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

2016/08/31 by Matthieu Mambrini, Román Orús, Roman Orus +1 · 57 citations
Mathematics · Physics and Astronomy · #Advanced Condensed Matter Physics #Algebraic structures and combinatorial models #Condensed matter physics #Geometry #Lattice (music) #Mathematics #Physics #Pure mathematics #Quantum many-body systems #Spin (aerodynamics) #Square (algebra) #Square lattice #Symmetry (geometry) #Tensor (intrinsic definition) #Theoretical physics #Thermodynamics #cond-mat.str-el #hep-lat #quant-ph

paper · pdf · doi:10.1103/physrevb.94.205124

published in Physical review. B./Physical review. B 94(20) (American Physical Society) · 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

openalex publication_date 2016/11/14 · arxiv created 2016/11/15 · arxiv updated 2016/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We elaborate a simple classification scheme of all rank-5 SU(2) spin rotational symmetric tensors according to (i) the onsite physical spin S, (ii) the local Hilbert space V^\ensuremath\bigotimes4 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\ensuremath≤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 (D\ensuremath-1)-dimensional manifold of spin liquids (potentially) preserving lattice symmetries and defined in terms of 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 N'eel 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.

Citations

• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Colloquium: Area laws for the entanglement entropy
• Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations
• Global aspects of current algebra
• Non-Abelian Statistics in a Quantum Antiferromagnet
• Efficient Simulation of One-Dimensional Quantum Many-Body Systems
• Topological and entanglement properties of resonating valence bond wave functions
• Equivalence of the resonating-valence-bond and fractional quantum Hall states
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix product states represent ground states faithfully
• Topological Transitions from Multipartite Entanglement with Tensor Networks: A Procedure for Sharper and Faster Characterization
• Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems
• Fractional Quantum Hall Matrix Product States For Interacting Conformal Field Theories
• Nonabelions in the fractional quantum hall effect
• Classification of trivial spin-1 tensor network states on a square lattice

Cited by

• Quantum spin liquid phases in the bilinear-biquadratic two-SU(4)-fermion Hamiltonian on the square lattice