Chiral condensate in the Schwinger model with matrix product operators

Tensor network (TN) methods, in particular the matrix product states (MPS) ansatz, have proven to be a useful tool in analyzing the properties of lattice gauge theories. They allow for a very good precision, much better than standard Monte Carlo (MC) techniques for the models that have been studied so far, due to the possibility of reaching much smaller lattice spacings. The real reason for the interest in the TN approach, however, is its ability, shown so far in several condensed matter models, to deal with theories which exhibit the notorious sign problem in MC simulations. This makes it prospective for dealing with the nonzero chemical potential in QCD and other lattice gauge theories, as well as with real-time simulations. In this paper, using matrix product operators, we extend our analysis of the Schwinger model at zero temperature to show the feasibility of this approach also at finite temperature. This is an important step on the way to deal with the sign problem of QCD. We analyze in detail the chiral symmetry breaking in the massless and massive cases and show that the method works very well and gives good control over a broad range of temperatures, essentially from zero to infinite temperature.

Figure 1

Examples of the $D$-dependence of the ground state chiral condensate for $m/g=0.125$, $x=200$ and five system sizes (left). The right plot shows a zoom into the region $D\in [80,160]$ for $N=368$. The red band represents the uncertainty related to the bond dimension, taken as explained in the text.

Figure 2

Examples of the $D$-dependence of the ground state chiral condensate for $m/g=0.125$, $x=10$ and five system sizes (left). The right plot shows a zoom into the region $D\in [80,160]$ for $N=84$. See comments in the text about the irregular approach to the $1/D=0$ limit. The red band represents the uncertainty related to the bond dimension, taken as explained in the text.

Figure 3

Examples of the infinite volume extrapolations of the $T=0$ chiral condensate for $x=10$ (left) and $x=200$, at $m/g=0.125$. Lines are fits of Eq. (4.1). The points have error bars, but they are too small to be seen.

Figure 4

Continuum limit extrapolations of the $T=0$ chiral condensate for all our fermion masses. Shown are the data for the subtracted infinite volume condensate and the fits of Eq. (4.2) in the interval $1/\sqrt{x}\in [0,0.23]$ ($x\in [20,600]$, i.e. left of the dashed line). These fits are example fits that enter our procedure of extraction of the systematic uncertainty related to the choice of the fitting range. Note that the rightmost two points are not included in the fit, but are still rather well described by it.

Figure 5

Examples of the $D$-dependence of the chiral condensate for $m/g=0.25$, $g\beta =0.5$, $L_{\mathrm{cut}}=10$, $x=9$, $N=60$ and $\delta =0.0020833$ (upper left), $\delta =0.0041667$ (upper right) and $\delta =0.0083333$ (lower left). The red bands represent the uncertainty related to the bond dimension, taken as explained in the text. Example of a $\delta$-extrapolation for $m/g=0.25$, $g\beta =0.5$, $L_{\mathrm{cut}}=10$, $x=9$ and three values of $N=48$, 60, 72 (lower right). Lines are fits of Eq. (4.3). The data points have error bars, but they are too small to be seen.

Figure 6

Examples of the $D$-dependence of the chiral condensate for $m/g=0.25$, $g\beta =0.5$, $L_{\mathrm{cut}}=10$, $x=1024$, $N=640$ and $\delta =0.00019531$ (upper left), $\delta =0.00039063$ (upper right) and $\delta =0.00078125$ (lower left). The red bands represent the uncertainty related to the bond dimension, taken as explained in the text. Example of a $\delta$-extrapolation for $m/g=0.25$, $g\beta =0.5$, $L_{\mathrm{cut}}=10$, $x=1024$ and three values of $N=512$, 640, 768 (lower right). Lines are fits of Eq. (4.3). The data points have error bars, but they are too small to be seen.

Figure 7

Examples of the infinite volume extrapolations of the chiral condensate for $g\beta =0.5$ (left) and $g\beta =4$ and five values of $x$, with $L_{\mathrm{cut}}=10$. Lines are fits of Eq. (4.1).

Figure 8

Examples of the continuum limit extrapolations of the chiral condensate for $m/g=0.25$ and temperatures $g\beta =6.0$ (upper left), $g\beta =2.0$ (upper right), $g\beta =0.4$ (lower left) and $g\beta =0.2$ (lower right).

Figure 9

Inverse temperature dependence of the continuum limit extrapolated chiral condensate for all our fermion masses. Shown is also the result obtained at $T=0$ and the result of the approximation of Ref. [30].