Dyonic Lieb-Schultz-Mattis theorem and symmetry protected topological phases in decorated dimer models

Abstract

We consider 2+1D lattice models of interacting bosons or spins, with both magnetic flux and fractional spin in the unit cell. We propose and prove a modified Lieb-Shultz Mattis (LSM) theorem in this setting, which applies even when the spin in the enlarged magnetic unit cell is integral. There are two nontrivial outcomes for gapped ground states that preserve all symmetries. In the first case, one necessarily obtains a symmetry protected topological (SPT) phase with protected edge states. This allows us to readily construct models of SPT states by decorating dimer models of Mott insulators to yield SPT phases, which should be useful in their physical realization. In the second case, exotic bulk excitations, i.e. topological order, is necessarily present. While both scenarios require fractional spin in the lattice unit cell, the second requires that the symmetries protecting the fractional spin is related to that involved in the magnetic translations. Our discussion encompasses the general notion of fractional spin (projective symmetry representations) and magnetic flux (magnetic translations tied to a symmetry generator). The resulting SPTs display a dyonic character in that they associate charge with symmetry flux, allowing the flux in the unit cell to screen the projective representation on the sites. We provide an explicit formula that encapsulates this physics, which identifies a specific set of allowed SPT phases.CONTENTS arXiv:1705.05421v1 [cond-mat.str-el]

Intrinsic and emergent anomalies at deconfined critical points

“…We proceed to discuss the physical interpretation of the bulk action (9). The first term in this action,…”

“…From a lattice perspective, 5 We could have chosen a more general manifold S 1 x × Y 3 with odd x flux along S 1 x to recover (9). The choice of a three-torus for Y 3 is made for ease of visualization and physical interpretation.…”

“…In fact, this perturbation is very likely irrelevant: Unitarity implies that the scaling dimension of an operator with angular momentum = 0 satisfies l + D − 2, where D is the space-time dimension [43], so in our case, =2 3. 9 It is unlikely that an operator other than the energy-momentum tensor exactly saturates the unitarity bound (if it does, it gives rise to a conserved = 2 current). The numerically observed emergent U (1) VBS symmetry of the deconfined critical point [29,44] is also consistent with the irrelevancy of (28).…”

“…Crucially, the boundary pin(2) − bundle need not extend to the "physical" bulk X 3 , so in general X∪Ȳ x 3 = 0. Indeed, when X 3 has no boundary, (A2) reduces to (9), which is the topological response of a Z 2 -protected SPT. This tells us that the surface theory has a Z ∈ Z is not zero, we cannot extend a from M to some Y 3 as a u(1) gauge field.…”

“…In recent years, this result has been generalized to a variety of cases where one relies on lattice symmetries other than translation-e.g., rotation, reflection, or glide-in combination with SO(3) s , or replaces SO(3) s by time-reversal symmetry, to rule out a trivial gap [4][5][6][7][8][9]. Furthermore, it was noted that the impossibility of a trivial gap is very reminiscent of the situation occurring on the boundary of a topological insulator, or a more general symmetry-protected topological (SPT) phase.…”