Notes on 8 Majorana Fermions

Tong, David; Turner, Carl

doi:10.21468/scipostphyslectnotes.14

Cited by 15 publications

(15 citation statements)

References 57 publications

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“…Following [27][28][29][30], our goal is to see to when such interactions can lead to a trivially gapped state preserving the chiral symmetry. Our treatment closely parallels the review [38]. For two fermions, the first case where this interaction is possible, the free theory describes a c ¼ 1 model and the quartic fermion interaction is the unique exactly marginal operator.…”

Section: A Z 2 Invariant ð1 + 1þd Fermionsmentioning

confidence: 88%

Decorated Z2 symmetry defects and their time-reversal anomalies

et al. 2020

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We discuss an isomorphism between the possible anomalies of (d þ 1)-dimensional quantum field theories with Z 2 unitary global symmetry, and those of d-dimensional quantum field theories with timereversal symmetry T. This correspondence is an instance of symmetry defect decoration. The worldvolume of a Z 2 symmetry defect is naturally invariant under T, and bulk Z 2 anomalies descend to T anomalies on these defects. We illustrate this correspondence in detail for ð1 þ 1Þd bosonic systems where the bulk Z 2 anomaly leads to a Kramers degeneracy in the symmetry defect Hilbert space and exhibits examples. We also discuss ð1 þ 1Þd fermion systems protected by Z 2 global symmetry where interactions lead to a Z 8 classification of anomalies. Under the correspondence, this is directly related to the Z 8 classification of ð0 þ 1Þd fermions protected by T. Finally, we consider ð3 þ 1Þd bosonic systems with Z 2 symmetry where the possible anomalies are classified by Z 2 × Z 2. We construct topological field theories realizing these anomalies and show that their associated symmetry defects support anyons that can be either fermions or Kramers doublets.

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Section: A Z 2 Invariant ð1 + 1þd Fermionsmentioning

confidence: 88%

Decorated Z2 symmetry defects and their time-reversal anomalies

et al. 2020

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“…This SPT phase arises, for example, as the infra-red limit of two Majorana fermions with masses of opposite sign and, in the condensed matter literature, it is better known as the topological phase of the Kitaev chain [5,8] . Recent applications of this topological field theory can found [9][10][11][12][13][14][15][16][17][18][19][20]. However, on a Riemann surface with boundary, the Arf topological field theory is not well defined: it suffers the same mod 2 anomaly that we saw above.…”

Section: The Mod 2 Anomalymentioning

confidence: 97%

“…Acting with these operators changes the θ -angles that, as we saw in (13), are needed to characterise the boundary state.…”

Section: Marginal Operatorsmentioning

confidence: 97%

“…These phases arise even for the simplest boundary conditions, where the reflection of a single left-moving fermion into a right-moving fermion can, in general, be implemented by the boundary condition ψ L = e iθ ψ R . The N phases θ i that appear in the boundary state (13) are generalisation to multiple fermions with a chiral boundary condition R.…”

Section: Constructing Boundary Statesmentioning

confidence: 99%

“…These we label as L n , though they are distinct from the bulk holomorphic generators. The usual Cardy trick is to relate this "open string" partition function to the "closed string" partition function of free fermions on a cylinder which, after the conformal map shown along the top of Figure 1, becomes the annulus Consider two boundary states A = |θ , R〉 and B = |θ , R〉 of the form (13). Note that these states share the same R matrix, but differ in the theta angles.…”

Section: The Partition Functionmentioning

confidence: 99%

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Boundary RG flows for fermions and the mod 2 anomaly

Smith

Tong

2021

SciPost Phys.

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Boundary conditions for Majorana fermions in d=1+1d=1+1 dimensions fall into one of two SPT phases, associated to a mod 2 anomaly. Here we consider boundary conditions for 2N2N Majorana fermions that preserve a U(1)^NU(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 gg. 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}gIR2/gUV2=dim𝒪, the dimension of the perturbing boundary operator. Any relevant operator necessarily has \mathrm{dim} \, \mathcal{O} < 1dim𝒪<1, ensuring that the central charge decreases in accord with the gg-theorem.

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Fun with F24

Harrison

Paquette

Persson

et al. 2021

J. High Energ. Phys.

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We study some special features of F24, the holomorphic c = 12 superconformal field theory (SCFT) given by 24 chiral free fermions. We construct eight different Lie superalgebras of “physical” states of a chiral superstring compactified on F24, and we prove that they all have the structure of Borcherds-Kac-Moody superalgebras. This produces a family of new examples of such superalgebras. The models depend on the choice of an $$ \mathcal{N} $$ N = 1 supercurrent on F24, with the admissible choices labeled by the semisimple Lie algebras of dimension 24. We also discuss how F24, with any such choice of supercurrent, can be obtained via orbifolding from another distinguished c = 12 holomorphic SCFT, the $$ \mathcal{N} $$ N = 1 supersymmetric version of the chiral CFT based on the E8 lattice.

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Notes on 8 Majorana Fermions

Cited by 15 publications

References 57 publications

Decorated Z2 symmetry defects and their time-reversal anomalies

Decorated Z2 symmetry defects and their time-reversal anomalies

Boundary RG flows for fermions and the mod 2 anomaly

Fun with F24

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