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Physical review. B, Condensed matter

Hole motion in a quantum Néel state

Subir Sachdev A

Published Jun 1, 1989• 135 citations
Publisher

Abstract

The spin-wave expansion is used to motivate a simple trial wave function for the N\'eel ground state of a spin-1/2 Heisenberg antiferromagnet on a square lattice; the wave function yields an upper bound on the ground-state energy per bond of (-0.3317\ifmmode\pm\else\textpm\fi{}0.0002)J, where J is the exchange constant. The wave function is easily generalizable to the case where a single hole, with hopping matrix element t, is present. The hole energy-momentum relationship is determined for J/t>0.25; the minimum hole energy is always on the zone boundary at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2). The hole bandwidth, W, has a maximum value of 1.24t at J=0.73t. The small parameter which makes all of these calculations possible is 1/(2Z-2), where Z=4 is the coordination number of the square lattice.

DOI
10.1103/physrevb.39.12232

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The spin-wave expansion is used to motivate a simple trial wave function for the N'eel ground state of a spin-1/2 Heisenberg antiferromagnet on a square lattice; the wave function yields an upper bound on the ground-state energy per bond of (-0.3317\ifmmode\pm\else\textpm\fi{}0.0002)J, where J is the exchange constant. The wave function is easily generalizable to the case where a single hole, with hopping matrix element t, is present. The hole energy-momentum relationship is determined for J/t>0.25; the minimum hole energy is always on the zone boundary at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2). The hole bandwidth, W, has a maximum value of 1.24t at J=0.73t. The small parameter which makes all of these calculations possible is 1/(2Z-2), where Z=4 is the coordination number of the square lattice.
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 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.
The ground-state properties of the spin-(1/2 Heisenberg antiferromagnet on a square lattice are studied by using a simple variational wave function that interpolates continuously between the N\'eel state and short-range resonating-valence-bond states. Exact calculations of the variational energy for small systems show that the state with the lowest energy has long-range antiferromagnetic order. The staggered magnetization in this state is approximately 70% of its maximum possible value. The variational estimate of the ground-state energy is substantially lower than the value obtained for the nearest-neighbor resonating-valence-bond wave function.
We have investigated one-hole motion in the anisotropic three-dimensional (A3D) quantum antiferromagnet by use of the A3D t-J model in a simple cubic lattice. The hopping matrix element and the Heisenberg exchange energy are t and J in the xy plane. They are ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ and ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ in the z direction. Within the self-consistent mean-field approximation, the hole spectral function has been calculated by use of an iterative technique. We find that the quasiparticle picture still holds and the lowest-energy quasiparticle state has momentum either k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2,\ensuremath{\pi}/2) or k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2,0) depending on the parameters of the model. Holding J and ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ constant and allowing ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ to increase, the spectral weight and the peak height of the quasiparticle decreases. In the quasi-two-dimensional cases with small ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ and ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, the spectral weight and the peak height of the quasiparticle are smaller than those in the exact two-dimensional case.
Ground state properties of two holes in the antiferromagnetic state in a square lattice are studied by using the t-J model and a variational method where the Hilbert space is restricted. It is shown that two holes are bound to each other and the ground state has d-symmetry for J/∣t∣≳ 0.18 with the momentum K = (0, 0). The binding energy of two holes is -0.7 ∼ -0.5 × J/∣t∣ and the average distance between two holes is 1.6 ∼ 1.8 × a, where a is the lattice constant, for realistic parameter values. Static correlation function Szz(Q) shows a broad peak around Q = (π/2, π/2) which originates from the motion of holes. By making use of the results for one and two holes, an effective Hamiltonian for the quasi-particles is proposed.
By using Majorana fermion representations for spin 1/2 and hole operators on the square lattice antiferromagnet, the Neel ground state energy is calculated by a variational method.When a hole is doped on the lattice, the spin gap is formed as a condensate of the spin and the conduction hole, and becomes an elementary excitation.When more holes are doped and they exchange spin Majorana fermions between them, the spin gap becomes unstable.The instability of the spin gap leads to high Tc superconductivity, and a superconductive gap appears instead of the spin gap, where a superconductive hole is described by the Nambu representation of a massive Dirac fermion.§ 1. Introduction