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Drude and Superfluid Weights in Variational Theory: Application to the Gutzwiller Wavefunction

Balázs Hetényi

Published Aug 3, 2012• 0 citations
OpenAlex

Abstract

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.

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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 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 weight D and the dc conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$(T) of strongly correlated electrons are investigated theoretically. Analytic results are derived for the homogeneous phase of the Hubbard model at d=\ensuremath{\infty}, and for spinless fermions in this limit with 1/d corrections systematically included to lowest order. It is found that ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$(T) is finite for all T>0, displaying Fermi-liquid behavior, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$\ensuremath{\propto}1/T2{\mathit{T}}^{2}T2, at low temperatures. The validity of this result for finite dimensions is examined by investigating the importance of umklapp scattering processes and vertex corrections. A finite dc conductivity for T>0 is argued to be a generic feature of correlated lattice electrons in not too low dimensions.
The Drude weight, the quantity which distinguishes metals from insulators, is proportional to the second derivative of the ground state energy with respect to a flux at zero flux.The same expression also appears in the definition of the Meissner weight, the quantity which indicates superconductivity, as well as in the definition of non-classical rotational inertia of bosonic superfluids.It is shown that the difference between these quantities depends on the interpretation of the average momentum term, which can be understood as the expectation value of the total momentum (Drude weight), the sum of the expectation values of single momenta (rotational inertia of a superfluid), or the sum over expectation values of momentum pairs (Meissner weight).This distinction appears naturally when the current from which the particular transport quantity is derived is cast in terms of shift operators.
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.
We apply well-established finite-temperature quantum Monte Carlo techniques to one-dimensional Bose systems with soft- and hard-core constraints, as well as to spinless fermion systems. We give clear and robust numerical evidence that, as expected, no superfluid density for bosons or Meissner fraction for fermions is possible at any nonzero temperature in one-dimensional interacting Bose or Fermi lattice models, whereas a finite Drude weight is generally observed in gapless systems, in partial disagreement with previous expectations.