Subsystem non-invertible symmetry operators and defects

Abstract

We explore non-invertible symmetries in two-dimensional lattice models with subsystem \mathbb Z_2ℤ2 symmetry. We introduce a subsystem \mathbb Z_2ℤ2-gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wannier transformation. The corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion rules with subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss its subtleties.

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

“…Unlike some other references (such as [14][15][16][17][18][26][27][28]), we will insist that the operator D acts as an operator on the Hilbert space of the theory, rather than being a map from one Hilbert space to another. This will allow us to examine its algebra, and in particular, to compute D 2 , as in (4).…”

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

“…The two lattice translation symmetries T and GT are related by parity or time reversal, as do the two continuum symmetries that emanate from them. 8 Various authors [67][68][69][70][71][72][73][74] have discussed similar lattice operators, which roughly involve "translation by half the lattice spacing." (See also [75].)…”

“…So one either has an internal non-invertible algebra (242) on a non-tensor-factorized Hilbert space, or a non-invertible algebra mixing with the lattice translation (241) on a tensor product Hilbert space. Our noninvertible operator D is also different from the one in [68,[70][71][72][73]117]. The authors of these papers considered a map N from one Hilbert space of spins on the sites, to another one with spins on the links.…”

Symmetry TFT for subsystem symmetry

“…Another famous example is the boson-fermion duality [55][56][57], where the Ising CFT is dual to a free Majorana fermion by first stacking a topological phase given by the Arf-invariant (Kitaev Majorana chain) and then gauging the diagonal Z 2 symmetry. Recently, generalizations of Kramers-Wannier (KW) and Jordan-Wigner (JW) duality has been studied in the context of subsystem symmetry [58][59][60][61], where a new subsystem non-invertible symmetry has been found.…”

“…In this section, we will review the subsystem Z 2 symmetry in (2 + 1)d regularized on a 2d square lattice and the duality transformations including the subsystem Kramers-Wannier (KW) transformation [60] and the subsystem Jordan-Wigner (JW) transformation [61].…”