Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries

Abstract

We study the symmetries of closed Majorana chains in 1+1d, including the translation, fermion parity, spatial parity, and time-reversal symmetries. The algebra of the symmetry operators is realized projectively on the Hilbert space, signaling anomalies on the lattice, and constraining the long-distance behavior. In the special case of the free Hamiltonian (and small deformations thereof), the continuum limit is the 1+1d free Majorana CFT. Its continuum chiral fermion parity (-1)^{F_L}(−1)FL emanates from the lattice translation symmetry. We find a lattice precursor of its mod 8 ’t Hooft anomaly. Using a Jordan-Wigner transformation, we sum over the spin structures of the lattice model (a procedure known as the GSO projection), while carefully tracking the global symmetries. In the resulting bosonic model of Ising spins, the Majorana translation operator leads to a non-invertible lattice translation symmetry at the critical point. The non-invertible Kramers-Wannier duality operator of the continuum Ising CFT emanates from this non-invertible lattice translation of the transverse-field Ising model.

Lattice regularization of reduced Kähler-Dirac fermions and connections to chiral fermions

“…However, the offsite nature of the shift symmetry makes it difficult to gauge the Z 2 . Recent work on Hamiltonian lattice systems with similar shift symmetries may offer a path forward [52,53].…”

Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space

“…Finally, we come to the non-invertible symmetry. It is implemented by a non-invertible operator D that satisfies the algebra [13] 1 lattice :…”

“…• Bosonizing the Majorana chain [13] (see Appendix C). It is also related to a sequential quantum circuit [23][24][25] (see Appendix D).…”

“…While we will mostly focus on the Ising Hamiltonian (10), most of our conclusions apply to more general Hamiltonians, gapped or gapless, with the non-invertible symmetry. In Section 4, we will consider deformations away from the critical Ising Hamiltonian (10) while preserving all the symmetries Many of the results in this section were known in the literature, such as in [13,14,105]. Here we emphasize the role of the lattice translation, and provide a streamlined discussion combining the operators and the defects in the setup of a Hamiltonian lattice model.…”

“…The non-invertible topological defect D corresponds to the Kramers-Wannier self-duality of the theory at the critical temperature. The defect Hamiltonian is given by [9,13,14,[105][106][107] 13 12 The transformation X j ⇝ Z j Z j+1 , Z j Z j+1 ⇝ X j+1 can be obtained from composing (21) with the lattice translation by one-site to the right T . 13 The defect Hamiltonian in (22) is related to the Hamiltonian H D of [13, (5.43)] as follows.…”