Tensor renormalization group study of the three-dimensional O(2) model

We calculate thermodynamic potentials and their derivatives for the three-dimensional $O(2)$ model using tensor-network methods to investigate the well-known second-order phase transition. We also consider the model at nonzero chemical potential to study the Silver Blaze phenomenon, which is related to the particle number density at zero temperature. Furthermore, the temperature dependence of the number density is explored using asymmetric lattices. Our results for both zero and nonzero magnetic field, temperature, and chemical potential are consistent with those obtained using other methods.

Figure 1

Representation of the initial tensor and its decomposition into triads for a three-dimensional system. The contracted indices are shown by red dashed lines.

Figure 2

The internal energy obtained using triad TRG with $D=50$, and finite differences with $\mathrm{\Delta }\beta =0.02$, agrees with the MC results on a lattice volume of ${32}^3$.

Figure 3

Specific heat capacity as a function of $\beta$ for a ${32}^3$ lattice volume. The triad data (orange) are computed with $D=72$ using a second-order finite difference of $\ln Z$ with step size $\mathrm{\Delta }\beta =0.01$. The HOTRG data (blue) used $D=15$ and are computed with a stabilized second-order finite-difference scheme with $\mathrm{\Delta }\beta ={10}^{-6}$. The peak of $C_{\mathrm{v}}$ suggests that the critical coupling is between $\beta =0.45$ and $\beta =0.46$. For reference, we show the infinite-volume MC result ${\beta }_{\mathrm{c}}=0.454165$ from [26] by the black dashed line.

Figure 4

Magnetization of the $O(2)$ model for different external magnetic fields. We see that for a sufficiently small symmetry-breaking field, the magnetization rises sharply around ${\beta }_c$. These results are obtained on a lattice of volume $(2^{13}{)}^3$ with $D=30$.

Figure 5

We compare the results obtained using triad TRG (symbols) with $D=50$ and worm algorithm (smooth lines) for the dependence of $\rho$ on $\mu$ for some values of the coupling $\beta$ on both sides (phases) of the critical coupling on a lattice of size ${64}^3$. We mark the threshold value ${\mu }_c$, which is related to the mass gap. It is clear that the mass gap decreases (and correlation length increases) as we go from $\beta =0.42$ to $\beta =0.45$ and would go to the CFT limit as $\beta \to {\beta }_c\approx 0.45417$.

Figure 6

We use the HOTRG with $D=13$, improved contraction order, and stabilized finite differences to compute the particle density $\rho$ for a ${64}^2\times N_t$ lattice with $N_t=2$, 4, 8, 16 (symbols) at $\beta =0.44$ and compare to the results obtained using the worm algorithm (smooth lines). There is clear indication that as we move towards zero temperature, the behavior we see in Fig. 5 starts to emerge.

Figure 7

Dependence of $\ln Z/V$ on the bond dimension $D$ on a lattice of volume $(2^{15}{)}^3$ at $\beta =0.45417$ for the three-dimensional $O(2)$ model obtained using the triad method. The red line shows the result of a linear fit using all data points.

Figure 8

Dependence of $\ln Z/V$ on $D$ on a $(2^{15}{)}^3$ lattice at $T=4.5115$ for the three-dimensional classical Ising model using triads. The red line shows the result of a quadratic fit using all data points. The current best estimate of $T_c$ is 4.5115247 [38].