Periodic Spinful 2D Fermi-Hubbard Model

• The periodic spinful 2D Fermi–Hubbard model is a lattice model of interacting spin-½ fermions, defined by nearest-neighbor hopping and on-site repulsion to capture strong correlation effects.
• Extensions incorporating next-nearest-neighbor hopping, artificial gauge fields, and decorated lattices enable the study of noncollinear magnetism, stripe order, and symmetry-protected topological phases.
• Robust numerical methodologies—such as XTRG, DQMC, and quantum circuit encodings—validate its predictions and benchmark many-body physics in experimental and simulation settings.

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 (Chen et al., 2020)","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 (Chen et al., 2020, Liang, 2 Jul 2025, Xie et al., 2024, Chen et al., 2016). 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 (Pereira et al., 6 Oct 2025, Pauw et al., 15 Sep 2025).

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 $t$0 controls filling (Chen et al., 2020). A closely related ground-state form replaces the chemical-potential term by fixed filling and may include next-nearest-neighbor hopping $t$1,

$t$2

which is the form used in periodic variational studies of stripe formation (Liang, 2 Jul 2025).

In the periodic setting, the defining feature is translationally invariant hopping on a two-dimensional lattice with explicit spin-$t$3 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 $t$4 tori (Xie et al., 2024), the periodic square-lattice variational study of stripe order under PBC (Liang, 2 Jul 2025), and the hard-core one-hole problem in the thermodynamic limit on lattices interpolating between square and triangular geometry (Pereira et al., 6 Oct 2025).

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 (Chen et al., 2016). Hofstadter-Hubbard variants retain the square lattice but introduce Peierls phases corresponding to a periodic magnetic flux pattern (Irsigler et al., 2018), while an extended Hofstadter-Fermi-Hubbard model supplements the on-site Hubbard repulsion by nearest-neighbor density repulsion $t$5 (Pauw et al., 15 Sep 2025). Rashba-type synthetic gauge couplings provide another periodic square-lattice generalization in which hopping matrices mix spin states (Minář et al., 2013).

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 (Chen et al., 2020). 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

$t$6

which yields particle-hole symmetry (Chen et al., 2020). In this limit the model has enhanced symmetry,

$t$7

and, in the large-$t$8 regime, it maps to a Heisenberg antiferromagnet with exchange

$t$9

Accordingly, low-temperature half-filled states develop strong antiferromagnetic correlations (Chen et al., 2020).

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 (Cheuk et al., 2016). In finite-temperature tensor-network calculations, the antiferromagnetic pattern is well developed for $U$0 and fades above $U$1 on the studied clusters, with $U$2 data agreeing well with the 2D optical-lattice experiment of Mazurenko et al. (Chen et al., 2020).

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

$U$3

so spin rotation symmetry is reduced from the isotropic case to a residual symmetry preserving $U$4 (Gukelberger et al., 2017, Xie et al., 2024). Because the symmetry is reduced to a discrete $U$5 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 (Xie et al., 2024).

The same anisotropic model has a controlled strong-coupling reduction to an XXZ Hamiltonian,

$U$6

with

$U$7

which explicitly favors Ising antiferromagnetism (Xie et al., 2024). Numerically, the critical temperature is nonmonotonic in $U$8, rising from weak coupling, reaching a maximum around $U$9, and then decreasing in the strong-coupling regime. For $\mu$0, the largest reported critical temperature is $\mu$1 with critical entropy $\mu$2 (Xie et al., 2024).

In weak coupling, the same model shows why the anisotropic case differs qualitatively from the isotropic one. Each spin species retains perfect nesting at $\mu$3, the longitudinal susceptibility diverges logarithmically as $\mu$4, and the dominant instability is in the longitudinal spin channel. The result is an Ising antiferromagnetic ground state for any infinitesimal repulsive interaction $\mu$5, together with a finite-temperature Ising transition for all $\mu$6; at $\mu$7, by contrast, spin $\mu$8 symmetry is restored and there is no finite-temperature magnetic order in purely 2D (Gukelberger et al., 2017).

