Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0210662

Quantum Orders and Spin Liquids in Cs$_2$CuCl$_4$

Yi Zhou, Xiao-Gang Wen

Published Oct 30, 2002• 10 citations
arXivHTMLPDF

Abstract

Motivated by experiments on Cs$_2$CuCl$_4$ samples, we studied and classified the symmetric spin liquids on triangular lattice. We identified 63 $Z_2$ spin liquids, 30 U(1) spin liquids and 2 SU(2) spin liquids. All those spin liquids have the same symmetry but different quantum orders. We calculated the spin spectral functions in some simple spin liquids and compared them to the one measured by experiments on Cs$_2$CuCl$_4$. We find that the $U1Cτ_{+}^0τ_{-}^0τ^1$ spin liquid or one of its relatives is consistent with observed properties of the spin liquid state in Cs$_2$CuCl$_4$. We discussed the fine distinctions among those possible spin liquids and the spin liquids proposed using slave-fermion approach, so that future experiments can determine which spin liquids actually describe the spin state in Cs$_2$CuCl$_4$.

DOI
10.48550/arxiv.cond-mat/0210662

Related papers

Symmetric Z2 spin liquids and their neighboring phases on triangular lattice

Yuan-Ming Lu

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).

arXiv:1505.06495•Physical review. B./Physical review. B•Apr 11, 2016•58 citations

Algebraic Vortex Liquid in Spin- 1 / 2 Triangular Antiferromagnets: Scenario for Cs 2 CuCl 4

Jason Alicea, Olexei I. Motrunich, Matthew P. A. Fisher

Motivated by inelastic neutron scattering data on Cs2CuCl4, we explore spin-1/2 triangular lattice antiferromagnets with both spatial and easy-plane exchange anisotropies, the latter due to an observed Dzyaloshinskii-Moriya interaction. Exploiting a duality mapping followed by a fermionization of the dual vortex degrees of freedom, we find a novel critical spin-liquid phase described in terms of Dirac fermions with an emergent global SU(4) symmetry minimally coupled to a noncompact U(1) gauge field. This "algebraic vortex liquid" supports gapless spin excitations and universal power-law correlations in the dynamical spin structure factor which are consistent with those observed in Cs2CuCl4. We suggest future neutron scattering experiments that should help distinguish between the algebraic vortex liquid and other spin liquids and quantum critical points previously proposed in the context of Cs2CuCl4.

arXiv:cond-mat/0508536•Physical Review Letters•Dec 6, 2005•54 citations

Projective symmetry group classification of chiral spin liquids

Samuel Bieri, C. Lhuillier, Laura Messio

We present a general review of the projective symmetry group classification of fermionic quantum spin liquids for lattice models of spin $S=1/2$. We then introduce a systematic generalization of the approach for symmetric ${\mathbb{Z}}_{2}$ quantum spin liquids to the one of chiral phases (i.e., singlet states that break time reversal and lattice reflection, but conserve their product). We apply this framework to classify and discuss possible chiral spin liquids on triangular and kagome lattices. We give a detailed prescription on how to construct quadratic spinon Hamiltonians and microscopic wave functions for each representation class on these lattices. Among the chiral ${\mathbb{Z}}_{2}$ states, we study the subset of U(1) phases variationally in the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{d}$ Heisenberg model on the kagome lattice. We discuss static spin structure factors and symmetry constraints on the bulk spectra of these phases.

arXiv:1512.00324•Physical review. B./Physical review. B•Mar 30, 2016•101 citations

Classification and Monte Carlo study of symmetric $Z_{2}$ spin liquids on the triangular lattice

Wayne Zheng, Jia‐Wei Mei, Yang Qi

We study different ways of symmetry fractionalization in $Z_{2}$ spin liquids on the triangular lattice. Our classification can be used to identify the symmetry fractionalization in the $Z_{2}$ spin liquid reported in recent density-matrix-renormalization-group simulations for $J_{1}-J_{2}$ spin model on the triangular lattice. We find 64 types of symmetry enriched $Z_{2}$ 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 $Z_{2}$ spin liquid for $J_{1}-J_{2}$ 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 $J_{1}-J_{2}$ spin model cannot open a gap for spinons on top of DSL to stabilize No.B5 state.

arXiv:1505.05351•May 20, 2015•13 citations

Classification of spin liquids on the square lattice with strong spin-orbit coupling

Johannes Reuther, Shu-Ping Lee, Jason Alicea

Spin liquids represent exotic types of quantum matter that evade conventional symmetry-breaking order even at zero temperature. Exhaustive classifications of spin liquids have been carried out in several systems, particularly in the presence of full SU(2) spin-rotation symmetry. Real magnetic compounds, however, generically break SU(2) spin symmetry as a result of spin-orbit coupling---which in many materials provides an ``order one'' effect. We generalize previous works by using the projective symmetry group method to classify ${\mathbb{Z}}_{2}$ spin liquids on the square lattice when SU(2) spin symmetry is maximally lifted. We find that, counterintuitively, the lifting of spin symmetry actually results in vastly more spin-liquid phases compared to SU(2)-invariant systems. A generic feature of the SU(2)-broken case is that the spinons naturally undergo $p+ip$ pairing; consequently, many of these ${\mathbb{Z}}_{2}$ spin liquids feature a topologically nontrivial spinon band structure supporting gapless Majorana edge states. We study in detail several spin-liquid phases with varying numbers of gapless edge states and discuss their topological protection. The edge states are often protected by a combination of time reversal and lattice symmetries and hence resemble recently proposed topological crystalline superconductors.

