Efficient Evaluation of Partition Functions of Inhomogeneous Many-Body Spin Systems

We present a numerical method to evaluate partition functions and associated correlation functions of inhomogeneous 2D classical spin systems and 1D quantum spin systems. The method is scalable and has a controlled error. We illustrate the algorithm by calculating the finite-temperature properties of bosonic particles in 1D optical lattices, as realized in current experiments.

Figure 1

(a) Free energy of the 2D Ising model ($J=1$) on a $50\times 50$ lattice. (b) and (c) Density and (quasi)-momentum distribution in the Tonks-Girardeau gas limit, plotted for $\beta J=1$, $L=40$, $N=21$, and $V_0/J=0.034$. In all parts, the numerical results for $\tilde{D}=2$ ($\tilde{D}=8$) are represented by dots (crosses) and the exact solution is illustrated by the solid line. From the insets, the error of the numerical results can be gathered. For comparison, in (a) the exact solution for the infinite-lattice case is also plotted (dashed line).

Figure 2

Density and (quasi)-momentum distributions for interaction strengths $U/J=4$ and 8. Here, $\beta J=1$, $L=40$, $N=21$, and $M=10$. Numerical results were obtained for $q=2$ (plus-signs), $q=3$ (crosses), and $q=4$ (solid line). For comparison, the distributions for $U/J=0$ (dotted lines) and $U/J\to \infty$ (dash-dotted lines) are also included.

Figure 3

FWHM of the (quasi)-momentum distribution as a function of $U/J$, calculated for temperatures $\beta J=0.5$ (plus-sign),$\beta J=1$ (crosses), and $\beta J=2$ (dots). The corresponding (quasi)-momentum distributions for $U/J=2$ and $U/J=8$ are illustrated in the plots at the right-hand side.