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arXiv:0903.3408 · Physical Review B

Classification of quantum phases for the star-lattice antiferromagnet via a projective symmetry group analysis

Ting-Pong Choy, Yong Baek Kim

Published Aug 13, 2009• 17 citations
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Abstract

We study possible quantum ground states of the Heisenberg antiferromagnet on the star lattice, which may be realized in the recently discovered polymeric iron acetate, ${\text{Fe}}_{3}({\ensuremath{\mu}}_{3}\text{-O}){(\ensuremath{\mu}\text{-OAc})}_{6}{({\text{H}}_{2}\text{O})}_{3}{[{\text{Fe}}_{3}({\ensuremath{\mu}}_{3}\text{-O}){(\ensuremath{\mu}\text{-OAc})}_{7.5}]}_{2}\ensuremath{\cdot}7{\text{H}}_{2}\text{O}$ [Y. Z. Zheng et al., Angew. Chem. Int. Ed. 46, 6076 (2007)]. Even though the ${\text{Fe}}^{\text{III}}$ moment in this material carries spin-5/2 and the system eventually orders magnetically at low temperatures, the magnetic ordering temperature is much lower than the estimated Curie-Weiss temperature, revealing the frustrated nature of the spin interactions. Anticipating that a lower spin analog of this material may be synthesized in future, we investigate the effect of quantum fluctuations on the star-lattice antiferromagnet using a large-$N$ $\text{Sp}(N)$ mean field theory and a projective symmetry group analysis for possible bosonic quantum spin liquid phases. It is found that there exist only two distinct gapped ${Z}_{2}$ spin liquid phases with bosonic spinons for nonvanishing nearest-neighbor valence-bond amplitudes. In particular, the spin liquid phase which has a lower energy in the nearest-neighbor exchange model can be stabilized for relatively higher spin magnitudes. Hence, it is perhaps a better candidate for the realization of quantum spin liquid state. We also determine the magnetic ordering patterns resulting from the condensation of the bosonic spinons in the two different spin liquid phases. We expect these magnetic ordering patterns would directly be relevant for the low temperature ordered phase of the iron acetate. The phase diagram containing all of these phases and various dimerized states are obtained for the nearest-neighbor exchange model and its implications are discussed.

