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Dynamics of the Hubbard model: weakly excited pairing states

Arianna Montorsi, Vittorio Penna

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

According to a recent proposal, we describe the quantum dynamics of the Hubbard model at the semiclassical level by implementing the Time-Dependent Variational Principle (TDVP) approach on appropriate macroscopic wave-functions constructed in terms of su(2)-coherent states. Within the TDVP procedure, such states turn out to include a time-dependent quantum phase, part of which can be recognized as Berry's phase. In particular, here we investigate the dynamics of the weakly excited states with non-vanishing pairing, away from half-filling and for repulsive Coulomb interaction.

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According to a recent proposal, we describe the quantum dynamics of the Hubbard model at the semiclassical level by implementing the Time-Dependent Variational Principle (TDVP) approach on appropriate macroscopic wave-functions constructed in terms of su(2)-coherent states. Within the TDVP procedure, such states turn out to include a time-dependent quantum phase, part of which can be recognized as Berry's phase. In particular, here we investigate the dynamics of the weakly excited states with non-vanishing pairing, away from half-filling and for repulsive Coulomb interaction.
We describe the quantum dynamics of the Hubbard model at the semiclassical level, by implementing the time-dependent variational principle (TDVP) procedure on appropriate macroscopic wave functions constructed in terms of SU(2)-coherent states. Within the TDVP procedure, such states turn out to include a time-dependent quantum phase, part of which can be recognized as Berry's phase. We derive two semiclassical model Hamiltonians for describing the dynamics in the paramagnetic, superconducting, antiferromagnetic and charge-density wave phases and solve the corresponding canonical equations of motion in various cases. Noticeably, a vortexlike ground-state phase dynamics is found to take place for U>0 away from half filling. Moreover, it appears that an oscillatorylike ground-state dynamics survives at the Fermi surface at half filling for any U. The low-energy dynamics is also exactly solved by separating fast and slow variables. The role of the time-dependent phase is shown to be particularly interesting in the ordered phases.
We propose a scheme for investigating the quantum dynamics of interacting electron models by means of a time-dependent variational principle and spin coherent states of space lattice operators. We apply such a scheme to the one-dimensional Hubbard model, and solve the resulting equations in different regimes. In particular, we find that at low densities the dynamics is mapped into two coupled nonlinear Schr\"odinger equations, whereas near half-filling the model is described by two coupled Josephson-junction arrays. Focusing then to the case in which only the phases of the spin variables are dynamically active, we examine a number of different solutions corresponding to the excitations of few macroscopic modes. Based on fixed-point equations of the simpler among them, we show that the standard one-band ground-state phase space is found.
Quantum dynamics of the Bose-Hubbard model is investigated through a semiclassical Hamiltonian picture provided by the time-dependent variational principle method. The system is studied within a factorized slow/fast dynamics. The semiclassical requantization procedure allows one to account for the strong quantum nature of the system when $t/U\ensuremath{\ll}1$. The phase diagram is in good agreement with quantum Monte Carlo results and third order strong coupling perturbative expansion.
We systematically extend Bogoliubov theory beyond the mean-field approximation of the Bose-Hubbard model in the superfluid phase. Our approach is based on the time-dependent variational principle applied to the family of all Gaussian states (i.e., Gaussian TDVP). First, we find the best ground-state approximation within our variational class using imaginary time evolution in 1D, 2D, and 3D. We benchmark our results by comparing to Bogoliubov theory and DMRG in 1D. Second, we compute the approximate one- and two-particle excitation spectrum as eigenvalues of the linearized projected equations of motion (linearized TDVP). We find the gapless Goldstone mode, a continuum of two-particle excitations and a doublon mode. We discuss the relation of the gap between Goldstone mode and two-particle continuum to the excitation energy of the Higgs mode. Third, we compute linear response functions for perturbations describing density variation and lattice modulation and discuss their relations to experiment. Our methods can be applied to any perturbations that are linear or quadratic in creation/annihilation operators. Finally, we provide a comprehensive overview how our results are related to well-known methods, such as traditional Bogoliubov theory and random phase approximation.
An effective Hamiltonian for the Kohn-Luttinger superconductor is constructed and solved in the BCS approximation. The method is applied to the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model in two dimensions with the following results: (i) The superconducting phase diagram at half filling is shown to provide a weak-coupling analog of the recently proposed spin liquid state in the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. (ii) In the parameter region relevant for the cuprates, we have found a nontrivial energy dependence of the gap function in the dominant d-wave pairing sector. The hot spot effect in the angular dependence of the superconducting gap is shown to be quite weak.
A simple and very flexible variational approach to the out-of-equilibrium quantum dynamics in strongly correlated electron systems is introduced through a time-dependent Gutzwiller wave function. As an application, we study the simple case of a sudden change of the interaction in the fermionic Hubbard model and find at the mean-field level an extremely rich behavior. In particular, a dynamical transition between small and large quantum quench regimes is found to occur at half-filling, in accordance with the analysis of Eckstein, Phys. Rev. Lett. 103, 056403 (2009)10.1103/PhysRevLett.103.056403, obtained by dynamical mean-field theory, that turns into a crossover at any finite doping.