Thermal transport, geometry, and anomalies

Abstract

The relation between thermal transport and gravity was highlighted in the seminal work by Luttinger in 1964, and has been extensively developed to understand thermal transport, most notably the thermal Hall effect. Here we review the novel concepts that relate thermal transport, the geometry of space-time and quantum field theory anomalies. We give emphasis to the cross-pollination between emergent ideas in condensed matter, notably Weyl and Dirac semimetals, and the understanding of gravitational and scale anomalies stemming from high-energy physics. We finish by relating to recent experimental advances and presenting a perspective of several open problems. CONTENTS I. Introduction II. Key concepts III. Thermal and electro-thermal transport A. Thermoelectric relations B. Luttinger theory of thermal transport coefficients 1. Tolman-Ehrenfest pioneering work: Thermodynamic equilibrium in gravitational fields C. Kubo formulas for the thermal and thermoelectric conductivities D. Formal derivation of generalized Luttinger relations IV. Transport and magnetization currents A. General remarks B. Luttinger, thermal transport and energy magnetization V. Transport and chiral anomalies A. Anomalies in QFT B. Anomalies in matter C. Anomalies and magnetotransport in Weyl semi-metals D. Anomaly induced magneto-electric and -thermal transport for Weyl semi-metals E. Weak magnetic field F. Thermoelectric transport relations of the anomaly-induced transport coefficients G. Large magnetic field H. Concluding remarks on magneto-transport VI. Axial gauge fields and the chiral anomaly. A. Axial gauge fields in Weyl semimetals. B. New anomaly induced transport phenomena in the presence of axial vector fields C. Curvature and elastic deformations VII. The scale anomaly and thermoelectromagnetic currents A. Scale anomaly in quantum electrodynamics B. Scale anomaly and transport effects in the bulk C. Anomalous thermoelectric current due to the conformal anomaly D. Scale electromagnetic effects at the boundary

Anomalous chiral transport with vorticity and torsion: Cancellation of two mixed gravitational anomaly currents in rotating chiral $p+ip$ Weyl condensates

“…The gravitational (i.e. thermal 39,48 ) CVE anomaly current at linear order in response has been related to the gravitational anomalies 15,47,49 that, at least naively, appear only at higher-orders in the hydrodynamic gradient expansion. The precise way how higher-order anomalies can contribute to the transport has lead to many useful insights about relativistic and non-relativistic hydrodynamics, e.g.…”

“…the simple Landau level approach to chiral torsional anomalies. In particular, recent papers 39,76 finding no torsional contribution can be easily incorporated with our framework by noting that when computing the linear response around flat space, torsion has been actually set to zero with only non-zero metric variations. This is equivalent to whether or not frame fields and torsion appear in the hydrodynamics and needs to be separately addressed at the level of the constitutive relations in the presence of sources.…”

Momentum relaxation of holographic Weyl semimetal from massive gravity

“…However, near M/b ≈ 0, its value is less than 1 and decreases with the increasing of graviton mass. 6 To characterise the effects of non-zero graviton mass terms on the properties of phase transition, we will also study the critical point (M/b) c as a function of graviton mass. The critical value of the phase transition is encoded in the anomalous Hall conductivity, which is the point with maximum | ∂σ AHE ∂(M/b) |.…”

“…Therefore, the quasiparticles behave like Weyl fermions which are anomalous quantum mechanically, leading to lots of exotic transport phenomena in Weyl semimetals. Thus, it has attracted numerous theoretical and experimental interest [1][2][3][4][5][6]. Theoretically, most studies on Weyl semimetals are based on the topological band theory or the weakly coupled field theory.…”

Effective electromagnetic actions for Lorentz violating theories exhibiting the axial anomaly

“…Following this idea, the Fujikawa prescription to obtain the chiral anomalies [12,13] has also been used to calculate the effective electromagnetic action of different materials in Refs. [14][15][16]. Nevertheless, as pointed out in Ref.…”