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arXiv:2604.02769

Semiclassical representation of the Hubbard model

Yuki Yamasaki, Hidemaro Suwa, Cristian D. Batista, Shintaro Hoshino

Published Apr 3, 2026• 0 citations
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Abstract

By revisiting the path-integral formulation of the Hubbard model, we propose a theoretical approach based on a semiclassical approximation employing an unconventional coherent-state representation. Within this framework, a subset of the dynamical variables is treated as static, yielding a nonperturbative scheme that is applicable at finite temperature, incorporates intersite correlations, and can be naturally extended to multiorbital systems. We assess the validity of the approximation by comparing its results with exact solutions for one- and two-site systems, focusing in particular on the particle number, double occupancy, hopping amplitude, and spin correlations, and find that the present approach qualitatively reproduces the exact behavior. Quantitatively, deviations arise, which is associated with the continuum (non-discretized) character of the underlying density of states. Furthermore, we derive the exact transformation associated with the coherent-state construction, thereby providing additional insight into the representation of the Hubbard model.

DOI
10.48550/arxiv.2604.02769

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By revisiting the path-integral formulation of the Hubbard model, we propose a theoretical approach based on a semiclassical approximation employing an unconventional coherent-state representation. Within this framework, a subset of the dynamical variables is treated as static, yielding a nonperturbative scheme that is applicable at finite temperature, incorporates intersite correlations, and can be naturally extended to multiorbital systems. We assess the validity of the approximation by comparing its results with exact solutions for one- and two-site systems, focusing in particular on the particle number, double occupancy, hopping amplitude, and spin correlations, and find that the present approach qualitatively reproduces the exact behavior. Quantitatively, deviations arise, which is associated with the continuum (non-discretized) character of the underlying density of states. Furthermore, we derive the exact transformation associated with the coherent-state construction, thereby providing additional insight into the representation of the Hubbard model.
We use a new formulation of the semiclassical coherent state propagator to derive and evaluate several different approximate representations of electron dynamics. For each representation we examine: (1) its ability to treat quantum effects and electron correlation, (2) its expected scaling with system size, and (3) the types of systems for which it can be used. We also apply two of the methods to a pair of model problems, namely the minimal basis electron dynamics in H2 and the magnetization dynamics in a cluster model of the Kagome lattice, in order to verify the feasibility of these approaches for realistic systems. Based on all these criteria, we find that the representation that takes the electron spins as the classical variables is particularly promising for the quantitative and qualitative description of large systems.
The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent-state path integral, leads in the large connectivity limit to the mean-field treatment of Fisher et al. [Phys. Rev. B 40, 546 (1989)] at the leading order, and to the bosonic dynamical mean field theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B 77, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary-time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite-dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.
In strongly correlated electron systems the constraint which prohibits the double electron occupation at local sites can be realized by either the infinite Coulomb interaction or the correlated hopping interaction described by the Hubbard operators, but they both render the conventional field theory inapplicable. Relaxing such the constraint leads to a class of correlated hopping models based on the deformed Hubbard operators which smoothly interpolate the locally free and strong coupling limits by a tunable interaction parameter 0≤λ≤10\leq λ\leq 10≤λ≤1. Here we propose a coherent state path integral approach appropriate to the deformed Hubbard operators for {\it arbitrary} λλλ. It is shown that this model system exhibits the correlated Fermi liquid behavior characterized by the enhanced Wilson ratio for all λλλ. It is further found that in the presence of on-site Coulomb interaction a finite Mott gap appears between the upper and lower Hubbard bands, with the upper band spectral weight being heavily reduced by λλλ. Our approach stands in general spatial dimensions and reveals an unexpected interplay between the correlated hopping and the Coulomb repulsion.
We consider the many-body spectra of interacting bosonic quantum fields on a lattice in the semiclassical limit of large particle number N. We show that the many-body density of states can be expressed as a coherent sum over oscillating long-wavelength contributions given by periodic, nonperturbative solutions of the, typically nonlinear, wave equation of the classical (mean-field) limit. To this end, we construct the semiclassical approximation for both the smooth and oscillatory parts of the many-body density of states in terms of a trace formula starting from the exact path integral form of the propagator between many-body quadrature states. We therefore avoid the use of a complexified classical limit characteristic of the coherent state representation. While quantum effects such as vacuum fluctuations and gauge invariance are exactly accounted for, our semiclassical approach captures quantum interference and therefore is valid well beyond the Ehrenfest time where naive quantum-classical correspondence breaks down. Remarkably, due to a special feature of harmonic systems with incommensurable frequencies, our formulas are generically valid also in the free-field case of noninteracting bosons.
We present the exact solution of the one-dimensional extended Hubbard model in the atomic limit within the Green's function and equations of motion formalism. We provide a comprehensive and systematic analysis of the model by considering all the relevant response and correlation functions as well as thermodynamic quantities in the whole parameters space. At zero temperature we identify four phases in the plane (U,n) ( U is the on-site potential and n is the filling) and relative phase transitions as well as different types of charge ordering. These features are endorsed by investigating at T=0 the chemical potential and pertinent local correlators, the particle and double occupancy correlation functions, the entropy, and by studying the behavior in the limit T-->0 of the charge and spin susceptibilities. A detailed study of the thermodynamic quantities is also presented at finite temperature. This study evidences that a finite-range order persists for a wide range of the temperature, as shown by the behavior of the correlation functions and by the two-peak structure exhibited by the charge susceptibility and by the entropy. Moreover, the equations of motion formalism, together with the use of composite operators, allows us to exactly determine the set of elementary excitations. As a result, the density of states can be determined and a detailed analysis of the specific heat allows for identifying the excitations and for ascribing its two-peak structure to a redistribution of the charge density.
We present a formalism for simulating quantum dynamics of lattice spin-1 systems by first introducing local hidden variables and then doing semiclassical (truncated Wigner) approximation in the extended phase space. In this way, we exactly take into account the local on-site Hamiltonian and approximately treat spin-spin interactions. In particular, we represent each spin with eight classical SU(3) variables. Three of them represent the usual spin components and five others are hidden variables representing local spin-spin correlations. We compare our formalism with exact quantum dynamics of fully connected spin systems and find very good agreement. As an application, we discuss quench dynamics of a Bose-Hubbard model near the superfluid-insulator transition for a 3D lattice system consisting of 1000 sites.