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arXiv:1805.11628 · Physical review. B./Physical review. B

Gapped SU(3) spin liquid with Z3 topological order

Ivana Kurečić, Laurens Vanderstraeten, Norbert Schuch

Published Jan 9, 2019• 14 citations
arXivHTMLPDF

Abstract

We construct a topological spin-liquid (TSL) model on the kagome lattice, with $\mathrm{SU}(3)$ symmetry with the fundamental representation at each lattice site, based on projected entangled pair states (PEPS). Using the PEPS framework, we can adiabatically connect the model to a fixed-point model (analogous to the dimer model for resonating valence bond states) which we prove to be locally equivalent to a ${\mathbb{Z}}_{3}$ quantum double model. Numerical study of the interpolation reveals no sign of a phase transition or long-range order, characterizing the model conclusively as a gapped TSL. We further study the entanglement spectrum of the model and find that while it is gapped, it exhibits branches with vastly different velocities, with the slow branch matching the counting of a chiral $\mathrm{SU}{(3)}_{1}$ Conformal Field Theory, suggesting that it can be deformed to a model with chiral $\mathrm{SU}{(3)}_{1}$ entanglement spectrum.

DOI
10.1103/physrevb.99.045116

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We construct a topological spin-liquid (TSL) model on the kagome lattice, with SU(3)\mathrm{SU}(3)SU(3) symmetry with the fundamental representation at each lattice site, based on projected entangled pair states (PEPS). Using the PEPS framework, we can adiabatically connect the model to a fixed-point model (analogous to the dimer model for resonating valence bond states) which we prove to be locally equivalent to a Z3{\mathbb{Z}}_{3}Z3​ quantum double model. Numerical study of the interpolation reveals no sign of a phase transition or long-range order, characterizing the model conclusively as a gapped TSL. We further study the entanglement spectrum of the model and find that while it is gapped, it exhibits branches with vastly different velocities, with the slow branch matching the counting of a chiral SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​ Conformal Field Theory, suggesting that it can be deformed to a model with chiral SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​ entanglement spectrum.
Motivated by the search for chiral spin liquids (CSL), we consider a simple model defined on the kagome lattice of interacting SU(3) spins (in the fundamental representation) including two-site and three-site permutations between nearest neighbor sites and on triangles, respectively. By combining analytical methods and various numerical techniques, namely, exact Lanczos diagonalizations and tensor network variational approaches, we find a rich phase diagram with short-range entangled (i.e., nontopological) and topological (possibly chiral) gapped spin liquids (SLs). Short-range entangled spin liquids include an Affleck-Kennedy-Lieb-Tasaki (AKLT)-like phase---argued to be a Symmetry-Protected Topological (SPT) phase---and a trimerized phase breaking the inversion center between the up and down triangles of the kagome lattice. A topological SL is stabilized in a restricted part of the phase diagram by the time-reversal symmetry breaking (complex) three-site permutation term. Analyzing the chiral edge modes of this topological SL on long cylinders or on finite disks, we have come up with two competing scenarios, either a CSL or a double Chern-Simons SL characterized by a single or by two counter-propagating Wess-Zumino-Witten SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​ chiral mode(s), respectively. In the vicinity of the extended ferromagnetic region, we have found a magnetic phase corresponding either to a modulated canted ferromagnet or to a uniform partially magnetized ferromagnet.
We construct and study quantum trimer models and resonating SU(3)-singlet models on the kagome lattice, which generalize quantum dimer models and the resonating valence bond wave functions to a trimer and SU(3) setting. We demonstrate that these models carry a Z(3) symmetry which originates in the structure of trimers and the SU(3) representation theory, and which becomes the only symmetry under renormalization. Based on this, we construct simple and exact parent Hamiltonians for the model which exhibit a topological ninefold degenerate ground space. A combination of analytical reasoning and numerical analysis reveals that the quantum order ultimately displayed by the model depends on the relative weight assigned to different types of trimers-it can display either Z(3) topological order or form a symmetry-broken trimer crystal, and in addition possesses a point with an enhanced U(1) symmetry and critical behavior. Our results accordingly hold for the SU(3) model, where the two natural choices for trimer weights give rise to either a topological spin liquid or a system with symmetry-broken order, respectively. Our work thus demonstrates the suitability of resonating trimer and SU(3)-singlet ansatzes to model SU(3) topological spin liquids on the kagome lattice.
We construct and study quantum trimer models and resonating SU(3)-singlet models on the kagome lattice, which generalize quantum dimer models and the resonating valence bond wave functions to a trimer and SU(3) setting.We demonstrate that these models carry a Z 3 symmetry which originates in the structure of trimers and the SU(3) representation theory, and which becomes the only symmetry under renormalization.Based on this, we construct simple and exact parent Hamiltonians for the model which exhibit a topological ninefold degenerate ground space.A combination of analytical reasoning and numerical analysis reveals that the quantum order ultimately displayed by the model depends on the relative weight assigned to different types of trimers-it can display either Z 3 topological order or form a symmetry-broken trimer crystal, and in addition possesses a point with an enhanced U(1) symmetry and critical behavior.Our results accordingly hold for the SU(3) model, where the two natural choices for trimer weights give rise to either a topological spin liquid or a system with symmetry-broken order, respectively.Our work thus demonstrates the suitability of resonating trimer and SU(3)-singlet ansatzes to model SU(3) topological spin liquids on the kagome lattice.
Quantum spin liquids can be faithfully represented and efficiently characterized within the framework of projected entangled pair states (PEPS). Guided by extensive exact diagonalization and density matrix renormalization group calculations, we construct an optimized symmetric PEPS for a SU(3)_{1} chiral spin liquid on the square lattice. Characteristic features are revealed by the entanglement spectrum (ES) on an infinitely long cylinder. In all three Z_{3} sectors, the level counting of the linear dispersing modes is in full agreement with SU(3)_{1} Wess-Zumino-Witten conformal field theory prediction. Special features in the ES are shown to be in correspondence with bulk anyonic correlations, indicating a fine structure in the holographic bulk-edge correspondence. Possible universal properties of topological SU(N)_{k} chiral PEPS are discussed.
Protected chiral edge modes are a well-known signature of topologically ordered phases, such as the fractional quantum Hall states (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, the authors construct a family of chiral spin-12\frac{1}{2}21​ quantum spin liquids with Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2)1{}_{1}1​), as is the case for the $\ensuremath{\nu}$=12\frac{1}{2}21​ (bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.
Projected entangled pair states (PEPS) provide exact representations for many nonchiral topologically ordered states whereas their range of applicability to interacting chiral topological phases remains largely unsettled. In this context, the symmetries of the local PEPS tensors are crucial for determining the characteristic topological features of the state. In this paper we examine the constraints that arise when different symmetries are imposed simultaneously on the local tensor such as internal SU(2)\mathrm{SU}(2)SU(2), point group, and translation symmetry. We show how the interplay of these symmetries manifests in the entanglement spectrum which is the main diagnostic tool for chiral topological order. We apply our results to a spin liquid PEPS introduced previously as a chiral generalization of the resonating valence bond state. Our findings explain the discrepancies observed between the entanglement spectrum of this state and the expected edge spectrum described by a chiral conformal field theory. Finally, in a certain parameter region where this PEPS possesses an additional U(1)U(1)U(1) symmetry we are able to resolve these discrepancies and obtain an entanglement spectrum with the expected state countings and conformal weight.