Spin-Spin Correlation at Half-Filling

• Spin-spin correlations are measures of magnetic moment alignments in lattice models at half-filling, revealing key electronic interactions and exchange mechanisms.
• Advanced numerical techniques like quantum Monte Carlo and DMRG rigorously capture the decay and spectral features of these correlations across dimensions.
• Studies in 1D, 2D, and 3D Hubbard models illustrate the balance between antiferromagnetic and ferromagnetic tendencies, informing our understanding of quantum phase transitions.

Spin-spin correlation at half-filling refers to the spatial and dynamical organization of local magnetic moments in interacting lattice models at a commensurate particle density—specifically, one particle per site (or equivalently, half-filling of the underlying band). This regime is foundational for Mott physics, quantum magnetism, and the emergence of exotic insulating, metallic, and superconducting states across one, two, and three dimensions. The form, sign structure, decay, and spectral content of the spin-spin correlation function encode the underlying electronic interactions, quantum statistics, exchange mechanisms, and dimensional constraints.

1. Definition of Spin-Spin Correlation Functions

The static (equal-time) spin-spin correlation function in lattice fermion models is

$$C(i,j) = \langle \mathbf{S}_i \cdot \mathbf{S}_j \rangle,$$

where $$\mathbf{S}_i = (1/2) \sum_{\sigma,\sigma'} c^\dagger_{i\sigma} \boldsymbol{\sigma}_{\sigma\sigma'} c_{i\sigma'}$$, and $\boldsymbol{\sigma}$ are Pauli matrices. In models with additional structure—such as layers or orbitals—the operator generalizes to $S_{i m \alpha}$, carrying indices for site $i$, layer $m$, and orbital $\alpha$ (Yang et al., 22 Aug 2024).

The real-space correlator $C(i,j)$ measures the spatial dependence of spin alignment between sites $i$ and $j$. Its momentum-space Fourier transform, the static spin structure factor, is

$$S(\mathbf{q}) = \frac{1}{N} \sum_{i,j} e^{i \mathbf{q}\cdot(\mathbf{r}_i-\mathbf{r}_j)} \langle \mathbf{S}_i \cdot \mathbf{S}_j \rangle.$$

Antiferromagnetic (AF) order is signaled by $S(\mathbf{q})$ peaking at the ordering vector (e.g., $\mathbf{q}=(\pi,\pi)$ in 2D and $(\pi,\pi,\pi)$ in 3D), while ferromagnetic phases peak at $\mathbf{q}=0$ (Yang et al., 22 Aug 2024, Patra et al., 2023).

2. Hubbard Models at Half-Filling: Dimensionality and Correlation Effects

The prototypical systems for studying spin-spin correlations at half-filling are Hubbard models in various dimensions:

• 1D Hubbard model: At half-filling, the ground state is a Mott insulator with power-law spin correlations decaying as $C(r) \sim (-1)^r r^{-\alpha}$, where $\alpha=1$ for spin-1/2 (Tiegel et al., 2012). Strong-coupling Kondo lattice extensions introduce composite spins and tunable exponents depending on spin quantum number and parity (Masui et al., 2022).
• 2D Hubbard model: The ground state at $n=1$ exhibits robust short-range AFM correlations. Auxiliary-field quantum Monte Carlo (AFQMC), diagrammatic Monte Carlo, and quantum gas microscopy have established that the nearest-neighbor correlation $C(1)$ is strongly negative (antiferromagnetic), increasing monotonically in magnitude with $U$, and approaching the Heisenberg limit as $U\to\infty$ (Qin et al., 2017, Kim et al., 2019, Cheuk et al., 2016).
• 3D Hubbard model: AFQMC and analytic continuation show a pronounced crossover in $C(1)$, with maximal AFM correlations inside the metal–insulator crossover regime at $U_c/t\simeq7.6$ and $T/t=0.36$, before falling off in the strong Mott limit (Song et al., 12 Apr 2024).
• Quasiflatband models / Ladders: Certain ladder models at half-filling realize long-range ferromagnetism, with strictly positive $C_{i,j}$ at all distances and a dominant $S(\mathbf{q}=0)$, provided the lowest band is nearly flat and the interaction band mixing is suppressed (Patra et al., 2023).

