Semiclassical representation of the Hubbard model

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.

Figures (5)

• Figure 1: Overview on the represenations of the Hubbard model considered in this paper.
• Figure 2: (Top row) Filling versus chemcal potential $\delta \mu = \mu - U/2$ for the (a,b) semiclassical and (c,d) exact results at temperatures $T=0.05, 1.0$. (Bottom row) Double occupancy versus filling for the (e,f) semiclassical and (g,h) exact results. The interaction parameters are chosen as (a,c,e,g) $U=0$ and (b,d,f,h) $U=8.0$.
• Figure 3: (Top row) Temperature dependence of the two-site spin correlation $\langle \bm S_1 \cdot \bm S_2 \rangle$ for several values of $U$, obtained by (a) the semiclassical method and (b) exact calculation. (Bottom row) Same as in the top row, but for the hopping amplitude: (c) semiclassical method and (d) exact calculation.
• Figure 4: Temperature depnendece of the spin correlation for semiclassical and quantum two-site spin models. The dotted line is a rescaled version of the semiclassical result.
• Figure 5: (Top row) Filling dependence of the two-site spin correlation at several temperatures, obtained by (a,b) the semiclassical method and (c,d) exact calculation. The interaction parameter is $U=0$ in (a,c) and $U=8$ in (b,d). (Middle row) Same as in the top row, but with the horizontal axis replaced by the chemical potential, defined as $\delta\mu = \mu - U/2$. Panels (e,f,g,h) correspond to panels (a,b,c,d), respectively. (Bottom row) Analogous to the top row, but with the vertical axis replaced by the hopping amplitude. The calculation methods and interaction strengths in panels (i,j,k,l) correspond to those in panels (a,b,c,d), respectively.