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arXiv:1010.1517 · Physical Review B

Classification of two-dimensional fermionic and bosonic topological orders

Zheng‐Cheng Gu, Zhenghan Wang, Xiao-Gang Wen

Published Mar 31, 2015• 62 citations
arXivHTMLPDF

Abstract

The string-net approach by Levin and Wen, and the local unitary transformation approach by Chen, Gu, and Wen, provide ways to classify topological orders with gappable edge in two-dimensional (2D) bosonic systems. The two approaches reveal that the mathematical framework for ($2+1)$-dimensional $(2+1)\mathrm{D}$ bosonic topological order with gappable edge is closely related to unitary fusion category theory. In this paper, we generalize these systematic descriptions of topological orders to 2D fermion systems. We find a classification of $(2+1)\mathrm{D}$ fermionic topological orders with gappable edge in terms of the following set of data $({N}_{k}^{ij},{F}_{k}^{ij},{F}_{jkn,\ensuremath{\chi}\ensuremath{\delta}}^{ijm,\ensuremath{\alpha}\ensuremath{\beta}},{d}_{i})$, which satisfy a set of nonlinear algebraic equations. The exactly soluble Hamiltonians can be constructed from the above data on any lattices to realize the corresponding topological orders. When ${F}_{k}^{ij}=0$, our result recovers the previous classification of $2+1\mathrm{D}$ bosonic topological orders with gappable edge.

