Hall number across a van Hove singularity

Abstract

In the context of the relaxation time approximation to Boltzmann transport theory, we examine the behavior of the Hall number, nH , of a metal in the neighborhood of a Lifshitz transition from a closed Fermi surface to open sheets. We find a universal non-analytic dependence of nH on the electron density in the high field limit, but a non-singular dependence at low fields. The existence of an assumed nematic transition produces a doping dependent nH similar to that observed in recent experiments in the high temperature superconductor YBa2Cu3O7−x.Introduction.-In the absence of superconductivity or exotic fractionalized phases, the low energy elementary excitations of a conducting system are typically the well-known quasiparticles of Fermi liquid theory. In sufficiently clean systems, much about the character of these excitations, and in particular, information concerning the geometry and topology of the Fermi surface, can be inferred most sensitively from transport experiments. Specifically, in many circumstances, the Hall number, n H ≡ (B/e)(1/ρ xy ), in the T → 0 limit can give information about the volume (area in 2D) enclosed by the Fermi surface. [1,2]. From this, one may extract insight concerning the existence of a putative broken symmetry state that "reconstructs" the Fermi surface. For example, density wave order that breaks translational symmetry, changes not only the topology of the Fermi surface, but the volume enclosed as well. In contrast, the constraints of Luttinger's theorem seemingly imply that Fermi surface changes produced by translation symmetry preserving orders, such as Ising nematic order, will be invisible to a measurement of the Hall number.There are, however, important caveats to using the Hall number as a proxy for the electron density of a metal. In the absence of Galilean invariance, it is only the B → ∞ limit of the Hall number that corresponds to the carrier density [2]. The B → 0 limit of the Hall number is sensitive to the momentum dependence of the Fermi velocity, and is related in a complicated way [3] to the dominant scattering processes and curvature of the Fermi surface. For open Fermi surfaces, the Hall number is in general a non-universal quantity, and is not related to the density in any simple fashion in either the strong or weak field limit. In fact, little is known about the critical behavior of the Hall number at the topological Lifshitz phase transition between open and closed Fermi surfaces. While there is intuitively no reason to expect singular behavior in the limit B → 0, since the Fermi surface is locally unchanged across the van Hove singularity, there is every reason to expect singular behavior at high fields, where quasiparticles exhibit many cyclotron orbits before being scattered, and so are sensitive to the global topology of the Fermi surface.In this Letter, we address these issues via exact solution of the Boltzmann equation in the relaxation time approximation for a two dimensional nearest-neighbor tight binding model, and by numerical solution of mode...

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