correlator
\[
g_2(r)=\frac{\langle \hat{m}_z^2(r)\hat{m}_z^2(0)\rangle}{\langle \hat{m}_z^2\rangle^2},
\]
revealing both antiferromagnetic correlations and the crossover from a Pauli-and-correlation hole at large doping to doublon-hole bunching near half filling [1606.04089]. The agreement with NLCE and DQMC is reported as excellent [1606.04089]. This experimental benchmark is especially valuable because away from half filling the sign problem sharply restricts unbiased classical simulations.

A second misconception is that all observed bunching in doped 2D Hubbard systems reflects attractive hole-hole interactions. The finite-temperature tensor and microscope results instead separate the channels: hole-hole correlations are repulsive, while hole-doublon correlations are attractive and dominate the positive antimoment or moment signal near half filling [2008.02179], [1606.04089].

## 6. Quantum-information formulations and hardware-oriented encodings

The periodic spinful 2D Fermi-Hubbard model has become a benchmark for local fermion-to-qubit mappings because two-dimensional spinful fermions are difficult to encode efficiently with Jordan-Wigner strings. A detailed gate-based route is provided by the Derby-Klassen compact mapping, which introduces primary qubits on lattice sites, secondary qubits on a checkerboard subset of faces, and local edge and vertex operators
\[
\tilde{V}_j = Z_j,
\]
together with local representations of the fermionic bilinears [2408.14543]. The physical Hilbert space is the simultaneous $+1$ eigenspace of plaquette stabilizers, directly related to $\mathbb{Z}_2$ topological order and toric-code structure [2408.14543]. For a $6\times8$ spinful example, the resource estimates are 132 qubits for all-to-all connectivity and 203 qubits for a diamond nearest-neighbor layout; for 10 second-order Trotter steps the reported depths are 409 CNOT layers in the all-to-all case and 733 in the diamond case [2408.14543].

The same model has also been proposed as a target for tailored quantum annealers. Using a low-weight encoding with maximum Pauli weight 3, the encoded Hubbard Hamiltonian contains only 3-local hopping terms and 2-local on-site interaction terms, while stabilizers enforce the logical subspace [2207.14374]. In the spinful case, the reported numerics focus on an open $4\times3$ lattice rather than periodic benchmarks, but the broader point is that the 2D spinful Hubbard Hamiltonian can be engineered with local $Z$ fields, 2-local $ZZ$ interactions, and 3-body couplers [2207.14374]. For the open $4\times3$ case at half filling and $U/t=8$, adding three Heisenberg $XX$ driver terms raises the fidelity from about $0.324$ to $0.981$ [2207.14374].

These encoding results do not yet amount to a full periodic-hardware solution for large 2D spinful systems. They do, however, establish a precise computational role for the model: it is difficult enough that locality-preserving mappings, stabilizer protection, and measurement design become central, yet structured enough that hopping and density terms fit naturally into compact encodings [2408.14543], [2207.14374]. This suggests that the periodic spinful 2D Fermi-Hubbard model will remain both a condensed-matter prototype and a hardware benchmark, with progress tied equally to many-body physics, numerical control, and the design of local fermionic simulators.

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