Anomalous Breaking of Anisotropic Scaling Symmetry in the Quantum Lifshitz Model

TL;DR

The paper investigates the anomalous breaking of Lifshitz-like scaling symmetry at z=2 in d=2+1 dimensions. It applies a field-theoretic heat-kernel treatment to the quantum Lifshitz model and a holographic renormalization approach via Einstein–Proca Lifshitz gravity, obtaining two central charges governing the anomaly. In both approaches, only one central charge is nonzero, yielding an anomaly built solely from time derivatives, and the holographic result agrees with the field-theory computation. The work highlights a potentially universal feature of non-relativistic scale anomalies and raises questions about deeper connections to lower-dimensional conformal structures and broader Lifshitz holography.

Abstract

In this note we investigate the anomalous breaking of anisotropic scaling symmetry in a non-relativistic field theory with dynamical exponent z=2. On general grounds, one can show that there exist two possible "central charges" which characterize the breaking of scale invariance. Using heat kernel methods, we compute these two central charges in the quantum Lifshitz model, a free field theory which is second order in time and fourth order in spatial derivatives. We find that one of the two central charges vanishes. Interestingly, this is also true for strongly coupled non-relativistic field theories with a geometric dual described by a metric and a massive vector field.