Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes

TL;DR

Using Lifshitz scaling in $1+1$ dimensions, the paper shows that algebras with exponents $z$ and $z^{-1}$ are isomorphic, yielding a duality between low and high temperatures on a cylinder of radius $l$. Under a spectral gap, this leads to a generalized Cardy-like formula for the asymptotic density of states, $S = 2 ext{π}l(z+1)ig[(( ext{Δ}_0)/z)^z extΔig]^{1/(z+1)}$, reducing to Cardy at $z=1$. The holographic realization identifies the soliton as the ground state and derives the black hole entropy from this Cardy-like counting; this is illustrated explicitly in BHT massive gravity with a Lifshitz black hole at $z=3$ and a gravitational soliton at $z= frac{1}{3}$, where the regularized Euclidean action yields a consistent entropy. Notably, the results do not rely on asymptotic symmetries or central charges, suggesting a robust, geometry-based mechanism that could extend to higher dimensions. This work thus links Lifshitz holography, solitons, and microscopic entropy in a concrete and testable way.

Abstract

Field theories with anisotropic scaling in 1+1 dimensions are considered. It is shown that the isomorphism between Lifshitz algebras with dynamical exponents z and 1/z naturally leads to a duality between low and high temperature regimes. Assuming the existence of gap in the spectrum, this duality allows to obtain a precise formula for the asymptotic growth of the number of states with a fixed energy which depends on z and the energy of the ground state, and reduces to the Cardy formula for z=1. The holographic realization of the duality can be naturally inferred from the fact that Euclidean Lifshitz spaces in three dimensions with dynamical exponents and characteristic lengths given by z, l, and 1/z, l/z, respectively, are diffeomorphic. The semiclassical entropy of black holes with Lifshitz asymptotics can then be recovered from the generalization of Cardy formula, where the ground state corresponds to a soliton. An explicit example is provided by the existence of a purely gravitational soliton solution for BHT massive gravity, which precisely has the required energy that reproduces the entropy of the analytic asymptotically Lifshitz black hole with z=3. Remarkably, neither the asymptotic symmetries nor central charges were explicitly used in order to obtain these results.