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arXiv:1208.0671 · Journal of the Physical Society of Japan

Current Response in Extended Systems as a Geometric Phase: Application to Variational Wavefunctions

Balázs Hetényi

Published Dec 4, 2012• 7 citations
arXivHTMLPDF

Abstract

The linear response theory for current is investigated in a variational context.Expressions are derived for the Drude and superfluid weights for general variational wavefunctions.The expression for the Drude weight highlights the difficulty in its calculation since it depends on the exact energy eigenvalues which are usually not available in practice.While the Drude weight is not available in a simple form, the linear current response is shown to be expressible in terms of a geometric phase, or alternatively in terms of the expectation value of the total position shift operator.The contribution of the geometric phase to the current response is then analyzed for some commonly used projected variational wavefunctions (Baeriswyl, Gutzwiller, and combined).It is demonstrated that this contribution is independent of the projectors themselves and is determined by the wavefunctions onto which the projectors are applied.

DOI
10.1143/jpsj.81.124711

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The linear response theory for current is investigated in a variational context.Expressions are derived for the Drude and superfluid weights for general variational wavefunctions.The expression for the Drude weight highlights the difficulty in its calculation since it depends on the exact energy eigenvalues which are usually not available in practice.While the Drude weight is not available in a simple form, the linear current response is shown to be expressible in terms of a geometric phase, or alternatively in terms of the expectation value of the total position shift operator.The contribution of the geometric phase to the current response is then analyzed for some commonly used projected variational wavefunctions (Baeriswyl, Gutzwiller, and combined).It is demonstrated that this contribution is independent of the projectors themselves and is determined by the wavefunctions onto which the projectors are applied.
The Drude and superfluid weights are investigated in the context of variational theory. It is shown that a commonly applied procedure to obtain the Drude weight actually corresponds to the superfluid weight. This realization changes previous interpretations of conductivity for variational wavefunctions, one result being that the Gutzwiller wavefunction displays a finite superfluid weight at finite interaction. To gain insight into DC conductivity an expression for the current response in terms of a geometric phase is derived. Applying to the Gutzwiller wavefunction it is shown that for any value of the interaction the Gutzwiller wavefunction is a perfect conductor.
The question of conductivity is revisited. Using the total momentum shift operator to construct the perturbed many-body Hamiltonian and ground state wave function the second derivative of the ground state energy with respect to the perturbing field is expressed in terms of the one and two-body momentum densities. The distinction between the adiabatic and envelope function derivatives, hence that between the Drude and superfluid weights, can be introduced in a straightforward manner. It is shown that a discontinuity in the momentum density leads to a contribution to the Drude weight, but not the superfluid weight, however a δ-function contribution in the two-body momentum density (such as in the BCS wave-funtion) contributes to both quantities. The connection between the discontinuity in the momentum density and localization is also demonstrated.
The question of conductivity is revisited. Using the total momentum shift operator to construct the perturbed many-body Hamiltonian and ground state wave function the second derivative of the ground state energy with respect to the perturbing field is expressed in terms of the one and two-body momentum densities. The distinction between the adiabatic and envelope function derivatives, hence that between the Drude and superfluid weights can be introduced in a straightforward manner. It is shown that a discontinuity in the momentum density leads to a contribution to the Drude weight, but not the superfluid weight, however a δδδ-function contribution in the two-body momentum density (such as in the BCS wave-funtion) contributes to both quantities. The connection between the discontinuity in the momentum density and localization is also demonstrated.
Taking the viewpoint that a complete solution of the Liouville equation contains much more information than is necessary in response theory (wherein one is concerned with a particular response), projection techniques are employed to introduce a shortcut in the problem. The treatment supplements and completes the formalism initiated in a previous publication and yields a general equation which is more relevant to response theory in the usual sense. The most natural application is to the theory of transport. The general transport equation is analyzed in the context of electrical conduction. Meaningful similarities to the simple classical Drude model of conduction are exhibited; the relationship of this exact formalism to the Kubo theory of linear response is shown, an exact solution for a step-function stimulus and approximation procedures which go beyond the linear regime are presented, and an expression for the electrical resistivity is explicitly calculated thus making contact with an earlier paper by Kenkre and Dresden. The response equation is also analyzed in the context of electric and magnetic polarization.
Abstract The superfluid weight is an important observable of superconducting materials since it is related to the London penetration depth of the Meissner effect. It can be computed from the change in the grand potential (or free energy) in response to twisted boundary conditions in a torus geometry. Here we review the Bardeen–Cooper–Schrieffer mean-field theory emphasizing its origin as a variational approximation for the grand potential. The variational parameters are the effective fields that enter in the mean-field Hamiltonian, namely the Hartree–Fock potential and the pairing potential. The superfluid weight is usually computed by ignoring the dependence of the effective fields on the twisted boundary conditions. However, it has been pointed out in recent works that this can lead to unphysical results, particularly in the case of lattice models with flat bands. As a first result, we show that taking into account the dependence of the effective fields on the twisted boundary conditions leads in fact to the generalized random phase approximation. Our second result is providing the mean-field grand potential as an explicit function of the one-particle density matrix. This allows us to derive the expression for the superfluid weight within the generalized random phase approximation in a transparent manner. Moreover, reformulating mean-field theory as a well-posed minimization problem in terms of the one-particle density matrix is a first step towards the application to superconducting systems of the linear scaling methods developed in the context of electronic structure theory.
Based on the method of hydrodynamic projections we derive a concise formula for the Drude weight of the repulsive Lieb-Liniger \delta δ -Bose gas. Our formula contains only quantities which are obtainable from the thermodynamic Bethe ansatz. The Drude weight is an infinite-dimensional matrix, or bilinear functional: it is bilinear in the currents, and each current may refer to a general linear combination of the conserved charges of the model. As a by-product we obtain the dynamical two-point correlation functions involving charge and current densities at small wavelengths and long times, and in addition the scaled covariance matrix of charge transfer. We expect that our formulas extend to other integrable quantum models.