Entanglement asymmetry in gauge theories: chiral anomaly in the finite temperature massless Schwinger model

Abstract

The entanglement asymmetry has emerged in recent years as a practical quantity to study phases of matter. We present the first study of entanglement asymmetry in gauge theories by considering the chiral anomaly of the analytically solvable massless Schwinger model at both zero and finite temperatures. At zero temperature, we find the asymmetry exhibits logarithmic growth with system size. At finite temperature, we show that it is parametrically more sensitive to chiral symmetry-breaking than the corresponding local order parameter: while the chiral condensate decays exponentially, the asymmetry decreases only logarithmically. This establishes the entanglement asymmetry as a promising tool to probe (finite-temperature) phase transitions in gauge theories.

Figures (4)

• Figure 1: Entanglement asymmetry (solid lines) and rescaled chiral condensate density $\langle\bar{\psi}\psi\rangle \cdot L$ (dashed lines) as functions of inverse temperature $\beta$ for different system sizes $L$. The logarithmic behavior of the asymmetry contrasts with the exponential suppression of the order parameter at high temperature. It also constrasts with the extensive behavior of the order parameter as a function of $mL$. The vertical dotted lines show the values of $1/mL$; we expect the thermodynamics limit to be reliable for $\beta m > 1/mL$.
• Figure 2: Second Rényi asymmetry $\Delta S^{(2)}$ and rescaled chiral condensate density $\langle\bar{\psi}\psi\rangle \cdot L$ as functions of $\beta$ for $mL=10$. Black dots show numerical results at finite $L$, while the blue curve shows the thermodynamic limit expression from \ref{['eq:rho5ovtr']}. The green dashed line indicates the small-$\beta$ asymptotic behavior $\Delta S^{(2)} \sim (\pi^2/12) L m^2 \beta$. The agreement between finite-$L$ and thermodynamic results is excellent for $\beta m > 1/(mL)$.
• Figure 3: Second Rényi asymmetry $\Delta S^{(2)}$ for different system sizes: numerical results at finite $mL$ (markers) versus thermodynamic limit analytics (solid curves) in the upper panel, and relative error in the lower panel. The results show rapid convergence as $mL$ increases. Colors indicate $mL=1$ (pink), $mL=2$ (blue), and $mL=10$ (black).
• Figure 4: Rényi entanglement asymmetries $\Delta S^{(n)}$ for $n=2,3,4$ as functions of $\beta$ for $mL=10$. Solid curves show numerical results, while dashed lines indicate the small-$\beta$ asymptotic behaviors, all of which are linear in $\beta$.