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Physical Review Letters

f -Sum Rules for Dissipative Systems

Xinxin Yang, Bohao Wu, Yu Chen, Wei Zhang

Published Sep 4, 2024• 2 citations
Publisher

Abstract

The f-sum rules set general constraints on the response of a quantum many-body system to an external probe and hold significant relevance in the realm of various spectroscopy measurements. In practical conditions, a system unavoidably couples with the environment and acquires effective dissipation. In this Letter, we derive and prove a set of f-sum rules for dissipative systems. Within the framework of linear response theory, we obtain the system response in both linear order of probe field and dissipation parameter. We formulate and prove one first-order and two second-order dissipative f-sum rules. These rules are validated numerically for some example models, and the realization schemes are proposed. In addition, the potential applications are discussed for two interacting many-body systems.

DOI
10.1103/physrevlett.133.100401

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The f-sum rules set general constraints on the response of a quantum many-body system to an external probe and hold significant relevance in the realm of various spectroscopy measurements. In practical conditions, a system unavoidably couples with the environment and acquires effective dissipation. In this Letter, we derive and prove a set of f-sum rules for dissipative systems. Within the framework of linear response theory, we obtain the system response in both linear order of probe field and dissipation parameter. We formulate and prove one first-order and two second-order dissipative f-sum rules. These rules are validated numerically for some example models, and the realization schemes are proposed. In addition, the potential applications are discussed for two interacting many-body systems.
Sum rules for response functions represent some of the few exact results in many-body physics, relating the dynamical response functions to equilibrium ground state properties. To date, however, sum rules have been examined primarily for linear response functions. In this work, the authors introduce a spectral density representation for second-order response functions, using it to systematically derive families of sum rules, including a nonlinear extension of the well-known fff-sum rule. This work places important constraints on future nonlinear spectroscopic experiments.
Various sum rules, accounting for the coupling between density and particle excitations and emphasizing in an explicit way the role of Bose–Einstein condensation, are discussed. Important consequences on the fluctuations of the particle operator as well as on the structure of elementary excitations are reviewed. These include a recent generalization of the Hohenberg–Mermin–Wagner theorem holding at zero temperature. Introduction The sum rule approach has been employed extensively in the literature in order to explore various dynamic features of quantum many body systems from a microscopic point of view (see [1] and references therein). An important merit of the method is its explicit emphasis on the role of conservation laws and of the symmetries of the problem. Furthermore, the explicit determination of sum rules is relatively easy and often requires only a limited knowledge of the system. Usually the sum rule approach is, however, employed without giving special emphasis to the possible occurrence of (spontaneously) broken symmetries. For example, the most famous f -sum rule [2] holding for a large class of systems is not affected by the existence of an order parameter in the system. The purpose of this paper is to discuss a different class of sum rules which are directly affected by the presence of a broken symmetry. These sum rules can be used to predict significant properties of the system which are the consequence of the existence of an order parameter.
Quantum dissipation involves both energy relaxation and decoherence, leading toward quantum thermal equilibrium. There are several theoretical prescriptions of quantum dissipation but none of them is simple enough to be treated exactly in real applications. As a result, formulations in different prescriptions are practically used with different approximation schemes. This review examines both theoretical and application aspects on various perturbative formulations, especially those that are exact up to second-order but nonequivalent in high-order system-bath coupling contributions. Discrimination is made in favor of an unconventional formulation that in a sense combines the merits of both the conventional time-local and memory-kernel prescriptions, where the latter is least favorite in terms of the applicability range of parameters for system-bath coupling, non-Markovian, and temperature. Also highlighted is the importance of correlated driving and dissipation effects, not only on the dynamics under strong external field driving, but also in the calculation of field-free correlation and response functions.
Sum rules have played an important role in the development of many branches of physics since the earliest days of quantum mechanics. We present examples of one-dimensional quantum mechanical sum rules and apply them to the infinite well and the single δ-function potential. These examples illustrate the different ways in which these sum rules can be realized and the varying techniques by which they can be confirmed. We use the same methods to evaluate the second-order energy shifts arising from the introduction of a constant external field, namely the Stark effect.
Optimal quantum control in dissipative environments is formulated combining the optimal control theory from the Wilson group with the projection operator technique. In the weak response limit, a set of formally exact equations defines the optimal weak field for arbitrary target states and arbitrary system-bath interactions. Perturbative approximations are then made to derive general equations for weakly dissipative non-Markovian systems, which are amenable to analytical and/or numerical solutions. Potential applications of these results are discussed.
While the theory of linear response functions in many body theory has developed into a substantial lore over the past decades, rather little is known about the properties and behavior of higher order (quadratic, etc.) response functions of many body systems. In this poster we first review essential elements of quadratic response theory such as the quadratic fluctuation dissipation theorems and sum rules for the quadratic density response function. Next we show how the correlational local field G(k,ω) in the linear response is fundamentally linked to the quadratic response function which opens a way to a non-perturbational analysis of the former. We proceed to present the results of the explicit calculation of the quadratic density response of 2D and 3D electron gas. Both, real and imaginary part of the latter reveal an extremely rich behavior. Of particular interest is the comparison between the two- and the three-dimensional case.