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arXiv:2305.13240 · Physical review. B./Physical review. B

Entanglement spectrum as a diagnostic of chirality of topological spin liquids: Analysis of SU(3) projected entangled pair states

Mark J. Arildsen, Ji-Yao Chen, Norbert Schuch, Andreas W. W. Ludwig

Published Dec 23, 2024• 1 citation
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Abstract

We address the key question of representation of chiral topological quantum states in $2+1$ dimensions (i.e., with nonzero chiral central charge) by projected entangled pair states (PEPS). A noted result (due to Wahl et al. [Phys. Rev. Lett. 111, 236805 (2013)], and Dubail and Read [Phys. Rev. B 92, 205307 (2015)] says that this is possible for noninteracting fermions, but the answer is as yet unknown for interacting systems. Characteristic counting of degeneracies of low-lying states in the entanglement spectrum (ES) at fixed transverse momentum of bipartitioned long cylinders (``Li-Haldane counting'') provides often-used supporting evidence for chirality. However, nonchiral PEPS (with zero chiral central charge), yet with strong breaking of time-reversal and reflection symmetries, with invariance under the product of these two operations (i.e., ``apparently'' chiral states), are known whose low-lying ES exhibits the same Li-Haldane counting as a chiral state in certain topological sectors [Kure\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \acute{c}\else \'{c}\fi{} et al., Phys. Rev. B 99, 045116 (2019); Arildsen et al., Phys. Rev. B 108, 245150 (2023)]. In this work, we identify a distinct indicator and hallmark of chirality in the ES of PEPS with global $\mathrm{SU}(3)$ symmetry: the splittings of conjugate irreps. We prove that in the ES of the chiral states conjugate irreps are exactly degenerate because the operators that would split them [related to the cubic Casimir invariant of $\mathrm{SU}(3)$] are forbidden. By contrast, in the ES of nonchiral states, conjugate splittings are demonstrably nonvanishing. Such a diagnostic provides an unambiguous and powerful tool to distinguish chiral and nonchiral topological states in $2+1$ dimensions via their entanglement spectra.

