Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries
Nathan Seiberg, Shu-Heng Shao
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} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:msup> </mml:math> 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.