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Physical review. B, Condensed matter

Hole propagation in correlated spin systems

Robert Joynt

Published May 1, 1988• 33 citations
Publisher

Abstract

An approximation method is presented for the calculation of the Green's function for a single hole moving in a correlated spin background. General formulas are obtained for calculating the moments of the Green's function, generalizing the Brinkman-Rice method to a general spin state. Numerical results are presented for a one-dimensional Gutzwiller resonating-valence-bond (RVB) state. In the two-dimensional case, the bandwidth is derived. The kinetic energy of a single hole in a correlated antiferromagnet or Gutzwiller RVB state is higher than in a N\'eel antiferromagnet.

DOI
10.1103/physrevb.37.7979

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We discuss the approximate calculation of a Green's function and its spectral density from its low-order moments by the use of ratio-extrapolation techniques to estimate and factor off the dominant shape of the band edges. The techniques are illustrated, following Brinkman and Rice, on the problem of a single hole propagating in ferromagnetic, antiferromagnetic, and random electron-spin configurations in the atomic limit of the Hubbard model; the methods are, however, of wider applicability.

Physical review. B, Solid state•May 1, 1972•24 citations
An approximation method is presented for the calculation of the Green's function for a single hole moving in a correlated spin background. General formulas are obtained for calculating the moments of the Green's function, generalizing the Brinkman-Rice method to a general spin state. Numerical results are presented for a one-dimensional Gutzwiller resonating-valence-bond (RVB) state. In the two-dimensional case, the bandwidth is derived. The kinetic energy of a single hole in a correlated antiferromagnet or Gutzwiller RVB state is higher than in a N'eel antiferromagnet.
The effective Hamiltonian of strongly correlated electrons on a square lattice is replaced by a renormalised Hamiltonian and the factors that renormalise the kinetic energy of holes and the Heisenberg spin-spin coupling are calculated using a Gutzwiller approximation scheme. The accuracy of this renormalisation procedure is tested numerically and found to be qualitatively excellent. Within the scheme a resonant valence bond (RVB) wavefunction is found at half-filling to be lower in energy than the antiferromagnetic state. If the wavefunction is expressed in fermion operators, local SU(2) and U(l) invariance leads to a redundancy in the representation. The introduction of holes removes these local invariances and we find that a d-wave RVB state is lowest in energy. This state has a superconducting order parameter whose amplitude is linear in the density of holes.
The effective Hamiltonian of strongly correlated electrons on a square lattice is replaced by a renormalised Hamiltonian and the factors that renormalise the kinetic energy of holes and the Heisenberg spin-spin coupling are calculated using a Gutzwiller approximation scheme. The accuracy of this renormalisation procedure is tested numerically and found to be qualitatively excellent. Within the scheme a resonant valence bond (RVB) wavefunction is found at half-filling to be lower in energy than the antiferromagnetic state. If the wavefunction is expressed in fermion operators, local SU(2) and U(1) invariance leads to a redundancy in the representation. The introduction of holes removes these local invariances and the authors find that a d-wave RVB state is lowest in energy. This state has a superconducting order parameter whose amplitude is linear in the density of holes.
An antiferromagnetic spin model (S=1/2) consisting of a chain of octahedra is considered. In a certain parameter regime of the Hamiltonian the exact ground state is a local resonating-valence-bond (RVB) state. We start with the t-J Hamiltonian for this model describing a single hole in the system and derive exact ground and excited states of the Hamiltonian when the hole is static. The local spin distortion around a hole is termed a spin polaron. The dynamics of the hole is studied through construction of exact eigenstates describing spin polarons in extended states. Coherent states for all momentum wave vectors in the chain direction exist for certain spin polarons. In the Nagaoka limit, the RVB polaronic states have energy greater than that of the ferromagnetic state. For exchange interaction J\ensuremath{\ne}0, the RVB states have lower energy. Other exact states obtained include states in which the hole motion is localized. Structural similarity of the present model to the arrangement of Cu spins in La2{\mathrm{La}}_{2}La2​CuO4{\mathrm{CuO}}_{4}CuO4​ is also highlighted.
We formulate a quasiparticle theory for a single hole in a quantum antiferromagnet in the limit that the Heisenberg exchange energy is much less than the hopping matrix element, J\ensuremath{\ll}t. We consider the ground state of the spins to be either a quantum N\'eel state or a d-wave resonating-valence-bond (RVB) state. We show in a self-consistent perturbation theory that the hole spectrum is strongly renormalized by the interactions with spin excitations. The hole can be described by a narrow quasiparticle band located at an energy of order -t with a quasiparticle residue of order J/t and a bandwidth of order J. Above the quasiparticle band is an incoherent band of width of order t. Our results indicate that the energy scale for any coherent phenomenon involving the holes is \ensuremath{\delta}J, where \ensuremath{\delta} is the doping concentration. In the N\'eel state we perform a spin-wave expansion on an anisotropic Heisenberg model. In the Ising limit we reproduce previously known results and then expand perturbatively about that limit. In this expansion we find that the holes have a quasiparticle residue of Jz{J}_{z}Jz​/t and a bandwidth of ${J}_{\ensuremath{\perp}}$. In the Heisenberg limit we employ a ``dominant pole'' approximation in which we ignore contributions to the self-energy from the incoherent part of the hole spectrum. A similar technique is used to study the d-wave RVB state. The relevance of our results to recent optical experiments is discussed.
An accurate variational ansatz is proposed for a single quasihole excitation in a two-dimensional quantum antiferromagnet. The trial state includes spin-hole and spin-hole-spin correlations, in addition to background antiferromagnetic spin-spin correlations. Background spin-spin correlations are induced by the Heisenberg antiferromagnetic exchange interaction and describe the zero-point motion of spin waves around the Ne\ifmmode\acute\else\textasciiacute\fi{}el state. Spin-hole and spin-hole correlations describe ``strings'' of spins displaced by one lattice site along the hole path. This is a generalization of the Brinkman-Rice approach to include quantum spin fluctuations. We have used the Monte Carlo method to compute the variational energy and the spectral weight of the quasihole state in the t-J model on the square lattice, with the proposed ansatz. Comparison with exact results for the 4\ifmmode\times\else\texttimes\fi{}4 lattice shows very good agreement for both energy and spectral weight, at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) in the range 0\ensuremath{\le}t/J\ensuremath{\le}5. We report results for lattices of several sizes up to 16\ifmmode\times\else\texttimes\fi{}16 in the above range of t/J and for the two values of k where the hole energy band attains its minimum [k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2)] and maximum [k=(0,0)].
We discuss the approximate calculation of a Green's function and its spectral density from its low-order moments by the use of ratio-extrapolation techniques to estimate and factor off the dominant shape of the band edges. The techniques are illustrated, following Brinkman and Rice, on the problem of a single hole propagating in ferromagnetic, antiferromagnetic, and random electron-spin configurations in the atomic limit of the Hubbard model; the methods are, however, of wider applicability.