3. Measurement Techniques and Finite-Size Scaling

Quantum Monte Carlo and DMRG

• AFQMC: At half-filling, particle–hole symmetry eliminates the sign problem, enabling numerically exact computation of $C(r)$ in large supercells up to $L\sim22$ (2D) or $L\sim12$ (3D) with twist averaging to remove shell effects (Qin et al., 2017, Song et al., 12 Apr 2024). The nearest-neighbor antiferromagnetic correlator $c_L$ is extrapolated to the thermodynamic limit by fits $c_L = c_\infty + a L^{-3}$ (Qin et al., 2017).
• DMRG: In 1D and ladders, DMRG resolves both ground-state properties and the decay exponents of $C(r)$, including the assessment of logarithmic corrections and the distinction between critical and gapped regimes (Tiegel et al., 2012, Masui et al., 2022, Patra et al., 2023).
• Maximum-Entropy and SAC Analytic Continuation: For dynamical quantities, DQMC yields imaginary-time correlators $S(\mathbf{q},\tau)$, analytically continued to real frequency by stochastic maximum entropy or SAC, extracting $S(\mathbf{q},\omega)$ (Yang et al., 22 Aug 2024, Song et al., 12 Apr 2024).

Finite-Size and Scaling Analysis

Order parameters and phase transitions are characterized by structure factor scaling:

$$S_{\rm AF}/N_c \rightarrow m_s^2 \quad \text{as} \; N_c\to\infty,$$

where $m_s$ is the staggered magnetization. Continuous transitions are located by scaling collapses, e.g.,

$$S_{\rm AF}/N_c = L^{-2\beta/\nu} F[L^{1/\nu}(U-U_c)/U_c],$$

with exponents (e.g., $(\beta, \nu)$) matching universality classes such as 3D O(3) Heisenberg (Yang et al., 22 Aug 2024).

4. Regimes, Decay Laws, and Universality

The behavior of $C(r)$ and $S(\mathbf{q})$ as a function of $U$, $T$, and model details maps out continuous crossovers and sharp phase transitions:

• Weak-coupling (metallic): $C(r)$ is weak, short-ranged, and can exhibit RKKY oscillations, decaying as $(-1)^r / r^d$ in $d$ dimensions (Kim et al., 2019).
• Intermediate-coupling (crossover): For $U$ above a model-dependent threshold, but below the Mott regime, spin correlations and correlation length $\xi$ increase rapidly. The peak in $C(1)$ (by magnitude) does not coincide with the most rapid suppression of double occupancy, reflecting that magnetic and charge crossovers are separated in the $(U,T)$ plane (Song et al., 12 Apr 2024).
• Strong-coupling (Mott insulating): $C(r)$ approaches the Heisenberg model result. In 2D, $C_{NN}(U\to\infty) \to -0.334718$, and the sublattice magnetization $m^2 \approx 0.094$ signals long-range antiferromagnetic order at $T=0$ (Qin et al., 2017). In 3D, bona fide Néel order appears for $T<T_N$.
• Flat-band ferromagnetism: In specialized lattice geometries, ferromagnetic $C_{i,j}>0$ dominates throughout, a manifestation of interaction-driven polarization in highly degenerate single-particle manifolds (Patra et al., 2023).
• 1D: Power-law antiferromagnetism: At half-filling in 1D, $C(r)\sim (-1)^r / r^\alpha$ ($\alpha=1$ for spin-1/2), including logarithmic corrections from marginal operators (Tiegel et al., 2012, Masui et al., 2022). The quantum numbers and parity (e.g., Haldane conjecture for spin chains) are essential for gapped vs. gapless behavior.

Table: Spin-Spin Correlator $C(r)$ at Half-Filling, Selected Results

Model/Dim.	Regime	$C_{NN}$ (Example)	Decay/Order
2D Hubbard (Qin et al., 2017)	$U/t=2$, $T=0$	$-0.0996$	Short-range, increases with $U$
2D Hubbard (Qin et al., 2017)	$U/t=12$, $T=0$	$-0.307$	Heisenberg limit
2D Hubbard (Cheuk et al., 2016)	$U/t=7.2$, $T/t\sim1.2$	$-0.09$	Strong AFM, finite-T
3D Hubbard (Song et al., 12 Apr 2024)	$U/t=7.6$, $T/t=0.36$	$-0.147$	Strong AF, $\xi\sim3a$
KH Model (Masui et al., 2022)	1D, Spin-1, strong $J_K$	$C(r)\sim(-1)^r/r$	Gapless, power-law
Ladder Flat Band (Patra et al., 2023)	Special geometry	$+0.22$ (FM)	Long-range FM