DOI
10.1103/physrevb.110.235147

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We address the key question of representation of chiral topological quantum states in 2+12+12+1 dimensions (i.e., with nonzero chiral central charge) by projected entangled pair states (PEPS). A noted result (due to Wahl et al. [Phys. Rev. Lett. 111, 236805 (2013)], and Dubail and Read [Phys. Rev. B 92, 205307 (2015)] says that this is possible for noninteracting fermions, but the answer is as yet unknown for interacting systems. Characteristic counting of degeneracies of low-lying states in the entanglement spectrum (ES) at fixed transverse momentum of bipartitioned long cylinders (Li-Haldane counting'') provides often-used supporting evidence for chirality. However, nonchiral PEPS (with zero chiral central charge), yet with strong breaking of time-reversal and reflection symmetries, with invariance under the product of these two operations (i.e., apparently'' chiral states), are known whose low-lying ES exhibits the same Li-Haldane counting as a chiral state in certain topological sectors [Kure\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \acute{c}\else '{c}\fi{} et al., Phys. Rev. B 99, 045116 (2019); Arildsen et al., Phys. Rev. B 108, 245150 (2023)]. In this work, we identify a distinct indicator and hallmark of chirality in the ES of PEPS with global SU(3)\mathrm{SU}(3)SU(3) symmetry: the splittings of conjugate irreps. We prove that in the ES of the chiral states conjugate irreps are exactly degenerate because the operators that would split them [related to the cubic Casimir invariant of SU(3)\mathrm{SU}(3)SU(3)] are forbidden. By contrast, in the ES of nonchiral states, conjugate splittings are demonstrably nonvanishing. Such a diagnostic provides an unambiguous and powerful tool to distinguish chiral and nonchiral topological states in 2+12+12+1 dimensions via their entanglement spectra.
The Li-Haldane correspondence [Phys. Rev. Lett. 101, 010504 (2008)] is often used to help identify wave functions of (2+1)(2+1)(2+1)-dimensional chiral topological phases (i.e., with nonzero chiral central charge) by studying low-lying entanglement spectra (ES) on long cylinders of finite circumference. Here we consider such ES of states [in fact, of wave functions of certain projected entangled pair states (PEPSs)] that are not chiral (i.e., having zero chiral central charge) but which strongly break time-reversal as well as reflection symmetry, while preserving their product, thus exhibiting the same symmetry properties as a chiral topological state. This leads to ES which have branches of both right- and left-moving chiralities, but with vastly different velocities. For circumferences much smaller than the inverse entanglement gap scale, the low-lying ES appear chiral in some topological sectors, and precisely follow the Li-Haldane state counting of a truly chiral phase. On its face, this could lead one to mistakenly identify the phase as chiral. However, by considering the ES in all possible sectors, one can observe distinct differences from a chiral phase. We explore this phenomenon in the setting of an SU(3) spin liquid PEPS studied by Kure\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \acute{c}\else \'{c}\fi{} et al. [Phys. Rev. B 99, 045116 (2019)], where the topologically trivial sector appeared to have the characteristic Li-Haldane counting of a chiral SU(3)\mathrm{SU}(3)SU(3)-level-one [SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​] conformal field theory (CFT). In fact, the PEPS has D(Z3)D({\mathbb{Z}}_{3})D(Z3​) topological order, with nine sectors. We compute the ES in minimally entangled states corresponding to all these sectors, which map to the nine anyon types of doubled SU(3)\mathrm{SU}(3)SU(3)-level-one Chern-Simons topological field theory. The state countings of the ES in all sectors are exactly consistent with our expectation: specifically, the ES contain representations of global SU(3) symmetry from the tensor products of the (lowest-lying) multiplet of primary states of a ``high-velocity'' chiral SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​ CFT with the full content of a ``low-velocity'' chiral SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​ CFT sector, a nonchiral structure beyond that observable in the topologically trivial sector of the ES.
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.
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.
Li and Haldane observed that low-lying entanglement spectra of ground states of (2+1)D chiral topological phases display characteristic degeneracies of the (1+1)D conformal field theory (CFT) appearing at a physical boundary. This work explains the finite-size splittings of these degeneracies by irrelevant deformations of a (1+0)D boundary of the CFT, representing the ground state. These intriguingly arise from operators of fractional dimension in certain entanglement spectra. This approach has applications for diagnosing chirality in topological projected entangled pair states (PEPS).
Doubts have been raised on the representation of chiral spin liquids exhibiting topological order in terms of projected entangled pair states (PEPSs). Here, starting from a simple spin-1/2 chiral frustrated Heisenberg model, we show that a faithful representation of the chiral spin liquid phase is in fact possible in terms of a generic PEPS upon variational optimization. We find a perfectly chiral gapless edge mode and a rapid decay of correlation functions at short distances consistent with a bulk gap, concomitant with a gossamer long-range tail originating from a PEPS bulk-edge correspondence. For increasing bond dimension, (i) the rapid decrease of spurious features-SU(2) symmetry breaking and long-range tails in correlations-together with (ii) a faster convergence of the ground state energy as compared to state-of-the-art cylinder matrix-product state simulations involving far more variational parameters, prove the fundamental relevance of the PEPS ansatz for simulating systems with chiral topological order.
A simple spin-12\frac{1}{2}21​ frustrated antiferromagnetic Heisenberg model (AFHM) on the square lattice---including chiral plaquette cyclic terms---was argued [A. E. B. Nielsen, G. Sierra, and J. I. Cirac, Nat. Commun. 4, 2864 (2013)] to host a bosonic Kalmeyer-Laughlin (KL) fractional quantum Hall ground state [V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59, 2095 (1987)]. Here, we construct generic families of chiral projected entangled pair states (chiral PEPS) with low bond dimension (D=3,4,5D=3,4,5D=3,4,5) which, upon optimization, provide better variational energies than the KL Ansatz. The optimal D=3D=3D=3 PEPS exhibits chiral edge modes described by the Wess-Zumino-Witten SU(2)1\text{SU}{(2)}_{1}SU(2)1​ model, as expected for the KL spin liquid. However, we find evidence that, in contrast to the KL state, the PEPS spin liquids have power-law dimer-dimer correlations and exhibit a gossamer long-range tail in the spin-spin correlations. We conjecture that these features are genuine to local chiral AFHM on bipartite lattices.