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Physical review. B, Condensed matter

Variational description of a quasihole excitation in a quantum antiferromagnet

Massimo Boninsegni, Efstratios Manousakis

Published Mar 1, 1992• 36 citations
Publisher

Abstract

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)].

DOI
10.1103/physrevb.45.4877

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The aim of the present paper is twofold. The first goal is to show that high resolution angle-resolved photoelectron spectra from cuprates indicate the presence of string-like excitations of the quasihole excitation in a quantum antiferromagnet. In order to compare with the experimental intensity plots, we calculate the spectral function of the $t\ensuremath{-}J$ and the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}J$ models within the self-consistent Born approximation for widely accepted values of the parameters of these models. The main features of the high resolution photoelectron spectra are in general agreement with the results based on the above models and can be understood by considering not only the lowest-energy quasiparticle peak but, also, the higher-energy string-like excitations of the hole. These features are (a) the energy-momentum dispersion of the string-like excitations, (b) the momentum dependence of the spectral weight of the quasiparticle peak, and (c) the way in which the spectral weight of the lowest-energy quasiparticle peak is transferred to higher-energy string excitations, including the fact that it vanishes near the $\ensuremath{\Gamma}$ point. The second goal of the present paper is to make the case that a proper analysis of both the numerical results obtained from these models and of the experimental results, suggests a theoretical picture for the internal structure of the hole-quasiparticle and a string-exchange pairing mechanism due to strong antiferromagnetic correlations among the background spins. We find that, using a simple model in which the Hilbert space is restricted to states of only unbroken strings attached to the holes and in which the holes are connected with unbroken strings, we can provide a good quantitative description of the most accurate numerical results. We find that the holes experience an effective interaction due to a string-exchange mechanism which is characterized by a rather large string tension and this can provide pairing energy scales much larger than those suggested by spin-fluctuation-mediated pairing models. In addition, it is argued that such string-exchange interaction tends to bind holes in a bound state with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry.

arXiv:cond-mat/0702288•Physical Review B•Jan 8, 2007•79 citations
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 study a single quasihole excitation in a quantum antiferromagnet using a variational wave function that includes spin-spin as well as spin-hole correlations. The wave function can be cast in the form of a complex correlation operator acting on a Marshall-type wave function for the spin background. Such a correlation operator, featuring momentum-dependent spin-hole correlations, is shown to correspond to a coherent local rotation of the spins, which can be also interpreted as a ``spin-backflow'' current, analogous to the Feynman-Cohen backflow current in helium liquids. The variational Monte Carlo method is used to calculate the hole excitation spectrum on a square lattice for the t-J model with this wave function. Comparing our results with available exact results on a 4\ifmmode\times\else\texttimes\fi{}4 square lattice, we find qualitative and semiquantitative agreement. We provide results obtained on much larger lattices. We find that the quasihole energy band attains its minimum at k=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) while the hole mass is strongly anisotropic in different directions of k space. The quasihole excitation creates both a planar long-range distortion of the antiferromagnetic (AF) moment of the background \ensuremath{\delta}${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(r\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\sim}(k\ensuremath{\cdot}r)/r2{\mathit{r}}^{2}r2 and a ferromagnetic moment localized in the immediate neighborhood of the hole, pointing in the perpendicular direction. The magnitude of the ferromagnetic moment depends on the momentum of the excitation, and at the minimum of the hole band only the long-range AF planar distortion is present.
We study the properties of a mobile hole in the $t\ensuremath{-}J$ model on the square lattice by means of variational Monte Carlo simulations based on the entangled-plaquette ansatz. Our energy estimates for small lattices reproduce available exact results. We obtain values for the hole energy dispersion curve on large lattices in quantitative agreement with earlier findings based on the most reliable numerical techniques. Accurate estimates of the hole spectral weight are provided.
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.
We show that the entangled-plaquette variational Ansatz can be adapted to study the two-dimensional t-J model in the presence of two mobile holes. Specifically, we focus on a square lattice comprising up to N=256N=256N=256 sites in the parameter range $0.4\ensuremath{\le}J/t\ensuremath{\le}2.0$. Ground state energies are obtained via the optimization of a wave function in which the weight of a given configuration is expressed in terms of variational coefficients associated with square and linear entangled plaquettes. Our estimates are in excellent agreement with exact results available for the N=16N=16N=16 lattice. By extending our study to considerably larger systems we find, based on the analysis of the long-distance tail of the probability of finding two holes at spatial separation rrr, and on our computed two-hole binding energies, the existence of a two-hole bound state for all the values of J/tJ/tJ/t explored here. It is estimated that ddd-wave binding of the two holes does not occur for $J/t<{J}_{c}/t\ensuremath{\simeq}0.19$.
The motion of a single hole in an arbitrary magnetic background is investigated for the two-dimensional $t\ensuremath{-}J$ model. The wave function of the hole is described within a generalized string picture which leads to a modified concept of spin polarons. We calculate the one-hole spectral function using a large-string basis for the limits of a N\'eel ordered and a completely disordered background. In addition we use a simple approximation to interpolate between these cases. For the antiferromagnetic background we reproduce the well-known quasiparticle band. In the disordered case the shape of the spectral function is found to be strongly momentum dependent, the quasiparticle weight vanishes for all hole momenta. Finally, we discuss the relevance of results for the lowest energy eigenvalue and its dispersion obtained from calculations using a polaron of minimal size as found in the literature.
The aim of the present paper is twofold. The first goal is to show that high resolution angle-resolved photoelectron spectra from cuprates indicate the presence of string-like excitations of the quasihole excitation in a quantum antiferromagnet. In order to compare with the experimental intensity plots, we calculate the spectral function of the $t\ensuremath{-}J$ and the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}J$ models within the self-consistent Born approximation for widely accepted values of the parameters of these models. The main features of the high resolution photoelectron spectra are in general agreement with the results based on the above models and can be understood by considering not only the lowest-energy quasiparticle peak but, also, the higher-energy string-like excitations of the hole. These features are (a) the energy-momentum dispersion of the string-like excitations, (b) the momentum dependence of the spectral weight of the quasiparticle peak, and (c) the way in which the spectral weight of the lowest-energy quasiparticle peak is transferred to higher-energy string excitations, including the fact that it vanishes near the $\ensuremath{\Gamma}$ point. The second goal of the present paper is to make the case that a proper analysis of both the numerical results obtained from these models and of the experimental results, suggests a theoretical picture for the internal structure of the hole-quasiparticle and a string-exchange pairing mechanism due to strong antiferromagnetic correlations among the background spins. We find that, using a simple model in which the Hilbert space is restricted to states of only unbroken strings attached to the holes and in which the holes are connected with unbroken strings, we can provide a good quantitative description of the most accurate numerical results. We find that the holes experience an effective interaction due to a string-exchange mechanism which is characterized by a rather large string tension and this can provide pairing energy scales much larger than those suggested by spin-fluctuation-mediated pairing models. In addition, it is argued that such string-exchange interaction tends to bind holes in a bound state with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry.