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.
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.
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.
Symmetry protected topological (SPT) states are short-range entangled states with symmetry. The boundary of a SPT phases has either gapless excitations or degenerate ground states, around a gapped bulk. Recently, we proposed a systematic construction of SPT phases in interacting bosonic systems, however it is not very clear what is the form of the low-energy excitations on the gapless edge. In this paper, we answer this question for two-dimensional (2D) bosonic SPT phases with
ZN and
U(1) symmetry. We find that while the low-energy modes of the gapless edges are nonchiral, symmetry acts on them in a ``chiral'' way, i.e., acts on the right movers and the left movers differently. This special realization of symmetry protects the gaplessness of the otherwise unstable edge states by prohibiting a direct scattering between the left and right movers. Moreover, understanding of the low-energy effective theory leads to experimental predictions about the SPT phases. In particular, we find that all the 2D
U(1) SPT phases have even integer quantized Hall conductance.
Abstract Boundary conditions for a massless Dirac fermion in 2+1 dimensions where the space is a half-plane are discussed in detail. It is argued that linear boundary conditions that leave the Hamiltonian Hermitian generically break C P and T symmetries as well as Lorentz and conformal symmetry. We show that there is essentially one special case where a single species of fermion has C PT and the full Poincare and conformal symmetry of the boundary. We show that, with doubled fermions, there is a second special case which respects C PT but still violates Lorentz and conformal symmetry. This second special case is essentially the unique boundary condition where the Dirac operator has fermion zero mode edge states. We discuss how the edge states lead to exotic representations of scale, phase and translation symmetries and how imposing a symmetry requirement leads to edge ferromagnetism of the system. We prove that the exotic ferromagnetic representations are indeed carried by the ground states of the system perturbed by a class of interaction Hamiltonians which includes the non-relativistic Coulomb interaction.
In (1+1)-dimensional topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group
Gf of the ground state splits as ${G}_{f}={G}_{b}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}$, where
Z2f is generated by fermion parity ${(\ensuremath{-}1)}^{F}$. In contrast, (2+1)-dimensional [(2+1)D] topological superconductors (TSC) can host unpaired MZMs at defects even when
Gf is not of the form ${G}_{b}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}$. In this paper we study how
Gf together with the chiral central charge ${c}_{\ensuremath{-}}$ strongly constrain the existence of unpaired MZMs and the quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of (2+1)D invertible fermionic topological states, which provides a nonperturbative approach based on topological quantum field theory, beyond free fermions. We study physically relevant groups such as $\mathrm{U}{(1)}^{f}\ensuremath{\rtimes}H,\phantom{\rule{0.28em}{0ex}}\mathrm{SU}{(2)}^{f}\ifmmode\times\else\texttimes\fi{}H,\phantom{\rule{0.28em}{0ex}}\mathrm{U}{(2)}^{f}\ensuremath{\rtimes}H$, generic Abelian groups, as well as more general compact Lie groups, antiunitary symmetries, and crystalline symmetries. We present an algebraic formula for the fermionic crystalline equivalence principle, which gives an equivalence between states with crystalline and internal symmetries. In light of our theory, we discuss several previously proposed realizations of unpaired MZMs in TSC materials such as
Sr2RuO4, transition metal dichalcogenides, and iron superconductors, in which crystalline symmetries are often important; in some cases we present additional predictions for the properties of these models.
We study the classification of interacting fermionic symmetry-protected topological (SPT) phases with both rotation symmetry and Abelian internal symmetries in one, two, and three dimensions. By working out this classification, on the one hand, we demonstrate the recently proposed correspondence principle between crystalline topological phases and those with internal symmetries through explicit block-state constructions. We find that for the precise correspondence to hold it is necessary to change the central extension structure of the symmetry group by the
Z2 fermion parity. On the other hand, we uncover new classes of intrinsically fermionic SPT phases that are only enabled by interactions, both in 2D and 3D. Several 3D Majorana-type fermionic SPT phases are identified. Moreover, several new instances of Lieb-Schultz-Mattis-type theorems for Majorana-type fermionic SPT phases are obtained and we discuss their interpretations from the perspective of bulk-boundary correspondence.