Lifshitz entanglement entropy from holographic cMERA

TL;DR

This work uses Nozaki–Ryu–Takayanagi’s cMERA-inspired bulk metric to study entanglement entropy in free Lifshitz scalar theories, applying a holographic RT-like extremisation to compute strip and disc entanglement for massless and massive cases across dimensions. It shows that massless Lifshitz entanglement in 1D reduces to a z-dependent rescaling of the relativistic result, with a transition to a volume law at large $z$, and it provides higher-dimensional area-law behavior with a consistent large-$z$ volume-law limit. Mass deformations introduce a correlation length $\xi$ and reveal a phase structure between connected and disconnected minimal surfaces, yielding a Lifshitz-generalised Cardy–Calabrese-type log for large $\xi/\varepsilon$. An RG flow analysis demonstrates a monotonic decrease of entanglement along flows from UV Lifshitz fixed points to IR CFTs, supporting a Lifshitz analogue of entanglement monotonicity, though the overall normalization $c_z$ remains undetermined. Overall, the paper offers a pragmatic, predictive framework for Lifshitz entanglement using cMERA-inspired geometries, with potential lattice and replica-trick checks to benchmark these holographic-like results.

Abstract

We study entanglement entropy in free Lifshitz scalar field theories holographically by employing the metrics proposed by Nozaki, Ryu and Takayanagi in \cite{Nozaki:2012zj} obtained from a continuous multi-scale entanglement renormalisation ansatz (cMERA). In these geometries we compute the minimal surface areas governing the entanglement entropy as functions of the dynamical exponent $z$ and we exhibit a transition from an area law to a volume law analytically in the limit of large $z$. We move on to explore the effects of a massive deformation, obtaining results for any $z$ in arbitrary dimension. We then trigger a renormalisation group flow between a Lifshitz theory and a conformal theory and observe a monotonic decrease in entanglement entropy along this flow. We focus on strip regions but also consider a disc in the undeformed theory.

Figures (4)

• Figure 1: Types of geodesic for $J_2=1/2$, $J_1=1/5$. The blue and red curves are the two possible connected geodesics and the pair of green lines is the disconnected geodesic. All three geodesics end at $mr_\textrm{UV}/c=1/\sqrt{26}$ and the dashed line is the IR cut-off $mr_\textrm{IR}/c=1$.
• Figure 2: Ratio of the connected geodesic length to the disconnected geodesic length for $J_1=1/5$ as a function of $J_2$. The red, green and blue dots correspond to the curves plotted in Figure \ref{['fig:CFTCurvePlot']} for $J_2=1/2$ and the black dot marks the critical value of $J_2$ for this $J_1$. The region to the left of the vertical dashed line is unphysical since $J_2<J_1$ therein.
• Figure 3: Phase diagram for $d=1$. The curves correspond to $z=1$ (blue), $z=2$ (red), $z=3$ (green) and $z=4$ (yellow). To the left of each curve the shortest geodesic is connected, whereas to the right the shortest geodesic is disconnected. The dashed lines mark the boundaries of the physical region $J_1<1$ and $J_2>J_1$. The black dots are extracted directly at $J_1=0$.
• Figure 4: Entanglement entropy for a flow between a CFT in the IR and $z=2$ Lifshitz theory in the UV. Left: Length as a function of $\log_{10}K$ for $\ell/\varepsilon=2,5,10,20$ (bottom to top). These curves interpolate monotonically between the correct IR (left) and UV (right) limits. Right: Length as a function of $\ell/\varepsilon$ for $K=0.5,5,50$ (bottom to top, solid). The dashed lines correspond to the IR (black) and UV (red) limits.