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Physical review. B, Condensed matter

t- Jz model on the Cayley tree and the square lattice

Peter Prelovšek, Igor Sega

Published Jun 1, 1994• 16 citations
Publisher

Abstract

Candidates for the ground state of the anisotropic t-${\mathit{J}}_{\mathit{z}}$ model at low hole doping are studied within two related lattices: the Cayley tree and the square lattice. The Cayley tree allows for the evaluation of the single hole as well as the hole-pair ground states to arbitrary precision. Threshold values for the formation of bound hole pairs appear to be similar for both lattices. We show that the main contributions to the binding of a hole pair on the Cayley tree comes from the excess number of frustrated bonds (``spinons''). Considering systems with a larger number of holes, we establish high stability of configurations with holes forming straight domain walls between nonequivalent antiferromagnetic domains. In particular, for the Cayley tree we find that domain-wall-like configurations have lower energy than separate hole pairs. For the square lattice, however, there exists at low but finite doping a parameter interval 0.18J/t0.56 where, in the ground state, holes are paired.

DOI
10.1103/physrevb.49.15241

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Candidates for the ground state of the anisotropic t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model at low hole doping are studied within two related lattices: the Cayley tree and the square lattice. The Cayley tree allows for the evaluation of the single hole as well as the hole-pair ground states to arbitrary precision. Threshold values for the formation of bound hole pairs appear to be similar for both lattices. We show that the main contributions to the binding of a hole pair on the Cayley tree comes from the excess number of frustrated bonds (``spinons''). Considering systems with a larger number of holes, we establish high stability of configurations with holes forming straight domain walls between nonequivalent antiferromagnetic domains. In particular, for the Cayley tree we find that domain-wall-like configurations have lower energy than separate hole pairs. For the square lattice, however, there exists at low but finite doping a parameter interval 0.18J/t0.56 where, in the ground state, holes are paired.
Hole-density correlation functions in the t-J model on lattices with square geometry are studied by the method of exact diagonalization for systems of various sizes with up to 26 sites. For the largest sizes we employ only a restricted basis, whereby for the extrapolation we introduce and test a scaling procedure. For two holes in an antiferromagnet our results confirm their binding for J>Jc{\mathit{J}}_{\mathit{c}}Jc​\ensuremath{\sim}0.2t. For four holes the two-point and four-point density correlations, calculated for N=18,20 sites, show clear evidence for hole-clustering instabilities for J>Js{\mathit{J}}_{\mathit{s}}Js​\ensuremath{\sim}1.5t; a first instability is likely to be towards a striped phase, with holes forming domain walls along the (0,1) or (1,0) direction. Signs for the phase separation into a hole-rich and a hole-free phase appear only at J>Js∗{\mathit{J}}_{\mathit{s}}^{\mathrm{*}}Js∗​\ensuremath{\sim}2.5t. Results in the intermediate regime 0.4J/t1.2 are consistent with the pairing picture, although there are also indications for a large weight of configurations representing domain walls along the (1,1) direction.
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.
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 present analytic and numerical results for the ground state of a dynamical hole in a quantum Ising antiferromagnet (the t-Jz{J}_{z}Jz​ model) on a two-dimensional square lattice. We find that the ground state has a localized region of reduced N\'eel order associated with the hole, with an energy that is well described by ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{h}}$/t\ensuremath{\approxeq}-3.63+2.93(Jz{\mathit{J}}_{\mathit{z}}Jz​/t)23{)}^{23})23 for 6\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$\ensuremath{\lesssim}Jz{\mathit{J}}_{\mathit{z}}Jz​/t\ensuremath{\lesssim}1. This resolves a disagreement in the literature regarding the coefficient of (Jz{J}_{z}Jz​/t)2/3{)}^{2/3})2/3. A simple string picture gives a reasonable description of the extent of the ground-state polaron in this intermediate-coupling regime. Disruption of long-range antiferromagnetic order in the bulk limit is estimated to occur at Jz{J}_{z}Jz​/t\ensuremath{\approxeq}0.3 for a hole-doping fraction of x=(1/16.
The $t\ensuremath{-}{J}_{z}$ model is the strongly anisotropic limit of the $t\ensuremath{-}J$ model which captures some general properties of doped antiferromagnets (AF's). The absence of spin fluctuations simplifies the analytical treatment of hole motion in an AF background, and allows us to calculate single- and two-hole spectra with a high accuracy using a regular diagram technique combined with a real-space approach. At the same time, numerical studies of this model via exact diagonalization on small clusters show negligible finite-size effects for a number of quantities, thus allowing a direct comparison between analytical and numerical results. Both approaches demonstrate that the holes have a tendency to pair in p- and d-wave channels at realistic values of t/J.t/J.t/J. Interactions leading to pairing and effects selecting p and d waves are thoroughly investigated. The role of transverse spin fluctuations is considered using perturbation theory. Based on the results of the present study, we discuss the pairing problem in the realistic $t\ensuremath{-}J$-like model. Possible implications for preformed pairs formation and phase separation are drawn.
The phase diagram of the planar t-J model at small hole doping is investigated by finite-size scaling of exact diagonalization data of \ensuremath{\surd}N \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}N clusters (N\ensuremath{\le}26). Hole-droplet binding energies, compressibility, and static spin and charge correlations are calculated. Short-range antiferromagnetic correlations can produce attractive forces between holes leading to a very rich phase diagram including a liquid of d-wave hole pairs (for J/t\ensuremath{\gtrsim}0.2), a liquid of hole droplets (quartets) for larger J/t ratios (J/t\ensuremath{\gtrsim}0.5), and, at even larger coupling J/t, an instability towards phase separation.