DOI
10.1103/physrevb.80.064404

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We study possible quantum ground states of the Heisenberg antiferromagnet on the star lattice, which may be realized in the recently discovered polymeric iron acetate, Fe3(\ensuremathμ3-O)(\ensuremathμ-OAc)6(H2O)3[Fe3(\ensuremathμ3-O)(\ensuremathμ-OAc)7.5]2\ensuremath⋅7H2O{\text{Fe}}_{3}({\ensuremath{\mu}}_{3}\text{-O}){(\ensuremath{\mu}\text{-OAc})}_{6}{({\text{H}}_{2}\text{O})}_{3}{[{\text{Fe}}_{3}({\ensuremath{\mu}}_{3}\text{-O}){(\ensuremath{\mu}\text{-OAc})}_{7.5}]}_{2}\ensuremath{\cdot}7{\text{H}}_{2}\text{O}Fe3​(\ensuremathμ3​-O)(\ensuremathμ-OAc)6​(H2​O)3​[Fe3​(\ensuremathμ3​-O)(\ensuremathμ-OAc)7.5​]2​\ensuremath⋅7H2​O [Y. Z. Zheng et al., Angew. Chem. Int. Ed. 46, 6076 (2007)]. Even though the FeIII{\text{Fe}}^{\text{III}}FeIII moment in this material carries spin-5/2 and the system eventually orders magnetically at low temperatures, the magnetic ordering temperature is much lower than the estimated Curie-Weiss temperature, revealing the frustrated nature of the spin interactions. Anticipating that a lower spin analog of this material may be synthesized in future, we investigate the effect of quantum fluctuations on the star-lattice antiferromagnet using a large-NNN Sp(N)\text{Sp}(N)Sp(N) mean field theory and a projective symmetry group analysis for possible bosonic quantum spin liquid phases. It is found that there exist only two distinct gapped Z2{Z}_{2}Z2​ spin liquid phases with bosonic spinons for nonvanishing nearest-neighbor valence-bond amplitudes. In particular, the spin liquid phase which has a lower energy in the nearest-neighbor exchange model can be stabilized for relatively higher spin magnitudes. Hence, it is perhaps a better candidate for the realization of quantum spin liquid state. We also determine the magnetic ordering patterns resulting from the condensation of the bosonic spinons in the two different spin liquid phases. We expect these magnetic ordering patterns would directly be relevant for the low temperature ordered phase of the iron acetate. The phase diagram containing all of these phases and various dimerized states are obtained for the nearest-neighbor exchange model and its implications are discussed.
We investigate various competing paramagnetic ground states of the Heisenberg antiferromagnet on the two-dimensional star lattice which exhibits geometric frustration. Using slave particle mean-field theory combined with a projective symmetry group analysis, we examine a variety of candidate spin liquid states on this lattice, including chiral spin liquids, spin liquids with Fermi surfaces of spinons, and nematic spin liquids which break lattice rotational symmetry. Motivated by connection to large-NNN SU(N)\text{SU}(N)SU(N) theory as well as numerical exact diagonalization studies, we also examine various valence-bond solid (VBS) states on this lattice. Based on a study of energetics using Gutzwiller projected states, we find that a fully gapped spin liquid state is the lowest-energy spin liquid candidate for this model. We also find from a study of energetics using Gutzwiller projected wave functions and bond operator approaches that this spin liquid is unstable toward two different VBS states---a VBS state which respects all the Hamiltonian symmetries and a VBS state which exhibits $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order---depending on the ratio of the Heisenberg exchange couplings on the two inequivalent bonds of the lattice. We compute the triplon dispersion in both VBS states within the bond operator approach and discuss possible implications of our work for future experiments on candidate materials.
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.
A symmetry-based analysis (projective symmetry group) is used to study spin-liquid phases on the triangular and Kagom\'e lattices in the Schwinger boson framework. A maximum of eight distinct Z2{Z}_{2}Z2​ spin-liquid states are found for each lattice, which preserve all symmetries. Out of these only a few have nonvanishing nearest-neighbor amplitudes, which are studied in greater detail. On the triangular lattice, only two such states are present---the first (zero-flux state) is the well-known state introduced by Sachdev, which on condensation of spinons leads to the 120\ifmmode^\circ\else\textdegree\fi{} ordered state. The other solution, which we call the $\ensuremath{\pi}$-flux state has not previously been discussed. Spinon condensation leads to an ordering wave vector at the Brillouin zone edge centers, in contrast to the 120\ifmmode^\circ\else\textdegree\fi{} state. While the zero-flux state is more stable with just nearest-neighbor exchange, we find that the introduction of either next-neighbor antiferromagnetic exchange or four-spin ring exchange (of the sign obtained from a Hubbard model) tends to favor the $\ensuremath{\pi}$-flux state. On the Kagom\'e lattice four solutions are obtained---two have been previously discussed by Sachdev, which on spinon condensation give rise to the q=0q=0q=0 and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ spin-ordered states. In addition we find two states with significantly larger values of the quantum parameter at which magnetic ordering occurs. For one of them this even exceeds unity ${\ensuremath{\kappa}}_{c}\ensuremath{\approx}2.0$ in a nearest-neighbor model, indicating that if stabilized, could remain spin disordered for physical values of the spin. This state is also stabilized by ring-exchange interactions with signs as derived from the Hubbard model.
Motivated by recent numerical discovery of a gapped spin liquid phase in spin-1/21/21/2 triangular-lattice ${J}_{1}\ensuremath{-}{J}_{2}$ 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).
We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling J1>0{J}_{1}>0J1​>0 and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.
The nature of quantum spin liquids is studied for the spin-1/21/21/2 antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, J1J_1J1​, and those between next-nearest-neighbor sites, J2J_2J2​. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region 0.4<J2/J1≤0.60.4< J_2/J_1\le 0.60.4<J2​/J1​≤0.6 sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of 0.4<J2/J1≤0.50.4 < J_2/J_1 \le 0.50.4<J2​/J1​≤0.5, while the gapped valence-bond-crystal (VBC) phase is stabilized for 0.5<J2/J1≤0.60.5 < J_2/J_1 \le 0.60.5<J2​/J1​≤0.6. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the C4vC_{4v}C4v​ symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation ⟨S(0)S(r)⟩∝1/rz+η\langle S(0)S(r)\rangle \propto 1/r^{z+η}⟨S(0)S(r)⟩∝1/rz+η at a long distance rrr appears to increase from z+η∼1z+η\sim 1z+η∼1 at J2/J1∼0.4J_2/J_1\sim0.4J2​/J1​∼0.4 toward the continuous transition to the VBC phase at J1/J1∼0.5J_1/J_1\sim0.5J1​/J1​∼0.5. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.