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arXiv:1902.11241 · Physical Review Letters

Galois Conjugated Tensor Fusion Categories and Nonunitary Conformal Field Theory

Laurens Lootens, Robijn Vanhove, Jutho Haegeman, Frank Verstraete

Published Mar 23, 2020• 28 citations
arXivHTMLPDF

Abstract

We provide a generalization of the matrix product operator formalism for string-net projected entangled pair states (PEPS) to include nonunitary solutions of the pentagon equation. These states provide the explicit lattice realization of the Galois conjugated counterparts of (2+1)-dimensional topological quantum field theories, based on tensor fusion categories. Although the parent Hamiltonians of these renormalization group fixed point states are gapless, these states can still be the topological ground states of a gapped non-Hermitian Hamiltonian. We show by example that the topological sectors of the Yang-Lee theory (the nonunitary counterpart of the Fibonacci fusion category) can be constructed, even in the absence of closure under Hermitian conjugation of the basis elements of the Ocneanu tube algebra. The topological sector construction is demonstrated by applying the concept of strange correlators to the Yang-Lee model, giving rise to a nonunitary version of the classical hard hexagon model in the Yang-Lee universality class and obtaining all generalized twisted boundary conditions on a finite cylinder of the Yang-Lee edge singularity. Finally, we construct the PEPS transfer matrix and show that taking the Hermitian conjugate changes the topological phase for these nonunitary string-net models.