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$9

thereby introducing holes with density denoted by $$H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},$$0 (Chen et al., 2020). In the canonical square-lattice model, the spin sector is commonly characterized by the normalized correlator

$$H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},$$1

together with the structure factor

$$H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},$$2

and the finite-size staggered magnetization

$$H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},$$3

These observables resolve nearest-neighbor, diagonal, and next-nearest-neighbor antiferromagnetic correlations as well as their finite-size melting with temperature (Chen et al., 2020).

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 $$H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},$$4 finite-size AF order is essentially lost in the $$H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},$$5 data (Chen et al., 2020). The site-resolved optical-lattice experiment reported the same qualitative evolution: antiferromagnetic spin correlations are strongest at half filling and weaken monotonically upon doping (Cheuk et al., 2016).

A more specific doped signature is the sign reversal of longer-range spin correlations. In the finite-temperature square-lattice study, the diagonal correlator $$H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},$$6 changes sign around $$H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},$$7 and the next-nearest-neighbor correlator $$H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},$$8 reverses sign around $$H = -t\sum_{\langle i,j\rangle,\sigma} \left(\hat c^\dagger_{i\sigma}\hat c_{j\sigma} + \text{h.c.}\right) + U\sum_i \hat n_{i\uparrow}\hat n_{i\downarrow} - \mu \sum_{i,\sigma}\hat n_{i\sigma},$$9 (Chen et al., 2020). 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 (Chen et al., 2020).

The charge sector is naturally expressed in terms of the hole projector

$\hat c^\dagger_{i\sigma}$0

the doublon projector

$\hat c^\dagger_{i\sigma}$1

and the antimoment operator

$\hat c^\dagger_{i\sigma}$2

In the finite-temperature square-lattice calculations, hole-hole correlations are repulsive or anticorrelated, $\hat c^\dagger_{i\sigma}$3, while hole-doublon correlations are attractive or bunched, $\hat c^\dagger_{i\sigma}$4. Near half filling, hole-doublon attraction dominates and gives antimoment bunching; at larger doping, hole-hole repulsion dominates and antimoments become antibunched (Chen et al., 2020). 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 $\hat c^\dagger_{i\sigma}$5, corresponding to a doping $\hat c^\dagger_{i\sigma}$6, and the positive signal is attributed primarily to neighboring doublon-hole pairs (Cheuk et al., 2016).

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 $\hat c^\dagger_{i\sigma}$7 and $\hat c^\dagger_{i\sigma}$8 on the $\hat c^\dagger_{i\sigma}$9 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 $\hat c_{i\sigma}$0, and the spin structure factor peaks at $\hat c_{i\sigma}$1 (Liang, 2 Jul 2025). The same work reports that results at $\hat c_{i\sigma}$2 and $\hat c_{i\sigma}$3 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 (Liang, 2 Jul 2025).

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 $\hat c_{i\sigma}$4-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	$\hat c_{i\sigma}$5 Laughlin-like FCI; skyrmions near $\hat c_{i\sigma}$6

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 $\hat c_{i\sigma}$7-density-wave charge order that doubles the unit cell (Chen et al., 2016). 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 (Chen et al., 2016).

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 $\hat c_{i\sigma}$8 realizes the Nagaoka ferromagnetic ground state with maximal total spin, while the triangular-lattice endpoint $\hat c_{i\sigma}$9 yields a spin-singlet state associated with a $\sigma\in\{\uparrow,\downarrow\}$0 spiral (Pereira et al., 6 Oct 2025). The ferromagnet becomes unstable not to a conventional spin wave but to a spin spiral with wavevector $\sigma\in\{\uparrow,\downarrow\}$1, and the exact critical point is

$\sigma\in\{\uparrow,\downarrow\}$2

At the triangular limit the spiral reaches the $\sigma\in\{\uparrow,\downarrow\}$3 noncollinear state with $\sigma\in\{\uparrow,\downarrow\}$4 (Pereira et al., 6 Oct 2025).

