Detecting the critical point through entanglement in the Schwinger model

Using quantum simulations on classical hardware, we study the phase diagram of the massive Schwinger model with a $\theta$ term at finite chemical potential $\mu$. We find that the quantum critical point in the phase diagram of the model can be detected through the entanglement entropy and entanglement spectrum. As a first step, we chart the phase diagram using conventional methods by computing the dependence of the charge and chiral condensates on the fermion mass $m$, coupling constant $g$, and the chemical potential $\mu$. At zero density, the Schwinger model possesses a quantum critical point at $\theta =\pi$ and $m/g\simeq 0.33$. We find that the position of this quantum critical point depends on the chemical potential. Near this quantum critical point, we observe a sharp maximum in the entanglement entropy. Moreover, we find that the quantum critical point can be located from the entanglement spectrum by detecting the position of the gap closing point.

Figure 1

Illustration of the lattice setup. Red dots are fermions, and blue dots are antifermions. We take the boundary condition as $L_0=-Q/2$, which leads to $L_N=+Q/2$, with $Q$ being the total charge. See text for explanation.

Figure 2

From left to right: density plot of total charge, chiral condensate, charge fluctuation, and ground-state left-right entanglement entropy for different values of fermion mass and chemical potential, with $N=20$, $L=4/g$, and $\theta =0$. The entanglement entropy is for the ground state, corresponding to $T=0$, whereas the other panels are for $T=g/10$. The first-order phase transition lines (black dashed contours) are overlaid on top of every panel. See text for more details.

Figure 3

From top to bottom: fermion mass dependence of chiral condensate, vector charge susceptibility, chiral susceptibility, and vacuum entanglement entropy, with $\theta =0$ and $\mu =0$. In the top three panels, red, blue, and purple curves represent $T=g/3$, $g/5$, and $g/10$, respectively.

Figure 4

From left to right: spatial distribution of chiral condensate (upper panel) and charge density (lower panel) for $Q=1$, 2, and 3 states, with shown values of fermion mass.

Figure 5

Central charge versus lattice spacing. Purple, blue, orange, and red symbols correspond to lengths $L=4/g$, $5/g$, $6/g$, and $7/g$, respectively. The dashed curves are second-order polynomial fits to guide the eye. They come close to the expected central charge $c=1$ in the continuum limit $a\to 0$ within 15% accuracy. Results are evaluated at vanishing chemical potential.

Figure 6

Entanglement spectrum at $\theta =0$ (upper panel) and $\theta =\pi /2$ (lower panel). Other parameters are set to $N=10$, $a=0.5/g$, and $\mu =0$.

Figure 7

Entanglement entropy versus the fermion mass with the number of lattice sites being $N=8$ (purple), 12 (blue), 16 (orange), and 20 (red). Lattice spacing is fixed to be $a=0.5/g$.

Figure 8

Mass (top) and length (bottom) dependence of entanglement entropy. Curves in the middle panel are fitted results using Eq. (7). Purple, blue, orange, and red curves correspond to total numbers of lattice sites $N=8$, 12, 16, and 20, respectively. Results are evaluated at vanishing chemical potential.