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International Journal of Quantum Chemistry

Response theory of open quantum systems

C. H. Obcemea

Published Mar 1, 1984• 1 citation
Publisher

Abstract

We discuss in this paper the various objections to the Kubo formalism initiated by van Kampen. We motivate the need to embed the system of interest in a larger reservoir to resolve these objections using the van Hove limit. In particular, in the context of spin magnetic systems, we show that it is precisely the spin–lattice coupling together with the above limit which is responsible for the validity of the linearization ansatz.

DOI
10.1002/qua.560260807

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We discuss in this paper the various objections to the Kubo formalism initiated by van Kampen. We motivate the need to embed the system of interest in a larger reservoir to resolve these objections using the van Hove limit. In particular, in the context of spin magnetic systems, we show that it is precisely the spin–lattice coupling together with the above limit which is responsible for the validity of the linearization ansatz.
A critical discussion of linear response theory is given. It is argued that in the formalism as it stands no dissipation is manifest. A physical reinterpretation for the case of a system in weak interaction with a reservoir is given. Mathematically this means that the van Hove limit, as well as the large system limit, is applied to the time-dependent Heisenberg operators of the Kubo formalism. The reduced operators can be put in a very compact form, viz.,BRα(t) =[exp(−Λdt)]Bα, where Bα is a Schrödinger operator and Λd is the Liouville space superoperator corresponding to the transition operator of the master equation. In this form the relaxation character of the transport expressions, and the approach to equilibrium is at once evident. New expressions for the generalized susceptibility and conductivity in this limit are presented. Also, the Onsager relations and other symmetry properties are confirmed.
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 static limit of the Kubo susceptibility describes the adiabatic response of an isolated quantum system, in which energy-level occupations remain unchanged, and generally differs from the isothermal susceptibility, where these occupations readjust to thermal equilibrium. In this work, we introduce a single dynamical framework in which the adiabatic and isothermal responses arise naturally as two distinct limiting regimes. This can be done by considering explicitly an open system version of the Kubo approach in which the dissipation is governed by a typical thermalization rate of the system. By exchanging the order of the static limit and of the vanishing thermalization rate limit, the protocol yields either the adiabatic or isothermal response. This protocol has a solid physical motivation in terms of the thermalization dynamics and is valid for generic systems. We apply it to the case of the magnetic response of the electron gas and reproduce the expected results.
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.