5. Dynamical Aspects: Spin Structure Factor and Collective Modes

The dynamical spin structure factor,

$$S(\mathbf{q},\omega) = (1/\pi) \Im \int_0^\infty dt\, e^{i\omega t} \langle S_\mathbf{q}(t) S_{-\mathbf{q}}(0) \rangle,$$

distinguishes between paramagnetic, quantum-disordered, and magnetically ordered regimes (Yang et al., 22 Aug 2024). In weakly insulating and metallic phases, $S(\mathbf{q},\omega)$ shows broad continua without sharp modes, while in ordered AFM Mott phases, sharp magnon branches with Goldstone modes at the ordering wavevector appear. Spin gaps can be extracted from the threshold of spectral response at selected $\mathbf{q}$.

Notably, in a simplified bilayer two-orbital Hubbard model, increasing $U$ drives the closure of the spin gap at $(\pi,\pi)$, the emergence of a divergent quasistatic peak, and the development of magnon-like excitations, consistent with a 3D O(3) universality-class transition (Yang et al., 22 Aug 2024).

6. Special Topics: Kondo Lattices, XXZ Mapping, and Flat-Band Ladders

• Kondo-Heisenberg Models: At half-filling, strong Kondo coupling $J_K$ binds conduction electrons to local moments into “Kondo doublets,” mapping the system onto an effective Heisenberg chain with spin $S_{\text{eff}}=S-1/2$. For integer $S$, critical Luttinger-liquid correlations with $C(r)\sim(-1)^r/r$ are observed; half-integer $S$ yields gapped states (Masui et al., 2022).
• Bose-Hubbard to XXZ Mapping: At half-integer filling, the 1D Bose-Hubbard model maps onto an XXZ chain, with exponents and amplitudes of $C(r)$ fully controlled by bosonization and finite-U perturbation theory. The algebraic decay and prefactor, as well as finite-size corrections due to open boundaries, are quantitatively recovered (Giuliano et al., 2012).
• Flat-Band Ferromagnetism: On two-leg ladders with modulated hopping and flux, the system can realize a ferromagnetic Mott insulator at half-filling, evidenced by positive and slowly decaying $C_{i\alpha;j\beta}$ for all separations, a sharply peaked $S(k=0)$, and a finite charge gap (Patra et al., 2023).

7. Experimental Realizations and Observations

Quantum gas microscopy and ultracold atom emulators have directly measured $C_s(r)$ in the 2D Hubbard model. At $U/t=7.2$ and $T/t\approx1.2$, the nearest-neighbor correlator at half-filling is $C_s(1)\approx -0.09$ (30% of ground-state value). Extended correlations to several lattice spacings have been observed, confirming theoretical results and benchmarking analytic and numerical approaches (Cheuk et al., 2016).

Observation of such correlations as a function of temperature, doping, and higher $U/t$ demonstrates the formation of local moments, development of AFM correlations, and the crossover from correlated metal to Mott insulator (Kim et al., 2019, Cheuk et al., 2016).

8. Universality and Phase Transitions

Continuous quantum phase transitions in models with two or more orbitals or layers have been characterized by analyzing the onset of long-range magnetic order. For example, in bilayer systems, the critical exponents $(\beta, \nu) \approx (0.36, 0.71)$ extracted from the scaling of $S_{\rm AF}^\alpha$ align precisely with the 3D O(3) Heisenberg universality class, and the critical interaction is located at $U_c/t_1^x \approx 4.15$ (Yang et al., 22 Aug 2024). The weakly insulating, nonmagnetic phase below $U_c$ has short-ranged $C(i,j)$, while the AFM Mott phase above $U_c$ supports a finite $m_s^\alpha$ and robust static and dynamical AFM features.

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Spin–spin correlations at half-filling serve as a principal diagnostic of quantum magnetism, Mott physics, and metal–insulator transitions in correlated lattice systems. The phenomenology encompasses a wide range of universal behaviors—algebraic, exponential, and long-range ordered correlations—depending on dimensionality, band structure, spin quantum number, interaction strength, and temperature. Comprehensive understanding relies on the interplay between numerically exact many-body computation, finite-size scaling, analytic continuation, and experimental quantum simulation, as manifest in recent large-scale QMC, DMRG, and quantum gas microscopy studies (Yang et al., 22 Aug 2024, Song et al., 12 Apr 2024, Qin et al., 2017, Cheuk et al., 2016, Masui et al., 2022, Tiegel et al., 2012, Patra et al., 2023, Giuliano et al., 2012).