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International Journal of Modern Physics B

NUMERICAL STUDY OF HOLE CORRELATION IN THE t-J MODEL WITH FOUR HOLES ON A 32-SITE LATTICE

P. W. Leung

Published Jul 30, 2007• 1 citation
Publisher

Abstract

We solve the t-J model with four holes on a 32-site lattice using exact diagonalization. This is the largest scale calculation ever reported for this model. The size of the matrix involved in this problem is about 3×109 by 3×109. We find that at the realistic parameter J/t = 0.3, holes in the ground state of this system are unbound. They have short range repulsion due to lowering of kinetic energy. The electron momentum distribution function resembles hole pockets. We show that in case antiferromagnetic order is strong, holes form d-wave bound pairs and there is mutual repulsion among the holes. This presumably will occur at low doping level.

DOI
10.1142/s0217979207044007

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We solve the t-J model with four holes on a 32-site lattice using exact diagonalization. This is the largest scale calculation ever reported for this model. The size of the matrix involved in this problem is about 3×109 by 3×109. We find that at the realistic parameter J/t = 0.3, holes in the ground state of this system are unbound. They have short range repulsion due to lowering of kinetic energy. The electron momentum distribution function resembles hole pockets. We show that in case antiferromagnetic order is strong, holes form d-wave bound pairs and there is mutual repulsion among the holes. This presumably will occur at low doping level.
We study the $t\text{\ensuremath{-}}J$ model with four holes on a 32-site square lattice using exact diagonalization. This system corresponds to doping level x=18x=\frac{1}{8}x=81​. At the ``realistic'' parameter J∕t=0.3J∕t=0.3J∕t=0.3, holes in the ground state of this system are unbound. They have short range repulsion due to the lowering of the kinetic energy. There is no antiferromagnetic spin order and the electron momentum distribution function resembles hole pockets. Furthermore, we show evidence that in case antiferromagnetic order exists, holes form ddd-wave bound pairs and there is mutual repulsion among hole pairs. This presumably will occur at low doping levels. This scenario is compatible with a checkerboard-type charge density state proposed to explain the ``18\frac{1}{8}81​ anomaly'' in the lanthanum cuprate family, except that it is the ground state only when the system possesses strong antiferromagnetic order.
We have calculated the hole-hole correlation function for two holes doped into a S=1/2 antiferromagnetic insulator described by the t-J model on a 16-site \ensuremath{\surd}8\ifmmode\times\else\texttimes\fi{}\ensuremath{\surd}32 lattice and a 24-site \ensuremath{\surd}18\ifmmode\times\else\texttimes\fi{}\ensuremath{\surd}32 lattice using an exact diagonalization technique. We predict that the carriers described by this model, in the parameter range relevant to CuO2{\mathrm{CuO}}_{2}CuO2​ planes, tend to stay as far apart as possible. This result is in direct contradiction to previously published exact diagonalization data [e.g., D. Poilblanc, Phys. Rev. B 49, 1477 (1994)]. We suggest that if the carriers do tend to stay as far apart from one another as possible, then the electron momentum distribution function shows evidence for hole pockets at k\ensuremath{\rightarrow}=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2), consistent with many recent studies of these models, including recent photoemission work.
We report on a detailed examination of numerical results and analytical calculations devoted to a study of two holes doped into a two-dimensional, square lattice described by the $t\ensuremath{-}J$ model. Our exact diagonalization numerical results represent the first solution of the exact ground state of two holes in a 32-site lattice. Using this wave function, we have calculated several important correlation functions, notably the electron momentum distribution function and the hole-hole spatial correlation function. Further, by studying similar quantities on smaller lattices, we have managed to perform a finite-size scaling analysis. We have augmented this work by endeavouring to compare these results to the predictions of analytical work for two holes moving in an infinite lattice. This analysis relies on the canonical transformation approach formulated recently for the $t\ensuremath{-}J$ model. From this comparison we find excellent correspondence between our numerical data and our analytical calculations. We believe that this agreement is an important step helping to justify the quasiparticle Hamiltonian, and, in particular, the quasiparticle interactions that result from the canonical transformation approach. Also, the analytical work allows us to critique the finite-size scaling ansatzes used in our analysis of the numerical data. One important feature that we can infer from this successful comparison involves the role of higher harmonics in the two-particle, d-wave symmetry bound state---the conventional $[\mathrm{cos}{(k}_{x})\ensuremath{-}\mathrm{cos}{(k}_{y})]$ term is only one of many important contributions to the d-wave symmetry pair wave function.
We study the ``t-J'' model with strongly correlated electrons (infinite-U Hubbard-model kinetic energy plus superexchange J) by exact diagonalization on small two-dimensional clusters of six, eight, and twelve sites on a square lattice with periodic boundary conditions. We find an attractive pairing potential for two holes when J/t\ensuremath{\ge}(J/t)c{)}_{\mathit{c}})c​\ensuremath{\simeq}0.2, with t the hopping matrix element. Ground-state calculations show that extended s-wave and d-wave pairing susceptibilities and equal-time pairing correlations grow weakly with increasing J and system size at fixed filling n. Pairing correlations may be enhanced for (J/t)>(J/t)c{)}_{\mathit{c}})c​\ensuremath{\approxeq}0.2. We compare equal-time pairing correlations to a mean-field theory for J=0 and note potential difficulties with interpreting numerical results for such small systems. The widths of one- and two-hole energy bands in low-spin states are of order J; the one-hole effective mass is larger than the two-hole effective mass in such states. Spin-charge correlation functions for a single hole in the eight- and twelve-site lattices are consistent with the spin-charge decoupling scenario for resonating-valence-bond states.
Using numerical results from density matrix renormalization group (DMRG) calculations for the t-J model, on systems as large as 10\ifmmode\times\else\texttimes\fi{}7, we examine the structure of the one and two hole ground states in ladder systems and in two dimensional clusters. A simple theoretical framework is used to explain why holes bind in pairs in two-dimensional antiferromagnets. For the case J/t=0.5, which we have studied, the hole pairs reside predominantly on a 2\ifmmode\times\else\texttimes\fi{}2 core plaquette with the probability that the holes are on diagonal sites greater than nearest-neighbor sites. There is a strong singlet bond connecting the spins on the two remaining sites of the plaquette. We find that a general characteristic of dynamic holes in an antiferromagnet is the presence of frustrating antiferromagnetic bonds connecting next-nearest-neighbor sites across the holes. Pairs of holes bind in order to share the frustrating bonds. At low doping, in addition to hole pairs, there are two additional low-energy structures which spontaneously form on certain finite systems. The first is an undoped L\ifmmode\times\else\texttimes\fi{}2 spin-liquid region, or ladder. The second is a hole moving along a one dimensional chain of sites. At higher doping we expect that hole pairing is always favored.
We study the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}{t}^{\ensuremath{''}}\text{\ensuremath{-}}J$ model with parameters chosen to model an electron-doped high-temperature superconductor. The model with one, two, and four charge carriers is solved on a 32-site lattice using exact diagonalization. Our results demonstrate that at doping levels up to x=0.125x=0.125x=0.125 the model possesses robust antiferromagnetic correlation. When doped with one charge carrier, the ground state has momenta $(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi},0)$ and $(0,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi})$. On further doping, charge carriers are unbound and the momentum distribution function can be constructed from that of the single-carrier ground state. The Fermi surface resembles that of small pockets at single-charge-carrier ground-state momenta, which is the expected result in a lightly doped antiferromagnet. This feature persists upon doping up to the largest doping level we achieved. We therefore do not observe the Fermi surface changing shape at doping levels up to 0.125.