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arXiv:1002.1607

A statistically consistent variational approach to the renormalized mean-field theory of the t-J model: critical hole concentrations for a paired state

Jakub Jȩdrak, Józef Spałek

Published Feb 8, 2010• 0 citations
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Abstract

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 $x_{c} \approx 0.27$ for the disappearance of the d-wave superconductivity. Also, the hole concentration $x \approx 0.125$ is obtained for the optimal doping. These results are in rough accordance with experimental results for high-$T_c$ superconducting cuprates.

DOI
10.48550/arxiv.1002.1607

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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=3t/J =3t/J=3, an upper critical hole concentration xc≈0.27x_{c} \approx 0.27xc​≈0.27 for the disappearance of the d-wave superconductivity. Also, the hole concentration x≈0.125x \approx 0.125x≈0.125 is obtained for the optimal doping. These results are in rough accordance with experimental results for high-TcT_cTc​ superconducting cuprates.
High-temperature superconductivity in copper oxides (cuprates, e.g. La2−xSrxCuO4, YBa2Cu3O6+δ or Bi2Sr2CaCu2O8+δ) remains among the most spectacular phenomena in condensed matter physics. Since its discovery in 1986, an enormous number (∼ 105) of papers on the subject have appeared. So far, there is no single, commonly accepted theory of high-temperature superconductivity. However, it is widely believed that a basic description of this phenomenon can be provided by a single-band Hubbard model or its derivative, the t-J model. The latter model is regarded as a minimal microscopic model, capable of describing the essential aspects of the complex physics of the cuprates. Unfortunately, in general case, neither of those two models can be solved exactly and therefore various approximate methods are used. Among them, the so-called mean-field methods provide a simple, yet fairly reasonable description of the cuprates. In particular, some of the main qualitative features of the phase diagram and the essential features of electronic spectrum are roughly reproduced. A standard mean-field approach to the t-J model, known under the name of renormalized mean-field theory (RMFT) goes beyond the Hartree-Fock approximation. Consequently, its fully consistent treatment requires a novel theoretical approach. This has been our original motivation to develop a general approach to the mean-field models, which is based on the maximum entropy (MaxEnt) principle. The method is presented in detail in Part II of this Thesis, and in Part III it is applied to study RMFT of the t-J model. First, we compare the results obtained within our formalism with those of the frequently used non-variational approach based entirely on the self-consistent equations. Also, various versions of RMFT are compared, and the most satisfactory of them is selected. This optimal version is subsequently used to study different versions of the original t-J Hamiltonian. As a result, upper critical concentration and doping dependence of the selected physical quantities (e.g. the superconducting gap and the Fermi velocity) is determined at low temperatures and in the absence of external magnetic field. We compare our findings both with theoretical results obtained from the Variational Monte Carlo (VMC) methods, as well as with the experimental data for selected cuprates. We show that the version of RMFT approach formulated in this Thesis provides a reasonable qualitative and in some cases semiquantitative rationalization of the principal characteristics of the hole-doped high-temperature superconductors at the optimal doping and in the overdoped regime. Possible extensions of the proposed analysis are mentioned at the end.
Using an advanced version of the renormalized mean-field theory for the ttt-JJJ 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 ttt-JJJ 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)].
We review the resonating valence bond (RVB) theory of high-temperature superconductivity using Gutzwiller projected wave functions that incorporate strong correlations. After a general overview of the phenomenon of high-temperature superconductivity, we discuss Anderson's RVB picture and its implementation by renormalized mean-field theory (RMFT) and variational Monte Carlo (VMC) techniques. We review RMFT and VMC results with an emphasis on recent developments in extending VMC and RMFT techniques to excited states. We compare results obtained from these methods with angle-resolved photoemission spectroscopy (ARPES) and scanning tunnelling microscopy (STM). We conclude by summarizing recent successes of this approach and discuss open problems that need to be solved for a consistent and complete description of high-temperature superconductivity using Gutzwiller projected wave functions.
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 review the Resonating Valence Bond (RVB) theory of high temperatur e superconductivity using Gutzwiller projected wave functions that incorporate strong correlations. After a general overview of the phenomenon of high temperature superconductivity, we discuss Anderson's RVB picture and its implementation by renormalised mean field theory (RMFT) and variational Monte Carlo (VMC) techniques. We review RMFT and VMC results with an emphasis on recent development s in extending VMC and RMFT techniques to excited states. We compare results obtained from these methods with angle resolved photoemission spectroscopy (ARPES) and scanning tunnelling microscopy (STM). We conclude by summarising recent successes of this approach and discuss open problems that need to be solved for a consistent and complete description of high temperature superconductivity using Gutzwiller projected wave functions.
We discuss the coexistence of antiferromagnetism and ddd-wave superconductivity within the so-called statistically consistent Gutzwiller approximation (SGA) applied to the ttt--JJJ--UUU 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∣=13J/|t|=\frac{1}{3}J/∣t∣=31​, coexistence of antiferromagnetism (AF) and superconductivity (SC) appears only for U/∣t∣>10.6U/|t|>10.6U/∣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.