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Physical review. B, Condensed matter

Ground state with soliton in one-dimensionalt-t’-J-J’ model

Ippei Doi, Kazuhiro Sano, Ken'ichi Takano

Published Jan 1, 1992• 9 citations
Publisher

Abstract

The one-dimensional t-t'-J-J' model is examined by numerical diagonalization in the case that one hole is doped to the half-filled band. In the presence of t', two types of solitons are found; one is a bound object of the hole and a singlet electron pair in a ferromagnetic background (t'>0, J0), and the other is that of the hole and a triplet electron pair in a singlet background (t'0, J>0). It seems that the solitons remain in the ground state when the system size becomes infinite. The formation of the solitons is ascribed to a kinematic exchange effect due to the t' term.

DOI
10.1103/physrevb.45.274

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Physical review. B, Condensed matter•May 1, 1993•4 citations
The one-dimensional t-t'-J-J' model is examined by numerical diagonalization in the case that one hole is doped to the half-filled band. In the presence of t', two types of solitons are found; one is a bound object of the hole and a singlet electron pair in a ferromagnetic background (t'>0, J0), and the other is that of the hole and a triplet electron pair in a singlet background (t'0, J>0). It seems that the solitons remain in the ground state when the system size becomes infinite. The formation of the solitons is ascribed to a kinematic exchange effect due to the t' term.
We investigate the one-dimensional t - J - J ′ model by the numerical diagonalization where a few holes are doped in the half-filled band. By introducing a finite-size scaling, we estimate the spin gap at J ′ / J =0.5 and J / t =2. When one hole is doped, the spin gap decreases to 0.190 J in comparison with 0.236 J which is the spin gap of the half-filled band. The spin gap further reduces to 0.141 J for two-hole doping and to about 0.05 J for three-hole doping. This result shows that the value of the spin gap as a function of the electron density is discontinuous at half-filling.
The ground-state phase diagram of the two-dimensional t-J model is investigated in the context of the tensor network algorithm in terms of the graded Projected Entangled-Pair State representation of the ground-state wave functions. There is a line of phase separation between the Heisenberg anti-ferromagnetic state without hole and a hole-rich state. For both J=0.4t and J=0.8t, a systematic computation is performed to identify all the competing ground states for various dopings. It is found that, besides a possible Nagaoka's ferromagnetic state, the homogeneous regime consists of four different phases: one phase with charge and spin density wave order coexisting with a p_x (p_y)-wave superconducting state, one phase with the symmetry mixing of d+s-wave superconductivity in the spin-singlet channel and p_x (p_y)-wave superconductivity in the spin-triplet channel in the presence of an anti-ferromagnetic background, one superconducting phase with extended s-wave symmetry, and one superconducting phase with p_x (p_y)-wave symmetry in a ferromagnetic background.
We introduced variational wave functions to describe dimer states in the one-dimensional t-J-J' model, when one hole is doped and J'=J/2. For finite J/\ensuremath{\Vert}t\ensuremath{\Vert} in the first excited state with S=1, there exists a bound state of the hole and an unpaired electron. For the choice of J/\ensuremath{\Vert}t\ensuremath{\Vert}=2, the spin gap is ${\mathrm{\ensuremath{\Delta}}}^{\mathrm{var}}$=0.181J and is smaller than the value of the corresponding spin system, ${\mathrm{\ensuremath{\Delta}}}_{0}^{\mathrm{var}}$=0.25J. The existence of the bound state suggests that the excited state as a function of the electron density is discontinuous at half-filling if J/\ensuremath{\Vert}t\ensuremath{\Vert} is finite.
In the spin-wave approximation the energy spectrum of one-hole states of the two-dimensional t-J model is considered in the parameter range 0.01\ensuremath{\le}J/t\ensuremath{\le}1, where J is the exchange constant and t the hopping-matrix element. Attention is focused on single-particle states (in contrast to hole-magnon continua) that form energy bands characterized by different values of the z component of the total spin Sz{\mathit{S}}_{\mathit{z}}Sz​, corresponding to ferromagnetically ordered regions of different size around the hole. For an infinite lattice, the shapes and relative positions of the lowest bands are computed by means of the Lanczos method. The formation process of the ferromagnetically ordered region in the one-hole ground state is discussed. It is connected with a characteristic dependence of the band energy on J/t for different Sz{\mathit{S}}_{\mathit{z}}Sz​. The shapes of the magnon cloud around the hole and the point symmetries of the eigenfunctions are determined for various points in the Brillouin zone and for different bands of interest.
We show that a hole and a triplet spin form a bound state in a nearly half-filled band of the one- and two-dimensional ${t}_{1}\ensuremath{-}{t}_{2}\ensuremath{-}{J}_{1}\ensuremath{-}{J}_{2}$ models. Numerical calculation indicates that the bound state is a spatially small object and moves as a composite particle with spin 1 and charge +e+e+e in the spin-gapped background. Two bound states repulsively interact with each other in a short distance and move independently as long as they keep their distance. If a finite density of bound states behaves as bosons, the system undergoes the Bose-Einstein condensation, which means a superconductivity with charge +e+e+e.
The influence of the staggered magnetic field h on properties of the one-dimensional t-J and Hubbard models with nearly half-filled bands are studied. Using the exact diagonalization method for small systems as well as analytical approaches possible in the anisotropic model and for h\ensuremath{\gg}t, we determine the phase diagram of the t-J-h model at low doping, which displays three different regimes: the phase with repelling holes for JJc{\mathit{J}}_{\mathit{c}}Jc​, the phase with paired holes for Jc{\mathit{J}}_{\mathit{c}}Jc​JJs{\mathit{J}}_{\mathit{s}}Js​, and the phase separation for J>Js{\mathit{J}}_{\mathit{s}}Js​. Employing for h\ensuremath{\ne}0 the analogy with the universal behavior of one-dimensional interacting fermions within the Luther-Emery regime, we calculate the charge exponent ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, which can be ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$>1 in the pairing regime, indicating dominant superconducting fluctuations. An analogous analysis, in particular the perturbation expansion in t/h, performed for the Hubbard t-U-h model shows a phase of stable hole pairs for UUc{\mathit{U}}_{\mathit{c}}Uc​. Again ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$ seems to approach a nonuniversal value at half-filling.