DOI
10.1103/physrevlett.124.120601

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We provide a generalization of the matrix product operator formalism for string-net projected entangled pair states (PEPS) to include nonunitary solutions of the pentagon equation. These states provide the explicit lattice realization of the Galois conjugated counterparts of (2+1)-dimensional topological quantum field theories, based on tensor fusion categories. Although the parent Hamiltonians of these renormalization group fixed point states are gapless, these states can still be the topological ground states of a gapped non-Hermitian Hamiltonian. We show by example that the topological sectors of the Yang-Lee theory (the nonunitary counterpart of the Fibonacci fusion category) can be constructed, even in the absence of closure under Hermitian conjugation of the basis elements of the Ocneanu tube algebra. The topological sector construction is demonstrated by applying the concept of strange correlators to the Yang-Lee model, giving rise to a nonunitary version of the classical hard hexagon model in the Yang-Lee universality class and obtaining all generalized twisted boundary conditions on a finite cylinder of the Yang-Lee edge singularity. Finally, we construct the PEPS transfer matrix and show that taking the Hermitian conjugate changes the topological phase for these nonunitary string-net models.
We extend the concept of strange correlators, defined for symmetry-protected phases in You et al. [Phys. Rev. Lett. 112, 247202 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.247202], to topological phases of matter by taking the inner product between string-net ground states and product states. The resulting two-dimensional partition functions are shown to be either critical or symmetry broken, since the corresponding transfer matrices inherit all matrix product operator symmetries of the string-net states. For the case of critical systems, these nonlocal matrix product operator symmetries are the lattice remnants of topological conformal defects in the field theory description. Following Aasen et al. [J. Phys. A 49, 354001 (2016)JPAMB51751-811310.1088/1751-8113/49/35/354001], we argue that the different conformal boundary conditions can be obtained by applying the strange correlator concept to the different topological sectors of the string net obtained from Ocneanu's tube algebra. This is demonstrated on the lattice by calculating the conformal field theory spectra in the different topological sectors for the Fibonacci (hard-hexagon) and Ising string net. Additionally, we provide a complementary perspective on symmetry-preserving real-space renormalization by showing how known tensor network renormalization methods can be understood as the approximate truncation of an exactly coarse-grained strange correlator.
We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models. Given such a PEPS representation, we show that the consistency conditions of its MPO symmetries amount to a set of six coupled equations that can be identified with the pentagon equations of a bimodule category. This allows us to classify all equivalent PEPS representations and build MPO intertwiners between them, synthesising and generalising the wide variety of tensor network representations of topological phases. Furthermore, we use this generalisation to build explicit PEPS realisations of domain walls between different topological phases as constructed by Kitaev and Kong [Commun. Math. Phys. 313 (2012) 351-373]. While the prevailing abstract categorical approach is sufficient to describe the structure of topological phases, explicit tensor network representations are required to simulate these systems on a computer, such as needed for calculating thresholds of quantum error-correcting codes based on string-nets with boundaries. Finally, we show that all these string-net PEPS representations can be understood as specific instances of Turaev-Viro state-sum models of topological field theory on three-manifolds with a physical boundary, thereby putting these tensor network constructions on a mathematically rigorous footing.
Galois conjugation relates unitary conformal field theories and topological quantum field theories (TQFTs) to their nonunitary counterparts. Here we investigate Galois conjugates of quantum double models, such as the Levin-Wen model. While these Galois-conjugated Hamiltonians are typically non-Hermitian, we find that their ground-state wave functions still obey a generalized version of the usual code property (local operators do not act on the ground-state manifold) and hence enjoy a generalized topological protection. The key question addressed in this paper is whether such nonunitary topological phases can also appear as the ground states of Hermitian Hamiltonians. Specific attempts at constructing Hermitian Hamiltonians with these ground states lead to a loss of the code property and topological protection of the degenerate ground states. Beyond this, we rigorously prove that no local change of basis can transform the ground states of the Galois-conjugated doubled Fibonacci theory into the ground states of a topological model whose Hermitian Hamiltonian satisfies Lieb-Robinson bounds. These include all gapped local or quasilocal Hamiltonians. A similar statement holds for many other nonunitary TQFTs. One consequence is that these nonunitary TQFTs do not describe physical realizations of topological phases. In particular, this implies that the ``Gaffnian'' wave function can not be the ground state of a gapped fractional quantum Hall state.
The Levin-Wen string-net formalism provides a canonical mapping from spherical fusion categories to local Hamiltonians defining Topological Quantum Field Theories (TQFTs). While the topological invariance of the ground state is guaranteed by the pentagon identity, the realization of the model on a physical Hilbert space requires the category to be unitary. In this work, we investigate the obstructions arising when this construction is applied to non-unitary spherical categories, specifically the Yang-Lee model (the non-unitary minimal model M(2,5)\mathcal{M}(2,5)M(2,5)). We first validate our framework by explicitly constructing and verifying the Hamiltonians for rank-3 (Rep(D3)\text{Rep}(D_3)Rep(D3​)), rank-5 (TY(Z4)\text{TY}(\mathbb{Z}_4)TY(Z4​)), and Abelian (Z7\mathbb{Z}_7Z7​) unitary categories. We then apply this machinery to the non-unitary Yang-Lee model. Using a custom gradient-descent optimization algorithm on the manifold of FFF-symbols, we demonstrate that the Yang-Lee fusion rules admit no unitary solution to the pentagon equations. We explain this failure analytically by proving that negative quantum dimensions impose an indefinite metric on the string-net space, realizing a Krein space rather than a Hilbert space. Finally, we invoke the theory of PT\mathcal{PT}PT-symmetric quantum mechanics to interpret the non-Hermitian Hamiltonian, establishing that the obstruction is intrinsic to the fusion ring and cannot be removed by unitary gauge transformations.
Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of Projected Entangled Pair States(PEPS), topological order is characterized by an entanglement symmetry of the local tensor which describes the model. This symmetry can take the form of a tensor product of group representations, or in the more general case a correlated symmetry action in the form of a Matrix Product Operator(MPO), which encompasses all string-net models. Among other things, these entanglement symmetries allow for the description of ground states and anyon excitations. Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net and its Galois conjugate, the non-hermitian double Yang-Lee (DYL) string-net. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and show that these can be mapped to certain restricted solid on solid(RSOS) models, which are equivalent to the ((5±5)/2)((5\pm\sqrt{5})/2)((5±5​)/2)-state Potts model, respectively. The known exact solutions of the statistical models allow us to locate the critical points, and to predict the critical exponents for the order parameters. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the order parameters for condensation and deconfinement.
We revisit a class of non-Hermitian topological models that are Galois conjugates of their Hermitian counter parts. Particularly, these are Galois conjugates of unitary string-net models. We demonstrate that these models necessarily have real spectra, and that topological numbers are recovered as matrix elements of operators evaluated in appropriate biorthogonal basis, that we conveniently reformulate as a concomitant Hilbert space here. We also compute in the biorthogonal basis the topological entanglement entropy, demonstrating that its real part is related to the quantum dimension of the topological order. While we focus mostly on the Yang-Lee model, the results in the paper apply generally to Galois conjugates.