Entanglement Entropy of a Massive Fermion on a Torus

TL;DR

The paper investigates entanglement and Renyi entropies of a free massive Dirac fermion on a torus at finite temperature and chemical potential, combining analytic bosonization methods with lattice numerics. It delivers analytic results for the massless, multi-interval case and computes leading small-mass corrections via the sine-Gordon mapping, validated by lattice simulations; it also analyzes high- and low-temperature regimes and mutual information. A key finding is the exponential suppression of thermal corrections in the gapped regime and a noncommutativity of the $m\to 0$ and $T\to 0$ limits in degenerate ground states. The work highlights both the power and limitations of perturbative bosonization in finite-volume, finite-temperature settings and points to nonperturbative and lattice techniques as productive avenues for further progress.

Abstract

The Renyi entropies of a massless Dirac fermion on a circle with chemical potential are calculated analytically at nonzero temperature by using the bosonization method. The bosonization of a massive Dirac fermion to the sine-Gordon model lets us obtain the small mass corrections to the entropies. We numerically compute the Renyi entropies by putting a massive fermion on the lattice and find agreement between the analytic and numerical results. In the presence of a mass gap, we show that corrections to Renyi and entanglement entropies in the limit m >> T scale as exp(-m/T). We also show that when there is ground state degeneracy in the gapless case, the limits m to zero and T to zero do not commute.

Figures (6)

• Figure 1: The mutual Rényi informations of two intervals $A$ and $B$ of width $\ell_1 = \ell_2 = L/10$ with $n=2$ in the $\nu=2$ [Left], $\nu=3$ [Middle] and $\nu=4$ [Right] sectors. $\ell_3$ is the distance between the two intervals. The blue dashed and orange solid curves are for $\beta = 10,1/5$, respectively.
• Figure 2: The $\ell$ dependence of the $O(m^2)$ correction to the $n=2$ Rényi entropy for $\nu=2$. The curves are produced by numerical integration of (\ref{['Cndef']}). The points are from a lattice computation. From top to bottom, $\beta = 1/2$, 1, and 2.
• Figure 3: The Rényi entropy for $n=2$ of two intervals of width $\ell_1 = \ell_2=L/10$ whose distance is $\ell_3$. The $\nu=2$ [Left], $\nu=3$ [Middle] and $\nu=4$ [Right] sectors are depicted. The curves are analytic and the dots are numerical. The blue dotted and orange solid curves are for $\beta=1/5$ and $10$, respectively.
• Figure 4: The Rényi entropy for $n=2$ of two intervals of width $\ell_1=L/10$ and $\ell_2$. The distance between the intervals is fixed to $\ell_3=L/10$ and $\ell_2$ is varied. The $\nu=2$ [Left], $\nu=3$ [Middle] and $\nu=4$ [Right] sectors are depicted. The curves are analytic and the dots are numerical. The orange solid, blue dotted and black dashed curves are for $\beta=1/10,1/5$ and $1$.
• Figure 5: The single interval Rényi entropy for $\nu =2$: $n=2$ [Left] and $n=3$ [Right]. In both cases, the $\lim_{T\to 0} \lim_{m \to 0}$ curve (orange) lies above and the $\lim_{m\to 0} \lim_{T \to 0}$ curve (blue) lies below. The points were computed using the lattice.