Entanglement in Lifshitz-type Quantum Field Theories

TL;DR

This work analyzes how Lifshitz scaling with dynamical exponent $z$ modifies quantum entanglement in free scalar field theories, using the correlator method to compute entanglement entropy and related measures in $(1+1)$ and $(2+1)$ dimensions. The authors show that in the massless case the entanglement entropy transitions from area- to volume-law scaling as $z$ grows, with a proposed leading form $S^{(z)}(\,\ell\,) \sim \# (\\ell/\\epsilon)^{1-1/z}$ in 1D (recovering logarithmic behavior at $z=1$) and analogous volume-dominated behavior in higher dimensions; in $(2+1)$-D they find disk regions dominated by a volume term and squares featuring shape-dependent corner contributions that become non-additive at large $z$, while mutual information increases with $z$ and tripartite information remains positive. For massive theories, the results indicate a reversion to area-law scaling and a suppression of nonlocal entanglement, with no volume-law regime for small subregions. The study provides insight into the nonlocal entanglement structure at Lifshitz critical points and offers data relevant for holographic considerations and future investigations of quenches and mixed states.

Abstract

We study different aspects of quantum entanglement and its measures, including entanglement entropy in the vacuum state of a certain Lifshitz scalar theory. We present simple intuitive arguments based on "non-local" effects of this theory that the scaling of entanglement entropy depends on the dynamical exponent as a characteristic parameter of the theory. The scaling is such that in the massless theory for small entangling regions it leads to area law in the Lorentzian limit and volume law in the $z\to\infty$ limit. We present strong numerical evidences in (1+1) and (2+1)-dimensions in support of this behavior. In (2+1)-dimensions we also study some shape dependent aspects of entanglement. We argue that in the massless limit corner contributions are no more additive for large enough dynamical exponents due to non-local effects of Lifshitz theories. We also comment on possible holographic duals of such theories based on the sign of tripartite information.

Figures (16)

• Figure 1: The nearest points inside a disk entangling region to its boundary are shown in green. Those points which are correlated to the green ones due to the discrete kinetic term of this theory are shown with a red shadow. The left panel belongs to the Lorentzian case $z=1$. This shows how we intuitively understand the area law of entanglement for regions which their characteristic length, say $\ell$, is much bigger than the lattice spacing (the inverse of the UV cut-off). Moving from the left panel to the right, the dynamical exponent is increasing. One can see that for a disk with a radius $\sim 3.5$ in units of lattice spacing, for $z> 6$, since all points inside the entangling region are correlated with the green ones, we expect entanglement measures to scale with the volume (instead of the area) of the entangling region. We will show numerically that this is the correct scaling of entanglement entropy for large enough $z$ in the following of this paper. Although this figure belongs to $(2+1)$-dimensions, its horizontal (vertical) slices describe what happens in $(1+1)$-dimensions and it can be generalized to higher dimensions straightforwardly.
• Figure 2: EE as a function of $\ell$ for different values of $z$. Here we consider Dirichlet boundary condition for a massless scalar with $N=500$ (left) and $N=20000$ (right).
• Figure 3: Left: $\frac{\ell S'}{S}$ as a function of $z$ for different values of $\ell$. For large values of $z$ this quantity approaches to a constant value exhibiting volume law behavior. right: Fitting data for EE as a function of $\ell$ according to Eq.\ref{['fitfunction']} for different values of $z$. Here we consider Dirichlet boundary condition for a massless scalar with $N=20000$.
• Figure 4: Renyi entropy as a function of $\ell$ for large values of $z$ with $m=0$ and $N=20000$.
• Figure 5: EE for large regions $\ell \sim N$ showing $S_A=S_{\bar{A}}$.