arXiv:1407.4124•Physical Review B•Nov 14, 2014•28 citations

Statistics of spinons in the spin-liquid phase of Cs2CuCl4

Chung‐Hou Chung, Klaus Voelker, Yong Baek Kim

Motivated by a recent experiment on ${\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}$ [Coldea et al., Phys. Rev. Lett. 86, 1335 (2001)], we study the spin dynamics of the spin-liquid phase of the $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ frustrated Heisenberg antiferromagnet on the anisotropic triangular lattice. There have been two different proposals for the spin-liquid phase of ${\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}.$ These spin-liquid states support different statistics of spinons; the bosonic $\mathrm{Sp}(N)$ large-N mean-field theory predicts bosonic spinons, while the SU(2) slave-boson mean-field theory leads to fermionic spinons. We compute the dynamical spin structure factor for both types of spin-liquid state at zero and finite temperatures. While at zero temperature both theories agree with experiment on a qualitative level, they show substantial differences in the temperature dependence of the dynamical spin structure factor.

arXiv:cond-mat/0302452•Physical review. B, Condensed matter•Sep 12, 2003•39 citations
Quantum Orders and Spin Liquids in Cs2_22​CuCl4_44​
Motivated by experiments on Cs2_22​CuCl4_44​ samples, we studied and classified the symmetric spin liquids on triangular lattice. We identified 63 Z2Z_2Z2​ spin liquids, 30 U(1) spin liquids and 2 SU(2) spin liquids. All those spin liquids have the same symmetry but different quantum orders. We calculated the spin spectral functions in some simple spin liquids and compared them to the one measured by experiments on Cs2_22​CuCl4_44​. We find that the U1Cτ+0τ−0τ1U1Cτ_{+}^0τ_{-}^0τ^1U1Cτ+0​τ−0​τ1 spin liquid or one of its relatives is consistent with observed properties of the spin liquid state in Cs2_22​CuCl4_44​. We discussed the fine distinctions among those possible spin liquids and the spin liquids proposed using slave-fermion approach, so that future experiments can determine which spin liquids actually describe the spin state in Cs2_22​CuCl4_44​.
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).
Motivated by inelastic neutron scattering data on Cs2CuCl4, we explore spin-1/2 triangular lattice antiferromagnets with both spatial and easy-plane exchange anisotropies, the latter due to an observed Dzyaloshinskii-Moriya interaction. Exploiting a duality mapping followed by a fermionization of the dual vortex degrees of freedom, we find a novel critical spin-liquid phase described in terms of Dirac fermions with an emergent global SU(4) symmetry minimally coupled to a noncompact U(1) gauge field. This "algebraic vortex liquid" supports gapless spin excitations and universal power-law correlations in the dynamical spin structure factor which are consistent with those observed in Cs2CuCl4. We suggest future neutron scattering experiments that should help distinguish between the algebraic vortex liquid and other spin liquids and quantum critical points previously proposed in the context of Cs2CuCl4.
We present a general review of the projective symmetry group classification of fermionic quantum spin liquids for lattice models of spin S=1/2S=1/2S=1/2. We then introduce a systematic generalization of the approach for symmetric Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids to the one of chiral phases (i.e., singlet states that break time reversal and lattice reflection, but conserve their product). We apply this framework to classify and discuss possible chiral spin liquids on triangular and kagome lattices. We give a detailed prescription on how to construct quadratic spinon Hamiltonians and microscopic wave functions for each representation class on these lattices. Among the chiral Z2{\mathbb{Z}}_{2}Z2​ states, we study the subset of U(1) phases variationally in the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{d}$ Heisenberg model on the kagome lattice. We discuss static spin structure factors and symmetry constraints on the bulk spectra of these phases.
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.
Spin liquids represent exotic types of quantum matter that evade conventional symmetry-breaking order even at zero temperature. Exhaustive classifications of spin liquids have been carried out in several systems, particularly in the presence of full SU(2) spin-rotation symmetry. Real magnetic compounds, however, generically break SU(2) spin symmetry as a result of spin-orbit coupling---which in many materials provides an ``order one'' effect. We generalize previous works by using the projective symmetry group method to classify Z2{\mathbb{Z}}_{2}Z2​ spin liquids on the square lattice when SU(2) spin symmetry is maximally lifted. We find that, counterintuitively, the lifting of spin symmetry actually results in vastly more spin-liquid phases compared to SU(2)-invariant systems. A generic feature of the SU(2)-broken case is that the spinons naturally undergo p+ipp+ipp+ip pairing; consequently, many of these Z2{\mathbb{Z}}_{2}Z2​ spin liquids feature a topologically nontrivial spinon band structure supporting gapless Majorana edge states. We study in detail several spin-liquid phases with varying numbers of gapless edge states and discuss their topological protection. The edge states are often protected by a combination of time reversal and lattice symmetries and hence resemble recently proposed topological crystalline superconductors.
Motivated by a recent experiment on Cs2CuCl4{\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}Cs2​CuCl4​ [Coldea et al., Phys. Rev. Lett. 86, 1335 (2001)], we study the spin dynamics of the spin-liquid phase of the $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ frustrated Heisenberg antiferromagnet on the anisotropic triangular lattice. There have been two different proposals for the spin-liquid phase of Cs2CuCl4.{\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}.Cs2​CuCl4​. These spin-liquid states support different statistics of spinons; the bosonic Sp(N)\mathrm{Sp}(N)Sp(N) large-N mean-field theory predicts bosonic spinons, while the SU(2) slave-boson mean-field theory leads to fermionic spinons. We compute the dynamical spin structure factor for both types of spin-liquid state at zero and finite temperatures. While at zero temperature both theories agree with experiment on a qualitative level, they show substantial differences in the temperature dependence of the dynamical spin structure factor.