A variational treatment of the Gutzwiller---renormalized $t\text{\ensuremath{-}}J$ Hamiltonian combined with the mean-field (MF) approximation is proposed, with a simultaneous inclusion of additional consistency conditions. Those conditions guarantee that the averages calculated variationally coincide with those calculated from the self-consistent equations. This is not ensured a priori because the effective Hamiltonian contains renormalization factors which depend explicitly on the mean-field averages. A comparison with previous mean-field treatments is made for both superconducting $(d\text{-RVB})$ and normal states and encompasses calculations of the superconducting gap and the renormalized hopping amplitudes, as well as the electronic structure. The ${C}_{4v}$-symmetry breaking in the normal phase---the Pomeranchuk instability---is also analyzed.
A variational treatment of the Gutzwiller---renormalized t\ensuremath-J Hamiltonian combined with the mean-field (MF) approximation is proposed, with a simultaneous inclusion of additional consistency conditions. Those conditions guarantee that the averages calculated variationally coincide with those calculated from the self-consistent equations. This is not ensured a priori because the effective Hamiltonian contains renormalization factors which depend explicitly on the mean-field averages. A comparison with previous mean-field treatments is made for both superconducting (d-RVB) and normal states and encompasses calculations of the superconducting gap and the renormalized hopping amplitudes, as well as the electronic structure. The C4v-symmetry breaking in the normal phase---the Pomeranchuk instability---is also analyzed.
Recently, Fukushima [Phys. Rev. B \textbf{78} 115105 (2008)] proposed a systematic derivation of the Gutzwiller approximation for the t-J model. In the present paper, using this approach we construct an effective single-particle Hamiltonian, which leads to a renormalized mean-field theory (RMFT). We also use the method proposed by us recently and based on the maximum entropy principle (MaxEnt), which in turn, yields a consistent statistical description of the problem. On the examples of non-magnetic superconducting d-wave resonating valence bond (dRVB) and normal staggered-flux (SF) solutions, we compare two selections of the Gutzwiller renormalization schemes, i.e. the one proposed by Fukushima with that used earlier by Sigrist et al. [Phys. Rev. B \textbf{49}, 12 058 (1994)]. We also confront the results coming from our variational solutions with the self-consistency conditions build in, with those of the non-variational approach based on the Bogoliubov-de Gennes self-consistent equations. Combination of the present variational approach with the new renormalization scheme (taken from Fukushima's work) provides, for t/J=3, an upper critical hole concentration xc≈0.27 for the disappearance of the d-wave superconductivity. Also, the hole concentration x≈0.125 is obtained for the optimal doping. These results are in rough accordance with experimental results for high-Tc superconducting cuprates.
We discuss the coexistence of antiferromagnetism and d-wave superconductivity within the so-called statistically consistent Gutzwiller approximation (SGA) applied to the t--J--U model. In this approach, the averages calculated in a self-consistent manner coincide with those determined variationally. Such consistency is not guaranteed within the standard renormalized mean-field theory. With the help of SGA, we show that for the typical value J/∣t∣=31, coexistence of antiferromagnetism (AF) and superconductivity (SC) appears only for U/∣t∣>10.6 and in a very narrow range of doping $(\ensuremath{\delta}\ensuremath{\lesssim}0.006)$ in the vicinity of the Mott insulating state, in contrast to some previous reports. In the coexistent AF+SC phase, a staggered spin-triplet component of the superconducting gap appears also naturally; its value is very small.
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.
We propose a new method of solving a class of mean-field (MF) models, which is based on the Maximum Entropy (MaxEnt) principle with additional constraints included. Next, we show equivalence of our method when applied to the Gutzwiller approximation (GA), with the mean-field slave-boson (SB) formalism (on the example of the single-band Hubbard model). This equivalence provides thus an alternative justification of the results obtained within the SB approach which, however, contains ad hoc assumptions to position it in agreement with GA. Our approach implies that all predictions of the MF SB method can be obtained in a simpler, transparent, and controllable manner within GA when supplemented with the statistical-consistency conditions. We call the method as the Statistically-consistent Gutzwiller Approximation (SGA). Explicitly, the present formulation does not require introducing the condensed amplitudes of auxiliary Bose fields, which do not have a direct physical meaning and do not appear in the present formulation. Although the results of SGA are in the present case equivalent to SB, one can improve them further by utilizing more advanced schemes of calculating averages beyond the standard GA. To illustrate our approach, as well as to outline its advantages over alternative treatments of GA, we select the case of almost localized Fermi liquid (ALFL) in two dimensions and analyze it in detail within the tight-binding approximation. We also comment on significance of our method for describing correlated fermions. Namely, the reasoning used here can be applied to the corresponding MF treatment of the multiband Hubbard, the periodic Anderson, the t−J, and the t−J−U models. In this manner our method may be applied for strongly correlated electron systems, optical lattices, and other related situations.
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.
Using an advanced version of the renormalized mean-field theory for the t-J model, we examine the spin-singlet superconducting (SC) state of ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry. Overall doping dependence of the SC gap magnitude is in good agreement with experimental results for ${\text{Bi}}_{2}{\text{Sr}}_{2}\text{Ca}{\text{Cu}}_{2}{\text{O}}_{8+\ensuremath{\delta}}$ and ${\text{La}}_{2\ensuremath{-}x}{\text{Sr}}_{x}\text{Cu}{\text{O}}_{4}$ compounds at the optimal doping and in the overdoped regime. We also calculate the dispersion relation for the Bogoliubov quasiparticles and compare our findings both with the angle-resolved photoemission data for the cuprates as well as with the variational Monte Carlo and other mean-field studies. Within the method proposed by Fukushima [Phys. Rev. B 78, 115105 (2008)], we analyze different forms of the t-J Hamiltonian, i.e., modifications caused by the form of exchange interaction, and by the presence of three-site terms. It is shown that although the former has a small influence, the latter suppresses strongly the superconductivity. We also analyze the temperature dependence of the gap magnitude and compare the results with those of the recently introduced finite-temperature renormalized mean-field theory of Wang et al. [Phys. Rev. B 82, 125105 (2010)].