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

TL;DR

The paper analyzes the symmetry structure of a 1+1d Majorana chain, revealing lattice 't Hooft anomalies realized as projective algebras among translation, fermion parity, parity, and time-reversal; it connects these lattice anomalies to the continuum Majorana CFT via emanant symmetries, notably showing that lattice translation gives rise to the chiral parity (-1)^{F_L}. By performing a GSO projection (summing over spin structures) on the lattice, the Majorana system bosonizes to the Ising model, where the lattice translation becomes a non-invertible symmetry whose continuum limit is the Kramers-Wannier duality D. The work develops explicit lattice realizations of RR/NSNS/NSR/RNS sectors, analyzes defects and their impact on partition functions, and demonstrates how the non-invertible lattice translation D reproduces the continuum non-invertible duality symmetry emanating from the fermionic origin. Overall, it establishes a precise lattice–continuum dictionary for emanant and non-invertible symmetries in 1+1d, with a concrete route from Majorana chains to Ising CFTs via GSO projection and bosonization.

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 $(-1)^{F_\text{L}}$ 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.

Figures (1)

• Figure 1: The spectrum of the free fermion Hamiltonian \ref{['Hamiltonian4']} for even $L=2N$ and odd $L=2N+1$, and that for the twisted Hamiltonian \ref{['twHamiltonian4']} for even $L=2N$. (The figures are for $N=20$.) The Hamiltonians in momentum space are given in \ref{['kspaceHs']}, \ref{['oddHmom']}, and \ref{['eventwHmom']}. In each case, the low-lying modes near $k=0$ and $k=N$ correspond to the right- and left-moving modes in the continuum Majorana CFT. More specifically, the three cases correspond to the RR, NSR, and NSNS theories in the continuum, respectively. The black dots represent the nonzero modes, while the red dots represent the zero modes. The gray line plots the energy function by treating the momentum $k$ as a continuous variable.