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arXiv:0910.2203 · Physical Review Letters

Supersolidity in the Triangular Lattice Spin- 1/2 XXZ Model: A Variational Perspective

Dariush Heidarian, Arun Paramekanti

Published Jan 4, 2010• 47 citations
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Abstract

We study the spin-1/2 XXZ model on the triangular lattice with a nearest neighbor antiferromagnetic Ising coupling J(z) > 0 and unfrustrated (J(perpendicular) < 0) or frustrated (J(perpendicular) >0) kinetic terms in a zero magnetic field. Incorporating long-range Jastrow correlations over a mean-field spin state, we obtain the variational phase diagram of this model on large lattices for arbitrary J(z) and either sign of J(perpendicular). For J(perpendicular) < 0, we find a square root(3) x square root(3) supersolid for J(z)/J(perpendicular)| approximately > 4.7, in excellent agreement with quantum Monte Carlo data. For J(perpendicular) > 0, a distinct square root(3) x square root(3) supersolid is found to emerge for J(z)/J(perpendicular) > or = 1. Both supersolids exhibit a spontaneous density deviation from half-filling. At J(z)/J(perpendicular) = infinity, the crystalline order parameters of these two supersolids are nearly identical, consistent with exact results.

DOI
10.1103/physrevlett.104.015301

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We study the spin-1/2 XXZ model on the triangular lattice with a nearest neighbor antiferromagnetic Ising coupling J(z) > 0 and unfrustrated (J(perpendicular) < 0) or frustrated (J(perpendicular) >0) kinetic terms in a zero magnetic field. Incorporating long-range Jastrow correlations over a mean-field spin state, we obtain the variational phase diagram of this model on large lattices for arbitrary J(z) and either sign of J(perpendicular). For J(perpendicular) < 0, we find a square root(3) x square root(3) supersolid for J(z)/J(perpendicular)| approximately > 4.7, in excellent agreement with quantum Monte Carlo data. For J(perpendicular) > 0, a distinct square root(3) x square root(3) supersolid is found to emerge for J(z)/J(perpendicular) > or = 1. Both supersolids exhibit a spontaneous density deviation from half-filling. At J(z)/J(perpendicular) = infinity, the crystalline order parameters of these two supersolids are nearly identical, consistent with exact results.
We study hard-core bosons with unfrustrated hopping (t) and nearest neighbor repulsion (U) (spin S=1/2 XXZ model) on the triangular lattice. At half filling, the system undergoes a zero temperature (T) quantum phase transition from a superfluid phase at small U to a supersolid at Uc approximately 4.45 in units of 2t. This supersolid phase breaks the lattice translation symmetry in a characteristic sqrt[3] x square root of 3 pattern, and is remarkably stable--indeed, a smooth extrapolation of our results indicates that the supersolid phase persists for arbitrarily large U/t.
Recent work has shown that the triangular lattice spin-$\frac{1}{2} {J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg and XXZ antiferromagnets may exhibit coplanar or supersolid orders proximate to a gapless Dirac spin-liquid phase. We explore a distinct $\text{SU}(2N)\ifmmode\times\else\texttimes\fi{}\text{SU}(M)$ fermionic parton approach, complemented by variational Monte Carlo calculations for the spin-12\frac{1}{2}21​ model, to study the phase diagram of these models. We also calculate their dynamical spin response including parton interactions within a random-phase approximation, and discuss implications for neutron scattering on triangular lattice cobaltates Ba3CoSb2O9,Na2BaCo(PO4)2,K2Co(SeO3)2,Rb2Co(SeO3)2{\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}, {\mathrm{Na}}_{2}\mathrm{BaCo}{({\mathrm{PO}}_{4})}_{2}, {\mathrm{K}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}, {\mathrm{Rb}}_{2}\mathrm{Co}{({\mathrm{SeO}}_{3})}_{2}Ba3​CoSb2​O9​,Na2​BaCo(PO4​)2​,K2​Co(SeO3​)2​,Rb2​Co(SeO3​)2​, and Yb-based magnet KYbSe2{\mathrm{KYbSe}}_{2}KYbSe2​.
We determine the phase diagram of hard-core bosons on a triangular lattice with nearest-neighbor repulsion, paying special attention to the stability of the supersolid phase. Similar to the same model on a square lattice we find that for densities rho<1/3 or rho>2/3 a supersolid phase is unstable and the transition between a commensurate solid and the superfluid is of first order. At intermediate fillings 1/3<rho<2/3 we find an extended supersolid phase even at half filling rho=1/2. The emergence of the supersolid on the triangular lattice reflects a novel and interesting way for a quantum system to avoid classical frustration, similar to an order-by-disorder mechanism. It also offers an exciting possibility of realizing such phenomena in ultracold atoms on optical lattices.
Motivated by recent experiments, we investigate the spin- 1 2 XXZ model on the triangular lattice with strong Ising anisotropy, combining large-scale numerical simulations and analytical methods to uncover unconventional spin dynamics at T = 0 . First, we compute the dynamical spin structure factor using density matrix renormalization group (DMRG) simulations and find excellent agreement with inelastic neutron scattering data on the layered compound K 2 Co ( SeO 3 ) 2 . The low-energy spectrum reveals a rotonlike minimum at the M point, absent in linear spin-wave theory, accompanied by peak intensity and a broad continuum above it. Near the Γ point, we observe an approximately linear dispersion with vanishing spectral weight. Second, we compare two analytical frameworks that reproduce the observed features. The first is a hard-core boson approach, which includes (i) an effective staggered boson model at zero magnetic field, (ii) perturbation theory applied to the one-third magnetization plateau, and (iii) a self-consistent mean-field Schwinger boson theory (SBT). The second framework is based on a variational supersolid quantum dimer model (QDM) ansatz, combined with a single-mode approximation. The SBT captures the broad continuum, the M -point minimum, and linear dispersion at Γ , whereas the QDM reproduces the roton minimum and linear dispersion at finite momentum near Γ . Remarkably, both the QDM wave function and the DMRG ground state exhibit nearly identical structure factors with pronounced transverse photonlike excitations. Together, our comprehensive theoretical and numerical analysis elucidates the microscopic origin of supersolid excitations in the XXZ triangular-lattice model and their proximity to a spin-liquid phase observed experimentally.
We study a model of hard-core bosons with frustrated nearest-neighbor hopping (t) and repulsion (V) on the triangular lattice. We argue for a supersolid ground state in the large repulsion (V>>|t|) limit where a dimer representation applies, by constructing a unitary mapping to the well-understood unfrustrated hopping case. This generalized "Marshall sign rule" allows us to establish the precise nature of the supersolid order by utilizing a recently proposed dimer variational wave function, whose correlations can be efficiently calculated using the Grassmann approach. By continuity, a supersolid is predicted over the wide parameter range, V>-2t>0. This also establishes a simple phase diagram for the triangular lattice spin 1/2 XXZ antiferromagnet.
We numerically demonstrate that a supersolid phase exists in a frustrated hard-core boson system on a triangular lattice over a wide range of interaction strength. In the infinite repulsion (Ising) limit, we establish a mapping to the same problem with unfrustrated hopping, which connects the supersolid to the known results in that case. The weak superfluidity can be destroyed or strongly enhanced by a next-nearest-neighbor hopping term, which provides valuable information for experimental realization of a supersolid phase on optical lattice. A phase diagram for the frustrated anisotropic Heisenberg model on triangular lattice is also established.