Anomaly Induced Current in Boundary Lifshitz Field Theory

TL;DR

This work investigates how anisotropic Lifshitz scale anomalies generate boundary currents in boundary Lifshitz field theories (BLFTs) coupled to a background U(1) gauge field. It first classifies the possible anomaly structures for Lifshitz theories with gauge fields, identifying an anomaly density in 5D given by $\mathcal{I}=c_T\,F_{ti}F^{ti}+c_S\,F_{ij}\nabla^2 F^{ij}$ and relating the central charges to the anomaly. The authors then show that, upon a Lifshitz Weyl rescaling mapped from flat BLFT to the Lifshitz geometry via $\sigma=\ln x$, the anomaly induces a boundary current with distinct leading-distance behavior for the temporal and spatial components, controlled by $c_T$ and $c_S$. A holographic dual description is developed using a Lifshitz bulk with an end-of-the-world brane and a Maxwell field; near the boundary, the currents obey $\langle J_t\rangle=-\alpha_d^T F_{tn}/x^{d-z-2}$ and $\langle J_i\rangle=-\alpha_d^S F_{in}/x^{d+z-4}$, with explicit results for the dual 5D BLFT (where $\alpha_5^T=1$ and $\alpha_5^S=\frac{1}{2(1-\tanh(\rho_*/L))}$). The holographic results reproduce the field-theory scaling and yield central charges $c_T=-\tfrac{1}{4}$ and $c_S=-\alpha_5^S/4$, providing a nontrivial consistency check. Overall, the paper establishes a direct link between Lifshitz scale anomalies and boundary-induced transport in non-relativistic QFTs, highlighting qualitative differences from relativistic BCFTs and offering multiple avenues for future exploration and potential experimental relevance.

Abstract

We study quantum transport phenomena induced by anisotropic Lifshitz scale anomaly in a boundary Lifshitz field theory (BLFT) coupled to an external electromagnetic background. In this context, we obtain the anisotropic scale anomaly in Lifshitz field theories coupled to a background $U(1)$ gauge field and subsequently compute the anomaly induced near boundary current in a BLFT. Focusing on 5D BLFTs, we find that the temporal and spatial components of the induced current exhibit distinct power law dependencies on the distance from the boundary, reflecting the intrinsic time-space anisotropy of the theory. We further derive this anomalous current holographically from the bulk dual of BLFT and find that the temporal component is independent of the boundary conditions while the spatial component depends explicitly on them. The distance dependence is in exact agreement with the dual field theory result.