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arXiv:2306.10845 · SciPost Physics

Boundary condition and reflection anomaly in $2+1$ dimensions

Jiunn-Wei Chen, Chang-Tse Hsieh, Ryutaro Matsudo

Published Aug 26, 2024• 1 citation
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Abstract

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.

DOI
10.21468/scipostphys.17.2.068

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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.
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.
Andreev reflection converts an incoming fermion into an outgoing hole and is usually tied to a superconducting interface. We show that an analogous charge-conjugating boundary condition emerges from a purely lattice duality defect. Starting from a Majorana representation of the transverse-field Ising chain, we construct a folded lattice model in which a boundary Majorana impurity implements a one-site translation of a staggered Majorana chain. In the continuum, this translation becomes a chiral fermion-parity defect: it flips the sign of the only left-moving Majorana mode while leaving the right-moving mode unchanged. When the two Majorana modes are recombined into a complex fermion in the folded geometry, this sign flip becomes the Andreev-like boundary condition. Our lattice formulation gives a microscopic interpretation of the Emery--Kivelson boundary of the two-channel Kondo problem and of Maldacena--Ludwig monopole scattering, while identifying the boundary as the interface between a Kitaev-chain SPT phase and a gapless chain. The same Majorana translation defect also provides a lattice realization of an axial U(1)AU(1)_AU(1)A​-symmetric charge-flip boundary.
We study a relationship between conformally invariant boundary conditions and anomalies of conformal field theories (CFTs) in 1+11+11+1 dimensions. For a given CFT with a global symmetry, we consider symmetric ``gapping potentials'', which are relevant perturbations to the CFT. If a gapping potential is introduced only in a subregion of the system, it provides a certain boundary condition to the CFT. From this equivalence, if there exists a Cardy boundary state, which is invariant under a symmetry, then the CFT can be gapped with a unique ground state by adding the corresponding gapping potential. This means that the symmetry of the CFT is anomaly free. Using this approach, we systematically deduce the anomaly-free conditions for various types of CFTs with several different symmetries. They include the free compact boson theory, Wess-Zumino-Witten models, and unitary minimal models. When the symmetry of the CFT is anomalous, it implies a Lieb-Schultz-Mattis type ingappability of the system. Our results are consistent with, where available, known results in the literature. Moreover, we extend the discussion to other symmetries including spin groups and generalized time-reversal symmetries.
We construct a dual-bosonized description of a massless Majorana fermion in (2+1)ddd. In contrast to Dirac fermions, for which a bosonized description can be constructed using a flux attachment procedure, neutral Majorana fermions call for a different approach.We argue that the dual theory is an SO(N)1\mathrm{SO}{(N)}_{1}SO(N)1​ Chern-Simons gauge theory with a critical SO(N)\mathrm{SO}(N)SO(N) vector bosonic matter field ($N\ensuremath{\ge}3$). The monopole of the SO(N)\mathrm{SO}(N)SO(N) gauge field is identified with the Majorana fermion. We provide evidence for the duality by establishing the correspondence of adjacent gapped phases and by a parton construction. We also propose a generalization of the duality to Nf{N}_{f}Nf​ flavors of Majorana fermions, and discuss possible resolutions of a caveat associated with an emergent global Z2{Z}_{2}Z2​ symmetry. Finally, we conjecture a dual description of an N=1\mathcal{N}=1N=1 supersymmetric fixed point in (2+1)ddd, which is realized by tuning a single flavor of Majorana fermions to an interacting (Gross-Neveu) critical point.
Charge conjugation (C), mirror reflection (R), and time reversal (T) symmetries, along with internal symmetries, are essential for massless Majorana and Dirac fermions. These symmetries are sufficient to rule out potential fermion bilinear mass terms, thereby establishing a gapless free fermion fixed point phase, pivotal for symmetric mass generation (SMG) transition. In this work, we systematically study the anomaly of C-R-T-internal symmetry in all spacetime dimensions by analyzing the projective representation (i.e. the fractionalization) of the C-R-T-internal symmetry group in the quantum many-body Hilbert space on the lattice. By discovering the fermion-flavor-number-dependent C-R-T-internal symmetry's anomaly structure, we demonstrate an alternative way to derive the minimal flavor number for SMG, which shows consistency with known results from Kähler-Dirac fermion or cobordism classification. Our findings reveal that, in general spatial dimensions, either 8 copies of staggered Majorana fermions or 4 copies of staggered Dirac fermions admit SMG. By directly searching for 4-fermion interactions that form commuting stabilizers respecting all symmetry constraints, we can prove the explicit SMG gapping retained a unique ground state in the codespace. Furthermore, we establish the correspondence between the symmetry operators of staggered fermions and free fermions, which is instrumental in facilitating the analysis of symmetry fractionalization at the field theory level.
A bstract We study wavepackets of exotic excitations after two-dimensional fermions are scattered by the boundary condition constructed by Maldacena and Ludwig, which turns elementary excitations into exotic fractionally-charged objects. They are of interest in the s-wave approximation of the fermion-monopole scattering in four-dimensional QED and of the multi-channel Kondo effect. Our main result is an explicit expression for the outgoing state of a pair of such particles. We show analytically and numerically that the expectation value 〈 N 〉 of the total number of fermions and anti-fermions diverges when the wavepacket is localized to a point. The energy and charge densities remain localized, and the charge integrated locally around each wavepacket is finite and fractional, as expected from the known action of the boundary condition.