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arXiv:2204.10333 · Physical review. B./Physical review. B

Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension

Ömer M. Aksoy, Christopher Mudry

Published Jul 12, 2022• 9 citations
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Abstract

Invertible fermionic topological (IFT) phases are gapped phases of matter with nondegenerate ground states on any closed spatial manifold. When open boundary conditions are imposed, nontrivial IFT phases support gapless boundary degrees of freedom. Distinct IFT phases in one-dimensional space with an internal symmetry group ${G}_{f}$ have been characterized by a triplet of indices $([(\ensuremath{\nu},\ensuremath{\rho})],[\ensuremath{\mu}])$. Our main result is an elementary derivation of the fermionic stacking rules of one-dimensional IFT phases for any given internal symmetry group ${G}_{f}$ from the perspective of the boundary, i.e., we give an explicit operational definition for the boundary representation $([({\ensuremath{\nu}}_{\ensuremath{\wedge}},{\ensuremath{\rho}}_{\ensuremath{\wedge}})],[{\ensuremath{\mu}}_{\ensuremath{\wedge}}])$ obtained from stacking two IFT phases characterized by the triplets of boundary indices $([({\ensuremath{\nu}}_{1},{\ensuremath{\rho}}_{1})],[{\ensuremath{\mu}}_{1}])$ and $([({\ensuremath{\nu}}_{2},{\ensuremath{\rho}}_{2})],[{\ensuremath{\mu}}_{2}])$, respectively.

DOI
10.1103/physrevb.106.035117

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Invertible fermionic topological (IFT) phases are gapped phases of matter with nondegenerate ground states on any closed spatial manifold. When open boundary conditions are imposed, nontrivial IFT phases support gapless boundary degrees of freedom. Distinct IFT phases in one-dimensional space with an internal symmetry group Gf{G}_{f}Gf​ have been characterized by a triplet of indices ([(\ensuremathν,\ensuremathρ)],[\ensuremathμ])([(\ensuremath{\nu},\ensuremath{\rho})],[\ensuremath{\mu}])([(\ensuremathν,\ensuremathρ)],[\ensuremathμ]). Our main result is an elementary derivation of the fermionic stacking rules of one-dimensional IFT phases for any given internal symmetry group Gf{G}_{f}Gf​ from the perspective of the boundary, i.e., we give an explicit operational definition for the boundary representation ([(\ensuremathν\ensuremath∧,\ensuremathρ\ensuremath∧)],[\ensuremathμ\ensuremath∧])([({\ensuremath{\nu}}_{\ensuremath{\wedge}},{\ensuremath{\rho}}_{\ensuremath{\wedge}})],[{\ensuremath{\mu}}_{\ensuremath{\wedge}}])([(\ensuremathν\ensuremath∧​,\ensuremathρ\ensuremath∧​)],[\ensuremathμ\ensuremath∧​]) obtained from stacking two IFT phases characterized by the triplets of boundary indices ([(\ensuremathν1,\ensuremathρ1)],[\ensuremathμ1])([({\ensuremath{\nu}}_{1},{\ensuremath{\rho}}_{1})],[{\ensuremath{\mu}}_{1}])([(\ensuremathν1​,\ensuremathρ1​)],[\ensuremathμ1​]) and ([(\ensuremathν2,\ensuremathρ2)],[\ensuremathμ2])([({\ensuremath{\nu}}_{2},{\ensuremath{\rho}}_{2})],[{\ensuremath{\mu}}_{2}])([(\ensuremathν2​,\ensuremathρ2​)],[\ensuremathμ2​]), respectively.
A bstract We study the interplay of duality and stacking of bosonic and fermionic symmetry-protected topological phases in one spatial dimension. In general the classifications of bosonic and fermionic phases have different group structures under the operation of stacking, but we argue that they are often isomorphic and give an explicit isomorphism when it exists. This occurs for all unitary symmetry groups and many groups with antiunitary symmetries, which we characterize. We find that this isomorphism is typically not implemented by the Jordan-Wigner transformation, nor is it a consequence of any other duality transformation that falls within the framework of topological holography. Along the way to this conclusion, we recover the fermionic stacking rule in terms of G \mathcal{G} G G -pin partition functions, give a gauge-invariant characterization of the twisted group cohomology invariant, and state a procedure for stacking gapped phases in the formalism of symmetry topological field theory.
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.
Integer quantum Hall states and their generalization to topological insulators and superconductors are paradigmatic examples of invertible fermionic topological states of matter. Here, the authors develop a comprehensive characterization and classification of invertible fermionic topological phases of matter in two spatial dimensions for general symmetry groups. The results are nonperturbative and apply to strongly interacting systems. In particular, they extend previous classification results to account for the possible chiral nature of invertible phases.
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Despite great successes in the study of gapped phases, a comprehensive understanding of the gapless phases and their transitions is still under development.In this paper, we study a general phenomenon in the space of (1 + 1)-dimensional critical phases with fermionic degrees of freedom described by a continuous family of conformal field theories (CFTs), also known as the conformal manifold.Along a one-dimensional locus on the conformal manifold, there can be a transition point, across which the fermionic CFTs on the two sides differ by stacking an invertible fermionic topological order (IFTO), point by point along the locus.At every point on the conformal manifold, the order and disorder operators have power-law two-point functions, but their critical exponents cross over with each other at the transition point, where stacking the IFTO leaves the fermionic CFT unchanged.We call this continuous transition on the fermionic conformal manifold a topological transition.By gauging the fermion parity, the IFTO stacking becomes a Kramers-Wannier duality between the corresponding bosonic CFTs.Both the IFTO stacking and the Kramers-Wannier duality are induced by the electromagnetic duality of the (2 + 1)-dimensional Z 2 topological order.We provide several examples of topological transitions, including the familiar Luttinger model of spinless fermions (i.e., the c = 1 massless Dirac fermion with the Thirring interaction) and a class of c = 2 examples describing U(1) × SU(2)-protected gapless phases.
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.