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arXiv:1610.07849 · Physical review. B./Physical review. B

Fermionic matrix product states and one-dimensional topological phases

Nick Bultinck, Dominic J. Williamson, Jutho Haegeman, Frank Verstraete

Published Feb 3, 2017• 102 citations
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Abstract

We develop the formalism of fermionic matrix product states (fMPS) and show how irreducible fMPS fall in two different classes, related to the different types of simple ${\mathbb{Z}}_{2}$ graded algebras, which are physically distinguished by the absence or presence of Majorana edge modes. The local structure of fMPS with Majorana edge modes also implies that there is always a twofold degeneracy in the entanglement spectrum. Using the fMPS formalism, we make explicit the correspondence between the ${\mathbb{Z}}_{8}$ classification of time-reversal-invariant spinless superconductors and the modulo 8 periodicity in the representation theory of real Clifford algebras. Studying fMPS with general onsite unitary and antiunitary symmetries allows us to define invariants that label symmetry-protected phases of interacting fermions. The behavior of these invariants under stacking of fMPS is derived, which reveals the group structure of such interacting phases. We also consider spatial symmetries and show how the invariant phase factor in the partition function of reflection-symmetric phases on an unorientable manifold appears in the fMPS framework.

DOI
10.1103/physrevb.95.075108

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We develop the formalism of fermionic matrix product states (fMPS) and show how irreducible fMPS fall in two different classes, related to the different types of simple Z2{\mathbb{Z}}_{2}Z2​ graded algebras, which are physically distinguished by the absence or presence of Majorana edge modes. The local structure of fMPS with Majorana edge modes also implies that there is always a twofold degeneracy in the entanglement spectrum. Using the fMPS formalism, we make explicit the correspondence between the Z8{\mathbb{Z}}_{8}Z8​ classification of time-reversal-invariant spinless superconductors and the modulo 8 periodicity in the representation theory of real Clifford algebras. Studying fMPS with general onsite unitary and antiunitary symmetries allows us to define invariants that label symmetry-protected phases of interacting fermions. The behavior of these invariants under stacking of fMPS is derived, which reveals the group structure of such interacting phases. We also consider spatial symmetries and show how the invariant phase factor in the partition function of reflection-symmetric phases on an unorientable manifold appears in the fMPS framework.
We extend the formalism of matrix product states (MPS) to describe one-dimensional gapped systems of fermions with both unitary and antiunitary symmetries. Additionally, systems with orientation-reversing spatial symmetries are considered. The short-ranged entangled phases of such systems are classified by three invariants, which characterize the projective action of the symmetry on edge states. We give interpretations of these invariants as properties of states on the closed chain. The relationship between fermionic MPS systems at a renormalization group fixed point and equivariant algebras is exploited to derive a group law for the stacking of fermionic phases. The result generalizes known classifications to symmetry groups that are nontrivial extensions of fermion parity and time-reversal.
In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of one-dimensional systems. We focus on the time-reversal-invariant Majorana chain (BDI symmetry class). While the band classification yields an integer topological index kkk, it is known that phases characterized by values of kkk in the same equivalence class modulo 8 can be adiabatically transformed one to another by adding suitable interaction terms. Here we show that the eight equivalence classes are distinct and exhaustive, and provide a physical interpretation for the interacting invariant modulo 8. The different phases realize different Altland-Zirnbauer classes of the reduced density matrix for an entanglement bipartition into two half chains. We generalize these results to the classification of all one-dimensional gapped phases of fermionic systems with possible antiunitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.
