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Physical review. B, Condensed matter

Loop-expansion study of the single-hole spectral function in thet-Jmodel

Zhiping Liu, Efstratios Manousakis

Published Feb 1, 1995• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.51.3156

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Rotationally invariant approximation for the two-dimensional t−J model

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Using the description in terms of the Hubbard operators hole and spin Green's functions of the two-dimensional $t\ensuremath{-}J$ model are calculated in an approximation that retains the rotation symmetry of the spin susceptibility in the paramagnetic state and has no predefined magnetic ordering. In this approximation, Green's functions are represented by continued fractions that are interrupted with the help of the decoupling corrected by the constraint of zero site magnetization in the paramagnetic state. Results obtained in this approach for an undoped $32\ifmmode\times\else\texttimes\fi{}32$ lattice (the Heisenberg model) and for one hole in a $4\ifmmode\times\else\texttimes\fi{}4$ lattice are in good agreement with Monte Carlo and exact-diagonalization data, respectively. In the limit of heavy doping the hole spectrum described by the obtained formulas acquires features of the spectrum of weakly correlated excitations.

arXiv:cond-mat/0201526•Physical review. B, Condensed matter•Mar 26, 2002•31 citations
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 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 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 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.
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.
The motion of a single hole in a two-dimensional Heisenberg antiferromagnet (hAF) is studied in a representation where holes are described as spinless fermions (holons) and spins as normal bosons. Assuming long-range AF order the spin dynamics is treated in linear spin-wave theory. The formulation highlights the close relation with the conventional polaron problem. The holon Green's function is calculated self-consistently within the Born approximation using finite-cluster geometries for the numerical solution. As a remarkable result we find close agreement with the spectral function A(k,\ensuremath{\omega}) of a hole calculated by exact diagonalization methods. A(k,\ensuremath{\omega}) is characterized by a narrow quasiparticle (QP) peak at the low-energy side of the spectrum, which is well separated from the incoherent part for large enough J values. A complete characterization of our solution is given, including the spectral weight, the dispersion relation, and effective masses of the QP state. A finite-size-scaling study gives a nonvanishing spectral weight of the QP in the thermodynamic limit for values J/t typical for copper oxide superconductors. Our calculations indicate that the self-consistent Born approximation is a valuable scheme for characterizing the dynamics of a hole in the t-J model, even in the strong-coupling regime.
Using the description in terms of the Hubbard operators hole and spin Green's functions of the two-dimensional $t\ensuremath{-}J$ model are calculated in an approximation that retains the rotation symmetry of the spin susceptibility in the paramagnetic state and has no predefined magnetic ordering. In this approximation, Green's functions are represented by continued fractions that are interrupted with the help of the decoupling corrected by the constraint of zero site magnetization in the paramagnetic state. Results obtained in this approach for an undoped $32\ifmmode\times\else\texttimes\fi{}32$ lattice (the Heisenberg model) and for one hole in a $4\ifmmode\times\else\texttimes\fi{}4$ lattice are in good agreement with Monte Carlo and exact-diagonalization data, respectively. In the limit of heavy doping the hole spectrum described by the obtained formulas acquires features of the spectrum of weakly correlated excitations.