Hamiltonian simulation of the Schwinger model at finite temperature

Using matrix product operators, the Schwinger model is simulated in thermal equilibrium. The variational manifold of gauge-invariant matrix product operators is constructed to represent Gibbs states. As a first application, the chiral condensate in thermal equilibrium is computed, and agreement with earlier studies is found. Furthermore, as a new application, the Schwinger model is probed with a fractional charged static quark-antiquark pair separated infinitely far from each other. A critical temperature beyond which the string tension is exponentially suppressed is found and is in qualitative agreement with analytical studies in the strong coupling limit. Finally, the $CT$ symmetry breaking is investigated, and our results strongly suggest that the symmetry is restored at any nonzero temperature.

Figure 1

$m/g=0.25$, $x=200$, and $\alpha =0$. (a) $p_{\min }$ and $p_{\max }$ for $\epsilon ={10}^{-6}$ and $\epsilon =5\times {10}^{-6}$. Inset: maximum bond dimension over all the charge sectors. (b): Electric field per site $E_0(\beta ,x)$. Because $\alpha =0$, we should have $E_0(\beta ,x)=0$.

Figure 2

$m/g=0.5$, $x=100$, and $\alpha =0.25$. (a) $p_{\min }$ and $p_{\max }$ for $\epsilon ={10}^{-7}$, $d\beta =0.01$ and $\epsilon ={10}^{-6}$, $d\beta =0.05$. Inset: maximum bond dimension over all the charge sectors. (b): Electric field per site $E_{\alpha }(\beta ,x)/g$. Inset: 10-base logarithm of the difference of $E_{\alpha }(\beta ,x)/g$ between simulations with $\epsilon ={10}^{-7}$, $d\beta =0.01$ and $\epsilon ={10}^{-6}$, $d\beta =0.05$.

Figure 3

$m/g=0$. Our continuum result $\mathrm{\Sigma }(\beta )$ (blue) compared with the analytical result ${\mathrm{\Sigma }}_{\mathrm{SW}}(\beta )$ (magenta). (a) Zooming in on the interval $\beta g\in [0,2]$. (b) Results for $\beta g\in [0,10]$ and convergence toward the result at $\beta g=+\infty$ (red). Inset: zooming in on the interval $\beta g\in [7,10]$.

Figure 4

Subtracted chiral condensate ${\mathrm{\Sigma }}_{\text{sub}}(m/g)$ for $m/g\ne 0$ for different values of $m/g$. Our results (blue) are compared with the simulations of Bañuls et al. [47] (magenta). The red line shows the results at $\beta g=+\infty$ obtained in Ref. [32]. In the inset, we zoom in on the convergence toward this result for larger values of $\beta g$. (a): $m/g=0.125$. (b) $m/g=0.25$. (c) $m/g=0.5$. (d) $m/g=1$.

Figure 5

$x=100$. $\alpha =0.05$ (full line), $\alpha =0.1$ (dashed line), $\alpha =0.25$ (dotted line), and $\alpha =0.4$ (dashed dotted line). (a) ${\sigma }_{\alpha }(\beta ,x)/g^2{\alpha }^2$. (b) $E_{\alpha }(\beta ,x)/g\alpha$. (c) $\mathrm{\Delta }{\mathrm{\Sigma }}_{\alpha }(\beta ,x)/g{\alpha }^2$. (d) ${\mathcal{S}}_{\alpha }(\beta ,x)/g^2{\alpha }^2$. The stars in (a) and (b) are the values at $\beta g=+\infty$ for $\alpha =0.25$.

