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Physical review. B, Condensed matter

Perturbation theory of thet-Jmodel fort/J≪1

M. Uchinami

Published Dec 1, 1990• 10 citations
Publisher

Abstract

We apply a finite-temperature functional-integral approach by means of which Arovas and Auerbach analyzed low-dimensional Heisenberg spin models to the t-J model in the slave-fermion representation. Using a perturbation theory of the t-J model for t\ensuremath{\ll}J, self-consistent equations are derived for finite hole concentration that reduce to those in the case of the pure Heisenberg spin model, i.e., to those of Arovas and Auerbach, and Takahashi as a special case. Numerical solution of the corresponding self-consistent equations for the one-dimensional t-J model at sufficiently low temperature suggests that, if the transfer t is not too large, there is a phase-separation state into a hole-rich and a no-hole phase as conjectured recently by Emery et al.

DOI
10.1103/physrevb.42.10178

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We apply a finite-temperature functional-integral approach by means of which Arovas and Auerbach analyzed low-dimensional Heisenberg spin models to the t-J model in the slave-fermion representation. Using a perturbation theory of the t-J model for t\ensuremath{\ll}J, self-consistent equations are derived for finite hole concentration that reduce to those in the case of the pure Heisenberg spin model, i.e., to those of Arovas and Auerbach, and Takahashi as a special case. Numerical solution of the corresponding self-consistent equations for the one-dimensional t-J model at sufficiently low temperature suggests that, if the transfer t is not too large, there is a phase-separation state into a hole-rich and a no-hole phase as conjectured recently by Emery et al.
We investigate the charge density correlation function in the slave fermion representation of the two-dimensional t - J model using the self-consistent perturbation approach developed for that model by Li et al (1992 Phys. Rev. B 45 5428), which to lowest order in (t, J) gives a diagrammatic equivalent to the usual mean field theory. At the lowest order the equal-time correlation function (interpreted as the holon density correlation function) shows similar features to the results of recent high-temperature series calculations. However the loop corrections, representing scattering of holes by the antiferromagnetic spin fluctuations, lead to a divergent compressibility at low temperatures for low hole dopings and small J/t, which we interpret as indicating a tendency towards phase separation at a temperature .
Motion of a single hole is studied at finite temperatures in the t-J model on a slave-fermion Schwinger-boson representation. The spin fluctuation is treated with the mean-field theory of Arovas and Auerbach. The Green's function for the slave fermion is calculated within the self-consistent Born approximation. A sharp quasiparticle peak is found to be separated from a broad spectrum of incoherence in the spectral function for low temperatures. The Green's function for the physical hole is calculated by taking account of the multiple scattering between the slave fermion and the Schwinger boson. A bound state of the slave fermion and the Schwinger boson is found at low temperatures, suggesting that the spin and the charge cannot be separated into a simple form. The energy of the bound state is minimized at momenta (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2, \ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2), indicating that a small pocketlike Fermi surface is formed around the momenta for low concentrations of dopant holes.
Based on the U(1) slave-boson functional integral approach to the $t\ensuremath{-}J$ model Hamiltonian, we investigate the phase diagram of phase separation for the hole-doped two-dimensional system of antiferromagnets. We discuss the phase separation (into a hole-rich and a hole-free phase) as a function of J/tJ/tJ/t where J is the Heisenberg (antiferromagnetic) coupling constant and t the hopping energy. For the physically interesting region of relatively small Heisenberg coupling, 0<J/t<0.2,0<J/t<0.2,0<J/t<0.2, we find that the phase separation occurs in a region of hole doping concentration below ${x}_{c}\ensuremath{\sim}0.1$ and that in the same region the critical hole doping concentrations remain robust with nearest-neighbor hole-hole repulsive interactions. On the other hand, in the region of larger J/tJ/tJ/t the phase separation boundary is found to be highly sensitive to the variation of the repulsive hole-hole interactions. In agreement with other studies, we find that the phase separation occurs at a critical value of ${J}_{c}\ensuremath{\approx}3.4t$ corresponding to the critical hole concentration xc=1.{x}_{c}=1.xc​=1.
The t-J model in the large-t/J and low-doping limit is approximated by an effective Hamiltonian that treats spin degrees of freedom within the linear-spin-wave approximation and the hole-hopping term with a hole-spin interaction vertex that is linear in the spin-wave operators. The Dyson's equation for the single-hole Green's function is solved numerically on finite- but sufficiently large-size square lattices. Our calculation is based on an approximate equation for the hole motion that has been obtained using a self-consistent perturbation approach where crossing diagrams are neglected. We have studied the contribution of vertex corrections and found that the leading two-loop crossing diagram is exactly zero. Thus, the leading nonzero contribution to the proper self-energy from crossing diagrams is a three-loop correction. In two dimensions, it is possible to solve Dyson's equation numerically including these three-loop corrections. The results demonstrate that the contribution of such vertex corrections is indeed small. Our solution for the hole spectral function, obtained by neglecting such small contributions from vertex corrections, shows that most features describing the hole motion are in close agreement with the results of the exact diagonalization on the 42{4}^{2}42 lattice in the region of J/t\ensuremath{\le}0.2. The results obtained on sufficiently large-size lattices suggest that certain important features of the spectral function remain while others change due to finite-size effects. In particular, we find well-defined peaks above the lowest-energy peak that survive in the thermodynamic limit. We analyze these higher-energy peaks of the spectral functions and find that they can be understood as ``string'' excitations, which are created by the hole moving in a nearly linear potential.
It is shown that, for the t-J model, dilute holes in an antiferromagnet are unstable against phase separation into a hole-rich and a no-hole phase. When the spin-exchange interaction J exceeds a critical value Jc{J}_{c}Jc​, the hole-rich phase has no electrons. It is proposed that for J slightly less than Jc{J}_{c}Jc​ the hole-rich phase is a low-density superfluid of electron pairs. A brief discussion of phase separation in other related models is given.
We consider a single hole in the t-J model where the Heisenberg and the hole-hopping terms are expanded in powers of the spin-deviation operators. The lowest order corresponds to the linear spin-wave approximation and leads to an effective Hamiltonian studied earlier by means of self-consistent perturbation theory. In this work we shall include all terms generating up to two-loop diagrams in the perturbative expansion of the single-hole Green's function. The hard-core constraints are partially respected and the overall effect gives a renormalization factor of 1.158 to the hopping parameter t. The vertex that describes the process where a spin wave is absorbed and emitted at the same time by the hole is far more important than others. We find that the numerical solutions of Dyson's equation including all two-loop diagrams lower considerably the single-hole ground-state energies over the previous one-loop results and our results on the 4\ifmmode\times\else\texttimes\fi{}4 lattice are in better agreement with those obtained by exact diagonalization in the physical regime (small J/t). Our method can provide numerical solutions to the hole Green's function on very large lattices. We find that with the inclusion of the two-loop diagrams the spectral function retains all its features found in the one-loop calculation and the bandwidth, the quasiparticle residues, and the ``string'' excitations feel very small corrections.