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Physical review. B, Condensed matter

Two-holed-wave binding in the physical region of thet-Jmodel: A Green’s-function Monte Carlo study

Massimo Boninsegni, Efstratios Manousakis

Published May 1, 1993• 60 citations
Publisher

Abstract

We investigate numerically the ground state of two holes with ${\mathit{d}}_{\mathit{x}}^{2}$-${\mathit{y}}^{2}$ symmetry in the two-dimensional t-J model, on lattices of significantly larger sizes than the ones studied so far. The Green's-function Monte Carlo method, in conjunction with suitable initial states and guidance functions, is used to estimate the ground-state energy of one and two holes, as well as to compute the two-hole distribution function. Our results show a significant decrease of the two-hole binding energy as the size of the lattice is increased from 4\ifmmode\times\else\texttimes\fi{}4. A critical value ${\mathit{J}}_{\mathit{c}}$\ensuremath{\sim}0.27t is found such that d-wave hole binding no longer occurs for J${\mathit{J}}_{\mathit{c}}$.

DOI
10.1103/physrevb.47.11897

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We investigate numerically the ground state of two holes with dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 symmetry in the two-dimensional t-J model, on lattices of significantly larger sizes than the ones studied so far. The Green's-function Monte Carlo method, in conjunction with suitable initial states and guidance functions, is used to estimate the ground-state energy of one and two holes, as well as to compute the two-hole distribution function. Our results show a significant decrease of the two-hole binding energy as the size of the lattice is increased from 4\ifmmode\times\else\texttimes\fi{}4. A critical value Jc{\mathit{J}}_{\mathit{c}}Jc​\ensuremath{\sim}0.27t is found such that d-wave hole binding no longer occurs for JJc{\mathit{J}}_{\mathit{c}}Jc​.
We show how the Green's function Monte Carlo method can be used to compute accurately the ground-state energy of two-dimensional t-J model for lattices of large size. This is demonstrated for the case of a single hole, with the aid of suitable initial and guidance functions. Our transient estimates for the ground-state energy of the 4\ifmmode\times\else\texttimes\fi{}4 lattice are in remarkable agreement with the exact values; we report results for lattices of significantly larger size than 4\ifmmode\times\else\texttimes\fi{}4.
We calculated the binding energy of two holes with antiparallel spin in the t-J model using a variational wave function based on the string or spin-bag picture. The symmetry of the bound state is dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2. Although the size of matrices to be diagonalized is only 3\ifmmode\times\else\texttimes\fi{}3 we could reproduce the results of various extensive numerical calculations with reasonable accuracy. We show that a sizeable contribution of the binding energy is due to the quantum fluctuations in the ground state of the Heisenberg antiferromagnet. In contrast to exact diagonalization results on small clusters in the thermodynamical limit the binding energy decreases rapidly with increasing t/J. A detailed discussion of the various finite-size effects in a small cluster is also included.
Ground state properties of two holes in the antiferromagnetic state in a square lattice are studied by using the t-J model and a variational method where the Hilbert space is restricted. It is shown that two holes are bound to each other and the ground state has d-symmetry for J/∣t∣≳ 0.18 with the momentum K = (0, 0). The binding energy of two holes is -0.7 ∼ -0.5 × J/∣t∣ and the average distance between two holes is 1.6 ∼ 1.8 × a, where a is the lattice constant, for realistic parameter values. Static correlation function Szz(Q) shows a broad peak around Q = (π/2, π/2) which originates from the motion of holes. By making use of the results for one and two holes, an effective Hamiltonian for the quasi-particles is proposed.
Clusters of the two-dimensional t-J model with two holes and up to 26 sites are diagonalized using a Lanczos algorithm. The behavior of the binding energy with system size suggests the existence of a finite critical value of J above which binding occurs in the bulk. Only the d-wave pair-field operator acting on the Heisenberg ground state has a finite overlap with the two-hole ground state for all the clusters considered. The related spectral function associated with the propagation of a d-wave (singlet) pair of holes in the antiferromagnetic background is calculated. The quasiparticle peak at the bottom of the spectrum as well as some structure appearing above the peak survive with increasing cluster size. Although no simple scaling law was found for the quasiparticle weight the data strongly suggest that this weight is finite in the bulk limit and is roughly proportional to the antiferromagnetic coupling J (for J1).
We investigate the phase diagram of the two-dimensional t-J model using a recently developed Green's function Monte Carlo method for lattice fermions. We use the technique to calculate exact ground-state energies of the model on large lattices. In contrast to many previous studies, we find the model phase separates for all values of J/tJ/tJ/t. In particular, it is unstable at the hole dopings and interaction strengths at which the model was thought to describe the cuprate superconductors.
We have numerically evaluated the energy of several kinds of wave functions considered to be candidate ground states of the two-dimensional t-J model at various hole densities. We searched a parameter space which includes d-wave and s-wave superconductivity and spin-density-wave ordering as well as the projected Fermi-liquid state. Coexistence of different orderings, such as the s+id state and d-wave spin-density-wave state, were found to be stable states. We find a phase diagram in the density-t/J plane which has coexistence of antiferromagnetism and superconductivity at very low hole concentrations and superconductivity up to rather high values of density---about 40%. At intermediate concentrations, the time-reversal symmetry-breaking s+id state is found.