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arXiv:2304.11716 · Physical Review Research

Quantum spin supersolid as a precursory Dirac spin liquid in a triangular lattice antiferromagnet

Haichen Jia, Bowen Ma, Z. D. Wang, Gang Chen

Published Jul 8, 2024• 21 citations
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Abstract

Based on the recent experiments on the triangular lattice antiferromagnet Na 2 BaCo ( PO 4 ) 2 , we propose the easy-axis XXZ spin-1/2 model on the triangular lattice, that exhibits a quantum spin supersolid, to be a precursory Dirac spin liquid. Despite the presence of a three-sublattice magnetic order as a spin supersolid, we suggest that this system is close to a Dirac spin liquid by exploring its spectroscopic response. The physical consequence is examined from the spectroscopic response, and we establish the continuous spectra near the M point in addition to the K point excitation from the spinon continuum on top of the three-sublattice order. Moreover, the satellite peaks were predicted at the mid-points connecting the Γ and K points. This proposal offers a plausible understanding of the recent inelastic neutron scattering measurement in Na 2 BaCo ( PO 4 ) 2 and could inspire further research in relevant models and materials, such as K 2 Co ( SeO 3 ) 2 and Rb 2 Co ( SeO 3 ) 2 , and even more anisotropic magnets like PrMgAl 11 O 19 . Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.033031

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Based on the recent experiments on the triangular lattice antiferromagnet Na 2 BaCo ( PO 4 ) 2 , we propose the easy-axis XXZ spin-1/2 model on the triangular lattice, that exhibits a quantum spin supersolid, to be a precursory Dirac spin liquid. Despite the presence of a three-sublattice magnetic order as a spin supersolid, we suggest that this system is close to a Dirac spin liquid by exploring its spectroscopic response. The physical consequence is examined from the spectroscopic response, and we establish the continuous spectra near the M point in addition to the K point excitation from the spinon continuum on top of the three-sublattice order. Moreover, the satellite peaks were predicted at the mid-points connecting the Γ and K points. This proposal offers a plausible understanding of the recent inelastic neutron scattering measurement in Na 2 BaCo ( PO 4 ) 2 and could inspire further research in relevant models and materials, such as K 2 Co ( SeO 3 ) 2 and Rb 2 Co ( SeO 3 ) 2 , and even more anisotropic magnets like PrMgAl 11 O 19 . Published by the American Physical Society 2024
Abstract Prototypical models and their material incarnations are cornerstones to the understanding of quantum magnetism. Here we show theoretically that the recently synthesized magnetic compound Na 2 BaCo(PO 4 ) 2 (NBCP) is a rare, nearly ideal material realization of the S = 1/2 triangular-lattice antiferromagnet with significant easy-axis spin exchange anisotropy. By combining the automatic parameter searching and tensor-network simulations, we establish a microscopic model description of this material with realistic model parameters, which can not only fit well the experimental thermodynamic data but also reproduce the measured magnetization curves without further adjustment of parameters. According to the established model, the NBCP hosts a spin supersolid state that breaks both the lattice translation symmetry and the spin rotational symmetry. Such a state is a spin analog of the long-sought supersolid state, thought to exist in solid Helium and optical lattice systems, and share similar traits. The NBCP therefore represents an ideal material-based platform to explore the physics of supersolidity as well as its quantum and thermal melting.
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.
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 present a systematic numerical study of the spin-1 antiferromagnetic Heisenberg model on the triangular lattice in an out-of-plane magnetic field, using Density Matrix Renormalization Group (DMRG) methods. By mapping out the quantum phase diagram as a function of the single-ion anisotropy DzD_zDz​ and magnetic field, we identify distinct dipolar and quadrupolar spin supersolid states, characterized by spontaneous U(1) symmetry breaking with finite spin superfluid stiffness coexisting with longitudinal translational symmetry breaking. At zero field, the dipolar spin supersolid with a 'Y'-type spin configuration persists down to Dz=0D_z = 0Dz​=0, whereas the quadrupolar spin supersolid prevails at large DzD_zDz​. At intermediate fields, the phase diagram is dominated by an up-up-down phase. At high fields below saturation, a quadrupolar spin superfluid emerges in the large-DzD_zDz​ regime, whereas a dipolar spin supersolid with a 'V'-type spin configuration dominates at small DzD_zDz​. These phases are characterized through their order parameters and spin superfluid stiffness using calculations on various system sizes. Furthermore, the dynamical spin structure factor is obtained across the phase diagram, where characteristic spectral signatures of different phases are observed, including the gapless Goldstone mode and the roton-like minima. These features are directly accessible to inelastic neutron scattering experiments. Our results provide a theoretical understanding of the interplay between frustrations, anisotropy, and Zeeman interactions in driving distinct spin supersolid phases in the spin-1 system, which are relevant to various triangular-lattice antiferromagnets such as Na2_22​BaNi(PO4_44​)2_22​ and K2_22​Ni(SeO3_33​)2_22​.
Motivated by recent experimental evidence for spin supersolid states in triangular-lattice compounds, we numerically investigate the dynamical properties of magnetic field-induced phases in the spin-1/2 easy-axis triangular antiferromagnetic Heisenberg model in the presence of magnetic impurities. In both weak- and strong-field spin supersolid states, the gapless Goldstone mode at the KKK points remains robust against impurities, which is a direct manifestation of spin superfluidity. By contrast, at the same impurity density, impurities induce a splitting of the magnon bands in the conventional magnetic state, the so-called up-up-down state. In addition, the finite superfluid stiffness probed by the twisted phase in the spin supersolid states is consistent with the excitation spectrum. We argue that the excitation spectrum with impurities provides direct spectroscopic evidence for dissipationless spin dynamics in the spin supersolid states, which is experimentally accessible via inelastic neutron scattering.
Dynamics and stability of U(1) spin liquids beyond mean-field theory: Triangular-lattice J1J_1J1​-J2J_2J2​ Heisenberg model
Quantum spin liquids (QSLs) are long-range entangled phases of frustrated magnets exhibiting fractionalised spin excitations. In two dimensions, there is limited analytical understanding of their excitation spectra beyond parton mean-field theories, which fail to capture many features of the finite frequency dynamical response. We use a self-consistent mean-field theory combined with the random phase approximation (RPA) for the J1J_1J1​-J2J_2J2​ Heisenberg model on the triangular lattice to describe the strong spinon-spinon interactions of the U(1) Dirac QSL. We obtain quantitative results for the dynamical spin structure factor and phase diagram compatible with comprehensive numerical efforts. We show the continuum response at the Brillouin zone corners is described by a spinon-exciton hybridised with monopole gauge excitations. We further show that the transition from the QSL to 120-degree coplanar order is in the QED3_33​ O(3)-GN universality class. We extend the method to chiral QSLs and XXZ anisotropy, and discuss its broad range of applicability to other models and for describing inelastic neutron scattering experiments in KYbSe2_22​.