We introduce the concept of fermionic matrix product operators, and show that they provide a natural representation of fermionic fusion tensor categories. This allows for the classification of two dimensional fermionic topological phases in terms of matrix product operator algebras. Using this approach we give a classification of fermionic symmetry protected topological phases with respect to a group GGG in terms of three cohomology groups: H1(G,Z2)H^1(G,\mathbb{Z}_2)H1(G,Z2​), describing which matrix product operators are of Majorana type, H2(G,Z2)H^2(G,\mathbb{Z}_2)H2(G,Z2​), describing the fermionic nature of the fusion tensors that arise when two matrix product operators are multiplied, and the supercohomolgy group Hˉ3(G,U(1))\bar{H}^3(G,U(1))Hˉ3(G,U(1)) which corresponds to the associator that changes the order of fusion. We also generalize the tensor network description of the string-net ground states to the fermionic setting, yielding simple representations of a class that includes the fermionic toric code.
Crystalline symmetry and time-reversal symmetry are commonly present in real superconducting materials. However, the topological classification of systems respecting these symmetries, particularly for interacting fermions, remains incomplete. In this work, we systematically classify time-reversal symmetry-protected crystalline topological superconductors in two-dimensional interacting fermionic systems using an explicit real-space construction. Among the resulting phases, we identify intrinsically interacting fermionic topological superconductors, i.e., phases that cannot be realized in either free-fermion or interacting bosonic systems. For spinless fermions with protecting symmetry group C4×Z2TC_4 \times Z_2^TC4​×Z2T​ or D4×Z2TD_4 \times Z_2^TD4​×Z2T​ (plus fermion parity), the intrinsic sector has a Z4Z_4Z4​ classification. The corresponding root phases generating this Z4Z_4Z4​ classification admit a transparent real-space construction in terms of decorated 1D blocks. These blocks are 1D fermionic symmetry-protected topological (FSPT) phases, realizable as double Majorana chains. We further find the corresponding Z4Z_4Z4​ spinless intrinsic phases for wallpaper groups p4p4p4, p4mp4mp4m, and p4gp4gp4g. We also find an additional Z2Z_2Z2​ intrinsically interacting phase for spinless fermions with wallpaper group pmpmpm, which is absent with the corresponding point-group symmetry alone. Moreover, these intrinsic phases naturally give rise to higher-order FSPT phases that support corner zero modes. Finally, we verify the crystalline equivalence principle for generic 2D interacting FSPT systems with both crystalline and internal symmetries.
In the Fock representation, we propose a framework to construct the generalized matrix product states (MPS) for topological phases with Zp{\mathbb{Z}}_{p}Zp​ parafermions. Unlike the Z2{\mathbb{Z}}_{2}Z2​ Majorana fermions, the Zp{\mathbb{Z}}_{p}Zp​ parafermions form intrinsically interacting systems. Here we explicitly construct two topologically distinct classes of irreducible Z3{\mathbb{Z}}_{3}Z3​ parafermionic MPS wave functions, characterized by one or two parafermionic zero modes at each end of an open chain. Their corresponding parent Hamiltonians are found as the fixed point models of the single Z3{\mathbb{Z}}_{3}Z3​ parafermion chain and two-coupled parafermion chains with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ symmetry. Our results thus pave the road to investigate all possible topological phases with Zp{\mathbb{Z}}_{p}Zp​ parafermions within the matrix product representation in one dimension.
In the Fock representation, we construct matrix product states (MPS) for one-dimensional gapped phases for Zp{\mathbb{Z}}_{p}Zp​ parafermions. From the analysis of irreducibility of MPS, we classify all possible gapped phases of Zp{\mathbb{Z}}_{p}Zp​ parafermions without extra symmetry other than Zp{\mathbb{Z}}_{p}Zp​ charge symmetry, including topological phases, spontaneous symmetry breaking phases, and a trivial phase. For all phases, we find the irreducible forms of local matrices of MPS, which span different kinds of graded algebras. The topological phases are characterized by the nontrivial simple Zp{\mathbb{Z}}_{p}Zp​ graded algebras with the characteristic graded centers, yielding the degeneracies of the full transfer matrix spectra uniquely. However, the spontaneous symmetry breaking phases correspond to the trivial semisimple Zp/n{\mathbb{Z}}_{p/n}Zp/n​ graded algebras, which can be further reduced to the trivial simple Zp/n{\mathbb{Z}}_{p/n}Zp/n​ graded algebras, where nnn is the divisor of ppp. So the present results provide the complete classification of the parafermionic gapped phases and deepen our understanding of topological phases in one dimension.