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:2501.09567 · Physical review. B./Physical review. B

Bridging conformal field theory and parton approaches to SU ( n ) k chiral spin liquids

Tong Liu, Ying-Hai Wu, Hong-Hao Tu, Tao Xiang

Published May 27, 2025• 3 citations
arXivHTMLPDF

Abstract

We employ the $\mathrm{SU}{(n)}_{k}$ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions. The spins on all lattice sites are chosen to transform under the $\mathrm{SU}(n)$ irreducible representation with a single row and $k$ boxes in the Young tableau. It is demonstrated that the wave functions can be reinterpreted as parton states, which enables efficient conversion to matrix product states, such that many physical properties can be evaluated directly. In one dimension, these wave functions describe critical spin chains whose universality classes are in one-to-one correspondence with the WZW models used in the construction. In two dimensions, our constructions yield model wave functions for chiral spin liquids, and we show how to find all topological sectors of them in a systematic way. Using the null vectors of Kac-Moody algebras, parent Hamiltonians of the $\mathrm{SU}{(3)}_{k}$ series are derived. The $\mathrm{SU}{(3)}_{k}$ chiral spin liquids are lattice analogs of non-Abelian spin-singlet fractional quantum Hall states, and the $k=2$ member hosts Fibonacci anyons.

DOI
10.1103/physrevb.111.205137

Related papers

Quantum spin models for the SU ( n ) 1 Wess–Zumino–Witten model

Hong-Hao Tu, Anne E. B. Nielsen, Germán Sierra

We propose 1D and 2D lattice wave functions constructed from the SU(n)1 Wess–Zumino–Witten (WZW) model and derive their parent Hamiltonians. When all spins in the lattice transform under SU(n) fundamental representations, we obtain a two-body Hamiltonian in 1D, including the SU(n) Haldane–Shastry model as a special case. In 2D, we show that the wave function converges to a class of Halperin's multilayer fractional quantum Hall states and belongs to chiral spin liquids. Our result reveals a hidden SU(n) symmetry for this class of Halperin states. When the spins sit on bipartite lattices with alternating fundamental and conjugate representations, we provide numerical evidence that the state in 1D exhibits quantum criticality deviating from the expected behaviors of the SU(n)1 WZW model, while in 2D they are chiral spin liquids being consistent with the prediction of the SU(n)1 WZW model.

arXiv:1405.2950•Nuclear Physics B•Jul 5, 2014•46 citations

Lattice spin models for non-Abelian chiral spin liquids

P. Lecheminant, Alexei M. Tsvelik

We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids---spin analogs of fractional non-Abelian quantum Hall states---with gapped bulk and gapless chiral edge excitations described by the SU(2)${}_{n}$ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-$n/2$ ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)${}_{n}$ gapless edge excitations.

arXiv:1608.05977•Physical review. B./Physical review. B•Apr 26, 2017•39 citations

Resonating valence bond realization of spin-1 non-Abelian chiral spin liquid on the torus

Hua-Chen Zhang, Ying-Hai Wu, Hong-Hao Tu +1 more

We propose resonating valence bond wave functions for a spin-1 system on the torus that realize a non-Abelian chiral spin liquid. The wave functions take the form of infinite-dimensional matrix product states constructed from conformal blocks of the $\mathrm{SO}{(3)}_{1}$ Wess-Zumino-Witten model. This means that they are lattice analogs of the bosonic Moore-Read state introduced in fractional quantum Hall systems. The topological order of this system is revealed by explicit construction of threefold-degenerate ground states and analytical computation of the modular $S$ and $T$ matrices.

arXiv:2009.04911•Physical review. B./Physical review. B•Feb 17, 2021•8 citations

Conformal field theory and the non-abelian SU (2) k chiral spin liquid

Thomas Quella, Abhishek Roy

Abstract We construct a family of 1D and 2D long-range SU (2) spin models as parent Hamiltonians associated with infinite dimensional matrix product states that arise from simple current correlation functions in SU (2) k WZW models. Our results provide a conformal field theory foundation for recent proposals by Greiter and coauthors regarding the realization of non-abelian chiral spin liquids. We explain, in particular, how the symmetrization procedure for the amplitudes of the ground state wave function suggested by Greiter M and Thomale R (2009 Phys. Rev. Lett. 102 207203) originates from the conformal field theory description.

arXiv:1911.01505•Journal of Statistical Mechanics Theory and Experiment•May 1, 2020•4 citations

Projected BCS states and spin Hamiltonians for the SO( n ) 1 Wess-Zumino-Witten model

Hong-Hao Tu

We propose a class of projected BCS wave functions and derive their parent spin Hamiltonians. These wave functions can be formulated as infinite matrix product states constructed by chiral correlators of Majorana fermions. In one dimension, the spin Hamiltonians can be viewed as SO($n$) generalizations of Haldane-Shastry models. We numerically compute the spin-spin correlation functions and R\'enyi entropies for $n=5$ and 6. Together with the results for $n=3$ and 4, we conclude that these states are critical and their low-energy effective theory is the SO($n$)${}_{1}$ Wess-Zumino-Witten model. In two dimensions, we show that the projected BCS states are chiral spin liquids, which support non-Abelian anyons for odd $n$ and Abelian anyons for even $n$.

