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arXiv:cond-mat/9712015

New Paired-Wavefunction for the Frustrated Antiferromagnetic Spin-Half Chain

Zou Ha

Published Dec 1, 1997• 0 citations
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Abstract

I propose a new paired-wavefunction with a parameter that continuously interpolates from the 1D Jastrow-product to the Majumdar-Ghosh dimer-wavefunction appropriate for the frustrated Heisenberg $S = 1/2$ antiferromagnet. This spin paired-state constructed in $S_z$ basis is an alternative to the well-known resonating-valence-bond basis state for describing the $S = 0$ ground-state with no apparent long-range spin order. Some numerical evidences are presented.

DOI
10.48550/arxiv.cond-mat/9712015

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I propose a new paired-wavefunction with a parameter that continuously interpolates from the 1D Jastrow-product to the Majumdar-Ghosh dimer-wavefunction appropriate for the frustrated Heisenberg S=1/2S = 1/2S=1/2 antiferromagnet. This spin paired-state constructed in SzS_zSz​ basis is an alternative to the well-known resonating-valence-bond basis state for describing the S=0S = 0S=0 ground-state with no apparent long-range spin order. Some numerical evidences are presented.
Starting from the exact result of Majumdar and Ghosh for the dimerized ground state of a Heisenberg linear-chain antiferromagnet with competing next-nearest-neighbor interaction J2{\mathit{J}}_{2}J2​=1/2J1{\mathit{J}}_{1}J1​, it is shown that this state is an exact eigenstate of the same Hamiltonian with couplings extended up to fourth-nearest neighbors, if the relations J1{\mathit{J}}_{1}J1​-2J2{\mathit{J}}_{2}J2​+J3{\mathit{J}}_{3}J3​=0 and J3{\mathit{J}}_{3}J3​=2J4{\mathit{J}}_{4}J4​ hold. Furthermore, it is proved to all orders in perturbation theory that this state is the ground state of this Hamiltonian in a finite interval of values of the parameter \ensuremath{\lambda}==1/2-j2{\mathit{j}}_{2}j2​ around \ensuremath{\lambda}=0. The actual extension of this interval is computed numerically for finite chains and is shown to be significant.
We introduce a family of Jastrow pair product states for quasi-one-dimensional spin systems. Depending on a parameter they interpolate between the resonating valence-bond ground state of the Haldane-Shastry model describing a spin liquid and the (dimerized) valence-bond solid ground states of the Majumdar-Ghosh spin chain. These states are found to form an excellent basis for variational studies of Heisenberg chains with next-nearest-neighbor interactions and bond alternation as realized in the spin-Peierls system CuGeO3{}_{3}3​.
Models of interacting quantum spins have contributed significantly to our understanding of magnetism. The Heisenberg model on square lattice, which exhibits semiclassical N´eel order, is one of the canonical models. However, with frustration introduced by competing interactions, the system becomes computationally intractable. Exotic quantum phases that have no magnetic long range order may be present due to the frustration. The quantum phases do not have any classical analogue, and a valence bond crystal state with translational and/or rotational symmetry breaking as well as a quantum spin liquid state have been proposed as potential candidates. We construct several trial wavefunctions in the resonating valence bond basis and apply large-scale unbiased calculations to examine the possible descriptions of the ground state in the strongly frustrated region. An analytical master equation is also proposed to provide an approximate solution for the valence bond states. Our numerical and analytical studies suggest that the frustrated ground state exhibits a one-dimensional-like behaviour.
We make extensive simulations over a spin chain model that combines the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ spin chain and the long-range nonfrustrated ${(\ensuremath{-}1)}^{(r\ensuremath{-}1)}{r}^{\ensuremath{-}\ensuremath{\alpha}}$ decay interactions through the variational matrix product state method for both finite and infinite lengths. We study both the ground-state entanglement and phase diagram. We find that it is most entangled in the rotation invariant long-range ordered antiferromagnetic phase, where the entanglement scales approximately logarithmically. We determine the development of the Majudar-Ghosh point to a disorder line from entanglement. And we approximately determine the transition from the dimerized and incommensurate phase of the ${J}_{1}\ensuremath{-}{J}_{2}$ model to a decoupled phase by studying spin correlation and the dimerization order parameter. Some implications for entanglement in systems with long-range interactions are stated.
Twenty-five years after the first proposal, the question whether the ground state of a frustrated spin-half system is well described by a spin-liquid Resonating Valence Bond (RVB) wave function is still controversial. A physically transparent representation of a RVB state can be obtained in fermionic representation with a standard BCS-type pairing wave function, working in the subspace with fixed number of electrons and no double occupancies. In this work, we show that, using this variational wave function with a careful parameterization of the pairing function, it is possible to obtain an extremely accurate {\em ansatz} for the ground state of the Heisenberg antiferromagnet with next-nearest neighbors interactions (J1−J2J_{1}{-}J_{2}J1​−J2​ model) in the regime of strong frustration. Indeed, in the spin-half realization of this model, it is known that the combined effect of frustration and zero-point motion interferes with the mechanism of spontaneously broken symmetry, giving rise to a non-magnetic phase of purely quantum-mechanical nature (J2/J1≃0.5J_2/J_1\simeq 0.5J2​/J1​≃0.5). This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids.