We develop an extension of the Gutzwiller approximation to finite temperatures based on the Dirac-Frenkel variational principle. Our method does not rely on any entropy inequality, and is substantially more accurate than the approaches proposed in previous works. We apply our theory to the single-band Hubbard model at different fillings, and show that our results compare quantitatively well with dynamical mean field theory in the metallic phase. We discuss potential applications of our technique within the framework of first-principle calculations.
A new approach is presented for variational Monte Carlo simulations of Hubbard-Anderson models using Gutzwiller projection operators to restrict charge fluctuations in noninteracting wave functions. The Gutzwiller matrix elements are mapped onto a statistical model, which is evaluated using a stochastic algorithm that is normally applied to finite-temperature simulation. The method has been successfully tested for one-dimensional (1D) Hubbard and 1D Anderson models, and results are presented for the 2D Hubbard model. We find a region close to the half-filled limit where the Gutzwiller states of different magnetic ordering are very close in energy.
We introduce Gutzwiller-correlated wave functions for the variational investigation of general multi-band Hubbard models. We set up a diagrammatic formalism which allows us to evaluate analytically ground-state properties in the limit of infinite spatial dimensions. In this limit recent results obtained within the Gutzwiller approximation are seen to become exact for these wave functions. We further show that the slave-boson mean-field theory for degenerate bands becomes variationally controlled at zero temperature in infinite dimensions. Lastly, we briefly comment on the variational approach to the Anderson transition in strongly correlated electron systems.
We formulate a finite-temperature scheme of the variational cluster approximation (VCA) particularly suitable for an exact-diagonalization cluster solver. Based on the analytical properties of the single-particle Green's function matrices, we explicitly show the branch-cut structure of logarithm of the complex determinant functions appearing in the self-energy-functional theory (SFT) and whereby construct an efficient scheme for the finite-temperature VCA. We also derive the explicit formulas for entropy and specific heat within the framework of the SFT. We first apply the method to explore the antiferromagnetic order in a half-filled Hubbard model by calculating the entropy, specific heat, and single-particle excitation spectrum for different values of on-site Coulomb repulsion
U and temperature
T. We also calculate the
T dependence of the single-particle excitation spectrum in the strong coupling region, and discuss the overall similarities to and the fine differences from the spectrum obtained by the spin-density-wave mean-field theory at low temperatures and the Hubbard-I approximation at high temperatures. Moreover, we show a necessary and sufficient condition for the third law of thermodynamics in the SFT. On the basis of the thermodynamic properties, such as the entropy and the double occupancy, calculated via the
T and/or
U derivative of the grand potential, we obtain a crossover diagram in the
(U,T) plane, which separates a Slater-type insulator and a Mott-type insulator. Next, we demonstrate the finite-temperature scheme in the cluster-dynamical-impurity approximation (CDIA), i.e., the VCA with noninteracting bath orbitals attached to each cluster, and study the paramagnetic Mott metal-insulator transition in the half-filled Hubbard model. Formulating the finite-temperature CDIA, we first address a subtle issue regarding the treatment of the artificially introduced bath degrees of freedom which are absent in the originally considered Hubbard model. We then apply the finite-temperature CDIA to calculate the finite-temperature phase diagram in the
(U,T) plane. Metallic, insulating, coexistence, and crossover regions are distinguished from the bath-cluster hybridization-variational-parameter dependence of the grand-potential functional. We find that the Mott transition at low temperatures is discontinuous, and the coexistence region of the metallic and insulating states persists down to zero temperature. The result obtained here by the finite-temperature CDIA is complementary to the previously reported zero-temperature CDIA phase diagram.
We review the recently proposed extension of the Gutzwiller approximation, M. Schiro' and M. Fabrizio, Phys. Rev. Lett. 105, 076401 (2010), designed to describe the out-of-equilibrium time-evolution of a Gutzwiller-type variational wave function for correlated electrons. The method, which is strictly variational in the limit of infinite lattice-coordination, is quite general and flexible, and it is applicable to generic non-equilibrium conditions, even far beyond the linear response regime. As an application, we discuss the quench dynamics of a single-band Hubbard model at half-filling, where the method predicts a dynamical phase transition above a critical quench that resembles the sharp crossover observed by time-dependent dynamical mean field theory. We next show that one can actually define in some cases a multi-configurational wave function combination of a whole set of mutually orthogonal Gutzwiller wave functions. The Hamiltonian projected in that subspace can be exactly evaluated and is equivalent to a model of auxiliary spins coupled to non-interacting electrons, closely related to the slave-spin theories for correlated electron models. The Gutzwiller approximation turns out to be nothing but the mean-field approximation applied to that spin-fermion model, which displays, for any number of bands and integer fillings, a spontaneous
Z2 symmetry breaking that can be identified as the Mott insulator-to-metal transition.
For weakly interacting many-body quantum systems one can apply the extensive framework of perturbative quantum field theory. On the other hand, strong interactions can not be treated practically in perturbation theory because of the absence of a small control parameter. In this context variational methods are a powerful tool to obtain reliable informations regarding the properties of the ground state of the strongly correlated electron system. In this thesis we present novel formulations of two widely used methods: the Gutzwiller wave function and the slave-spin theory. These techniques share a similar philosophy and the extensions are based on the common attitude towards a more detailed description of the high-energy incoherent excitations characterizing interacting electron systems together with low-energy quasiparticles excitations, captured by standard approaches. Our task is not only crucial for a faithful description of the insulating phase but also improves the variational characterization of the low-energy quasiparticle excitations. We apply the novel ghost-Gutzwiller wave function technique to tackle an intriguing phenomenon, namely the exciton Mott transition in photoexcited semiconductors. Despite being a quite old topic that goes back to the 1970's, the nature of the exciton Mott transition still defies a complete understanding. By considering an idealised model of photoexcited semiconductors we unveil the important role of the exciton binding energy in determining the nature of the transition. Moreover, our results uncover rather anomalous electron-hole liquid phase next to the transition, which still sustains excitonic excitations although being a degenerate Fermi liquid of heavy mass quasiparticles. By means of the Dirac-Frenkel variational principle, we generalize the ghost-Gutzwiller wave function to study the out of equilibrium dynamics of strongly correlated electron systems. Numerical results on the single-band Hubbard model show a remarkable agreement with those obtained with time-dependent dynamical mean field theory. We believe that the method opens the way to several promising developments and future applications. Concerning the slave-particle approach, we consider Anderson impurity models and we show that, within our formulation, the constraint on the slave-spin variable can be removed, leaving thermal average free by any projection on the "physical" subspace of the enlarged Hilbert space. To the best of our knowledge, this represents an exception and shows that our formulation is more convenient than other slave-particle approaches, where the constraint have to be implemented explicitly. The method, suited to deal with impurity models, finds direct application in studying time-dependent transport across an interacting quantum dot. To this aim we formulate, by means of the Keldysh Green's function, the slave-spin mean-field theory in the out of equilibrium framework and we apply the method to the dynamics of a driven quantum dot. Finally we study the current-voltage characteristic of an interacting quantum dot tunnel-coupled to the edge of a superconductive nanowire. The appearance of the topological Majorana edge mode is signalized by distinctive features in the charge transport properties of the junction. Apart from the well known half-integer zero bias conductance we find that the topological region is characterized by a vanishing Fano factor, that can be measured in experiments to detect the presence of a Majorana zero mode hybridized with the quantum dot.