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arXiv:2603.10144

Geometry of Contact Terms in Linear Response: Applications to Elasticity

Ian S. Osborne, Gustavo Bischir Monteiro, Barry Bradlyn

Published Mar 10, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2603.10144

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Abstract Linear response theory is a general formalism applicable to any physical system slightly departing from equilibrium. It allows us to express the linear response functions and the generalized susceptibilities in terms of the equilibrium correlation functions of the relevant dynamical variables. For instance, in the case of the response to an applied external field, one of the two relevant dynamical variables represents the physical quantity conjugate to the field whereas the other represents the physical quantity whose out-of-equilibrium average is measured. The computation of the linear response functions relies on a first-order perturbation expansion of the density operator (or of the phase space distribution function) of the system coupled to the field. The expressions thus obtained constitute the Kubo formulas of linear response theory.
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Quantum response theory is a cornerstone of statistical physics.However, the standard Kubo formalism is restricted to isolated equilibrium systems.We here generalize Kubo's results to open quantum systems in nonequilibrium steady states.We derive three different, but equivalent, forms of the quantum response function.We discuss for each of them the role of the noncommutativity of quantum operators and introduce a steady-state extension of the Kubo transformation.We show in particular that the equilibrium response vanishes for perturbations that commute with the unperturbed Hamiltonian, while the steady-state response does not, highlighting the profound difference between the two linear response theories.
The quantum geometric tensor and quantum Fisher information have recently been shown to provide a unified geometric description of the linear response of many-body systems. However, a similar geometric description of higher-order perturbative phenomena including nonlinear response in generic quantum systems is lacking. In this work, we develop a general framework for the time-dependent quantum geometry of many-body systems by treating external perturbing fields as coordinates on the space of density matrices. We use the Bures distance between the initial and time-evolved density matrix to define geometric quantities through a perturbative expansion. To lowest order, we derive a time-dependent generalization of the Bures metric related to the spectral density of linear response functions, unifying previous results for the quantum metric in various limits and providing a geometric interpretation of Fermi’s golden rule. At next order in the expansion, we define a time-dependent Bures-Levi-Civita connection for general many-body systems. We show that the connection is the sum of one contribution that is related to a second-order nonlinear response function, and a second contribution that captures the higher geometric structure of first-order perturbation theory. We show that in the quasistatic, zero-temperature limit for noninteracting fermions, this Bures connection reduces to the known expression for band-theoretic Christoffel symbols. Our work provides a systematic framework to explore many-body quantum geometry beyond the quantum metric and highlights how higher-order correlation functions can probe this geometry.
Introduction There are countless situations in physics when one is interested in calculating the response of a system to a small time-dependent perturbation acting on it. With some luck the response can be expanded in a power series of the strength of the perturbation, so that, to first order, it is a linear function of the latter. To compute this function is the objective of the linear response theory (LRT). Linear response theory has many important applications to the study of electronic matter. Virtually all interactions of electrons with experimental probes (electromagnetic fields, beams of particles) can be regarded as small perturbations to the system: if they were not, one would not be probing the system, but the system modified by the probe. Consequently, the results of these experiments can be expressed in terms of linear response functions, which are properties of the unperturbed system. In particular it will turn out that the analytic structure of these functions is entirely determined by the eigenvalues and eigenfunctions of the unperturbed system. Conversely, a measure of the linear response as a function of the frequency of the perturbation enables us to determine the excitation energies of the system. Beside being a cornerstone for the theory of single-particle properties to be developed in Chapter 8, the linear response functions also provide invaluable information in their own right. For example, as we will discuss in Chapter 5, the extent to which an external electrostatic potential is reduced by screening is controlled by the dynamical dielectric function which, in turn, is determined by the density–density response function.