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Physical review. B, Condensed matter

Detailed Lanczos study of one- and two-hole band structure and finite-size effects in thet-Jmodel

Ted Barnes, Allan E. Jacobs, Mitchel D. Kovarik, William G. Macready

Published Jan 1, 1992• 34 citations
Publisher

Abstract

We present accurate numerical results for low-lying one- and two-hole states in the t-J model on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. We find six level crossings in the one-hole ground state for 0t/J\ensuremath{\infty}; accurate t/J values of these crossings and the associated ground-state quantum numbers are given. A degeneracy of k=(0,0) S=1/2, and S=3/2 one-hole levels at t/J=1/2 is noted, which is consistent with a recent analytical result. For small t/J, the S=1/2 one-hole and S=0 two-hole bandwidths on the 4\ifmmode\times\else\texttimes\fi{}4 lattice are ${\mathit{W}}_{\mathit{h}}$=[1.190 445 7(1)]t and ${\mathit{W}}_{\mathrm{hh}}$=[2.575(4)]${\mathit{t}}^{2}$/J, respectively. The origin of these qualitatively different behaviors is discussed, and a simple relation is found between the small-(t/J) one-hole bandwidth and a static-hole ground-state matrix element. The linear-t term in ${\mathit{W}}_{\mathit{h}}$ is apparently a finite-lattice artifact. As a measure of finite-size effects we determined the rms hole-hole separation in the two-hole ground states; we find evidence of important finite-size effects for t/J\ensuremath{\gtrsim}1, for which the rms hole-hole separation is clearly constrained by the 4\ifmmode\times\else\texttimes\fi{}4 lattice. Intermediate-(t/J) hole separations and binding energies for 0.3\ensuremath{\lesssim}t/J\ensuremath{\lesssim}1, however, scale approximately as powers of t/J, and can be used to give bulk-limit estimates for t/J=3. In particular, we estimate that the bulk-limit ground-state rms hole-hole separation at t/J=3 is \ensuremath{\approxeq}1.8${\mathit{a}}_{0}$, corresponding to 7 \AA{} in the high-temperature superconductors. The similarity to the observed in-plane coherence length of ${\ensuremath{\xi}}_{\mathit{a}\mathit{b}}$\ensuremath{\approxeq}14 \AA{} supports the identification of t-J model hole pairs with the Cooper pairs of high-temperature superconductivity.

DOI
10.1103/physrevb.45.256

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We review in detail a recently proposed technique to extract information about dynamical correlation functions of many-body Hamiltonians with a few Lanczos iterations and without the limitation of finite size. We apply this technique to understand the low-energy properties and the dynamical spectral weight of a simple model describing the motion of a single hole in a quantum antiferromagnet: the t-${\mathit{J}}_{\mathit{z}}$ model in two spatial dimension and for a double-chain lattice. The simplicity of the model allows a well controlled numerical solution, especially for the two-chain case. Contrary to previous approximations, we have found that the single-hole ground state in the infinite system is continuously connected with the Nagaoka fully polarized state for ${\mathit{J}}_{\mathit{z}}$\ensuremath{\rightarrow}0. Analogously, we have obtained an accurate determination of the dynamical spectral weight relevant for photoemission experiments. For ${\mathit{J}}_{\mathit{z}}$=0 an argument is given that the spectral weight vanishes at the Nagaoka energy faster than any power law, as supported also by clear numerical evidence. It is also shown that spin-charge decoupling is an exact property for a single hole in the Bethe lattice but does not apply to the more realistic lattices where the hole can describe closed-loop paths.

