Certified extraction of anyon sectors and modular data from infinite PEPS: completeness of transfer-matrix fixed points and correlation-length scaling

Francuz, Dziarmaga, Vidal and Cincio [Phys. Rev. B 101, 041108(R) (2020)] introduced a pipeline that reads off the topological order of a two-dimensional system directly from its infinite projected entangled pair state (iPEPS): infinite matrix product operators (MPOs) acting as virtual symmetries project the boundary theory onto states of definite anyon flux, and overlaps between such states on modularly related tori yield the modular $S$ and $T$ matrices encoding anyon statistics. The non-Abelian extension requested in that paper has since been delivered by the same group, and the twist-degeneracy ambiguity it flagged was solved in its own appendix. Two reliability gaps acknowledged in the original work nevertheless remain open across the whole subsequent literature: the multiplicity of the degenerate leading eigenvalue of the PEPS transfer matrix — the number of anyon superselection sectors — is unknown a priori and can only be lower-bounded and then checked a posteriori through consistency tests; and the boundary-MPO bond dimension and computational cost required to restore the exact topological degeneracy and purity of the fixed points at correlation length $\xi$ are not established (accurate $F$-symbol extraction has been demonstrated only up to $\xi\approx 2.3$; the source's own near-critical demonstration of accurate $S$, $T$ at $\xi\approx 25$ in a single perturbed string-net model establishes no scaling of the required $\chi$ or cost with $\xi$). The problem asked here is to close these gaps: provide rigorous criteria or certified adaptive algorithms guaranteeing that the complete set of leading transfer-matrix fixed points has been found, together with the scaling relations that make certified extraction of anyon models from iPEPS with long correlation length feasible, including near continuous phase transitions where competing cylinder-based methods break down.

Background

A gapped two-dimensional quantum system is said to possess topological order when its distinguishing properties — a ground-state degeneracy depending only on the topology of the underlying surface, and the existence of anyons, quasiparticles whose exchange is described by representations of the braid group rather than by Bose or Fermi statistics — cannot be detected by any local order parameter. The standard algebraic fingerprint of a topological order is its anyon model: the list of superselection sectors, their fusion rules (which sectors combine into which when quasiparticles are brought together), and the modular $S$ and $T$ matrices, unitary matrices encoding respectively the mutual statistics of two anyons and the self-statistics (the topological spin $\theta_a$ of each anyon type $a$, with $T$ diagonal in the anyon basis); together they form a representation of the modular group of the torus. Well-known examples with $\mathbb{Z}_2$ topological order are the toric code and the double-semion state, both realized exactly by fixed-point wave functions such as string-net models (exactly solvable lattice constructions that realize a large class of anyon theories); the quantum double $D(G)$ of a finite group $G$ is the corresponding anyon model built from $G$.

Projected entangled pair states (PEPS) approximate ground states of two-dimensional local Hamiltonians by a network of tensors connected by virtual (bond) indices; in the infinite-size version (iPEPS) translation invariance is built in and no finite lattice needs to be truncated. For a topologically ordered state, contracting the bra and ket layers of the iPEPS produces a double-layer transfer matrix whose leading eigenvectors are matrix product states (MPS, one-dimensional variational states) living on the virtual boundary of a cut. These boundary fixed points correspond to the minimally entangled states — the degenerate, locally indistinguishable ground states labeled by the anyon flux through the cut — and the number of such sectors equals the number of anyon superselection sectors (four for the toric code, conventionally labeled $1$, $e$, $m$, $\epsilon$). Matrix product operators (MPOs), the operator analogue of MPS, can act as symmetries of the virtual indices: inserting an MPO symmetry along a non-contractible loop of the torus changes the boundary conditions and moves the state into a different anyon sector. Bultinck, Mariën, Williamson, Şahinoğlu, Haegeman and Verstraete [Ann. Phys. (NY) 378, 183 (2017)] showed that these MPO symmetries close an algebra isomorphic to the anyon fusion algebra, providing the algebraic backbone for such constructions.

The precursor method of Cincio and Vidal [Phys. Rev. Lett. 110, 067208 (2013)] extracted anyon statistics from degenerate ground states on infinite cylinders using the density matrix renormalization group (DMRG); it is confined to thin cylinders (circumference up to roughly a dozen sites) because its cost grows exponentially with the circumference, which excludes the long-correlation-length regime. Francuz, Dziarmaga, Vidal and Cincio [Phys. Rev. B 101, 041108(R) (2020)] removed that limitation by working directly with a single iPEPS on infinite tori: they construct, variationally, the MPO symmetry generators $Z_{\mathrm{v}}$, $\widetilde{Z}_{\mathrm{v}}$ that map boundary fixed points into one another, use them to project onto states of definite anyon flux, and evaluate overlaps between such states on three tori related by modular ($\mathfrak{s}$, $\mathfrak{t}$) transformations, following the overlap algebra of Zhang, Grover and Vishwanath [Phys. Rev. B 91, 035127 (2015)], to obtain $S$ and $T$. The method was demonstrated on the toric code, the double semion, the quantum double of $\mathbb{Z}_3$, a perturbed string-net toric code driven towards a ferromagnetic phase, and a variationally optimized iPEPS for the Kitaev honeycomb model in its toric-code phase.

