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Physical review. B, Condensed matter

Spectral function of a hole in thet-Jmodel

Zhiping Liu, Efstratios Manousakis

Published Aug 1, 1991• 98 citations
Publisher

Abstract

We give numerical solutions, on finite but large-size square lattices, of the equation for the single-hole Green's function obtained by the self-consistent approach of Schmitt-Rink et al. and Kane et al. The spectral function of the hole in a quantum antiferromagnet shows that most features describing the hole motion are in close agreement with the results of the exact diagonalization on the ${4}^{2}$ lattice in the region of J/t\ensuremath{\le}0.2. Our results obtained on sufficiently large-size lattices suggest that certain important features of the spectral function survive in the thermodynamic limit while others change due to finite-size effects. We find that the leading nonzero vertex correction is given by a two-loop diagram, which has a small contribution.

DOI
10.1103/physrevb.44.2414

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We give numerical solutions, on finite but large-size square lattices, of the equation for the single-hole Green's function obtained by the self-consistent approach of Schmitt-Rink et al. and Kane et al. The spectral function of the hole in a quantum antiferromagnet shows that most features describing the hole motion are in close agreement with the results of the exact diagonalization on the 42{4}^{2}42 lattice in the region of J/t\ensuremath{\le}0.2. Our results obtained on sufficiently large-size lattices suggest that certain important features of the spectral function survive in the thermodynamic limit while others change due to finite-size effects. We find that the leading nonzero vertex correction is given by a two-loop diagram, which has a small contribution.
The t-J model in the large-t/J and low-doping limit is approximated by an effective Hamiltonian that treats spin degrees of freedom within the linear-spin-wave approximation and the hole-hopping term with a hole-spin interaction vertex that is linear in the spin-wave operators. The Dyson's equation for the single-hole Green's function is solved numerically on finite- but sufficiently large-size square lattices. Our calculation is based on an approximate equation for the hole motion that has been obtained using a self-consistent perturbation approach where crossing diagrams are neglected. We have studied the contribution of vertex corrections and found that the leading two-loop crossing diagram is exactly zero. Thus, the leading nonzero contribution to the proper self-energy from crossing diagrams is a three-loop correction. In two dimensions, it is possible to solve Dyson's equation numerically including these three-loop corrections. The results demonstrate that the contribution of such vertex corrections is indeed small. Our solution for the hole spectral function, obtained by neglecting such small contributions from vertex corrections, shows that most features describing the hole motion are in close agreement with the results of the exact diagonalization on the 42{4}^{2}42 lattice in the region of J/t\ensuremath{\le}0.2. The results obtained on sufficiently large-size lattices suggest that certain important features of the spectral function remain while others change due to finite-size effects. In particular, we find well-defined peaks above the lowest-energy peak that survive in the thermodynamic limit. We analyze these higher-energy peaks of the spectral functions and find that they can be understood as ``string'' excitations, which are created by the hole moving in a nearly linear potential.
We consider a single hole in the t-J model where the Heisenberg and the hole-hopping terms are expanded in powers of the spin-deviation operators. The lowest order corresponds to the linear spin-wave approximation and leads to an effective Hamiltonian studied earlier by means of self-consistent perturbation theory. In this work we shall include all terms generating up to two-loop diagrams in the perturbative expansion of the single-hole Green's function. The hard-core constraints are partially respected and the overall effect gives a renormalization factor of 1.158 to the hopping parameter t. The vertex that describes the process where a spin wave is absorbed and emitted at the same time by the hole is far more important than others. We find that the numerical solutions of Dyson's equation including all two-loop diagrams lower considerably the single-hole ground-state energies over the previous one-loop results and our results on the 4\ifmmode\times\else\texttimes\fi{}4 lattice are in better agreement with those obtained by exact diagonalization in the physical regime (small J/t). Our method can provide numerical solutions to the hole Green's function on very large lattices. We find that with the inclusion of the two-loop diagrams the spectral function retains all its features found in the one-loop calculation and the bandwidth, the quasiparticle residues, and the ``string'' excitations feel very small corrections.
The spectral function of a single hole moving in a two-dimensional antiferromagnetic background is calculated, using the self-consistent Born approximation for the self-energy. For small systems, good agreement is found with the results of exact diagonalization studies. In the thermodynamic limit, the quasiparticle residue, the effective mass, and the incoherent multiple spin-wave background are determined as functions of the ratio of Heisenberg exchange J and hopping amplitude t. The relevance of the single-hole results to the physical problem of a finite hole concentration is briefly discussed.
We solved numerically the integral equation for the selfenergy which describes the motion of a single hole in a two-dimensional quantum antiferromagnet (AF) within the selfconsistent Born approximation. This formulation stresses the similarity of the AF-spin polaron with the standard polaron problem. We confine our calculation to finite cluster geometries and compare with results from previous exact diagonalization studies. The spectral function is characterized by a narrow quasiparticle (qp) peak at the low energy side of the spectra, which appears to be well separated from the incoherent band part for large enough J values. For small J we find a reduced width of ~7t for the incoherent band. The bottom of the coherent qp band always occurs at (±π/2, ±π/2). Its bandwidth initially increases with J until J≃t and then decreases as 2t2/J. A complete characterization of our solution is given, including the dispersion relation and effective masses of this quasiparticle. The comparison with exact diagonalization studies for a 4×4 cluster is remarkably good. From our results we see that a lattice of 16×16 sites describes adequately the thermodynamic limit. We conclude that the simple Born approximation is a valuable scheme to characterize the dynamics of one hole in the t-J model. both in the perturbative and the strong coupling regimes.
We present quantum Monte Carlo (QMC) simulations for a single hole in a $t\ensuremath{-}J$ model from J=0.4tJ=0.4tJ=0.4t to J=4tJ=4tJ=4t on square lattices with up to $24\ifmmode\times\else\texttimes\fi{}24$ sites. The lower edge of the spectrum is directly extracted from the imaginary time Green's function. In agreement with earlier calculations, we find flat bands around $(0,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}),$ $(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi},0)$ and the minimum of the dispersion at $(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2).$ For small J, both self-consistent Born approximation and series expansions give a bandwidth for the lower edge of the spectrum in agreement with the simulations, whereas for J/t>1,J/t>1,J/t>1, only series expansions agree quantitatively with our QMC results. This band corresponds to a coherent quasiparticle. This is shown by a finite-size scaling of the quasiparticle weight $Z(\stackrel{\ensuremath{\rightarrow}}{k})$ that leads to a finite result in the thermodynamic limit for the considered values of J/t.J/t.J/t. The spectral function $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ is obtained from the imaginary time Green's function via the maximum entropy method. Resonances above the lowest edge of the spectrum are identified, whose J dependence is quantitatively described by string excitations up to J/t=2.J/t=2.J/t=2.
We develop an efficient numerical method for the description of a single-hole motion in the antiferromagnetic background. The method is free of finite-size effects and allows the calculation of physical properties at an arbitrary wave vector. A methodical increase of the functional space leads to results that are valid in the thermodynamic limit. We found good agreement with cumulant expansion, exact-diagonalization approaches on finite lattices as well as self-consistent Born approximations. The method allows a straightforward addition of other inelastic degrees of freedom, such as lattice effects. Our results confirm the existence of a finite quasiparticle weight near the band minimum for a single hole and the existence of stringlike peaks in the single-hole spectral function.