DOI
10.1103/physrevb.91.125149

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The string-net approach by Levin and Wen, and the local unitary transformation approach by Chen, Gu, and Wen, provide ways to classify topological orders with gappable edge in two-dimensional (2D) bosonic systems. The two approaches reveal that the mathematical framework for (2+1)2+1)2+1)-dimensional (2+1)D(2+1)\mathrm{D}(2+1)D bosonic topological order with gappable edge is closely related to unitary fusion category theory. In this paper, we generalize these systematic descriptions of topological orders to 2D fermion systems. We find a classification of (2+1)D(2+1)\mathrm{D}(2+1)D fermionic topological orders with gappable edge in terms of the following set of data (Nkij,Fkij,Fjkn,\ensuremathχ\ensuremathδijm,\ensuremathα\ensuremathβ,di)({N}_{k}^{ij},{F}_{k}^{ij},{F}_{jkn,\ensuremath{\chi}\ensuremath{\delta}}^{ijm,\ensuremath{\alpha}\ensuremath{\beta}},{d}_{i})(Nkij​,Fkij​,Fjkn,\ensuremathχ\ensuremathδijm,\ensuremathα\ensuremathβ​,di​), which satisfy a set of nonlinear algebraic equations. The exactly soluble Hamiltonians can be constructed from the above data on any lattices to realize the corresponding topological orders. When Fkij=0{F}_{k}^{ij}=0Fkij​=0, our result recovers the previous classification of 2+1D2+1\mathrm{D}2+1D bosonic topological orders with gappable edge.
In recent years, fermionic topological phases of quantum matter has attracted a lot of attention. In a pioneer work by Gu, Wang, and Wen, the concept of equivalence classes of fermionic local unitary (FLU) transformations was proposed to systematically understand nonchiral topological phases in 2D fermion systems and an incomplete classification was obtained. On the other hand, the physical picture of fermion condensation and its corresponding super pivotal categories give rise to a generic mathematical framework to describe fermionic topological phases of quantum matter. In particular, it has been pointed out that in certain fermionic topological phases, there exists the so-called q-type anyon excitations, which have no analogues in bosonic theories. In this paper, we generalize the Gu, Wang, and Wen construction to include those fermionic topological phases with q-type anyon excitations. We argue that all nonchiral fermionic topological phases in 2+1D2+1\mathrm{D}2+1D are characterized by a set of tensors $({N}_{k}^{ij},{F}_{k}^{ij},{F}_{kln,\ensuremath{\chi}\ensuremath{\delta}}^{ijm,\ensuremath{\alpha}\ensuremath{\beta}},{n}_{i},{d}_{i})$, which satisfy a set of nonlinear algebraic equations parameterized by phase factors ${\mathrm{\ensuremath{\Xi}}}_{kl}^{ijm,\ensuremath{\alpha}\ensuremath{\beta}}$ and ${\mathrm{\ensuremath{\Xi}}}_{kln,\ensuremath{\chi}\ensuremath{\delta}}^{ij}$. Moreover, consistency conditions among algebraic equations give rise to additional constraints on these phase factors, which allow us to construct a topological invariant partition for an arbitrary triangulation of 3D spin manifold. Finally, several examples with q-type anyon excitations are discussed, including the fermionic topological phase from Tambara-Yamagami category for Z2N{\mathbb{Z}}_{2N}Z2N​, which can be regarded as the Z2N{\mathbb{Z}}_{2N}Z2N​ parafermion generalization of Ising fermionic topological phase.
We prove the conjectured classification of topological phases in two spatial dimensions with gappable boundary, in a simplified setting. Two gapped ground states of lattice Hamiltonians are in the same quantum phase of matter, or topological phase, if they can be connected by a constant-depth quantum circuit. It is conjectured that the Levin-Wen string-net models exhaust all possible gapped phases with gappable boundary, and these phases are labeled by unitary modular tensor categories. We prove this under the assumption that every phase has a representative state with zero correlation length satisfying the entanglement bootstrap axioms, or a strict form of area law. Our main technical development is to transform these states into string-net states using constant-depth quantum circuits.
Topological orders are a class of phases of matter that beyond the Landau symmetry breaking paradigm. The two (spatial) dimensional (2d) topological orders have been thoroughly studied. It is known that they can be fully classified by a unitary modular tensor category (UMTC) and a chiral central charge c. And a class of 2d topological orders whose boundary are gappable can be systematically constructed by Levin-Wen model whose ground states are string-net condensed states. Previously, the three spatial dimensional topological orders have been classified based on their canonical boundary described by some special unitary fusion 2-category, 2VecGω2\mathcal{V}ec_G^ω2VecGω​ or an EF 2-category. However, a lattice realization of a three spatial dimensional topological orders with both canonical boundary and arbitrary boundaries are still lacking. In this paper, we construct a 3d membrane-net model based on spherical fusion 2-category, which can be used to systematically study all general 3d topological order with gapped boundary. The partition function and lattice Hamiltonian of the membrane-net model is constructed based on state sum of spherical fusion 2-category. We also construct the 3d tube algebra of the membrane-net model to study excitations in the model. We conjecture all intrinsic excitations in membrane-net model have a one-to-one correspondence with the irreducible central idempotents (ICI) of the 3d tube algebra. We also provide a universal framework to study mutual statistics of all excitations in 3d topological order through 3d tube algebra. Our approach can be straightforwardly generalized to arbitrary dimension.
The string-net approach by Levin and Wen and the local unitary transformation approach by Chen, Gu and Wen provided ways to systematically label non-chiral topological orders in 2D. In those approaches, different topologically ordered many-body wave functions were characterized by different fixed-point tensors. Though extremely powerful, the resulting fixed-point tensors were mathematical abstractions and thus lacked a physical interpretation. As a result it was hard to judge if two different fixed-point tensors described the same quantum phase or not. We want to improve that approach by giving a more physical description of the topological orders. We find that the non-Abelian Berry's phases, TTT- and SSS-matrices, of the topological protected degenerate ground states on a torus give rise to a more physical description of topological orders. Using the Verlinde conjecture, we can even choose the canonical basis for the TTT- and SSS-matrices. It is conjectured that the TTT and SSS-matrices form a complete and one-to-one characterization of non-chiral topological orders and can replace the fixed-point tensor description to give us a more physical label for topological orders. As a result, all the topological properties can be obtained from the TTT- and SSS-matrices, such as number of quasiparticle types (from the dimension of TTT or SSS), the quasiparticle statistics (from the diagonal elements of TTT), the quantum dimensions of quasiparticles (from the first row of SSS), \etc.
String-net models allow us to systematically construct and classify (2+1)-dimensional [(2+1)D] topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wave function, which is described by a set of F-matrices [or more precisely, a unitary fusion category (UFC)], to study all the universal properties of such a topological order. In this paper, we describe a finite computational method, Q-algebra approach, that allows us to compute the non-Abelian statistics of the topological excitations [or more precisely, the unitary modular tensor category (UMTC)], from the string-net wave function (or the UFC). We discuss several examples, including the topological phases described by twisted gauge theory [i.e., twisted quantum double ${D}^{\ensuremath{\alpha}}(G)$]. Our result can also be viewed from an angle of holographic bulk-boundary relation. The (1+1)-dimensional [(1+1)D] anomalous topological orders, that can appear as edges of (2+1)D topological states, are classified by UFCs which describe the fusion of quasiparticles in (1+1)D. The (1+1)D anomalous edge topological order uniquely determines the (2+1)D bulk topological order (which are classified by UMTC). Our method allows us to compute this bulk topological order (i.e., the UMTC) from the anomalous edge topological order (i.e., the UFC).
Abstract This is the first part of a two-part work on a unified mathematical theory of gapped and gapless edges of 2d topological orders. We analyze all the possible observables on the 1+1D world sheet of a chiral gapless edge of a 2d topological order, and show that these observables form an enriched unitary fusion category, the Drinfeld center of which is precisely the unitary modular tensor category associated to the bulk. This mathematical description of a chiral gapless edge automatically includes that of a gapped edge (i.e. a unitary fusion category) as a special case. Therefore, we obtain a unified mathematical description and a classification of both gapped and chiral gapless edges of a given 2d topological order. In the process of our analysis, we encounter an interesting and reoccurring phenomenon: spatial fusion anomaly, which leads us to propose the Principle of Universality at RG fixed points for all quantum field theories. Our theory also implies that all chiral gapless edges can be obtained from a so-called topological Wick rotations. This fact leads us to propose, at the end of this work, a surprising correspondence between gapped and gapless phases in all dimensions.