Figure 6

$x=100$. Logarithm of the string tension at larger temperatures $T=1/\beta g$ as a function of $T$. (a) $m/g=0.125$, $\alpha =0.05$ (full line), and $\alpha =0.1$ (dashed line). Convergence of our results when improving the precision $\epsilon$. (b) $m/g=0.125$. Logarithm of string tension for $\alpha =0.05$ (full line), $\alpha =0.1$ (dashed line), $\alpha =0.25$ (dotted line), and $\alpha =0.4$ (dashed dotted line). (c) Same as (b), now for $m/g=0.25$. (d) Same as (b), now for $m/g=1$. The error bars in (b), (c), and (d) for $\beta g\ge 2$ are obtained by comparing the simulation for $\epsilon ={10}^{-9}$ with the simulation for $\epsilon =2\times {10}^{-9}$.

Figure 7

$x=100$. $\alpha =1/2-\delta$ for different values of $\delta$. Left: $m/g=0.125$. (a) Electric field. (c) String tension. Right: $m/g=0.25$. (b) Electric field. (d) String tension.

Figure 8

$x=100$. $\alpha =1/2-\delta$ for different values of $\delta$. Left: $m/g=0.5$. (a) Electric field. (c) String tension. Right: $m/g=1$. (b) Electric field. (d) String tension.

Figure 9

$x=100$. $\exp (-2{\beta }_{1/2}m)$ as a function of $\delta$. (a) $m/g=0.5$. (b) $m/g=1$.

Figure 10

$m/g=0.25$. Simulations for different values of $(\epsilon ,d\beta )$. (a–b) Electric field which should be zero for $\alpha =0$. (a) $x=100$. (b) $x=600$. (c–d) Chiral condensate $\mathrm{\Sigma }(\beta ,x)$ with respect to its value for $(\epsilon ,d\beta )=({10}^{-6},0.05)$. (c) $x=100$. (d) $x=600$.

Figure 11

$m/g=0.25,\alpha =0$. Evolution of the maximum bond dimension $D_{\max }={\max }_q{D}_q$ over the charge sectors for different values of $(\epsilon ,d\beta )$. Colors correspond to the legend of Fig. 10; dashed line: $d\beta =0.01$, full line: $d\beta =0.05$; red: $\epsilon ={10}^{-5}$, green: $\epsilon =5\times {10}^{-6}$, blue: $\epsilon ={10}^{-6}$. (a) $x=100$. (b) $x=600$.

Figure 12

$m/g=0$. Continuum extrapolation of $\mathrm{\Sigma }(\beta ,x)$ for different values of $\beta g$. The black line is the fit which determines the continuum estimate. The red line and dashed blue lines are the best fits for the other fitting functions $f_n$. The error bars on our data ($1/\sqrt{x}\ne 0$) represent the errors ${\mathrm{\Delta }}^{(\epsilon ,d\beta )}\mathrm{\Sigma }(\beta ,x)$ originating from taking nonzero $\epsilon$ and $d\beta$, as explained in Appendix pp1-s1. The blue error bar at $1/\sqrt{x}=0$ represents the error on our continuum estimate. The magenta star is the analytical result of Sachs and Wipf [71].

Figure 13

$m/g=0.25$. Continuum extrapolation of ${\mathrm{\Sigma }}_{\text{sub}}(\beta ,x)$ for different values of $\beta g$. The black line is the fit which determines the continuum estimate. The red line and dashed blue lines are the best fits for the other fitting functions $f_n$. The error bars on our data ($1/\sqrt{x}\ne 0$) represent the errors ${\mathrm{\Delta }}^{(\epsilon ,d\beta )}\mathrm{\Sigma }(\beta ,x)$ originating from taking nonzero $\epsilon$ and $d\beta$, as explained in Appendix pp1-s1. The blue error bar at $1/\sqrt{x}=0$ represents the error on our continuum estimate. The magenta star represents the result of Bañuls et al. and the magenta error bar represents the error that Bañuls et al. estimated on their result.

Figure 14

$m/g=0.25,\alpha =0.25$. Simulations for different values of $(\epsilon ,d\beta )$. (a–b) Electric field with respect to its value for $(\epsilon ,d\beta )=({10}^{-6},0.05)$. (a) $x=125$. (b) $x=250$. (c–d) Free energy with respect to its value for $(\epsilon ,d\beta )=({10}^{-6},0.05)$. (c) $x=125$. (d) $x=250$.

