Torsion and anomalies in the warped limit of Lifschitz theories
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Springer
Received: October Revised: January Accepted: January Published: January
21, 12, 17, 30,
2019 2020 2020 2020
Christian Copetti Instituto de F´ısica Te´ orica UAM/CSIC, Universidad Aut´ onoma de Madrid, c/Nicol´ as Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain
E-mail: [email protected] Abstract: We describe the physics of fermionic Lifschitz theories once the anisotropic scaling exponent is made arbitrarily small. In this limit the system acquires an enhanced (Carrollian) boost symmetry. We show, both through the explicit computation of the path integral Jacobian and through the solution of the Wess-Zumino consistency conditions, that the translation symmetry in the anisotropic direction becomes anomalous. This turns out to be a mixed anomaly between boosts and translations. In a Newton-Cartan formulation of the space-time geometry such anomaly is sourced by torsion. We use these results to give an effective field theory description of the anomalous transport coefficients, which were originally computed through Kubo formulas in [1]. Along the way we provide a link with warped CFTs. Keywords: Anomalies in Field and String Theories, Space-Time Symmetries, Conformal Field Theory ArXiv ePrint: 1909.01157
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)190
JHEP01(2020)190
Torsion and anomalies in the warped limit of Lifschitz theories
Contents 1 Introduction
1
2 The Lifschitz fermion 2.1 Fujikawa regularization for anisotropic translations
5 6 10 15 19
4 Warped CFTs 4.1 Free examples 4.2 Emergence of Carrollian symmetry
21 22 25
5 Conclusions and open questions
26
A Expansion of the regulated Jacobian
27
B Warped transport from Kubo formulas in the Lifschitz theory
28
C Carroll manifolds and Carrollian diffeomorphisms
29
D Consistency conditions
30
1
Introduction
Quantum critical points and their physics have attracted a huge amount of interest over the last decade [2]. In this paper we will focus on a nonrelativistic class of such theories and their effective description at low energies. We will focus on fermionic theories with an emergent Lifschitz scaling symmetry. A well known example is given by the low energy limit of the following four dimensional Lagrangian L = ψ¯ (iγ µ ∂µ − m + γ µ γ5 nµ ) ψ .
(1.1)
This is often interpreted, in condensed matter language [3], as describing the transition between a Weyl semimetal and a trivial insulator. The quantum critical point is reached upon tuning |m| = |n| (we take nµ to be a spatial vector) and its low energy excitations are characterized by an emergent anisotropic Lifschitz scaling symmetry with z = 1/2. This can be seen from the dispersion relation at criticality 2 2 (k) = k⊥ +
1 4 k + ... , 4m2 v
–1–
(1.2)
JHEP01(2020)190
3 Warped geometry and anomalies 3.1 The consistency condition 3.2 Transport and warped anomalies
with kv = kµ v µ and nµ v µ = 1. The Lifschitz symmetry is in this case a bit unconventional, since it scales anisotropically a space-like direction instead
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