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arXiv:2005.11314 · SciPost Physics

Boundary RG flows for fermions and the mod 2 anomaly

Philip Boyle Smith, David Tong

Published Jan 19, 2021• 15 citations
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Abstract

Boundary conditions for Majorana fermions in d=1+1 d = 1 + 1 dimensions fall into one of two SPT phases, associated to a mod 2 anomaly. Here we consider boundary conditions for 2N 2 N Majorana fermions that preserve a U(1)^N U ( 1 ) N symmetry. In general, the left-moving and right-moving fermions carry different charges under this symmetry, and implementation of the boundary condition requires new degrees of freedom, which manifest themselves in a boundary central charge g g . We follow the boundary RG flow induced by turning on relevant boundary operators. We identify the infra-red boundary state. In many cases, the boundary state flips SPT class, resulting in an emergent Majorana mode needed to cancel the anomaly. We show that the ratio of UV and IR boundary central charges is given by g^2_{IR} / g^2_{UV} = \mathrm{dim} \, \mathcal{O} g I R 2 / g U V 2 = d i m 𝒪 , the dimension of the perturbing boundary operator. Any relevant operator necessarily has \mathrm{dim} \, \mathcal{O} < 1 d i m 𝒪 < 1 , ensuring that the central charge decreases in accord with the g g -theorem.

DOI
10.21468/scipostphys.10.1.010

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Boundary conditions for Majorana fermions in d=1+1 d = 1 + 1 dimensions fall into one of two SPT phases, associated to a mod 2 anomaly. Here we consider boundary conditions for 2N 2 N Majorana fermions that preserve a U(1)^N U ( 1 ) N symmetry. In general, the left-moving and right-moving fermions carry different charges under this symmetry, and implementation of the boundary condition requires new degrees of freedom, which manifest themselves in a boundary central charge g g . We follow the boundary RG flow induced by turning on relevant boundary operators. We identify the infra-red boundary state. In many cases, the boundary state flips SPT class, resulting in an emergent Majorana mode needed to cancel the anomaly. We show that the ratio of UV and IR boundary central charges is given by g^2_{IR} / g^2_{UV} = \mathrm{dim} , \mathcal{O} g I R 2 / g U V 2 = d i m 𝒪 , the dimension of the perturbing boundary operator. Any relevant operator necessarily has \mathrm{dim} , \mathcal{O} < 1 d i m 𝒪 < 1 , ensuring that the central charge decreases in accord with the g g -theorem.
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Abstract We study boundary states for Dirac fermions in d = 1 + 1 dimensions that preserve Abelian chiral symmetries, meaning that the left- and right-moving fermions carry different charges. We derive simple expressions, in terms of the fermion charge assignments, for the boundary central charge and for the ground state degeneracy of the system when two different boundary conditions are imposed at either end of an interval. We show that all such boundary states fall into one of two classes, related to SPT phases supported by (−1)F , which are characterised by the existence of an unpaired Majorana zero mode.
Recently, Gaiotto (2012 arXiv:1201.0767) considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet (UV) ones for a given renormalization group flow. In this paper, we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the infrared (IR) conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect fields. For the boundary flows in minimal models triggered by ψ13 perturbation, we make an explicit proposal for the RG defect fields. We check our conjecture by a number of calculations done for the example of (p, 2) → (p − 1, 1)⊕(p + 1, 1) flows.
Recently, Gaiotto (2012 arXiv:1201.0767) considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet (UV) ones for a given renormalization group flow. In this paper, we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the infrared (IR) conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect fields. For the boundary flows in minimal models triggered by ψ 13 perturbation, we make an explicit proposal for the RG defect fields. We check our conjecture by a number of calculations done for the example of ( p , 2) → ( p − 1, 1)⊕( p + 1, 1) flows.
Boundary condition and reflection anomaly in 2+12+12+1 dimensions
It is known that the 2+1 2 + 1 d single Majorana fermion theory has an anomaly of the reflection, which is canceled out when 16 copies of the theory are combined. Therefore, it is expected that the reflection symmetric boundary condition is impossible for one Majorana fermion, but possible for 16 Majorana fermions. In this paper, we consider a reflection symmetric boundary condition that varies at a single point, and find that there is a problem with one Majorana fermion. The problem is the absence of a corresponding outgoing wave to a specific incoming wave into the boundary, which leads to the non-conservation of the energy. For 16 Majorana fermions, it is possible to connect every incoming wave to an outgoing wave without breaking the reflection symmetry. In addition, we discuss the connection with the fermion-monopole scattering in 3+1 3 + 1 dimensions.
Symmetry protected topological (SPT) phases of bosons in ddd spatial dimensions have been characterized by the action of the protecting global symmetry GGG on their boundary. The symmetry acts on the boundary in a way that would be impossible to realize in a purely d−1d-1d−1 dimensional system i.e., without the bulk. This is often formalized by saying the GGG symmetry is anomalous when the boundary theory is gauged. Simultaneously gauging the symmetry on the boundary and in the bulk yields a gauge-invariant composite system. One says there is an anomaly in-flow from the boundary to the bulk. Recently it has been appreciated that some anomalies are too severe to be regulated by an SPT bulk. These do not satisfy anomaly in-flow in the traditional sense. However, we show that these anomalous systems can be regulated as the symmetric boundaries of the newly discovered higher SPT phases. These higher SPT's are protected by a symmetry ddd-group, a higher categorical version of a group which can charge not just particles but also strings, volumes, etc. This structure emerges naturally from computation of the anomaly. One can interpret these severe anomalies as producing an emergent higher-form gauge field when GGG is gauged, and this gauge field participates in the bulk topological action.