arXiv:1210.1481•Physical Review B•Jan 14, 2013•41 citations

Possible SU(3) chiral spin liquid on the kagome lattice

Ying-Hai Wu, Hong-Hao Tu

We propose an SU(3) symmetric Hamiltonian with short-range interactions on the kagome lattice and show that it hosts an Abelian chiral spin liquid (CSL) state. We provide numerical evidence based on exact diagonalization to show that this CSL state is stabilized in an extended region of the parameter space and can be viewed as a lattice version of the Halperin 221 fractional quantum Hall state of two-component bosons. We also construct a parton wave function for this CSL state and demonstrate that its variational energies are in good agreement with exact diagonalization results. The parton description further supports that the CSL is characterized by a chiral edge conformal field theory of the $\mathrm{SU}{(3)}_{1}$ Wess-Zumino-Witten type.

arXiv:1601.02594•Physical review. B./Physical review. B•Nov 18, 2016•10 citations
We employ the SU(n)k\mathrm{SU}{(n)}_{k}SU(n)k​ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions. The spins on all lattice sites are chosen to transform under the SU(n)\mathrm{SU}(n)SU(n) irreducible representation with a single row and kkk boxes in the Young tableau. It is demonstrated that the wave functions can be reinterpreted as parton states, which enables efficient conversion to matrix product states, such that many physical properties can be evaluated directly. In one dimension, these wave functions describe critical spin chains whose universality classes are in one-to-one correspondence with the WZW models used in the construction. In two dimensions, our constructions yield model wave functions for chiral spin liquids, and we show how to find all topological sectors of them in a systematic way. Using the null vectors of Kac-Moody algebras, parent Hamiltonians of the SU(3)k\mathrm{SU}{(3)}_{k}SU(3)k​ series are derived. The SU(3)k\mathrm{SU}{(3)}_{k}SU(3)k​ chiral spin liquids are lattice analogs of non-Abelian spin-singlet fractional quantum Hall states, and the k=2k=2k=2 member hosts Fibonacci anyons.
We propose 1D and 2D lattice wave functions constructed from the SU(n)1 Wess–Zumino–Witten (WZW) model and derive their parent Hamiltonians. When all spins in the lattice transform under SU(n) fundamental representations, we obtain a two-body Hamiltonian in 1D, including the SU(n) Haldane–Shastry model as a special case. In 2D, we show that the wave function converges to a class of Halperin's multilayer fractional quantum Hall states and belongs to chiral spin liquids. Our result reveals a hidden SU(n) symmetry for this class of Halperin states. When the spins sit on bipartite lattices with alternating fundamental and conjugate representations, we provide numerical evidence that the state in 1D exhibits quantum criticality deviating from the expected behaviors of the SU(n)1 WZW model, while in 2D they are chiral spin liquids being consistent with the prediction of the SU(n)1 WZW model.
We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids---spin analogs of fractional non-Abelian quantum Hall states---with gapped bulk and gapless chiral edge excitations described by the SU(2)n{}_{n}n​ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-n/2n/2n/2 ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)n{}_{n}n​ gapless edge excitations.
We propose resonating valence bond wave functions for a spin-1 system on the torus that realize a non-Abelian chiral spin liquid. The wave functions take the form of infinite-dimensional matrix product states constructed from conformal blocks of the SO(3)1\mathrm{SO}{(3)}_{1}SO(3)1​ Wess-Zumino-Witten model. This means that they are lattice analogs of the bosonic Moore-Read state introduced in fractional quantum Hall systems. The topological order of this system is revealed by explicit construction of threefold-degenerate ground states and analytical computation of the modular SSS and TTT matrices.
Abstract We construct a family of 1D and 2D long-range SU (2) spin models as parent Hamiltonians associated with infinite dimensional matrix product states that arise from simple current correlation functions in SU (2) k WZW models. Our results provide a conformal field theory foundation for recent proposals by Greiter and coauthors regarding the realization of non-abelian chiral spin liquids. We explain, in particular, how the symmetrization procedure for the amplitudes of the ground state wave function suggested by Greiter M and Thomale R (2009 Phys. Rev. Lett. 102 207203) originates from the conformal field theory description.
We propose a class of projected BCS wave functions and derive their parent spin Hamiltonians. These wave functions can be formulated as infinite matrix product states constructed by chiral correlators of Majorana fermions. In one dimension, the spin Hamiltonians can be viewed as SO(nnn) generalizations of Haldane-Shastry models. We numerically compute the spin-spin correlation functions and R\'enyi entropies for n=5n=5n=5 and 6. Together with the results for n=3n=3n=3 and 4, we conclude that these states are critical and their low-energy effective theory is the SO(nnn)1{}_{1}1​ Wess-Zumino-Witten model. In two dimensions, we show that the projected BCS states are chiral spin liquids, which support non-Abelian anyons for odd nnn and Abelian anyons for even nnn.
We propose an SU(3) symmetric Hamiltonian with short-range interactions on the kagome lattice and show that it hosts an Abelian chiral spin liquid (CSL) state. We provide numerical evidence based on exact diagonalization to show that this CSL state is stabilized in an extended region of the parameter space and can be viewed as a lattice version of the Halperin 221 fractional quantum Hall state of two-component bosons. We also construct a parton wave function for this CSL state and demonstrate that its variational energies are in good agreement with exact diagonalization results. The parton description further supports that the CSL is characterized by a chiral edge conformal field theory of the SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​ Wess-Zumino-Witten type.