Approaching the Kosterlitz-Thouless transition for the classical $XY$ model with tensor networks

We apply variational tensor-network methods for simulating the Kosterlitz-Thouless phase transition in the classical two-dimensional $XY$ model. In particular, using uniform matrix product states (MPS) with non-Abelian $\mathrm{O}\left(2\right)$ symmetry, we compute the universal drop in the spin stiffness at the critical point. In the critical low-temperature regime, we focus on the MPS entanglement spectrum to characterize the Luttinger-liquid phase. In the high-temperature phase, we confirm the exponential divergence of the correlation length and estimate the critical temperature with high precision. Our MPS approach can be used to study generic two-dimensional phase transitions with continuous symmetries.

Figure 1

Free energy per site of the $XY$ model ($h=0$) as obtained from variational MPS simulations with bond dimension $D=30$ and without using any symmetries on the virtual legs.

Figure 2

Observables from MPS simulations with bond dimension $D=30$ (no virtual symmetries). In (a) we plot the order parameter, showing significant symmetry breaking in the critical region; the inset shows that this value (for $T=0.7$) of the order parameter decreases very slowly with the bond dimension. In (b) we plot the internal energy, showing no sign of a phase transition; here, the inset shows good convergence even for small bond dimensions ($T=0.7$). In (c) we plot the entropy per site, which is easily evaluated from the free and internal energy; we have renormalized the entropy as $s\left(\beta \right)\to s\left(\beta \right)-s\left(\infty \right)$, with $s\left(\infty \right)=log\left(\frac{1}{2\pi }\right)$, such that the zero-temperature limit yields a zero entropy and the entropy is positive everywhere.

Figure 3

The spin stiffness as a function of temperature for a set of different values of the magnetic field: $h={10}^{-3}$ (green), $h={10}^{-4}$ (purple), $h={10}^{-5}$ (yellow), $h={10}^{-6}$ (red), $h={10}^{-7}$ (blue). The solid lines were computed with a bond dimension $D=150$, and the dashed lines are $D=90$. We have also plotted the straight line $\frac{2T}{\pi }$, which is known to intersect the curve for the critical temperature [Eq. (1)]; for $h={10}^{-7}$ and $D=150$ we find an intersection at $T=0.899$.

Figure 4

The $XY$ model at $T=0.7$ as a Luttinger liquid. In (a) we have plotted the dispersion relation of the transfer matrix as defined by Eq. (7); the dashed line is a straight line with a slope equal to one, showing that we find a Luttinger velocity exactly equal to one. In (b) we have plotted the correlation length vs the entanglement entropy obtained in MPS approximations for the fixed point with increasing bond dimension ($D=50:25:175$); the fit is made from the three last points according to the form Eq. (8), yielding a value for the central charge $c=1.003$. In (c) we have plotted the response in the free energy and the expectation value for $Q_i$ with a quadratic and linear fit, respectively. The same value $\kappa \approx 1.10953$ is found from both fits.

Figure 5

Entanglement spectrum of the fixed-point MPS of the $XY$ transfer matrix at $T=0.8$. We have imposed the full $\mathrm{O}\left(2\right)$ symmetry on the MPS tensor, which implies that the entanglement spectrum is labeled by the irreps on the virtual bonds: two one-dimensional irreps with charge $q=0$ (blue, red), and two-dimensional irreps with charges $q=1,2,\cdots$. In (a) we plot the bare entanglement spectrum, and we find that the envelope of the entanglement spectrum follows a quadratic form (striped line). In (b) we have shifted the different sectors such that the lowest value is zero; this produces a nice free-boson boundary CFT spectrum. The inset provides an estimate for the Luttinger parameter (obtained from the rescaling of the spectrum) as a function of the inverse logarithm of the MPS correlation length.

Figure 6

The entanglement spectrum of the fixed-point MPS at $T=1.216$, where we impose $\mathrm{O}\left(2\right)$ invariance on the virtual legs (left) and without explicit symmetries (right). On the left, we label the different $\mathrm{O}\left(2\right)$ representations as follows: $n=0$ (blue crosses), $q=\pm 1$ (red), $q\pm 2$ (orange), and $q=\pm 3$ (purple), whereas on the right we do not have any labeling; we only show the Schmidt values above ${10}^{-4}$. The perfect correspondence of the entanglement spectra with and without explicit symmetries on the virtual legs shows that the MPS representation for the fixed point only contains integer representations of $\mathrm{O}\left(2\right)$.

Figure 7

In (a) we plot the extrapolation procedure for $T=0.93$, yielding a value for the correlation length $\xi =1463\left(4\right)$; the maximal dimension in each block was set at $D_{\max }=512$, yielding a total MPS bond dimension of $D=3754$. In (b) we plot the extrapolated correlation lengths as a function of temperature. We fit this to the KT form [Eq. (10)] (red line), yielding a value for the critical temperature $T_c=0.8930\left(1\right)$.

Figure 8

The spectrum $\omega \left(p\right)$ of the transfer matrix at temperature $T=1.3$. The blue line is the elementary excitation branch with charge $q=\pm 1$, and the red line is the edge of the two-particle continuum. The yellow dots are excitation energies that fall below the continuum edge, signaling a bound state in the $q=0$ sector.