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arXiv:2102.04562 · Mathematische Annalen

Projector matrix product operators, anyons and higher relative commutants of subfactors

Yasuyuki Kawahigashi

Published Nov 19, 2022• 7 citations
arXivHTMLPDF

Abstract

Abstract A bi-unitary connection in subfactor theory of Jones producing a subfactor of finite depth gives a 4-tensor appearing in a recent work of Bultinck–Mariën–Williamson–Şahinoğlu-Haegeman–Verstraete on two-dimensional topological order and anyons. In their work, they have a special projection called a projector matrix product operator. We prove that the range of this projection of lengthkis naturally identified with thekth higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between two-dimensional topological order and subfactor theory.

DOI
10.1007/s00208-022-02519-0

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Abstract A bi-unitary connection in subfactor theory of Jones producing a subfactor of finite depth gives a 4-tensor appearing in a recent work of Bultinck–Mariën–Williamson–Şahinoğlu-Haegeman–Verstraete on two-dimensional topological order and anyons. In their work, they have a special projection called a projector matrix product operator. We prove that the range of this projection of lengthkis naturally identified with thekth higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between two-dimensional topological order and subfactor theory.
Researchers in condensed matter physics recently study two-dimensional topological order in terms of tensor networks involving certain 3- and 4-tensors. Their 3-tensors satisfying the “zipper condition” play an important role there, and such 3-tensors can be made into certain 2-tensors by combining two wires into one. We identify their 4-tensors with bi-unitary connections in Jones’ subfactor theory in operator algebras with precise normalization constants. Then we prove that their 2-tensors satisfying the zipper condition are the same as flat fields of strings in subfactor theory which correspond to elements in the higher relative commutants of the subfactor arising from the bi-unitary connection. This is what we expect, since the zipper condition is a kind of pentagon relations, but we clarify what conditions are exactly needed for this—we do not need the flatness or the finite depth condition for the bi-unitary connection. We actually generalize their 4-tensors so that the four index sets of the 4-tensors can be all different and work on a “half-version” of the zipper condition.
Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the Z4\mathbb{Z}_4Z4​ toric code, we condense the boson e2m2e^2m^2e2m2 and identify the semion and anti-semion algebras. We then decompose each Z4\mathbb{Z}_4Z4​ qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the zzz direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a U(1)4U(1)_4U(1)4​ ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions d=4k−2d=4k{-}2d=4k−2, with one qubit or Z4\mathbb{Z}_4Z4​ qudit on each (2k−1)(2k{-}1)(2k−1)-cell for the semion or U(1)4U(1)_4U(1)4​ ISA, respectively.
We study planar two-dimensional quantum systems on a lattice whose Hamiltonian is a sum of local commuting projectors of bounded range. We consider whether or not such a system has a zero energy ground state. To do this, we consider the problem as a one-dimensional problem, grouping all sites along a column into "supersites"; using C∗C^*C∗-algebraic methods (Bravyi and Vyalyi), we can solve this problem if we can characterize the central elements of the interaction algebra on these supersite. Unfortunately, these central elements may be very complex, making brute force impractical. Instead, we show a characterization of these elements in terms of matrix product operators with bounded bond dimension. This bound can be interpreted as a bound on the number of particle types in lattice theories with bounded Hilbert space dimension on each site. Topological order in this approach is related to the existence of certain central elements which cannot be "broken" into smaller pieces without creating an end excitation. Using this bound on bond dimension, we prove that several special cases of this problem are in NP, and we give part of a proof that the general case is in NP. Further, we characterize central elements that appear in certain specific models, including toric code and Levin-Wen models, as either product operators in the Abelian case or matrix product operators with low bond dimension in the non-Abelian case; this matrix product operator representation may have practical application in engineering the complicated multi-spin interactions in the Levin-Wen models.
Abstract One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of G{\mathsf {G}}G G - and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.
Anyonic Chains – α\alpha α-Induction – CFT – Defects – Subfactors
Abstract Given a unitary fusion category, one can define the Hilbert space of a so-called “anyonic spin-chain” and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to 1+11+11+1 1+1 -dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bipartite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of 1+11+11+1 1+1 CFTs as constructed by Fröhlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called “double triangle algebra” arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by Böckenhauer et al. based on the braided structure of the categories and on α\alpha α α -induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.