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arXiv:2207.03246 · Physical review. B./Physical review. B

Entanglement spectra of nonchiral topological ( 2+1 )-dimensional phases with strong time-reversal symmetry breaking, Li-Haldane state counting, and PEPS

Mark J. Arildsen, Norbert Schuch, Andreas W. W. Ludwig

Published Dec 21, 2023• 1 citation
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Abstract

The Li-Haldane correspondence [Phys. Rev. Lett. 101, 010504 (2008)] is often used to help identify wave functions of $(2+1)$-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 $\mathrm{SU}(3)$-level-one [$\mathrm{SU}{(3)}_{1}$] conformal field theory (CFT). In fact, the PEPS has $D({\mathbb{Z}}_{3})$ 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 $\mathrm{SU}(3)$-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 $\mathrm{SU}{(3)}_{1}$ CFT with the full content of a ``low-velocity'' chiral $\mathrm{SU}{(3)}_{1}$ CFT sector, a nonchiral structure beyond that observable in the topologically trivial sector of the ES.

DOI
10.1103/physrevb.108.245150

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•Jan 1, 2021•0 citations
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 $\mathrm{SU}{(3)}_{1}$ 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.
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.
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).
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.
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.
Topological chiral phases are ubiquitous in the physics of the fractional quantum Hall effect. Nonchiral topological spin liquids are also well known. Here, using the framework of projected entangled pair states (PEPS), we construct a family of chiral spin liquids on the square lattice, which are generalized spin-1/2 resonating valence bond (RVB) states obtained from deformed local tensors with d+idd+i\phantom{\rule{0.16em}{0ex}}dd+id symmetry. On a cylinder, we construct four topological sectors with even or odd number of spinons on the boundary and even or odd number of (Z2{\mathbb{Z}}_{2}Z2​) fluxes penetrating the cylinder, which, we argue, remain orthogonal in the limit of infinite perimeter. The analysis of the transfer matrix provides evidence of short-range (long-range) triplet (singlet) correlations as for the critical (nonchiral) RVB state. The entanglement spectrum exhibits chiral edge modes, which we confront to predictions of the conformal field theory, and the corresponding entanglement Hamiltonian is shown to be long-ranged.
Topological phases of matter describe systems beyond the paradigm of spontaneous symmetry breaking, giving rise to unconventional phenomena with possible applications in topological quantum computing. Almost 40 years after their discovery, topological phases remain among the most active research fields of condensed matter physics. As evidenced by the recent discovery of higher order topological insulators (HOTIs), novel physics can emerge even in non-interacting systems. Theoretical investigations of strongly correlated systems are extremely difficult due to the exponential growth of the quantum-mechanical Hilbert space and frequently require the application of approximative methods such as model wave functions. Tensor network states (TNS) are a class of variational wave functions which allow an efficient encoding of relevant quantum many body states and which form the basis for very successful numerical algorithms. Projected entangled pair states (PEPS), a class of TNS in two and higher dimensions, permit the exact representation of many interacting topological phases with time reversal symmetry. However, the description of experimentally relevant chiral topological phases like the integer and fractional quantum Hall effects is much more subtle using the TNS framework. The first part of this thesis focuses on the description of chiral topological phases using PEPS. Firstly, we study a previously proposed chiral PEPS that was conjectured to possess anyonic excitations. By a careful analysis of its symmetries we are able to align important entanglement observables more closely with the expected universal behavior. This highlights that efficient PEPS can possess characteristic properties of chiral topological phases. Secondly, we focus on the issue that all known chiral PEPS have algebraic bulk correlation functions and therefore cannot be the ground states of gapped local Hamiltonians. Since this problem arises already for non-interacting chiral topological phases, we focus on two examples of such systems and show that their ground states can be represented exactly by efficient PEPS in a hybrid lattice with one momentum-space direction. After an inverse Fourier transform, the PEPS with only real-space coordinates requires an exponentially growing number of parameters for an exact representation. This provides a concrete illustration of the impossibility to encode gapped chiral phases exactly with efficient PEPS. Finally, we analyze a model wave function for a three-dimensional (3D) HOTI with strong intrinsic correlations using large-scale variational Monte Carlo simulations. This wave function is obtained by projection of two copies of a non-interacting HOTI with chiral hinge states. We characterize the gapless hinge states of the interacting system and show that they are of the same nature as the edge states of the 1/21/21/2 Laughlin state. Surprisingly, the gapped surfaces host a two-dimensional (2D) phase whose topological entanglement entropy is half of that of the 1/21/21/2 Laughlin state. Such a value cannot be obtained by any of the known 2D topological orders, showing a clear departure not only from the Laughlin 1/21/21/2 physics but from conventional 2D topological order. This demonstrates that 3D topological phases can host rich phenomena that are not yet fully understood.