Density Induced Phase Transitions in the Schwinger Model: A Study with Matrix Product States

We numerically study the zero temperature phase structure of the multiflavor Schwinger model at nonzero chemical potential. Using matrix product states, we reproduce analytical results for the phase structure for two flavors in the massless case and extend the computation to the massive case, where no analytical predictions are available. Our calculations allow us to locate phase transitions in the mass-chemical potential plane with great precision and provide a concrete example of tensor networks overcoming the sign problem in a lattice gauge theory calculation.

Figure 1

Continuum estimate for $\mathrm{\Delta }N$ versus ${\mu }_I/2\pi$, for volumes 2 (red solid), 6 (green dashed), and 8 (blue dash-dotted line). The vertical lines indicate the theoretical prediction for the phase transitions in the massless case. Upper inset: Close-up around the first transition for $Lg=8$, $x=16$, $m/g=0$, $D=160$. Shown are MPS results for $\mathrm{\Delta }N=0$ (blue crosses), $\mathrm{\Delta }N=2$ (red crosses), and the corresponding predictions (solid lines). Lower inset: Volume dependence of the continuous location of the transitions for the first (red crosses), second (green crosses), third (blue asterisks), and fourth (magenta dots) transition.

Figure 2

Continuum estimate for $\mathrm{\Delta }N$ versus ${\mu }_I/2\pi$, for volumes 2 (red solid), 6 (green dashed), and 8 (blue dash-dotted line). Inset: Isospin number versus ${\mu }_I/2\pi$ for $Lg=8$, $x=121$, $m/g=0.5$, $D=220$.

Figure 3

Phase diagram in the $m/g$—${\mu }_I/2\pi$ plane for $Lg=8$. The black crosses mark the computed data points, the different colors indicate the different phases.

Figure 4

$⟨C(y)⟩$ for $Lg=8$, $x=1024$, $m/g=0$, $D=160$, and different phases. The blue crosses represent $\mathrm{\Delta }N=0$, the red crosses $\mathrm{\Delta }N=2$, and the green asterisks $\mathrm{\Delta }N=4$.