Massless Lifshitz Field Theory for Arbitrary $z$

TL;DR

The paper constructs a massless Lifshitz scalar theory with arbitrary dynamical exponent z in (d+1) dimensions using fractional derivatives, establishing a Rokhsar–Kivelson ground state in (1+1)D and revealing a continuous ground-state degeneracy parameterized by a classical solution. It develops a 2d/1d correspondence for equal-time correlators via a 1D auxiliary path integral, and computes a suite of entanglement measures (entanglement entropy, mutual information, reflected entropy, Markov gap) for finite and disjoint interval configurations, all expressed through cross ratios that depend on z. The authors demonstrate a monotonic c-function along RG flow for z≥1 and connect the field theory results to holography by proposing a z-dependent Lifshitz radius in the bulk, deriving time-like entanglement entropy holographically, and discussing implications for a fundamental time-like entanglement definition beyond analytic continuation. Together, these results broaden the understanding of Lifshitz field theories, their entanglement structure, and their holographic descriptions, with implications for quantum critical systems and nonrelativistic holography.

Abstract

By using the notion of fractional derivatives, we introduce a class of massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary anisotropy index $z$. The Lifshitz scale invariant ground state of the theory is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show that there is a continuous family of ground states with degeneracy parameterized by the choice of solution to the equation of motion of an auxiliary classical system. The quantum mechanical path integral establishes a 2d/1d correspondence with the equal time correlation functions of the Lifshitz scalar field theory. We study the entanglement properties of the Lifshitz theory for arbitrary $z$ using the path integral representation. The entanglement measures are expressed in terms of certain cross ratio functions we specify, and satisfy the $c$-function monotonicity theorems. We also consider the holographic description of the Lifshitz theory. In order to match with the field theory result for the entanglement entropy, we propose a $z$-dependent radius scale for the Lifshitz background. This relation is consistent with the $z$-dependent scaling symmetry respected by the Lifshitz vacuum. Furthermore, the time-like entanglement entropy is determined using holography. Our result suggests that there should exist a fundamental definition of time-like entanglement other than employing analytic continuation as performed in relativistic field theory.

Figures (6)

• Figure 1: Schematic for the configuration consisting of intervals $A$ and $B$.
• Figure 2: Schematic of two adjacent intervals.
• Figure 3: Schematic of two disjoint intervals in a finite system.
• Figure 4: The Markov gap as function of $z$ for two disjoint intervals ${B_1}$ and ${B_2}$. It depends on the relative size of the separation interval $A$ in comparison with the minimum length of ${B_1}$ and ${B_2}$. When $l_A$ is not the minimum length of the three, the Markov gap, approaches to zero for large $z$ (green curve), otherwise it saturates to a finite value.
• Figure 5: Geodesics for holographic time-like entanglement entropy computed in the Poincaré patch. The two mirroring curves that reach the boundary provide the real contribution, while the other bulk curve provides the imaginary part of the time-like entanglement entropy.