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arXiv:cond-mat/0308252 · Physical Review Letters

Resonating Valence Bond Wave Function for the Two-Dimensional Fractional Spin Liquid

Seiji Yunoki, Sandro Sorella

Published Apr 15, 2004• 30 citations
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Abstract

The unconventional low-lying spin excitations, recently observed in neutron scattering experiments on $\mathrm{C}{\mathrm{s}}_{\mathrm{2}}\mathrm{CuC}{\mathrm{l}}_{\mathrm{4}}$, are explained with a spin liquid wave function. The dispersion relation as well as the wave vector of the incommensurate spin correlations are well reproduced within a projected BCS wave function with gapless and fractionalized spin-$1/2$ excitations around the nodes of the BCS gap function. The proposed wave function is shown to be very accurate for one-dimensional spin-$1/2$ systems and remains similarly accurate in the two-dimensional model corresponding to $\mathrm{C}{\mathrm{s}}_{\mathrm{2}}\mathrm{CuC}{\mathrm{l}}_{\mathrm{4}}$, thus representing a good ansatz for describing spin fractionalization in two dimensions.

DOI
10.1103/physrevlett.92.157003

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The unconventional low-lying spin excitations, recently observed in neutron scattering experiments on Cs2CuCl4\mathrm{C}{\mathrm{s}}_{\mathrm{2}}\mathrm{CuC}{\mathrm{l}}_{\mathrm{4}}Cs2​CuCl4​, are explained with a spin liquid wave function. The dispersion relation as well as the wave vector of the incommensurate spin correlations are well reproduced within a projected BCS wave function with gapless and fractionalized spin-1/21/21/2 excitations around the nodes of the BCS gap function. The proposed wave function is shown to be very accurate for one-dimensional spin-1/21/21/2 systems and remains similarly accurate in the two-dimensional model corresponding to Cs2CuCl4\mathrm{C}{\mathrm{s}}_{\mathrm{2}}\mathrm{CuC}{\mathrm{l}}_{\mathrm{4}}Cs2​CuCl4​, thus representing a good ansatz for describing spin fractionalization in two dimensions.
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.
The Gutzwiller projection allows the construction of an assortment of variational wave functions for strongly correlated systems. For quantum spin S=1/2S=1/2S=1/2 models, Gutzwiller-projected wave functions have resonating-valence-bond structure and may represent states with fractional quantum numbers for the excitations. Using insights obtained from field-theoretical descriptions of fractionalization in two dimensions, we construct candidate wave functions of fractionalized states by projecting specific superconducting states. We explicitly demonstrate the presence of topological order in these states.
Motivated by spin-wave continuum (SWC) observed in recent neutron scattering experiments in Herbertsmithite, we use Gutzwiller-projected wave functions to study dynamic spin structure factor S(q,ω)S(\mathbf{q},ω)S(q,ω) of spin liquid states on the kagome lattice. Spin-1 excited states in spin liquids are represented by Gutzwiller-projected two-spinon excited wave functions. We investigate three different spin liquid candidates, spinon Fermi-surface spin liquid (FSL), Dirac spin liquid (DSL) and random-flux spin liquid (RSL). FSL and RSL have low energy peaks in S(q,ω)S(\mathbf{q},ω)S(q,ω) at KKK points in the extended magnetic Brillouin zone, in contrast to experiments where low energy peaks are found at MMM points. There is no obviuos contradiction between DSL and neutron scattering measurements. Besides a fractionalized spin (i.e. spin-1/2), spinons in DSL carry a fractionalized crystal momentum which is potentially detectable in SWC in the neutron scattering measurements.
We study the ground-state phase diagram of the spin-1/21/21/2${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice with an accurate bosonic resonating-valence-bond (RVB) wave function. In contrast to the RVB ansatz based on Schwinger fermions, the representation based on Schwinger bosons, supplemented by a variational Monte Carlo technique enforcing the exact projection onto the physical subspace, is able to describe a fully gapped spin liquid in the strongly frustrated regime. In particular, a fully symmetric Z2{Z}_{2}Z2​ spin liquid is stable between two antiferromagnetic phases; a continuous transition at J2=0.4J1{J}_{2}=0.4{J}_{1}J2​=0.4J1​, when the Marshall sign rule begins to be essentially violated, and a first-order transition around J2=0.6J1{J}_{2}=0.6{J}_{1}J2​=0.6J1​ are present. Most importantly, the triplet gap is found to have a nonmonotonic behavior, reaching a maximum around J2=0.51J1{J}_{2}=0.51{J}_{1}J2​=0.51J1​, when the lowest spinon excitation moves from the $\ensuremath{\Gamma}$ to the MMM point, i.e., $k=(\ensuremath{\pi},0)$.