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Geometry of Contact Terms in Linear Response: Applications to Elasticity

Ian Osborne, Gustavo Monteiro, Barry Bradlyn

Abstract

Employing the Kubo linear response formalism to calculate the elasticity of anisotropic systems has been shown to yield odd elastic moduli. For Hamiltonian systems, this result seems to be contradictory as it would violate energy conservation. To resolve this discrepancy, we examine the predictions of quantum linear response in the context of our expectation from classical elasticity theory. Our framework reveals that the geometry of the space of strain perturbations introduces correction factors to the correspondence between the Kubo formula and the elastic moduli which resolves the contradiction. We use a two-dimensional gas of electrons in a magnetic field as a pedagogical example. We use generalized f-sum rules to demonstrate how contact terms may reveal themselves in experimental measurements. Finally, we discuss the implications of our results for interpreting more general linear response functions.

Geometry of Contact Terms in Linear Response: Applications to Elasticity

Abstract

Employing the Kubo linear response formalism to calculate the elasticity of anisotropic systems has been shown to yield odd elastic moduli. For Hamiltonian systems, this result seems to be contradictory as it would violate energy conservation. To resolve this discrepancy, we examine the predictions of quantum linear response in the context of our expectation from classical elasticity theory. Our framework reveals that the geometry of the space of strain perturbations introduces correction factors to the correspondence between the Kubo formula and the elastic moduli which resolves the contradiction. We use a two-dimensional gas of electrons in a magnetic field as a pedagogical example. We use generalized f-sum rules to demonstrate how contact terms may reveal themselves in experimental measurements. Finally, we discuss the implications of our results for interpreting more general linear response functions.
Paper Structure (11 sections, 99 equations, 2 figures)

This paper contains 11 sections, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of Classical and Quantum Elasticity Theory
  3. Elastic Modulus from Kubo
  4. Prior Formulation of the Elastic Contact Term
  5. Present Formulation of the Elastic Contact Term
  6. Example - Non-interacting Electron Fluids
  7. Stress-Strain Form of the Kubo Formula and Sum rules
  8. Discussion
  9. Review of Quantum Hydrodynamics and Elasticity
  10. Anisotropy
  11. Calculation of $\delta_\Lambda \hat{T}$

Figures (2)

  • Figure 1: The diffeomorphism $\phi$ from the material manifold $\mathcal{M}$ to the laboratory (spatial) manifold $\mathcal{S}$ represents the time-dependent flow of a continuous medium (hydrodynamics) or the action of a dynamical strain applied to an equilibrium configuration (elasticity).
  • Figure 2: a) The natural method of combining strains acting on a quantum mechanical system is by composition, which gives a representation of the $GL(d,\mathbb R)$ group structure. Multiple $\phi$ maps act one after the other where the lab manifold becomes the new reference manifold. b) The group structure of classical elasticity is abelian by relying on information about the reference manifold, i.e. $\phi$ acts on the constant $\mathcal{M}$. Because of this, successive strain maps $\phi_1$ and $\phi_2$ are both maps from $\mathcal{M}$ to $\mathcal{S}$ and so cannot be composed. However, exploiting the additivity of vectors, the point-wise addition $\phi_1+\phi_2$ is (for small strains) a well-defined strain. For arbitrarily large elements of the abelian strain space it is possible to create degenerate strains, i.e. a strain which collapses at least one dimension of the system. This pathology is a consequence of working in an abelian space (note this problem does not occur in the $GL(d,\mathbb R)$ strain group) and is mostly ignored in the literature by considering "small strains."