In this work, we analyse the variational problem emerging from the Gutzwiller approach to strongly correlated systems. This problem comprises two main steps: evaluation and minimization of the ground state energy for the postulated Gutzwiller Wave Function. We discuss the available methods for evaluating , in particular the recently proposed diagrammatic expansion method. We compare the two existing approaches to minimize : the standard approach based on the effective single-particle Hamiltonian and the so-called Statistically-consistent Gutzwiller Approximation (SGA). On the example of the superconducting phase analysis, we show that these approaches lead to the same minimum as it should be. However, the calculations within the SGA method are easier to perform and the two approaches allow for a simple cross-check of the obtained results. Finally, we show two ways of solving the equations resulting from the variational procedure, as well as how to incorporate the condition for a fixed number of particles.
In this work, we analyse the variational problem emerging from the Gutzwiller approach to strongly correlated systems. This problem comprises two main steps: evaluation and minimization of the ground state energy for the postulated Gutzwiller Wave Function. We discuss the available methods for evaluating , in particular the recently proposed diagrammatic expansion method. We compare the two existing approaches to minimize : the standard approach based on the effective single-particle Hamiltonian and the so-called Statistically-consistent Gutzwiller Approximation (SGA). On the example of the superconducting phase analysis, we show that these approaches lead to the same minimum as it should be. However, the calculations within the SGA method are easier to perform and the two approaches allow for a simple cross-check of the obtained results. Finally, we show two ways of solving the equations resulting from the variational procedure, as well as how to incorporate the condition for a fixed number of particles.
We study the superconducting phase of the Hubbard model using the Gutzwiller variational wave function (GWF) and the recently proposed diagrammatic expansion technique (DE-GWF). The DE-GWF method works on the level of the full GWF and in the thermodynamic limit. Here, we consider a finite-size system to study the accuracy of the results as a function of the system size (which is practically unrestricted). We show that the finite-size scaling used, e.g. in the variational Monte Carlo method can lead to significant, uncontrolled errors. The presented research is the first step towards applying the DE-GWF method in studies of inhomogeneous situations, including systems with impurities, defects, inhomogeneous phases, or disorder.
The principal purpose of this topical review is to single out some of the universal features of high-temperature (high-Tc) and related strongly-correlated systems, which can be compared with experiment in a quantitative manner. The description starts with the concept of exchange-interaction-mediated (real-space) pairing, combined with strong correlations among narrow band electrons, for which the reference state is that of the Mott–Hubbard insulator. The physical discussion of concrete properties relies on variational approach, starting from generalized renormalized mean-field theory (RMFT) in the form of statistically-consistent Gutzwiller approximation (SGA), and its subsequent generalization, i.e., a systematic Diagrammatic Expansion of the Variational (Gutzwiller-type) Wave Function (DE-GWF). The solution leads to the two energy scales, one involving unconventional quasiparticles close to the Fermi energy, and the other reflecting the fully correlated state, involving electrons deeper below the Fermi surface. Those two regimes are separated by a kink in the dispersion relation, which is observed in photoemission. As a result, one obtains both the doping dependent properties in the correlated state, and effective renormalized quasiparticles. The reviewed ground-state characteristics for high-Tc systems encompass high-Tc superconductivity, nematicity, charge- (and pair-) density-wave effects, as well as non-BCS kinetic energy gain in the paired state. The calculated dynamic properties are: the universal Fermi velocity, the Fermi wave-vector, the effective mass enhancement, the pseudogap; and the d-wave gap magnitude. We discuss that, within the variational approach, the minimal realistic model is represented by the so-called t-J-U Hamiltonian. Inadequacy of the t-J and Hubbard models, particularly of their RMFT versions, is discussed explicitly. For heavy fermion systems, modeled by the Anderson lattice model and with the DE-GWF approach, we discuss the phase diagram, encompassing superconducting, Kondo insulating, ferro- and anti-ferromagnetic states. The superconducting state is then a two-d-wave gap system. If the orbital degeneracy of f-electrons is included, the coexistent ferromagnetic- (spin–triplet) superconducting phases appear and match those observed for UGe2 in a semiquantitative manner. Finally, in the second part, we generalize our approach to the collective spin and charge fluctuations in high-Tc systems, starting from variational approach, defining the saddle-point state, and combining it with 1/N expansion. The present scheme differs essentially from that starting from the saddle-point Hartree–Fock approximation, and incorporating the fluctuations in the random phase approximation (RPA). The spectrum of collective spin and charge excitations is determined for the Hubbard and t-J-U models, and subsequently compared quantitatively with recent experiments. The Appendices provide formal details to make this review self-contained.
