Tensor renormalization group study of the non-Abelian Higgs model in two dimensions

We study the $SU(2)$ gauge-Higgs model in two Euclidean dimensions using the tensor renormalization group (TRG) approach. We derive a tensor formulation for this model in the unitary gauge and compare the expectation values of different observables between TRG and Monte Carlo simulations finding excellent agreement between the two methods. In practice we find the TRG method to be far superior to Monte Carlo simulation for calculations of the Polyakov loop correlation function which is used to extract the static quark potential.

Figure 1

An illustration of two $A$ tensors. Here the dashed lines represent the original links of the lattice. The $A$ tensors have two indices, each of which is a product state of three indices. The subscripts “a,” “b,” “l,” and “r” denote the relative position of the index to the link the tensor is attached to, corresponding to “above,” “below,” “left,” and “right.”

Figure 2

An illustration of a $B$ tensor. The black lines on the outside are Kronecker deltas, and the brown cross is a diagonal tensor which forces all four $r$s to be identical. The dashed lines are the original links of the lattice. The subscripts “a,” “b,” “l,” and “r” denote the relative position of the index with respect to the center of the plaquette, corresponding to “above,” “below,” “left,” and “right.”

Figure 3

Left: A graphical interpretation of the matrix factorization of the $A$ tensor into the product of two identical matrices, $A=L{L}^T$. The dashed line in between represents an intermediate sum over states. Right: The fundamental tensor $T_{ijkl}$ is built by contracting four $L$ matrices together with a single $B$ tensor at a plaquette. Forming this tensor sums over all the $r$ and $m$ variables on the lattice, and instead one is left with the sums over the intermediate indices from the $L$ matrices. See Eq. (29) for details.

Figure 4

The mass gap density as a function of $\kappa$ while taking the continuum limit. Here $\beta /N_s^2=0.01$ is held fixed as the volume is increased. We see the gap approaches the correct $\kappa =0$ value of $3/2$ as the continuum limit is taken.

Figure 5

Calculation of the average trace of a link variable, Eq. (43), using the TRG and comparing with MC. Here the continuum limit is approached by keeping the ratio $\beta /V$ fixed at 0.01 while increasing the volume. The colored markers are from the TRG calculations which were done with a local state space truncation at $r_{\max }=1$ and $D_{\mathrm{bond}}=50$. The Monte Carlo data are the black hollow markers.

Figure 6

The gauge-link susceptibility, Eq. (44), as a function of $\kappa$ as one approaches the continuum-limit. Here $\beta /V=0.01$ is held fixed as the volume is increased. We see convergence as the volume gets sufficiently large indicating a crossover behavior. The colored markers are from the TRG calculations which were done with a local state space truncation at $r_{\max }=1$ and $D_{\mathrm{bond}}=50$. The Monte Carlo data are the black hollow markers.

Figure 7

The average plaquette, Eq. (42), as a function of $\kappa$ as one approaches the continuum limit. Here $\beta /V=0.01$ is held fixed as the volume is increased. The colored markers are from the TRG calculations which were done with a local state space truncation at $r_{\max }=1$ and $D_{\mathrm{bond}}=50$. The Monte Carlo data are the black hollow markers.

Figure 8

The relative error of the Polyakov loop correlation function between the TRG and MC. This is for a larger value of $\kappa =2$, on a $8\times 1$ lattice with a separation of $d=4$ between the two loops. We see the TRG solution converges slowly to the MC value at larger final bond dimension. The TRG calculations were done with $r_{\max }=1$, and $\beta /V=0.01$ was held fixed.

Figure 9

The static charge potential, $\mathcal{V}$, from the Polyakov correlation function in the continuum limit. Here we took $\beta /V=0.01$ fixed as we increased the size of the lattice. These runs were done with the gauge-matter coupling $\kappa =1/2$, which is well in the confining regime of the model. This is characterized by a dominant linear potential across the whole system.

Figure 10

The static charge potential, $\mathcal{V}$, from the Polyakov correlation function in the continuum limit. Here we took $\beta /V=0.01$ fixed and varied the size of the lattice. The gauge-matter coupling $\kappa$ is 2, which is in the Higgs regime. This is marked by string breaking at small distances between the static charges.

Figure 11

The Polyakov loop calculated using the TRG compared with the exact analytic calculation for the case of infinite gauge coupling ($\beta =0$). The $x$-axis is the gauge-matter coupling $\kappa$, and the $y$-axis is the rescaled data compared with the exact dashed line from Eq. (53), with $I_1$, and $I_2$ being the modified Bessel functions. A few system sizes are plotted indicating a complete factorization of the Polyakov loop in this limit as expected.