Logarithmic Negativity in Lifshitz Harmonic Models

TL;DR

This work analyzes logarithmic negativity for Lifshitz harmonic lattices in $(1+1)$ and $(2+1)$ dimensions, focusing on vacuum and thermal states. Using Gaussian-state correlator methods and the partial transpose, it provides both numerical results for extended subregions and analytical results for a highly symmetric $p$-alternating sublattice, uncovering a robust linear-in-$z$ dependence of negativity in many regimes. In 1+1D, the vacuum negativity exhibits a CFT-like universal part with an effective central charge $c_{ ext{eff}}$, while in 2+1D area-law scaling remains valid for small $z$ but shows deviations at larger $z$ due to nonlocal Lifshitz couplings; thermal effects generally reduce negativity and lattice artifacts appear as sudden-death phenomena. Overall, the paper illuminates how Lifshitz scaling modifies entanglement structure in bosonic systems and highlights both universal features and lattice-specific artifacts, suggesting avenues for continuum-limit analyses and future quench studies.

Abstract

Recently generalizations of the harmonic lattice model has been introduced as a discrete approximation of bosonic field theories with Lifshitz symmetry with a generic dynamical exponent z. In such models in (1+1) and (2+1)-dimensions, we study logarithmic negativity in the vacuum state and also finite temperature states. We investigate various features of logarithmic negativity such as the universal term, its z-dependence and also its temperature dependence in various configurations. We present both analytical and numerical evidences for linear z-dependence of logarithmic negativity in almost all range of parameters both in (1+1) and (2+1)-dimensions. We also investigate the validity of area law behavior of logarithmic negativity in these generalized models and find that this behavior is still correct for small enough dynamical exponents.

Figures (13)

• Figure 1: Configurations we consider to studying negativity in $(1+1)$-dimensions. In $C_1$ we have two disjoint intervals while the complement $B$ is nonempty. In $C_2$ we have two adjacent intervals and the complement $B$ is nonempty again. In $C_3$ we have two adjacent intervals while the complement is empty. Note that in $C_1$ and $C_2$ the whole system can be either finite or infinite while in $C_3$ the system is finite.
• Figure 2: Left panel shows entanglement entropy as a function of the length of the entangling region for $0<z<2$. The middle panel shows logarithmic negativity for configuration $C_2$ as a function of $\ell_1$. In this plot we have fixed $\ell_1+\ell_2=200$. The solid lines in both left and middle panels are fit functions similar to CFT analytic expression. The right panel shows the effective 'c' read from entanglement entropy and logarithmic negativity. In all panels we have set the length of the chain to be $N_x=4000$ and $m=10^{-6}$ with periodic boundary condition.
• Figure 3: Left: Logarithmic negativity for the configuration $C_3$ as a function of $\ell_1$ in log-linear scale for a region with periodic BC for different values of $z$ and $m$. Right: Same plot for larger values of $z$ for $m=10^{-4}$ and $N_x=100$.
• Figure 4: The left panel shows logarithmic negativity for configuration $C_1$ versus the distance between $A_1$ and $A_2$ denoted by $d$. The parameters are $A_1=A_2=20$, $m=10^{-6}$ and $N_x=1000$. In the middle panel we have plotted the data regarding to configuration $C_2$ with the length of $A_1$ and $A_2$ being equal to each other on a chain of length $N_x=500$ and $m=10^{-4}$. In this figure as $z$ increases, the 'initial sleep' region happens for larger subregions. The right panel shows the data for different chain lengths and $z=45$ indicating that 'initial sleep' behavior disappears in the continuum limit.
• Figure 5: Logarithmic negativity as a function of temperature for different values of $z$ for a region with Dirichlet BC. Here we set $m=0$ and $N_x=100$. Left: $\mathcal{E}$ vanishes for $T>T_{\rm sd}$ due to lattice construction. Right: In small temperature limit the magnitude of $\Delta \mathcal{E}$ increases for larger values of $z$.