A systematic diagrammatic expansion for Gutzwiller wavefunctions (DE-GWFs) proposed very recently is used for the description of the superconducting (SC) ground state in the two-dimensional square-lattice t – J model with the hopping electron amplitudes t (and ) between nearest (and next-nearest) neighbors. For the example of the SC state analysis we provide a detailed comparison of the methodʼs results with those of other approaches. Namely, (i) the truncated DE-GWF method reproduces the variational Monte Carlo (VMC) results and (ii) in the lowest (zeroth) order of the expansion the method can reproduce the analytical results of the standard Gutzwiller approximation (GA), as well as of the recently proposed 'grand-canonical Gutzwiller approximation' (called either GCGA or SGA). We obtain important features of the SC state. First, the SC gap at the Fermi surface resembles a wave only for optimally and overdoped systems, being diminished in the antinodal regions for the underdoped case in a qualitative agreement with experiment. Corrections to the gap structure are shown to arise from the longer range of the real-space pairing. Second, the nodal Fermi velocity is almost constant as a function of doping and agrees semi-quantitatively with experimental results. Third, we compare the doping dependence of the gap magnitude with experimental data. Fourth, we analyze the k -space properties of the model: Fermi surface topology and effective dispersion. The DE-GWF method opens up new perspectives for studying strongly correlated systems, as it (i) works in the thermodynamic limit, (ii) is comparable in accuracy to VMC, and (iii) has numerical complexity comparable to that of the GA (i.e., it provides the results much faster than the VMC approach).
Abstract We propose a systematic approach to the systems of correlated electrons, the so-called -DE-GWF method, based on reciprocal-space ( -resolved) diagrammatic expansion of the variational Gutzwiller-type wave function for parametrized models of correlated fermions. The present approach, in contrast to either variational Monte-Carlo (VMC), or the recently developed real-space diagrammatic expansion of the Gutzwiller-type wave function (direct-space DE-GWF technique), is applicable directly in the thermodynamic limit and thus is suitable for describing selected singular features of the wave-vector-dependent quantities. We employ the -DE-GWF method to extract the non-analytic part of the two leading moments of the fermion spectral-density function across the (two-dimensional) Brillouin zone for the Hubbard model and away from the half-filling. Those moments are used to evaluate the nodal quasiparticle velocities and their spectral weights in the correlated superconducting state. The two velocities determined in that manner exhibit scaling with the electron concentration qualitatively different from that obtained earlier for the excited states of the high- T c cuprates within the projected quasi-particle ansatz, and the results are in a very good quantitative agreement with experimental data if interpreted as those characterizing the spectrum below and above the observed kink. We provide a detailed discussion of the two gaps and two excitation branches (two velocities) appearing naturally within our DE-GWF approach. The two separate sets of characteristics distinguish the renormalized quasiparticle states very close to the Fermi surface from the deeper correlated-state properties. Also, an enhancement of the -dependent magnetic susceptibility is shown to contain a spin-fluctuation contribution within our language. Finally, the -DE-GWF approach is compared to both the VMC and real-space DE-GWF results for the cases of Hubbard and t - J - U models.
This work covers three lectures on Gutzwiller wave functions for correlated electron systems. In the first lecture, some basic ideas on model Hamiltonians and their motivation are outlined, Gutzwiller's variational approach to Landau Fermi-liquid theory is sketched, and, as a first application, the theory is applied to the one-band Hubbard model where the Brinkman–Rice metal-to-insulator transition is found. The second lecture is devoted to the perturbative calculation of expectation values. In this rather technical part the variational analog of Wick's theorem, Feynman diagrams, and the linked-cluster theorem are introduced. The important differences between the variational approach and the standard Green function technique are the absence of a time-dependence and the flexibility to select the expansion parameter of the variational perturbation theory. When this flexibility is used appropriately, not a single self-energy diagram must be calculated in the limit of large dimensions, i.e. in the limit of large lattice coordination number the expectation values can be evaluated exactly for general Hubbard-type models for all interaction strengths and band fillings. The third lecture covers the application of the theory to ferromagnetism of nickel. After a discussion of the nickel problem, multi-band Hubbard models and Gutzwiller wave functions are introduced and parameterized. The favourable comparison between the results from Gutzwiller theory and a large number of experiments shows that correlations play an important role in itinerant ferromagnets.
A new analytic approach to the evaluation of ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function is presented. It is based on the observation that expectation values in terms of this wave function may be expressed by sums over different lattice sites. This makes the application of Wick's theorem and the resulting contractions extremely simple, since the latter involve only anticommuting numbers as in a Grassmann algebra. Expressions for the momentum distribution ${n}_{\mathrm{k}\ensuremath{\sigma}}$ and the Hubbard interaction in terms of a power series in a particular correlation parameter are derived which are valid for all dimensions. An explicit diagrammatic evaluation of the coefficients is described. In one dimension these coefficients may be determined to all orders which yields an approximation-free calculation of nk and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.