Boundary RG Flows for Fermions and the Mod 2 Anomaly

Abstract

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

arXiv comment47 pages. v2: references added

References in corpus (33)

• Unpaired Majorana fermions in quantum wires
• D-Branes And K-Theory
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• SO(32) Spinors of Type I and Other Solitons on Brane-Antibrane Pair
• Boson topological insulators: A window into highly entangled quantum phases
• On the Boundary Entropy of One-dimensional Quantum Systems at Low Temperature
• Boundary Deformation Theory and Moduli Spaces of D-Branes
• The g-theorem and quantum information theory
• A Web of 2d Dualities: ${\mathbf{Z}}_2$ Gauge Fields and Arf Invariants
• Non-BPS States and Branes in String Theory
• Conformal boundary states for free bosons and fermions
• Duality between $(2+1)d$ Quantum Critical Points
• Fusion of conformal interfaces
• Majorana Fermions, Exact Mapping between Quantum Impurity Fixed Points with four bulk Fermion species, and Solution of the ``Unitarity Puzzle''
• Developments in Topological Gravity
• A worldsheet extension of O(d,d;Z)
• Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD
• Boundary conformal field theory and symmetry protected topological phases in $2+1$ dimensions
• Decorated ${\mathbb{Z}}_2$ Symmetry Defects and Their Time-Reversal Anomalies
• Exceptional boundary states at c=1
• Topological Superconductors on Superstring Worldsheets
• Fermionic CFTs and classifying algebras
• Monopole-Catalysed Baryon Decay: A Boundary Conformal Field Theory Approach
• Fusion of Critical Defect Lines in the 2D Ising Model
• 't Hooft anomalies and boundaries
• Relationship between Symmetry Protected Topological Phases and Boundary Conformal Field Theories via the Entanglement Spectrum
• GSO projections via SPT phases
• Lessons from the Ramond sector
• Moonshine, Superconformal Symmetry, and Quantum Error Correction
• Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy
• Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy II -- Cutting through the boundary
• A study of truncation effects in boundary flows of the Ising model on a strip
• Notes on 8 Majorana Fermions

Cited by in corpus (10)

• Orbifold groupoids
• Remarks on Boundaries, Anomalies, and Noninvertible Symmetries
• Comments on Symmetric Mass Generation in 2d and 4d
• Symmetry-resolved Entanglement Entropy, Spectra & Boundary Conformal Field Theory
• Fermionization and boundary states in 1+1 dimensions
• Fermionization of conformal boundary states
• Monopole-fermion scattering and varying Fock space
• Information metric on the boundary
• Universal time evolution of string order parameter in quantum critical systems with boundary invertible or non-invertible symmetry breaking
• Total and Symmetry resolved Entanglement spectra in some Fermionic CFTs from the BCFT approach