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arXiv:1601.02028 · Physical review. B./Physical review. B

Braiding statistics and classification of two-dimensional charge- 2m superconductors

Chenjie Wang

Published Aug 16, 2016• 21 citations
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Abstract

We study braiding statistics between quasiparticles and vortices in two-dimensional charge-$2m$ (in units of $e$) superconductors that are coupled to a ${\mathbb{Z}}_{2m}$ dynamical gauge field, where $m$ is any positive integer. We show that there exist $16m$ types of braiding statistics when $m$ is odd, but only $4m$ types when $m$ is even. Based on the braiding statistics, we obtain a classification of topological phases of charge-$2m$ superconductors---or formally speaking, a classification of symmetry-protected topological phases, as well as invertible topological phases, of two-dimensional gapped fermions with ${\mathbb{Z}}_{2m}^{f}$ symmetry. Interestingly, we find that there is no nontrivial fermionic symmetry-protected topological phase with ${\mathbb{Z}}_{4}^{f}$ symmetry.

DOI
10.1103/physrevb.94.085130

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We study braiding statistics between quasiparticles and vortices in two-dimensional charge-2m2m2m (in units of eee) superconductors that are coupled to a Z2m{\mathbb{Z}}_{2m}Z2m​ dynamical gauge field, where mmm is any positive integer. We show that there exist 16m16m16m types of braiding statistics when mmm is odd, but only 4m4m4m types when mmm is even. Based on the braiding statistics, we obtain a classification of topological phases of charge-2m2m2m superconductors---or formally speaking, a classification of symmetry-protected topological phases, as well as invertible topological phases, of two-dimensional gapped fermions with Z2mf{\mathbb{Z}}_{2m}^{f}Z2mf​ symmetry. Interestingly, we find that there is no nontrivial fermionic symmetry-protected topological phase with Z4f{\mathbb{Z}}_{4}^{f}Z4f​ symmetry.
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We examine the interplay of symmetry and topological order in 2+12+12+1 dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the Gf{\cal G}^{\rm f}Gf fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry G=Gf/Z2fG= {\cal G}^{\rm f}/\mathbb{Z}_2^{\rm f}G=Gf/Z2f​ on the physical fermions is essentially the central extension of GGG by the Z2f\mathbb{Z}_2^{\rm f}Z2f​ fermion parity conservation that yields the fermionic symmetry Gf{\cal G}^{\rm f}Gf. We develop an algebraic theory of Gf{\cal G}^{\rm f}Gf symmetry defects for fermionic topological phases using GGG-crossed braided tensor category theory. This formalism allows us to fully characterize and classify 2+12+12+1 dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group Gf{\cal G}^{\rm f}Gf. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed Gf{\cal G}^{\rm f}Gf form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.
We construct a two-dimensional (2D) quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a ``symmetry-protected topological phase.'' We describe a simple physical construction that distinguishes this system from a conventional paramagnet: We couple the system to a Z2{\mathbb{Z}}_{2}Z2​ gauge field and then show that the $\ensuremath{\pi}$-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.
We study aspects of gapped phases of matter, focusing on their classification, including the group law under stacking, and their relation to topological quantum field theories (TQFT). In one spatial dimension, it is well-known that Matrix Product States (MPS) efficiently approximate ground states of gapped systems; by showing that these states arise naturally in 1 + 1-dimensional lattice TQFT, which in turn are closely related to continuum TQFT, we provide a concrete connection between ground states of lattice systems and TQFT in 1 + 1 dimensions. We generalize this to systems with symmetries and fermions, and obtain a classification and group law for the stacking of 1 + 1-dimensional symmetry-protected topological phases. Further, we study the effect of turning on/off interactions for the classification: the phase classification of a given symmetry class of Hamiltonians can be different depending on whether we allow interactions or not, and in low dimensions we provide some concrete formulas relating the phases under the non-interacting classification and those under the interacting classification. Lastly, we study the phases of the 2 + 1-dimensional topological superconductor, and show that for all 16 phases braiding statistics of vortices, which determine the underlying TQFT, can be obtained by stacking layers of the basic p + ip superconductor.
In free fermion systems with given symmetry and dimension, the possible topological phases are labeled by elements of only three types of Abelian groups, 0, Z2, or Z. For example, noninteracting one-dimensional fermionic superconducting phases with S(z) spin rotation and time-reversal symmetries are classified by Z. We show that with weak interactions, this classification reduces to Z4. Using group cohomology, one can additionally show that there are only four distinct phases for such one-dimensional superconductors even with strong interactions. Comparing their projective representations, we find that all these four symmetry-protected topological phases can be realized with free fermions. Further, we show that one-dimensional fermionic superconducting phases with Z(n) discrete S(z) spin rotation and time-reversal symmetries are classified by Z4 when n is even and Z2 when n is odd; again, all these strongly interacting topological phases can be realized by noninteracting fermions. Our approach can be applied to systems with other symmetries to see which one-dimensional topological phases can be realized with free fermions.
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.