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arXiv:1505.06495 · Physical review. B./Physical review. B

Symmetric Z2 spin liquids and their neighboring phases on triangular lattice

Yuan-Ming Lu

Published Apr 11, 2016• 58 citations
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Abstract

Motivated by recent numerical discovery of a gapped spin liquid phase in spin-$1/2$ triangular-lattice ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model, we classify symmetric ${Z}_{2}$ spin liquids on triangular lattice in the Abrikosov-fermion representation. We find 20 phases with distinct spinon symmetry quantum numbers, eight of which have their counterparts in the Schwinger-boson representation. Among them we identify two promising candidates (#1 and #20), which can realize a gapped ${Z}_{2}$ spin liquid with up to next nearest neighbor mean-field amplitudes. We analyze their neighboring magnetic orders and valence bond solid patterns, and find one state (#20) that is connected to 120-degree Neel order by a continuous quantum phase transition. We also identify gapped nematic ${Z}_{2}$ spin liquids in the neighborhood of the symmetric states and find three promising candidates (#1, #6, and #20).

DOI
10.1103/physrevb.93.165113

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Motivated by recent numerical discovery of a gapped spin liquid phase in spin-1/21/21/2 triangular-lattice J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ Heisenberg model, we classify symmetric Z2{Z}_{2}Z2​ spin liquids on triangular lattice in the Abrikosov-fermion representation. We find 20 phases with distinct spinon symmetry quantum numbers, eight of which have their counterparts in the Schwinger-boson representation. Among them we identify two promising candidates (#1 and #20), which can realize a gapped Z2{Z}_{2}Z2​ spin liquid with up to next nearest neighbor mean-field amplitudes. We analyze their neighboring magnetic orders and valence bond solid patterns, and find one state (#20) that is connected to 120-degree Neel order by a continuous quantum phase transition. We also identify gapped nematic Z2{Z}_{2}Z2​ spin liquids in the neighborhood of the symmetric states and find three promising candidates (#1, #6, and #20).
Classification and Monte Carlo study of symmetric Z2Z_{2}Z2​ spin liquids on the triangular lattice
We study different ways of symmetry fractionalization in Z2Z_{2}Z2​ spin liquids on the triangular lattice. Our classification can be used to identify the symmetry fractionalization in the Z2Z_{2}Z2​ spin liquid reported in recent density-matrix-renormalization-group simulations for J1−J2J_{1}-J_{2}J1​−J2​ spin model on the triangular lattice. We find 64 types of symmetry enriched Z2Z_{2}Z2​ spin liquid states on triangular lattice. Besides 8 states constructed in Schwinger-boson parton wavefunctions, 12 more states can be realized in Abrikosov-fermion parton construction. Within a larger gauge group than SU(2), the rest 44 states are also found in a spin-3/2 system. Among 20 types of Abrikosov-fermion Gutzwillerprojected wavefunctions, No.B5 state is a promising candidate for the Z2Z_{2}Z2​ spin liquid for J1−J2J_{1}-J_{2}J1​−J2​ spin model on the triangular lattice. No.B5 lies close to Dirac spin liquid (DSL). However, variational Monte Carlo simulation find that DSL has a good variational energy and J1−J2J_{1}-J_{2}J1​−J2​ spin model cannot open a gap for spinons on top of DSL to stabilize No.B5 state.
We study possible spin liquids on square lattice that respect all lattice symmetries and time-reversal symmetry within the framework of Schwinger boson (mean-field) theory. Such spin liquids have spin gap and emergent Z2{Z}_{2}Z2​ gauge field excitations. We classify them by the projective symmetry group method, and find six spin-liquid states that are potentially relevant to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. The properties of these states are studied under mean-field approximation. Interestingly we find a spin-liquid state that can go through continuous phase transitions to either the N\'eel magnetic order or magnetic orders of the wave vector at the Brillouin zone edge center. We also discuss the connection between our results and the Abrikosov fermion spin liquids.
We study the spin-12\frac{1}{2}21​ Heisenberg model on the triangular lattice with antiferromagnetic first- (J1)({J}_{1})(J1​) and second- (J2)({J}_{2})(J2​) nearest-neighbor interactions using density matrix renormalization group. By studying the spin correlation function, we find a ${120}^{\ensuremath{\circ}}$ magnetic order phase for ${J}_{2}\ensuremath{\lesssim}0.07{J}_{1}$ and a stripe antiferromagnetic phase for ${J}_{2}\ensuremath{\gtrsim}0.15{J}_{1}$. Between these two phases, we identify a spin-liquid region characterized by exponential decaying spin and dimer correlations, as well as large spin singlet and triplet excitation gaps on finite-size systems. We find two near degenerating ground states with distinct properties in two sectors, which indicates more than one spin-liquid candidate in this region. While the sector with spinons is found to respect time reversal symmetry, the even sector without spinons breaks such a symmetry for finite-size systems. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. Moreover, by tuning the anisotropic bond coupling, we explore the nature of the spin-liquid phase and find the optimal parameter region for gapped Z2{Z}_{2}Z2​ spin liquids.
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The ruby lattice has been the subject of much interest recently due its realization in Rydberg atom arrays, where its rich variety of frustrated interactions gives rise to topologically ordered quantum spin liquids. Similarly, numerical studies of ruby-lattice spin models, with both isotropic and anisotropic interactions, have provided evidence of gapped and gapless spin-liquid ground states with different low-energy gauge structures. Motivated by these findings, we perform a projective symmetry group (PSG) classification of U(1) and Z2\mathbb{Z}_{2}Z2​ fermionic spinon mean-field theories$\unicode{x2014}$respecting space-group and time-reversal symmetries$\unicode{x2014}$for S=1/2S=1/2S=1/2 spins. We obtain a total of 50 U(1) and 64 Z2\mathbb{Z}_{2}Z2​ PSGs, and upon restricting their realization via mean-field Ansa¨tze\textit{Ansätze}Ansa¨tze with up to second-nearest-neighbor singlet amplitudes (relevant to the models studied here), only 8 U(1) and 18 Z2\mathbb{Z}_{2}Z2​ distinct states are obtained. We present the singlet fields for all Ansa¨tze\textit{Ansätze}Ansa¨tze up to third-nearest-neighboring bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. Building on this information, we also obtain the phase diagram of the Heisenberg model in the presence of first (J1J_{1}J1​), second (J1′J_{1}'J1′​), and third (J2J_{2}J2​) neighbor antiferromagnetic couplings within a self-consistent mean-field approximation.
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., $k=(\ensuremath{\pi},0)$.