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Linear Response Function

Лев П. Питаевский, S. Stringari

Published Jan 1, 2016• 3 citations
Publisher

Abstract

Abstract This chapter develops the formalism of linear response function, dynamic structure factor, dynamic polarizability, and sum rules. The behaviour of the density response function in a superfluid is discussed and analytic results at finite temperatures are shown in the case of the ideal and weakly interacting Bose gases. Sum rules are used to derive general inequalities holding at both zero and finite temperatures. The Hohenberg–Mermin–Wagner theorem and its generalization to zero temperature, based on the use of the uncertainty inequality, are explicitly discussed in the case of Bose systems.

DOI
10.1093/acprof:oso/9780198758884.003.0007

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Abstract This chapter develops the formalism of linear response function, dynamic structure factor, dynamic polarizability, and sum rules. The behaviour of the density response function in a superfluid is discussed and analytic results at finite temperatures are shown in the case of the ideal and weakly interacting Bose gases. Sum rules are used to derive general inequalities holding at both zero and finite temperatures. The Hohenberg–Mermin–Wagner theorem and its generalization to zero temperature, based on the use of the uncertainty inequality, are explicitly discussed in the case of Bose systems.
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.
The formalism of sum rules is developed and applied to investigate the collective oscillations and the correlation effects exhibited by interacting quantum gases, with special emphasis on the role of dimensionality. General inequalities are derived and used to obtain rigorous results on the behaviour of long range order. Fully analytic expressions for the density response function are shown in some relevant cases, including the Bogoliubov and the Tonks-Girardeau gases. Applications are presented in the presence of harmonic trapping as well as of optical lattices.
Correlation functions related to the dynamic density response of the one-dimensional Bose gas in the model of Lieb and Liniger are calculated. An exact Bose-Fermi mapping is used to work in a fermionic representation with a pseudopotential Hamiltonian. The Hartree-Fock and generalized random phase approximations are derived and the dynamic polarizability is calculated. The results are valid to first order in $1∕\ensuremath{\gamma}$, where $\ensuremath{\gamma}$ is Lieb-Liniger coupling parameter. Approximations for the dynamic and static structure factor at finite temperature are presented. The results preclude superfluidity at any finite temperature in the large-$\ensuremath{\gamma}$ regime due to the Landau criterion. Due to the exact Bose-Fermi duality, the results apply for spinless fermions with weak ppp-wave interactions as well as for strongly interacting bosons.
A density-functional theory is established for inhomogeneous superfluids at finite temperature, subject to time-dependent external fields in isothermal conditions. After outlining parallelisms between a neutral superfluid and a charged superconductor, Hohenberg-Kohn-Sham-type theorems are proved for gauge-invariant densities and a set of Bogolubov-Popov equations including exchange and correlation is set up. Earlier results applying in the linear response regime are recovered. 1 Experiments on confined Bose-condensed gases have revealed a rich variety of dynamical behaviours. These include elementary excitations of lowlying shape deformation modes [1], propagation of sound waves in both the condensate and the thermal cloud [2], Josephson-like oscillations in double condensates [3], creation of vortices [4], phase dynamics from various atomlaser configurations [5] and Bloch oscillations of a condensate in an optical
The dynamics of a confined fluid of Bose atoms is treated within the linear response regime, with a view to establishing a current-density functional formalism for an inhomogeneous superfluid state. After evaluating in full detail a simplified case of an external coupling to the density and phase of the condensate, the theory is extended to include the coupling to the total current density. The Kohn-Sham response functions of the condensate and all the exchange-correlation kernels for the superfluid are introduced from the microscopic equations of motion and are expressed in a physically transparent way through functional derivatives of correlation functions. A microscopic formula for the superfluid density is derived and used to introduce a generalized hydrodynamic approach for a weakly inhomogeneous two-fluid model in isothermal conditions. Local-density expressions are thereby derived for the velocities of first and second sound in the weakly inhomogeneous superfluid and for visco-elastic functions describing the transition from the hydrodynamic to the collisionless regime. Landau's hydrodynamic theory and known results in Green's functions language are recovered in the limiting case of a homogeneous superfluid.