Signatures of a fracton spin liquid in a square lattice spin model

Gapless fracton quantum spin liquids are exotic phases of matter described by higher-rank U(1) gauge theories which host massive and immobile fracton matter excitations coupled to gapless photons. Despite well-known field theories, no spin models beyond purely classical systems have been proposed to realize these phases. Following a simple construction principle, here we introduce a square lattice spin model that shows signatures of a type-I fracton spin liquid in classical and quantum versions of our model. The unusual gauge constraint of the system leads to a significant Hilbert space fragmentation that prevents non-trivial quantum dynamics in the spin-1/2 case. This is mitigated in the quantum spin-1 model which has Hilbert sectors where the magnetic response, calculated with error-controlled quantum Monte-Carlo, shows characteristic patterns of suppressed pinch points consistent with a fractonic rank-2 U(1) field theory onto which our model can be mapped.