Thermal evolution of the Schwinger model with matrix product operators

We demonstrate the suitability of tensor network techniques for describing the thermal evolution of lattice gauge theories. As a benchmark case, we have studied the temperature dependence of the chiral condensate in the Schwinger model, using matrix product operators to approximate the thermal equilibrium states for finite system sizes with nonzero lattice spacings. We show how these techniques allow for reliable extrapolations in bond dimension, step width, system size and lattice spacing, and for a systematic estimation and control of all error sources involved in the calculation. The reached values of the lattice spacing are small enough to capture the most challenging region of high temperatures and the final results are consistent with the analytical prediction by Sachs and Wipf over a broad temperature range.

Figure 1

Left: convergence of condensate value with bond dimension, $D$, for $g\beta =0.4$, $x=6.25$, $N=40$ and $\delta =5\times {10}^{-4}$. Shown is the deviation with respect to the final value, $\mathrm{\Sigma }=0.01508769$. The inset shows additionally the results for $\delta =7\times {10}^{-4}$ (circles) and ${10}^{-3}$ (squares) on a common scale, to compare the magnitude of $D$ and $\delta$ errors. Right: linear $\delta$ extrapolation for the same $x$ and $g\beta$ and several system sizes.

Figure 2

Infinite volume (left) and continuum (right) extrapolations of the condensate, $\mathrm{\Sigma }$, at $g\beta =4.0$. In the right panel, extrapolations using $f_1$ (solid blue) and $f_2$ (dash-dotted red curve) are shown.

Figure 3

Condensate values with a truncation $L_{\mathrm{cut}}$ of the maximum electric flux per link. Left: finite volume extrapolation for $x=55$, $g\beta =0.1$ from $L_{\mathrm{cut}}=5$ (left-pointing red triangles), 8, 10, 12 and 15 (different shapes and colors, indistinguishable in the plot). The inset shows the extrapolation in Trotter parameter, $\delta$, for $N=170$ and $L_{\mathrm{cut}}=8-15$. Right: continuum limit for $g\beta =0.1$ (left-pointing purple triangles) and 0.2 (green diamonds). The corresponding exact results from [15] are indicated on the vertical axis. The inset shows explicitly the dependence on the cutoff for these $g\beta$ values at $x=55$.

Figure 4

Temperature dependence of the chiral condensate from data with $x\le 65$ and exact gauge sector (red triangles), compared to [15] (solid line). For the lowest $g\beta$ (right), the results deviate from the exact ones. Using the $L_{\mathrm{cut}}=10$ truncation, we reach smaller lattice spacings and recover the consistency with the analytical results (blue circles).

Figure 5

Schematic representation of the TN operations. Left: applying the first exponential for the first step of imaginary time evolution on the identity operator (which is an MPO with $D=1$). Center: vectorizing this first step allows us to work with the MPS formalism. Right: after each application of the exponential, the result is approximated by an MPS with bounded bond dimension.