arXiv:cond-mat/9501064•Physical review. B, Condensed matter•Jun 1, 1995•12 citations
We present accurate numerical results for low-lying one- and two-hole states in the t-J model on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. We find six level crossings in the one-hole ground state for 0t/J\ensuremath{\infty}; accurate t/J values of these crossings and the associated ground-state quantum numbers are given. A degeneracy of k=(0,0) S=1/2, and S=3/2 one-hole levels at t/J=1/2 is noted, which is consistent with a recent analytical result. For small t/J, the S=1/2 one-hole and S=0 two-hole bandwidths on the 4\ifmmode\times\else\texttimes\fi{}4 lattice are Wh{\mathit{W}}_{\mathit{h}}Wh​=[1.190 445 7(1)]t and Whh{\mathit{W}}_{\mathrm{hh}}Whh​=[2.575(4)]t2{\mathit{t}}^{2}t2/J, respectively. The origin of these qualitatively different behaviors is discussed, and a simple relation is found between the small-(t/J) one-hole bandwidth and a static-hole ground-state matrix element. The linear-t term in Wh{\mathit{W}}_{\mathit{h}}Wh​ is apparently a finite-lattice artifact. As a measure of finite-size effects we determined the rms hole-hole separation in the two-hole ground states; we find evidence of important finite-size effects for t/J\ensuremath{\gtrsim}1, for which the rms hole-hole separation is clearly constrained by the 4\ifmmode\times\else\texttimes\fi{}4 lattice. Intermediate-(t/J) hole separations and binding energies for 0.3\ensuremath{\lesssim}t/J\ensuremath{\lesssim}1, however, scale approximately as powers of t/J, and can be used to give bulk-limit estimates for t/J=3. In particular, we estimate that the bulk-limit ground-state rms hole-hole separation at t/J=3 is \ensuremath{\approxeq}1.8a0{\mathit{a}}_{0}a0​, corresponding to 7 \AA{} in the high-temperature superconductors. The similarity to the observed in-plane coherence length of \ensuremathξab{\ensuremath{\xi}}_{\mathit{a}\mathit{b}}\ensuremathξab​\ensuremath{\approxeq}14 \AA{} supports the identification of t-J model hole pairs with the Cooper pairs of high-temperature superconductivity.
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In this paper we present results of finite-lattice studies of the t-J and the t-t'-J model for 18 (20) sites with up to 4 (2) holes. A modified Lanczos algorithm allowed for the classification of the ground state according to total spin S. In the one-hole sector we find evidence for the existence of hole pockets. We calculate the effective quasiparticle band and show qualitative differences between the t-J and t-t'-J model for J\ensuremath{\gtrsim}0.2. As U increases we notice a substantial band narrowing, which is maximal at the Nagaoka transition, accompanied by S-level crossings rendering a quasiparticle description difficult. In the high-doping regime we present hole binding energies for up to six holes for the t-J model. Evidence for phase separation is only found for J>t. We also studied the magnetic structure of the ground state on the 18-site lattice in the limit J=0. We present a complete ground-state classification for all numbers of holes Nh{\mathit{N}}_{\mathit{h}}Nh​ and find S=Smax{\mathit{S}}_{\mathrm{max}}Smax​ at quarter filling in addition to the Nagaoka case.
The spectral functions of one hole in the t-J and one-band Hubbard models are calculated using exact diagonalization techniques on small lattices. Results for the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model are also presented. For the t-J model we found that there is a quasiparticle at the bottom of the hole spectrum with an energy well approximated by Eh{\mathit{E}}_{\mathit{h}}Eh​=-3.17+2.83J0.73{\mathit{J}}^{0.73}J0.73 (for 0.1\ensuremath{\le}J\ensuremath{\le}1.0, t=1) on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The rest of the spectrum is not incoherent: We identified at least two other peaks following a similar power-law behavior with J. We speculate that the J dependence of the results can be explained by a model where the hole is trapped in a confining potential as in the Ising limit. The bandwidth of the hole is linear in J in the region 0.1\ensuremath{\le}J\ensuremath{\le}0.4 although a power-law behavior is not excluded. The spectral weight of the quasiparticle grows like J0.5{\mathit{J}}^{0.5}J0.5 in the same region. We present new analytical results in the large J/t limit to understand the motion of the hole: In perturbation theory it can be shown that the momentum of the hole at large J/t is k=(\ensuremath{\pi},\ensuremath{\pi}) changing to k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) at intermediate J/t in agreement with numerical and spin-waves results. We show analytically and numerically that the bandwidth of the quasiparticle is of order t in the large J/t limit. This result corresponds to a spin-liquid state. The one-hole spectral function of the Hubbard model is obtained for lattices with 8 and 10 sites. A quasiparticle is also observed in this case. The bandwidth and the relation with the t-J model are discussed and a comparison with recent Monte Carlo results is made. We also review and extend previous results for the ground-state properties of the t-J model.
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 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 review in detail a recently proposed technique to extract information about dynamical correlation functions of many-body Hamiltonians with a few Lanczos iterations and without the limitation of finite size. We apply this technique to understand the low-energy properties and the dynamical spectral weight of a simple model describing the motion of a single hole in a quantum antiferromagnet: the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model in two spatial dimension and for a double-chain lattice. The simplicity of the model allows a well controlled numerical solution, especially for the two-chain case. Contrary to previous approximations, we have found that the single-hole ground state in the infinite system is continuously connected with the Nagaoka fully polarized state for Jz{\mathit{J}}_{\mathit{z}}Jz​\ensuremath{\rightarrow}0. Analogously, we have obtained an accurate determination of the dynamical spectral weight relevant for photoemission experiments. For Jz{\mathit{J}}_{\mathit{z}}Jz​=0 an argument is given that the spectral weight vanishes at the Nagaoka energy faster than any power law, as supported also by clear numerical evidence. It is also shown that spin-charge decoupling is an exact property for a single hole in the Bethe lattice but does not apply to the more realistic lattices where the hole can describe closed-loop paths.