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.
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Acknowledgments
JKB and DG acknowledge the support of the National Science and Technology Council (NSTC) of Taiwan with the Young Scholar Columbus Fellowship grant 112-2636-M-110-006. AC and CSC acknowledge support of this work by NCTS and the grant 110-2112-M-007-015-MY3 of NSTC. HP acknowledges the support of this work by NCTS.
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Basak, J.K., Chakraborty, A., Chu, CS. et al. Massless Lifshitz field theory for arbitrary z. J. High Energ. Phys. 2024, 284 (2024). https://doi.org/10.1007/JHEP05(2024)284
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DOI: https://doi.org/10.1007/JHEP05(2024)284