Lieb-Schultz-Mattis–type constraints on fractonic matter

The Lieb-Schultz-Mattis (LSM) theorem and its descendants impose strong constraints on the low-energy behavior of interacting quantum systems. In this paper, we formulate LSM-type constraints for lattice translation invariant systems with generalized U(1) symmetries which have recently appeared in the context of fracton phases: U(1) polynomial shift and subsystem symmetries. Starting with a generic interacting system with conserved dipole moment, we examine the conditions under which it supports a symmetric, gapped, and nondegenerate ground state, which we find requires that both the filling fraction and the bulk charge polarization take integer-values. Similar constraints are derived for systems with higher moment conservation laws or subsystem symmetries, in addition to lower bounds on the ground-state degeneracy when certain conditions are violated. Finally, we discuss the mapping between LSM-type constraints for subsystem symmetries and the anomalous symmetry action at boundaries of subsystem symmetric topological (SSPT) states.

Figure 1

Lowest-order dipole conserving processes in $d=2$. The motion of isolated charges is forbidden under these, while composite dipolar objects are allowed to hop. For subsystem symmetries, the only allowed process at this order is given by (a).