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arXiv:1610.04205 · Physical review. B./Physical review. B

Two holes in a two-dimensional quantum antiferromagnet: A variational study based on entangled-plaquette states

Fabio Mezzacapo, Adriano Angelone, Guido Pupillo

Published Oct 13, 2016• 10 citations
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Abstract

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 $N=256$ 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 $N=16$ 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 $r$, and on our computed two-hole binding energies, the existence of a two-hole bound state for all the values of $J/t$ explored here. It is estimated that $d$-wave binding of the two holes does not occur for $J/t<{J}_{c}/t\ensuremath{\simeq}0.19$.

DOI
10.1103/physrevb.94.155120

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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≤J/t\ensuremath≤2.00.4\ensuremath{\le}J/t\ensuremath{\le}2.00.4\ensuremath≤J/t\ensuremath≤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<Jc/t\ensuremath≃0.19J/t<{J}_{c}/t\ensuremath{\simeq}0.19J/t<Jc​/t\ensuremath≃0.19.
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.
We study, on the basis of the general entangled-plaquette variational ansatz, the ground-state (GS) properties of the spin- antiferromagnetic Heisenberg model on the triangular lattice. Our numerical estimates are in good agreement with available exact results and comparable, for large system sizes, to those computed via the best alternative numerical approaches, or by means of variational schemes based on specific (i.e. incorporating problem-dependent terms) trial wave functions (WFs). The extrapolation to the thermodynamic limit of our results for lattices comprising up to N =324 spins yields an upper bound of the GS energy per site (in units of the exchange coupling) of − 0.5458(2) [−0.4074(1) for the XX model], while the estimated infinite-lattice order parameter is 0.3178(5) (i.e. ∼64% of the classical value).
We propose a new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any 'sign' instability. We show results for various two-dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatze, is offered. In particular, our numerical estimates are in quite good agreement with exact ones for unfrustrated systems, and compare favorably to other methods for frustrated ones.
We propose a new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any 'sign' instability. We show results for various two-dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatze, is offered. In particular, our numerical estimates are in quite good agreement with exact ones for unfrustrated systems, and compare favorably to other methods for frustrated ones.
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.
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)].