Nonunitary string-net PEPS: tube-algebra central idempotents without C*-structure, gapped non-Hermitian parent Hamiltonians, convergent fixed-point algorithms, and extensions to logarithmic CFTs

The 2020 Physical Review Letter of Lootens, Vanhove, Haegeman and Verstraete extended the matrix-product-operator (MPO) formalism for string-net projected entangled pair states (PEPS) to nonunitary, Galois-conjugated solutions of the pentagon equation, thereby realizing on the lattice the nonunitary counterparts of (2+1)-dimensional topological quantum field theories, with the Yang-Lee model (the Galois conjugate of the Fibonacci fusion category) as the worked example: a strange-correlator map sends the Yang-Lee string-net PEPS to Baxter's hard-hexagon model at negative Boltzmann weights, critical at the Galois-conjugate fugacity $z_c=(\phi')^5$, with topological sectors matching Yang-Lee conformal characters. The construction leaves four open ends, which together define the completion of its program: (1) a rigorous theory of the central idempotents of the Ocneanu tube algebra when it is not a C*-algebra, so that the topological-sector projection is justified rather than pragmatic; (2) an explicit construction and physical characterization (spectrum, locality, stability) of the gapped non-Hermitian parent Hamiltonian whose existence is only asserted; (3) numerical algorithms that converge for the nonunitary PEPS transfer matrices, whose lowest eigenstate is non-normalizable and defeats variational uniform matrix product state (VUMPS)-type fixed-point iteration; and (4) extension of the construction beyond braided modular tensor categories to more general nonunitary categories, in particular nonsemisimple ones, so as to realize logarithmic conformal field theories such as percolation on the lattice. Of these, (3) and (4) are explicit future-work statements of the Letter's Conclusions ('we hope to develop new numerical methods that are able to deal with these theories in the future'; 'the obvious extension of this Letter is toward logarithmic CFTs ... such as percolation'), (1) is a gap the Letter explicitly acknowledges ('we have no a priori reason to assume the existence of idempotents'; it 'takes a pragmatic approach' with a constructive algorithm), and (2) is an assertion in the Letter's abstract — these states 'can still be the topological ground states of a gapped non-Hermitian Hamiltonian' — for which the Letter provides no construction.

Background

A large class of (2+1)-dimensional topologically ordered phases is realized exactly by Levin and Wen's string-net models: commuting-projector lattice Hamiltonians whose ground states condense fluctuating loops ('strings') fusing according to a unitary fusion category. The fusion data are encoded in $F$-symbols (basis-change coefficients for fusion spaces) satisfying the pentagon equation, and the ground states admit an exact representation as projected entangled pair states (PEPS) — tensor networks whose ancillary 'virtual' indices are contracted along bonds, with the fusion category living on the virtual level. The virtual symmetries of these PEPS are organized by matrix product operators (MPOs): one-dimensional tensor networks whose algebra reproduces the fusion algebra of the category. The emergent anyonic excitations ('topological sectors') are obtained by block-diagonalizing the Ocneanu tube algebra, the algebra of MPOs wrapped around a cylinder of the virtual lattice; for unitary fusion categories this algebra is a finite-dimensional C*-algebra (an algebra of operators closed under Hermitian conjugation), hence semisimple, and its central idempotents (basis elements that act as projectors onto superselection sectors) exist with clean orthogonality properties, with explicit formulas built from the modular $S$ matrix (which encodes the mutual braiding phases of pairs of anyon types) and the braid $R$ matrix (which encodes the exchange phases of two anyons) when the category is braided and modular (i.e., has nondegenerate braiding).

The $F$-symbols of e.g. the Fibonacci category are algebraic numbers, and applying a Galois conjugation to them yields another solution of the pentagon equation with the same fusion rules but a nonunitary category (the tensors are no longer isometries). Freedman, Gukelberger, Hastings, Trebst, Troyer and Wang showed that, up to local changes of basis, such Galois-conjugate string-net wavefunctions cannot be the ground states of a topological model whose Hermitian Hamiltonian satisfies Lieb-Robinson bounds (the speed limits on operator propagation, which cover all gapped local and quasi-local Hermitian Hamiltonians), which for a decade made their physical status unclear. The source Letter of Lootens, Vanhove, Haegeman and Verstraete reopened the question: it generalized the MPO formalism to nonunitary pentagon solutions whenever the input category is a braided modular tensor fusion category whose objects are self-dual (satisfied by all $\mathrm{su}(2)_k$ fusion rules), argued that topological sectors can still be constructed even though the tube algebra is no longer closed under Hermitian conjugation and hence not a C*-algebra, and demonstrated everything on the Yang-Lee theory, the Galois conjugate of the Fibonacci category. The Yang-Lee edge singularity is the critical point of the two-dimensional Ising model at imaginary magnetic field, described by the nonunitary minimal model with central charge $c=-22/5$ and a single nontrivial primary field of conformal weight $h=-1/5$; it had previously been realized in anyonic spin chains by Ardonne, Gukelberger, Ludwig, Trebst and Troyer. The Letter used the strange correlator — the overlap of a topologically ordered state with a trivial product state, whose squared norm equals the partition function of a two-dimensional classical model, a device introduced for symmetry-protected phases and adapted to topological phases by Vanhove, Bal, Williamson, Bultinck, Haegeman and Verstraete — to map the Yang-Lee string-net PEPS onto the hard-hexagon model — a classical lattice gas on the triangular lattice in which occupied sites exclude their neighboring sites — solved exactly by Baxter, but with negative Boltzmann weights, critical at the Galois-conjugate fugacity $z_c=(\phi')^5$ (with $\phi'$ the algebraic conjugate of the golden ratio $\phi$), and showed that the finite-size spectra, projected with the tube-algebra idempotents, reproduce the Yang-Lee conformal characters with the correct topological spin corrections. Finally, it analyzed the PEPS transfer matrix (the contraction operator whose eigenvalues govern correlations and entanglement): with the upper PEPS layer transposed ($T_1$) it is gapped with the MPO symmetries as exact fixed points, while with the usual Hermitian-conjugate layer ($T_2$) it is critical — the Hermitian-conjugated PEPS lies in a different topological phase because the MPO algebra is not closed under Hermitian conjugation. A natural tool for the $T_2$ fixed point would be VUMPS, the variational uniform matrix product state algorithm of Zauner-Stauber, Vanderstraeten, Fishman, Verstraete and Haegeman, which variationally optimizes infinite matrix product states and is standard for transfer-matrix fixed points; but in this nonunitary setting there is an excitation below the vacuum, the lowest transfer-matrix eigenstate is not normalizable, and VUMPS fails to converge, forcing finite-size exact diagonalization.