The authors of that paper were explicit, in the text and appendices, about the method's reliability limits. First, the degeneracy of the leading eigenvalue of a PEPS transfer matrix is in general unknown: in practice one can only bound it from below, proceed with the assumption that all leading eigenvectors were obtained, and verify this a posteriori through consistency checks such as unitarity of the computed $S$ matrix or the ability to find all symmetry generators $Z$. Second, because the leading eigenvectors are constrained to be MPOs of finite bond dimension $\chi$, the topological degeneracy — exact at the fixed point of a zero-correlation-length state — is lifted at finite correlation length $\xi$ and only restored as $\chi\to\infty$; the required scaling of $\chi$ (and hence of the computational cost) with $\xi$ was left undetermined. Third, the variational solution for the generators $Z_{\mathrm{v}}$, $\widetilde{Z}_{\mathrm{v}}$ must handle eigenvectors that are not invertible (they contain null blocks on the redundant part of the virtual space), and the optimization can involve near-singular environments.

Subsequent work by the same group resolved part of this agenda. Francuz and Dziarmaga [Phys. Rev. B 102, 235112 (2020)] extended the pipeline to non-Abelian order (Fibonacci and Ising string nets), extracting fusion rules, projectors, second-Rényi topological entanglement entropies (the universal subleading constant in the boundary-law scaling of the entanglement entropy across a cylinder), and $S$, $T$; they stress that the numerically obtained MPO symmetries carry zero modes, so the fusion algebra holds only for insertions between iPEPS layers. Francuz, Lootens, Verstraete and Dziarmaga [Phys. Rev. B 104, 195152 (2021)] went further: gauge ('zipper') tensors between MPO symmetries yield the $F$-symbols (the associativity data of the anyon fusion category), which can be brought to canonical gauge and fed into a computation of the Drinfeld center — the mathematical construction whose simple objects are precisely the anyon types of the doubled theory, and from which $S$ and $T$ follow; this route no longer requires the three-tori construction and its rotational-symmetry assumption. Yet that paper also reports that accurate $F$-symbols were recovered only for states with correlation length up to $\xi=2.3$, states explicitly that there is still much room for improvement, and its algorithm summary still begins with the unguaranteed step of finding all boundary fixed points.

Problem Statement

Given a translation-invariant iPEPS representing the (possibly variationally optimized) ground state of a gapped two-dimensional local Hamiltonian, with finite correlation length $\xi$ and unknown topological order, turn the extraction of its anyon model into a certified procedure, along the lines left open by Francuz, Dziarmaga, Vidal and Cincio [Phys. Rev. B 101, 041108(R) (2020)]. Two jointly needed deliverables:

(a) Completeness of sector enumeration. Provide a rigorous criterion, or an adaptive algorithm with a machine-checkable completeness certificate, that determines the multiplicity of the leading eigenvalue of the double-layer PEPS transfer matrix — equivalently, the number $n$ of anyon superselection sectors of the boundary theory — without assuming the anyon model in advance. The guarantee must cover non-fixed-point states at finite $\xi$, where the topological degeneracy is split by finite-$\chi$ effects, and must therefore state explicitly how the certified sector count relates to the boundary-MPO bond dimension $\chi$ and the splitting tolerance. This is to replace the current practice of lower-bounding the multiplicity and verifying a posteriori via consistency checks (unitarity of $S$, existence of the generators $Z$).

(b) Correlation-length scaling. Establish how the boundary-MPO bond dimension $\chi$ and the total computational cost required to restore the topological degeneracy and purity of the fixed points — and hence to extract $S$, $T$ (or the $F$-symbols) to a target accuracy — scale with the correlation length $\xi$, analytically or empirically with controlled extrapolation, including the asymptotic behavior as $\xi$ diverges when a continuous phase transition is approached from the gapped side (the states to be certified remain gapped with finite $\xi$; the critical point enters through the scaling limit, as in the source's perturbed string-net study at $\xi\approx 25$ below its transition).

These deliverables are two faces of one objective — reliable, certified identification of topological order from iPEPS: (b) delimits the regime of $\chi$ and $\xi$ in which the degeneracy assumption underlying the whole pipeline holds, and (a) certifies the enumeration within that regime; robustness of the variational construction of the MPO generators against null components and zero modes, acknowledged in the source, is the common technical obstacle both must overcome.

References

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