Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry

Abstract

We elaborate a simple classification scheme of all rank-5 SU(2)-spin rotational symmetric tensors according to i) the on-site physical spin-$S$, (ii) the local Hilbert space $V^{\otimes 4}$ of the four virtual (composite) spins attached to each site and (iii) the irreducible representations of the $C_{4v}$ point group of the square lattice. We apply our scheme to draw a complete list of all SU(2)-symmetric translationally and rotationally-invariant Projected Entangled Pair States (PEPS) with bond dimension $D⩽6$. All known SU(2)-symmetric PEPS on the square lattice are recovered and simple generalizations are provided in some cases. More generally, to each of our symmetry class can be associated a $(\mathcal{D}-1)$-dimensional manifold of spin liquids (potentially) preserving lattice symmetries and defined in terms of $\mathcal{D}$ independent tensors of a given bond dimension $D$. In addition, generic (low-dimensional) families of PEPS explicitly breaking either (i) particular point-group lattice symmetries (lattice nematics) or (ii) time reversal symmetry (chiral spin liquids) or (iii) SU(2)-spin rotation symmetry down to $U(1)$ (spin nematics or Néel antiferromagnets) can also be constructed. We apply this framework to search for new topological chiral spin liquids characterized by well-defined chiral edge modes, as revealed by their entanglement spectrum. In particular, we show how the symmetrization of a double-layer PEPS leads to a chiral topological state with a gapless edge described by a SU(2)${}_2$ Wess-Zumino-Witten model.

29 pages, 9 figures, 3 Appendices, revised version. Supplementary material with classification up to D=6 in the source file, and exact tensor expressions available as a Supplementary Material in the PRB published article

References in corpus (20)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Matrix product states represent ground states faithfully
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• The iTEBD algorithm beyond unitary evolution
• Tensor network states and algorithms in the presence of a global U(1) symmetry
• Resonating valence bond states in the PEPS formalism
• Spin Hamiltonian for which the Chiral Spin Liquid is the Exact Ground State
• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• Gauging quantum states: from global to local symmetries in many-body systems
• Non-Abelian Statistics in a Quantum Antiferromagnet
• Correlation Lengths and Topological Entanglement Entropies of Unitary and Non-Unitary Fractional Quantum Hall Wavefunctions
• Chiral topological spin liquids with projected entangled pair states
• Critical Correlations for Short-Range Valence-Bond Wave Functions on the Square Lattice
• Projected Entangled Pair States with non-Abelian gauge symmetries: an SU(2) study
• Topological transitions from multipartite entanglement with tensor networks: a procedure for sharper and faster characterization
• RVB superconductors with fermionic projected entangled pair states
• Constructing topological models by symmetrization: A PEPS study

Cited by in corpus (13)

• Investigation of the chiral antiferromagnetic Heisenberg model using PEPS
• Chiral spin liquids on the kagome lattice with projected entangled simplex states
• Simulating both parity sectors of the Hubbard Model with Tensor Networks
• Phase diagram of the chiral SU(3) antiferromagnet on the kagome lattice
• Stable and efficient differentiation of tensor network algorithms
• Matrix product state fixed points of non-Hermitian transfer matrices
• Efficient calculation of three-dimensional tensor networks
• Real-time dynamics of a critical Resonating Valence Bond spin liquid
• SU(4) topological RVB spin liquid on the square lattice
• Quantum state preparation of topological chiral spin liquids via Floquet engineering
• Tensor network variational optimizations for real-time dynamics: application to the time-evolution of spin liquids
• Floquet dynamical chiral spin liquid at finite frequency
• Entanglement Spectrum as a diagnostic of chirality of Topological Spin Liquids: Analysis of an $\mathrm{SU}(3)$ PEPS