Artificial gauge fields and spin imbalance generate additional ordered states. In the spin-imbalanced Hofstadter-Hubbard model with $\sigma\in\{\uparrow,\downarrow\}$5, the strong-coupling spin description contains frustrated exchange along the $\sigma\in\{\uparrow,\downarrow\}$6 direction, and the interplay of this frustration with longitudinal spin imbalance produces a transverse magnetization phase characterized by a nonzero in-plane net moment

$\sigma\in\{\uparrow,\downarrow\}$7

This phase is distinct from canted antiferromagnetism because the transverse components do not cancel globally (Irsigler et al., 2018).

The extended Hofstadter-Fermi-Hubbard model goes further by combining flux and nearest-neighbor repulsion. At flux $\sigma\in\{\uparrow,\downarrow\}$8 and $\sigma\in\{\uparrow,\downarrow\}$9, DMRG finds a spin-polarized $t$0 Laughlin-like fractional Chern insulator once the nearest-neighbor repulsion exceeds approximately $t$1, with bulk Chern number $t$2, a finite charge gap, short-distance suppression in density-density correlations, and hidden off-diagonal long-range order in the composite-boson basis (Pauw et al., 15 Sep 2025). At $t$3, the same model realizes a quantum Hall ferromagnet or Chern insulator with $t$4, and doping away from $t$5 produces skyrmionic excitations; for $t$6, both particle- and hole-skyrmions are stabilized, with fitted spin-flip numbers approximately $t$7 and $t$8 (Pauw et al., 15 Sep 2025).

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 $t$9, but increasing $U>0$0 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 (Minář et al., 2013). 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 $U>0$1 and, at half filling, $U>0$2, and reaches temperatures down to about $U>0$3; 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 (Chen et al., 2020). In their overlap regime the agreement is excellent (Chen et al., 2020).

At half filling in the anisotropic periodic model, finite-temperature AFQMC becomes numerically exact because an appropriate Hubbard-Stratonovich decomposition in the $U>0$4 or $U>0$5 channel makes the simulations sign-problem-free for both $U>0$6 and $U>0$7 (Xie et al., 2024). 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 (Xie et al., 2024). 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 $U>0$8-hole doping remains suggestive rather than conclusive (Xie et al., 2024).

Other periodic regimes are accessed by different tools. Exact diagonalization establishes the pseudospin ordering and the interacting SPT diagnosis in decorated-lattice Hubbard models (Chen et al., 2016). DMRG on cylinders and open squares, with bond dimension up to $U>0$9, resolves incompressibility, bulk Chern numbers, Tao-Thouless order, hidden composite-boson correlators, and skyrmionic textures in the extended Hofstadter-Fermi-Hubbard model (Pauw et al., 15 Sep 2025). Tensor-Backflow variational Monte Carlo targets large periodic lattices up to $t$00 and $t$01, discovers stripe order spontaneously under PBC, and reaches energies competitive with state-of-the-art tensor-network methods; for $t$02 and $t$03 on the $t$04 lattice under open boundary condition, the nearest-neighbor-backflow energy is only $t$05 above the fPEPS result with bond dimension $t$06 (Liang, 2 Jul 2025).

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

$t$07

revealing both antiferromagnetic correlations and the crossover from a Pauli-and-correlation hole at large doping to doublon-hole bunching near half filling (Cheuk et al., 2016). The agreement with NLCE and DQMC is reported as excellent (Cheuk et al., 2016). 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 (Chen et al., 2020, Cheuk et al., 2016).

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

$t$08

together with local representations of the fermionic bilinears (Jafarizadeh et al., 2024). The physical Hilbert space is the simultaneous $t$09 eigenspace of plaquette stabilizers, directly related to $t$10 topological order and toric-code structure (Jafarizadeh et al., 2024). For a $t$11 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 (Jafarizadeh et al., 2024).

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 (Levy et al., 2022). In the spinful case, the reported numerics focus on an open $t$12 lattice rather than periodic benchmarks, but the broader point is that the 2D spinful Hubbard Hamiltonian can be engineered with local $t$13 fields, 2-local $t$14 interactions, and 3-body couplers (Levy et al., 2022). For the open $t$15 case at half filling and $t$16, adding three Heisenberg $t$17 driver terms raises the fidelity from about $t$18 to $t$19 (Levy et al., 2022).

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 (Jafarizadeh et al., 2024, Levy et al., 2022). 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.