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Physical review. B, Condensed matter

Flux phases in thet-Jmodel

Menke U. Ubbens, Patrick A. Lee

Published Oct 1, 1992• 138 citations
Publisher

Abstract

We analyze a class of mean-field flux phases of the slave boson t-J model, and present a finite-temperature phase diagram. The mean-field theory arises from a decomposition of the four-fermion terms into exchange, pairing, and direct terms, using Hubbard-Stratonovich fields. We find numerically that at low doping staggered flux phases are energetically favorable over the d-wave-pairing state. Very close to half filling a \ensuremath{\pi}-flux phase becomes stable. The occurrence of flux phases suppresses the pairing gap transition temperature.

DOI
10.1103/physrevb.46.8434

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arXiv:cond-mat/9608122•Journal of the Physical Society of Japan•Mar 15, 1996•9 citations
We analyze a class of mean-field flux phases of the slave boson t-J model, and present a finite-temperature phase diagram. The mean-field theory arises from a decomposition of the four-fermion terms into exchange, pairing, and direct terms, using Hubbard-Stratonovich fields. We find numerically that at low doping staggered flux phases are energetically favorable over the d-wave-pairing state. Very close to half filling a \ensuremath{\pi}-flux phase becomes stable. The occurrence of flux phases suppresses the pairing gap transition temperature.
The supersymmetric formulation of the t - J model is studied in this paper at the mean-field level where the δ- T phase diagram is determined. We found that the slave-fermion–like spiral phase is stable at low doping concentration, and the slave-boson–like d -wave fermionic spin pairing state becomes energetically favourable when δ ⩾ 0.23. An improvement in free energy using Gutzwiller's method lowers the transition doping concentration to δ ≃ 0.06. We also point out the existence of new branches of excitations in the supersymmetric theory.
We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping which respects an SU(2)\mathrm{SU}(2)SU(2) symmetry---a symmetry previously known to be important at half filling. The mean field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain ddd-wave superconductor, spin gap, strange metal, and Fermi liquid phases. The spin gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The electron spectral function shows small Fermi pockets at low doping which continuously evolve into the large Fermi surface at high doping concentrations.
We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping that respects an SU(2) symmetry: a symmetry previously known to be important at half filling. The mean-field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain ddd-wave superconductor, spin-gap, strange metal, and Fermi-liquid phases. The spin-gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The physical electron spectral function shows small Fermi segments at low doping that continuously evolve into the large Fermi surface at high-doping concentrations. The close relation between the SU(2) and the U(1) slave-boson theory is discussed. The low-energy effective theory for the low-lying fluctuations is derived and additional lying modes [which were overlooked in the U(1) theory] are identified.
The competition between the staggered flux state, or the d-density wave state, and the d-wave pairing state is analyzed in a two-dimensional $t\ensuremath{-}J$ model based on the U(1) slave boson mean-field theory. Not only staggered flux of spinons but also staggered flux of holons are considered. In this formalism, the hopping order parameter of physical electron is described by the product of hopping order parameters of spinons and holons. The staggered flux amplitude of electrons is the difference of staggered flux amplitude of spinons and that of holons. In the $\ensuremath{\pi}$-flux phase of spinons, staggered fluxes of spinons and holons cancel completely and the staggered flux order of electrons does not exist. However, in the staggered flux phase of spinons whose staggered flux amplitude is not $\ensuremath{\pi},$ fluxes does not cancel completely and the staggered flux amplitude of electron remains. Thus the phase transition between these two phases, the $\ensuremath{\pi}$-flux phase and the staggered flux phase of spinons, becomes a second-order transition in the physical electron picture. The order parameter which characterizes this transition is the staggered flux order parameter of electrons. A mean-field phase diagram is shown. It is proved analytically that there is no coexistence of staggered flux and d-wave pairing. The temperature dependence of Fermi surface and excitation gap at $(0,\ensuremath{\pi})$ are shown. These behaviors are consistent with angle-resolved photoemission spectroscopy experiments.
Implication of our recent proposal [J. Phys. Soc. Jpn. 65 (1996) 687] to treat large-amplitude gauge-field fluctuations around the slave-boson mean-field theory for the t-J model has been explored in detail. By attaching gauge flux to spinons and holons and then treating them as free g-on's which respect the time-reversal symmetry, the optimum exclusion (g) and exchange (\a) statistics have been determined in the plane of doping rate and temperature. Two different relations between \a and g have been investigated, namely g=|\a| (Case1) and g=|\a|(2-|\a|) (Case2). The results indicate that slave fermion is favored at low doping while slave boson at high doping. For two dimension, in Case1 intermediate statistics are found in between, while in Case2 no intermediate statistics are found. The consequences of varying the dimensionality and strength of J have been studied also. The latter has no qualitative effect for both cases, while the former has a profound effect in Case1.