Figure 15

$x=100$, $\alpha =0.5$. Electric field $E_{1/2}(\beta ,x)$ for different values of the tolerance $\epsilon$. Inset: the maximum bond dimension over all charge sectors as a function of $\beta g$. (a): $m/g=0.125$. (b): $m/g=0.25$. (c) $m/g=0.5$. (d) $m/g=1$.

Figure 16

$x=100$, $\alpha =0.5$. String tension ${\sigma }_{1/2}(\beta ,x)$ for different values of the tolerance $\epsilon$. Inset: zooming in on the interval $\beta g\in [9,10]$. (a) $m/g=0.125$. (b) $m/g=0.25$. (c) $m/g=0.5$. (d) $m/g=1$.

Figure 17

$x=100$, $\alpha =0.495$. Electric field $E_{0.495}(\beta ,x)$ for different values of the tolerance $\epsilon$ and step $d\beta$ with respect to the simulation for $(\epsilon ,d\beta )=(5\times {10}^{-7},0.05)$. (a) $m/g=0.125$. (b) $m/g=0.25$. (c) $m/g=0.5$. (d) $m/g=1$.

Figure 18

$m/g=0.25$, $\alpha =0.25$. Continuum extrapolation of $\mathrm{\Delta }{\mathrm{\Sigma }}_{\alpha }(\beta ,x)$ for different values of $\beta g$. The black line is the fit which determines the continuum estimate. The red line and blue line are the most appropriate fits through the data for the other fitting functions. The error bars on our data for $1/\sqrt{x}\ne 0$ represent the errors ${\mathrm{\Delta }}^{(\epsilon ,d\beta )}{\mathrm{\Sigma }}_{\alpha }(\beta ,x)$ originating from taking nonzero $\epsilon$ and $d\beta$, as explained in Appendix pp2-s1. The error bar at $1/\sqrt{x}=0$ is the estimated error on the continuum result.

Figure 19

$m/g=0.25$, $\alpha =0.25$. Continuum extrapolation of ${\sigma }_{\alpha }(\beta ,x)$ for different values of $\beta g$. The black line is the fit which determines the continuum estimate. The red line and blue line are the most appropriate fits through the data for the other fitting functions. The error bars on our data for $1/\sqrt{x}\ne 0$ represent the errors ${\mathrm{\Delta }}^{(\epsilon ,d\beta )}{\sigma }_{\alpha }(\beta ,x)$ originating from taking nonzero $\epsilon$ and $d\beta$, as explained in Appendix pp2-s1. The error bar at $1/\sqrt{x}=0$ is the estimated error on the continuum result.

Figure 20

$m/g=0.125$. Continuum results with error bars for $\alpha =0.1$ (red), $\alpha =0.25$ (green), and $\alpha =0.45$ (blue). (a) String tension. The dashed lines are the results for $\beta g=+\infty$ obtained in Ref. [33]. (b) Electric field. (c) Renormalized entropy. (d) $\mathrm{\Delta }{\mathrm{\Sigma }}_{\alpha }(\beta )$.

Figure 21

$m/g=0.25$. Continuum results with error bars for $\alpha =0.1$ (red), $\alpha =0.25$ (green), and $\alpha =0.45$ (blue). (a) String tension. The dashed lines are the results for $\beta g=+\infty$ obtained in Ref. [33]. (b) Electric field. (c) Renormalized entropy. (d) $\mathrm{\Delta }{\mathrm{\Sigma }}_{\alpha }(\beta )$.

Figure 22

$\alpha =0.25$. $m/g=0.25$ (full line) and $m/g=0.5$ (dashed line). Quantities for $x=100,125,150,\dots ,300$. (a) String tension. (b) Electric field. (c) Renormalized chiral condensate. (d) Entropy.