Tensor renormalization group study of classical $XY$ model on the square lattice

Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the thermodynamic properties of the continuous $XY$ model on the square lattice. The temperature dependence of the free energy, the internal energy, and the specific heat agree with the Monte Carlo calculations. From the field dependence of the magnetic susceptibility, we find the Kosterlitz-Thouless transition temperature to be $0.8921\left(19\right)$, consistent with the Monte Carlo as well as the high temperature series expansion results. At the transition temperature, the critical exponent $\delta$ is estimated as $14.5$, close to the analytic value by Kosterlitz.

Figure 1

Variation of the modified Bessel function of the first kind $I_n\left(\beta \right)$ at three different temperatures on the logarithmic and linear (the inset) scale.

Figure 2

Temperature dependence of the free energy (dotted black line) and the internal energy (solid blue line) obtained by the HOTRG with $D=40$. The free energy (black squares) and the internal energy (blue circles) obtained by the Monte Carlo simulation on the $N=256\times 256$ lattice are also shown for comparison.

Figure 3

The entropy (dotted black line) and the specific heat (solid blue line) versus temperature from the HOTRG with $D=40$. Correspondingly, the black squares and the blue circles are from the Monte Carlo simulations with $N=256\times 256$ for comparison.

Figure 4

Magnetization versus temperature with $D=40$ under two applied magnetic fields $h=1\times {10}^{-4}$ (black dots) and $2\times {10}^{-4}$ (red squares). The magnetic susceptibility (blue circles) is obtained from the two magnetization curves using Eq. (9).

Figure 5

The peak temperature of the magnetic susceptibility versus the external magnetic field for $D=40$. The curve is a power fit to the peak temperature with Eq. (10). The critical temperature is extrapolated to be $0.8921\left(19\right)$.

Figure 6

Power-law fit for the field dependence of the magnetization at $T_c$ with formula (11).