The Letter ends by stating the difficulty of fixed-point algorithms for these nonunitary models and the hope to develop new numerical methods for them, and by identifying the extension toward logarithmic conformal field theories (CFTs) — CFTs whose representation category is not semisimple, so that correlation functions can contain logarithms, with percolation as the flagship open problem — as 'the obvious extension of this Letter'. Together with the two gaps in its own argument — the asserted but unconstructed gapped non-Hermitian parent Hamiltonian, and the pragmatic, unproven status of the central idempotents — these form the open ends addressed by this problem.

Problem Statement

Complete the program of the Galois-conjugate (nonunitary) string-net PEPS construction of Lootens, Vanhove, Haegeman and Verstraete, by resolving the four open ends the Letter leaves: two explicit future-work statements of its Conclusions (numerical methods for the nonunitary models; the extension to logarithmic CFTs), one gap the Letter explicitly acknowledges (the pragmatic, unproven central-idempotent construction), and one assertion in its abstract for which the Letter provides no construction (the gapped non-Hermitian parent Hamiltonian). The common objective is to place the lattice realization of nonunitary, Galois-conjugated topological quantum field theories and their conformal field theories on the same rigorous, physically interpretable, and computable footing as the unitary string-net/MPO theory, and to extend its range of validity; the four components are jointly needed for that objective, and each is independently verifiable:

(1) Rigorous idempotent foundations. Establish (or refute) the existence, uniqueness, and orthogonality properties of the central idempotents of the Ocneanu tube algebra for nonunitary braided modular tensor fusion categories — the setting in which the tube algebra is no longer a C*-algebra — including the precise role of the modular $S$ matrix and braid $R$ matrix in the nonunitary case, and the conditions under which the pragmatic constructive algorithm used in the Letter (and the generalization of the Koenig-Kuperberg-Reichardt formulas to nonunitary solutions) is justified. A characterization of which nonunitary categories admit the topological-sector construction, and why, is the target; the unitary C*-argument may not be invoked where it fails.

(2) Gapped non-Hermitian parent Hamiltonians. Construct explicitly, and physically characterize, the gapped non-Hermitian local Hamiltonians having the nonunitary string-net PEPS as topological ground states — as asserted but not demonstrated in the Letter's abstract — including the locality structure of its terms, the ground-state degeneracy and spectrum on the torus, and a notion of stability of the gapped non-Hermitian phase under perturbations, clarifying the relation to the gapped transpose-defined transfer matrix.

(3) Convergent algorithms for nonunitary tensor networks. Develop numerical algorithms that can compute fixed points, entanglement spectra, and thermodynamic-limit properties of the nonunitary PEPS and their transfer matrices — the regime where VUMPS-type variational fixed-point iteration fails because the lowest transfer-matrix eigenstate is not normalizable (there is an excitation below the vacuum) — demonstrated at least on the Yang-Lee/hard-hexagon negative-fugacity case of the source, including the critical Hermitian-conjugate transfer matrix, with results checkable against the Yang-Lee conformal data.

(4) Extension to logarithmic CFTs. Extend the MPO/PEPS/idempotent/strange-correlator construction beyond braided modular tensor categories to more general nonunitary categories — nonmodular or nonsemisimple — so as to produce lattice realizations of logarithmic conformal field theories, with percolation as the benchmark target identified by the source, including the identification of the logarithmic (indecomposable-representation) structure in the lattice data.

Each component carries its own completion criterion — the objective stated in it, completely attained: for (1), a proof of the existence, uniqueness, and orthogonality properties of the nonunitary tube-algebra central idempotents together with the claimed characterization of which nonunitary categories admit the topological-sector construction, or a refutation of that claim (a counterexample to any one of the listed properties is a refutation); for (2), the explicit construction of the gapped non-Hermitian parent Hamiltonian together with the locality, torus-spectrum and degeneracy, and stability characterization stated there; for (3), a convergent algorithm for the stated regime, demonstrated on the Yang-Lee/hard-hexagon negative-fugacity case with results checkable against the Yang-Lee conformal data; for (4), a lattice realization of a logarithmic CFT, with percolation as the benchmark and the logarithmic structure identified in the lattice data. A complete resolution of any single component is a complete result on that component, and partial resolutions of individual components constitute partial progress; but the problem as posed by the source's open ends is the conjunction: a complete answer supplies all four.

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