Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface

TL;DR

The work reframes the traditional LSM constraint as a bulk–boundary anomaly matching problem by linking 2D SET phases with translational and on-site symmetries to the surface of 3D SPT states. Central to the framework is the obstruction class [\mathscr{O}] ∈ \mathcal{H}^4[G,U(1)], whose components must match the weak SPT invariants arising from translational stacking via the Künneth decomposition, enabling precise restrictions on symmetry fractionalization patterns. A key result is the identification of anyonic spin-orbit coupling, where background anyons per unit cell and symmetry actions on quasiparticles constrain the possible SET orders and enforce the existence of a spinon under broad conditions (e.g., continuous G or certain lattice rotations). The paper further derives explicit conditions for when a spinon must exist, describes SOC physics in defect branch lines, and applies the framework to Laughlin states, Z_2 spin liquids, and toric code variants, providing a general, broadly applicable method to characterize SET phases in 2D with translational symmetry and to understand the bulk SPTs that support them.

Abstract

The Lieb-Schultz-Mattis theorem and its higher dimensional generalizations by Oshikawa and Hastings require that translationally invariant 2D spin systems with a half-integer spin per unit cell must either have a continuum of low energy excitations, spontaneously break some symmetries, or exhibit topological order with anyonic excitations. We establish a connection between these constraints and a remarkably similar set of constraints at the surface of a 3D interacting topological insulator. This, combined with recent work on symmetry-enriched topological phases (SETs) with on-site unitary symmetries, enables us to develop a framework for understanding the structure of SETs with both translational and on-site unitary symmetries, including the effective theory of symmetry defects. This framework places stringent constraints on the possible types of symmetry fractionalization that can occur in 2D systems whose unit cell contains fractional spin, fractional charge, or a projective representation of the symmetry group. As a concrete application, we determine when a topological phase must possess a "spinon" excitation, even in cases when spin rotational invariance is broken down to a discrete subgroup by the crystal structure. We also describe the phenomena of "anyonic spin-orbit coupling", which may arise from the interplay of translational and on-site symmetries. These include the possibility of on-site symmetry defect branch lines carrying topological charge per unit length and lattice dislocations inducing on-site symmetry protected degeneracies.

Figures (9)

• Figure 1: A 3D system of integer spins can enter a weak SPT phase equivalent to stacking together 1D AKLT chains (as indicated on the left). The weak SPT phase is characterized by a particular element of the cohomology class $\Omega_{\textrm{weak}} \in \mathcal{H}^{4}\left[ \mathbb{Z}^3_{\text{trans}} \times \textrm{SO}(3)_{\text{int}}, \textrm{U} (1) \right]$. Since each AKLT chain has degenerate emergent $S=1/2$ edge states, the effective Hilbert space of the 2D surface of the 3D SPT phase behaves like a $S=1/2$ magnet; it will have a projective (half integral) representation of SO(3) in each unit cell. Thus, studying surface phases of the 3D weak SPT is equivalent to studying a 2D $S=1/2$ magnet. Given a proposal for the braiding, statistics, and symmetry fractionalization of the 2D magnet, there is a procedure to calculate an obstruction class $\mathscr{O} \in \mathcal{H}^{4}\left[ \mathbb{Z}^2_{\text{trans}} \times \textrm{SO}(3)_{\text{int}}, \textrm{U} (1) \right]$, associated with defining an effective theory of the symmetry defects. The bulk-boundary correspondence requires $\mathscr{O}$ of the boundary and $\Omega_{\textrm{weak}}$ of the bulk to be compatible, constraining the allowed 2D SET orders.
• Figure 2: The types of 3D Weak SPT phases. a) Filling 0D SPT phases (charge density), $\nu_{xyz}$. b) Packing 1D SPT phases, $\nu_{xy}$. c) Stacking 2D SPT phases, $\nu_{z}$.
• Figure 3: Anyonic flux per unit cell $\textswab{b}(T_y,T_x)$.
• Figure 4: The toric code with an unconventional $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry that exhibits "anyonic spin-orbit coupling." Dots represent qubits on each bond; the dark blue ones transform as projective representations of the global $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry, while the light dots do not. The string operators which move the $e$-particles vertically (thick black) and $m$-particles horizontally (dashed black) involve operators on vertical links, so are charged under the symmetry operations.
• Figure 5: (a) Braiding an anyon $c_{\bf 0}$ around a unit length of ${\bf g}$-defect branch line is equivalent to the local action of $R_{T_i}^{-1} R_{\bf g}^{-1} R_{T_i} R_{\bf g}$ on the anyon (shown here for $i=x$). This implies that ${\bf g}$-defect branch lines carry topological charge $\textswab{b}(T_i , \mathbf{g})$ per unit length in the $\hat{i}$-direction. (b) The same result is obtained by comparing the results of braiding $c_{\bf 0}$ around two loops, each of which encloses the ${\bf g}$-defect, and where one is translated relative to the other in the $\hat{i}$-direction. This alternative derivation allows a generalization to the case where the ${\bf g}$-defect branch line is not localized.