Classical Holstein-Spin-Fermion Model

• The classical Holstein-spin-fermion model is a sign-problem-free formulation combining Holstein lattice coupling and spin-fermion interactions with quantum electrons.
• It employs a mean-field approximation to treat both lattice displacements and local spins as static, enabling efficient large-scale Monte Carlo simulations.
• The model reveals a phase transition from antiferromagnetic to charge-density-wave order, with surrogate machine learning methods enhancing sampling efficiency.

Searching arXiv for the specified papers and closely related Holstein / spin-fermion formulations. Searching for ([2509.05876](/papers/2509.05876)) classical Holstein spin fermion model. The classical Holstein-spin-fermion model is a sign-problem-free classical auxiliary-field formulation that combines a Holstein-type lattice coupling with a spin-fermion interaction, while retaining quantum itinerant electrons. In the formulation used to study its phase transition, the model is obtained from a Hubbard-Holstein starting point by a mean-field approximation, and its defining feature is that both the lattice displacement field and the local spin field are treated classically. In the adiabatic limit, the lattice field is static rather than a quantum phonon operator, and the local spins are classical unit vectors. The resulting model is designed to retain competing microscopic interactions—specifically antiferromagnetic and charge-order tendencies—while making large-scale Monte Carlo sampling feasible (Li, 7 Sep 2025).

1. Definition and Hamiltonian

The starting point is a Hubbard-Holstein-type Hamiltonian written as

$$H=H_0+H_{\mathrm{e\text{-}ph}}+H_{\mathrm{e\text{-}e}}+H_{\mathrm{lattice}},$$

with

$$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$

$$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$

$$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$

and

$$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$

Here $t$ is the nearest-neighbor hopping, $\mu$ the chemical potential, $g$ the electron-phonon coupling strength, $U$ the onsite Coulomb repulsion, $M$ the lattice mass, $$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$0 the elastic constant, $$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$1 the lattice displacement operator, and $$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$2 the fermion density operator. In the reported simulations, $$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$3 and $$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$4, and $$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$5 is adjusted to keep the electron density at 1 (Li, 7 Sep 2025).

To eliminate the sign problem of the full interacting fermion model, the mean-field approximation replaces the Hubbard interaction by a spin-fermion coupling and yields the classical Holstein-spin-fermion Hamiltonian

$$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$6

with

$$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$7

where $$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$8 is a classical local spin vector with fixed amplitude 1, $$H_0=-t\sum_{\langle i, j \rangle,\sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} - \mu\sum_{i,\sigma} \hat{n}_{i,\sigma},$$9 is the quantum electron spin operator, and $$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$0 is the spin-fermion coupling. The benchmark study sets $$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$1 (Li, 7 Sep 2025).

In the adiabatic limit $$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$2, the lattice kinetic term drops out and the lattice sector becomes

$$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$3

with $$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$4 treated as a classical variable. The Monte Carlo auxiliary fields are then

$$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$5

This formulation isolates a hybrid problem: electrons remain quantum mechanical, but the spin and lattice backgrounds are classical (Li, 7 Sep 2025).

2. Classical character and relation to the Holstein family

The model is “classical” in two distinct senses. First, the lattice field is classical: in the adiabatic limit, $$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$6 is not a dynamical quantum phonon operator but a classical displacement. Second, the local spins are classical: $$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$7 are classical unit vectors rather than quantum spin operators. This distinguishes the model from the full Hubbard-Holstein problem, where both the electron-electron and electron-phonon sectors are quantum, and from more conventional spin-fermion models without an explicit Holstein lattice sector (Li, 7 Sep 2025).

Its Holstein component is inherited from the standard local density-displacement structure. In the quantum Holstein model, the Hamiltonian is written as

$$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$8

with

$$H_{\mathrm{e\text{-}ph}}=g\sum_{i,\sigma} \hat{x}_i \left(\hat{n}_{i,\sigma}-\frac{1}{2}\right),$$9

After integrating out the phonons, the interaction becomes retarded and purely fermionic. That construction is explicitly quantum and retains full imaginary-time phonon dynamics, so it is not a classical Holstein-spin-fermion model in the static-background sense (Weber et al., 2016).

A related but distinct generalization appears in trapped-ion realizations of spin-Holstein physics. There, an effective spin model coupled to phonons can be mapped by a Jordan–Wigner transformation to a fermionic Holstein-like problem. The analogy is direct—$$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$0 becomes fermion density, spin-flip terms become hopping, and spin-phonon coupling becomes electron-phonon coupling—but the effective model contains long-range $$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$1 exchange, dispersive collective phonons, and nonlocal couplings $$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$2. It is therefore described as a generalized Holstein model rather than the textbook local one (Knörzer et al., 2021).

