Entanglement Entropy in Lifshitz Theories

TL;DR

The work analyzes entanglement entropy in 1+1D free Lifshitz scalar theories across arbitrary dynamical exponents $z$, using cMERA-based scaling, lattice discretization, and RG-deformation arguments. It establishes that EE scales linearly with $z$ for both subinterval and p-sublattice subsystems, and reveals an area-law to volume-law crossover when $z$ becomes comparable to subsystem size. The study also demonstrates that EE decreases under UV-to-IR RG flows in deformed Lifshitz theories, providing evidence for a weak Lifshitz c-theorem and suggesting a deeper link between non-relativistic scaling and entanglement structure. Together, these results illuminate how non-relativistic criticality governs entanglement and may inform holographic Lifshitz descriptions and other non-relativistic quantum systems.

Abstract

We discuss and compute entanglement entropy (EE) in (1+1)-dimensional free Lifshitz scalar field theories with arbitrary dynamical exponents. We consider both the subinterval and periodic sublattices in the discretized theory as subsystems. In both cases, we are able to analytically demonstrate that the EE grows linearly as a function of the dynamical exponent. Furthermore, for the subinterval case, we determine that as the dynamical exponent increases, there is a crossover from an area law to a volume law. Lastly, we deform Lifshitz field theories with certain relevant operators and show that the EE decreases from the ultraviolet to the infrared fixed point, giving evidence for a possible c-theorem for deformed Lifshitz theories.

Figures (8)

• Figure 1: Utilizing the assumption $N \to \infty$, we fixed $N_A=40,70$ and $J = 10^{-5}$ and plotted the vacuum EE $S_A$ as a function of $z$. We fitted the data using \ref{['LifshitzEE']} with $c_0 = 1.996$ for the $N_A = 40$ case and $c_0 = 2.080$ for the $N_A = 70$ case, as determined by Mathematica to be the best fit. We expect that the two values of $c_0$ should be the same, and the fit should become exact in the massless continuum limit $J \to 0$ and $N_A\to \infty$.
• Figure 2: Plot of vacuum EE as a function of $J$ for $N_A =20$ and different fixed values of $z$. Note that in the UV, the lattice spacing ${\epsilon}$ is very small, which implies $J \ll 1$, while in the IR, ${\epsilon}$ is very large, which implies $J \gg 1$. In this $J \gg 1$ regime, the correlation length, inversely proportional to $J$, becomes smaller than the lattice spacing, and we expect the EE to fall off. Thus, we see that regardless of $z$, as we flow from UV to IR, the vacuum EE $S_A$ decreases.
• Figure 3: Plot of vacuum EE as a function of the size $N_A$ of the subinterval $A$ for fixed $J = 10^{-5}$ and different fixed values of $z$. For the $z=1$ case, the plot matches with the Casini-Huerta prediction with central charge $c=1$. There also appears to be a crossover from logarithmic growth when $N_A > z$ to linear growth when $N_A < z$, as is best visible with the $z=8$ data points.
• Figure 4: One-dimensional periodic sublattices consisting of $N=12$ lattice sites, with subsystem $A$ consisting of the $N_A$ filled lattice points such that $N=pN_A$. On the left $N_A=6$ and $p=2$, while on the right $N_A=4$ and $p=3$. Figure taken from He:2016ohr.
• Figure 5: Vacuum EE density of $z=2$ and $z=4$ Lifshitz theories as a function of $p$ with $J=0$, or equivalently, $m = 0$. The vacuum EE density increases without bounds as $p$ increases.