We study possible gapless edge states of translation-symmetric
Z2 spin liquids. The gapless edge states emerge from dangling Majorana fermions at the boundary. We construct a series of mean-field Hamiltonians of
Z2 spin liquids on the square lattice; these models can be obtained by generalization of Wen's exactly solvable plaquette model. We also study the details of the edge theory of these
Z2 spin liquids and find their effective background field (BF) theory descriptions. The effective BF theories are shown to describe the crystal momenta of the ground states and their degeneracies and to predict the edge theories of these
Z2 spin liquids. As a by-product, we obtained a way to classify the BF theories reflecting the lattice symmetries. We discuss in closing three-dimensional
Z2 spin liquids with gapless surface states on the cubic lattice.
A projective symmetry group (PSG) has been regarded as a classification theory of spin liquids.However, it does not include a symmetry-protected topological order of fermionic spinon excitations, and thus the classification of gapped spin liquids is incomplete.We demonstrate the classification beyond PSG by utilizing the Kitaev model on the square-octagon lattice, where two gapped spin liquids are distinguished by a topological Z 2 invariant.This Z 2 invariant can be defined solely by the time-reversal and translation symmetries on condition that the time-reversal symmetry is implemented projectively.Thus, it is a hidden class of topological Kitaev spin liquids with helical edge states, which has been ignored for a long time.This suggests that there exists an unknown classification scheme of gapped spin liquids beyond PSG.
We introduce a clean cluster spin chain coupled to fully interacting spinless fermions, forming an unconstrained
Z2 lattice gauge theory (LGT), which possesses dynamical proximity effect controlled by the entanglement structure of the initial state. We expand the machinery of interaction-driven localization to the realm of LGTs such that for any starting product state, the matter fields exhibit emergent statistical bubble localization, which is driven solely by the cluster interaction, having no topologically trivial noninteracting counterpart, and thus is of a pure dynamical many-body effect. In this vein, our proposed setting provides possibly the minimal model dropping all the conventional assumptions regarding the existence of many-body localization. Through projective measurement of local constituting species, we also identify the coexistence of the disentangled nonergodic matter and thermalized gauge degrees of freedom, which stands completely beyond the standard established phenomenology of quantum disentangled liquids. As a byproduct of self-localization of the proximate fermions, the spin subsystem hosts the long-lived topological edge zero modes, which are dynamically decoupled from the thermalized background
Z2 charges of the bulk, and hence remains cold at arbitrary high-energy density. This provides a convenient platform for strong protection of the quantum bits of information, which are embedded at the edges of completely ergodic subsystem; the phenomenon that in the absence of such proximity-induced self-localization could, at best, come about with a prethermal manner in translational invariant systems. Finally, by breaking local
Z2 symmetry of the model, we argue that such admixture of particles no longer remains disentangled and the ergodic gauge degrees of freedom act as a ``small bath'' coupled to the localized components.