3. Competing interactions and phase structure

The central physical content of the classical Holstein-spin-fermion model is the competition between spin-driven antiferromagnetism and lattice-driven charge ordering. The spin-spin or AFM-favoring tendencies originate from the spin-fermion term, while the electron-lattice coupling favors a charge-density-wave state. This competition drives an AFM-to-CDW transition (Li, 7 Sep 2025).

At $$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$3, the reported numerical regime is divided as follows: AFM fluctuations occur for $$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$4, while CDW fluctuations occur for $$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$5. Accordingly, $$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$6 is referred to as the AFM region and $$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$7 as the CDW region (Li, 7 Sep 2025). The microscopic interpretation is explicit. In the AFM region, the electron-phonon term slightly raises the energy because it induces spin fluctuations away from the AFM ground state. In the CDW region, the coupling to the displacement field strongly lowers the energy by stabilizing a checkerboard charge modulation. The CDW picture is described in terms of atoms displaced by $$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$8, such that the density becomes 0 on one sublattice and 2 on the other, lowering the energy by

$$H_{\mathrm{e\text{-}e}}=U\sum_{i} \hat{n}_{i,\uparrow} \hat{n}_{i,\downarrow},$$9

This places the transition in the broader class of interacting-order competition problems (Li, 7 Sep 2025).

The broader Holstein literature provides a useful comparison. In one-dimensional half-filled quantum Holstein models, the Peierls transition is accompanied by phonon softening at $$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$0, and in the adiabatic regime this is consistent with a soft-mode transition. The ordered state is a Peierls insulating state with long-range charge-density-wave order at half filling (Weber et al., 2016). A plausible implication is that the classical Holstein-spin-fermion model retains the charge-ordering logic of the Holstein problem while replacing quantum phonon dynamics by static distortions and adding an explicit spin background.

The same charge-order tendency also appears in multicomponent generalizations. In the SU($$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$1) Holstein model at half filling, the low-temperature phase is an insulating CDW in which empty sites alternate with sites with $$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$2 particles; for $$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$3, the pattern is alternating empty and triply occupied sites. That ordered phase is treated as breaking a discrete $$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$4 symmetry and is analyzed using 2D Ising finite-size scaling (Feng et al., 23 Jun 2026). This suggests that classical or semiclassical Holstein-type charge order remains robust when the fermionic sector is generalized beyond the usual spin-$$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$5 case.

4. Monte Carlo formulation and self-learning acceleration

For a proposed field update $$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$6, the Monte Carlo acceptance probability is

$$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$7

The free energy is written as

$$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$8

with

$$H_{\mathrm{lattice}}=\sum_i \left[\frac{1}{2M}\left(\frac{d\hat{x}_i}{dt}\right)^2 + \frac{K^2}{2} \hat{x}_i^2\right].$$9

Here $t$0 are the electronic eigenenergies for a given auxiliary-field configuration, obtained by exact diagonalization with LAPACK’s zgeev routine. This exact-diagonalization step is the computational bottleneck: it scales as $t$1, and because it must be repeated through the update sequence, a standard QMC epoch scales as $t$2 (Li, 7 Sep 2025).

The self-learning quantum Monte Carlo strategy replaces the exact free-energy evaluation by a machine-learning surrogate $t$3. Trial updates are accepted with

$t$4

After a full epoch, a cumulative correction restores consistency through

$t$5

where $t$6 are the predicted free-energy changes accumulated over accepted trial updates. This reduces the per-epoch cost to $t$7 (Li, 7 Sep 2025).

Two surrogate classes are compared. The linear-regression model takes the form

$t$8

where the effective interactions are included up to the 7th nearest neighbor and the model has 24 terms excluding the bias. The onsite term is

$t$9

The neural-network model supplements this with two-point correlation functions and the linear-regression feature set. At $\mu$0 and $\mu$1, the neural network reduces the MSE to about $\mu$2, compared with $\mu$3 for linear regression (Li, 7 Sep 2025).

The simulations also employ annealing, cooling the system gradually from $\mu$4 to $\mu$5 with decay rate 0.9 and 2000 epochs per temperature. In this setting, both linear regression and neural-network surrogates reproduce the AFM-to-CDW transition, while the neural network provides improved free-energy prediction and better cumulative acceptance (Li, 7 Sep 2025).

5. Diagnostics, sampling efficiency, and finite-size effects

Phase identification is performed with magnetic and charge susceptibilities at momentum point $\mu$6,

$\mu$7

and

$\mu$8

Using these observables, the self-learning results from both surrogate classes agree well with standard QMC and recover AFM fluctuations at small $\mu$9 and CDW fluctuations at large $g$0 (Li, 7 Sep 2025).

