Lattice studies of entanglement entropy in $O(N)$ models at finite densities

Abstract

As a characteristic property of all quantum systems, entanglement participates in many important quantum phenomena. In this proceeding, we employ it in the study of quantum field theories at finite density. We incorporate evaluations of entanglement entropy using the replica trick into MC simulations of $O(N)$ models at finite density with the worm algorithm and present some initial results for the nonlinear $O(4)$ model in 3 dimensions.

Figures (6)

• Figure 1: An illustration of the worm update in the $k$-sector. The first step is the insertion of the external source-sink pair $\phi^+\phi^-$. Then the head $\phi^+$ is moved to a neighboring site $x'=x+\widehat{\nu}$ and $k_{x,\nu}$ is shifted by $\pm1$ (depending on whether $\nu$ is a positive or negative direction) to compensate for the charge displacement. This is continued until the head and tail are again on the same site when they can be removed.
• Figure 2: Illustration of the topologies of $Z$ and $Z(l,r=2)$. The fields are $1/T$ periodic in the temporal direction for $Z$. For $Z\mathopen{}\mathclose{\left(\ell,2\right)$, in region $A$ of width $\ell$, the fields are $2/T$ periodic. In region $B$, they are $1/T$ periodic. We have chosen to set $r=2$ here to provide a more clear example of $Z\mathopen{}\mathclose{\left(\ell,r}\right)$ and because it is the partition function used to approximate $\partial_\ell S_{\text{EE}}$.
• Figure 3: A visualization of the plaquette boundary update. Plaquette worms are worm updates that are constrained to move along a temporal plaquette. They change the values of the $k_i=k_{x_i-\widehat{t},t}$ and $\chi_i=\chi_{x_i-\widehat{t},t}$ such that $\Delta k=k_1-k_0=0$ and $\Delta \chi=\chi_1-\chi_0=0 \ (\text{mod} \ 2)$, which will allow the boundary conditions to be changed without producing defects. After the change, the same plaquettes are done in reverse to restore $\Delta k$ and $\Delta \chi (\text{mod} \ 2)$ to their original values.
• Figure 4: A pair of defects introduced by the change of temporal boundary conditions consists of a defect (purple disk) on sites $x_0$ and an anti-defect (purple circle) on site $x_1$ (middle panel). A worm (red) can move the defect from site $x_0$ around till it eventually reaches the site $x_1$ (right panel), after which the defects can be removed.
• Figure 5: Left: $\partial_\ell H_2\approx\partial_\ell S_{EE}$ as a function of $\ell$ for $N_t=\{5,8\}$ and $\mu=\{0.15;0.30\}$. Right: $\partial_\ell H_2$ as a function of $\mu$ for $N_t=\{5,6,7,8,9,10\}$ and $\ell=17.5$.