The detailed energy decomposition from the linear-regression model clarifies the transition. The total free energy changes smoothly across the crossover region; spin-spin interaction energies decrease rapidly in the AFM region; lattice-lattice terms become important and drop in the CDW region; and spin-lattice terms are smaller and change sign across the transition. A specific reported feature is that the second-nearest-neighbor spin-spin interaction vanishes in the CDW region, while nearest-neighbor spin correlations remain finite. The CDW state also suppresses some longer-range lattice correlations (Li, 7 Sep 2025).

The principal numerical limitation is a degradation of sampling efficiency near the AFM-CDW transition and on larger lattices. The cumulative update probability $g$1 is reasonably good in the AFM and CDW regimes far from criticality, but near $g$2 it becomes much smaller. For the linear-regression model at $g$3, $g$4 decreases from about 0.6 to 0.1 as $g$5 grows from 10 to 30. The neural network improves this behavior, but $g$6 still declines with size (Li, 7 Sep 2025).

The stated interpretation links Monte Carlo efficiency directly to surrogate accuracy. Near the AFM-CDW transition, simultaneous spin and lattice fluctuations increase the MSE and RMSE of the free-energy predictor. Since the acceptance depends on $g$7, the effective control parameter is $g$8. The finite-size effect is described in terms of the decreasing energy gap between the ground state and low-lying excited states: when that gap becomes comparable to or smaller than the machine-learning error, the sampled state is no longer close to a pure ground state but becomes a mixture of ground and excited states. The qualitative criterion reported is

$g$9

for some model-dependent exponent $U$0 (Li, 7 Sep 2025).

A related Monte Carlo issue appears in determinant QMC for the Holstein model, where the configuration space develops extremely long autocorrelation times despite being sign-problem free. A Wang–Landau flat-histogram sampling in configuration-weight space was introduced to reduce those autocorrelation times, with the key tradeoff that lower-probability configurations are sampled more often at some loss of sampling efficiency per step (Yao et al., 2021). This suggests that the classical Holstein-spin-fermion model belongs to a broader class of Holstein-type systems where the dominant obstacle is mixing time rather than the sign problem itself.

6. Related formulations and generalizations

Several neighboring models clarify what is specific to the classical Holstein-spin-fermion construction. The trapped-ion spin-Holstein model provides a generalized Holstein simulator in which effective spins, phonons, and laser-induced couplings emulate electron-phonon physics. After Jordan–Wigner transformation, it becomes a fermionic Holstein-like model with long-range hopping or exchange, dispersive phonons, and nonlocal couplings. Its phase diagram contains CDW order in the stiff regime, superconducting or pairing correlations in the soft regime, phase separation at strong coupling, and a mixed pCDW phase at quarter filling (Knörzer et al., 2021). These ingredients are absent from the classical Holstein-spin-fermion benchmark, whose purpose is instead to isolate competing AFM and CDW tendencies in a sign-problem-free setting.

A different generalization is the spin-dependent electron-transfer model motivated by radical-pair chemistry. There, the Holstein electron-transfer framework is extended by explicit electron spin, magnetic-field orientation, and hyperfine coupling to nuclear spins. The total Hamiltonian is $U$1, and the key physical conclusion is that the triplet-state reaction rate depends on the direction of the magnetic field, whereas the singlet-state rate does not (Yang et al., 2011). This is a spinful Holstein generalization, but it is not a lattice spin-fermion model and does not involve classical local-spin backgrounds.

The SU($U$2) Holstein model extends the fermionic sector from two species to $U$3 flavors coupled identically to a single local phonon mode per site. At half filling, defined by $U$4, it exhibits an insulating CDW phase in which empty sites alternate with fully occupied sites containing all $U$5 fermions. For $U$6, the reported critical temperature can be as high as twice the maximum attainable for $U$7, and the large-$U$8 trend approaches $U$9 at fixed representative parameters (Feng et al., 23 Jun 2026). This broader family emphasizes that the charge-order tendency of Holstein systems can be strengthened by increasing the number of fermionic components.

The quantum Holstein literature also contributes formal tools relevant to classical or semiclassical formulations. In continuous-time interaction-expansion Monte Carlo, bosonic observables can be reconstructed from fermionic correlation functions using generating functionals, and improved estimators can be built directly from the vertex distribution for total energy, phonon propagators, and fidelity susceptibility (Weber et al., 2016). A plausible implication is that classical Holstein-spin-fermion studies can draw on this formalism for diagnostics even when the bosonic sector is treated statically rather than quantum mechanically.