Translational Symmetry and Microscopic Constraints on Symmetry-Enriched Topological Phases: A View from the Surface

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 with on-site unitary symmetries, enables us to develop a framework for understanding the structure of symmetry-enriched topological phases 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 degeneracies protected by on-site symmetry.

Popular Summary

One of the central tenants of quantum physics is that many properties of particles come in discrete chunks. Much like a person climbing a ladder can stand only on the rungs and not in the space between, things like electric charge take on only certain values. But in some exotic quantum materials such as fractional quantum Hall insulators, where quantum mechanics is needed to explain macroscopic behavior, a striking phenomenon arises called fractionalization. Emergent excitations can carry just a fraction of an electron charge, for example, even though these excitations are ultimately built from electrons. We find that momentum of particles within a crystal, which is a conserved quantity, can also be fractionalized, and we lay out a general way to describe that behavior.

Understanding the behavior of interacting electrons in a crystal is a complex problem. An important consequence of our finding refines and expands the famous Lieb-Schultz-Mattis theorem, which stipulates that a Mott insulator (a material that behaves like an insulator even though conventional band theories predict that it should conduct electricity) must have fractional excitations. A basic property of any crystalline material is its “filling factor,” which specifies the amount of charge or spin contained in a unit volume and can be easily calculated just by knowing the chemistry. We establish a relation between this microscopic property and the possible fractionalization phenomena that can exist in the system.

The new perspective is to draw an analogy—a very precise one—between such a crystalline material and the surface states of 3D topological insulators. By contemplating the boundaries of 3D systems, we reveal new secrets about the physics of 2D crystalline systems. Our work might aid in the search for new quantum materials and in understanding how they behave.

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 ${\mathrm{\Omega }}_{\text{weak}}\in {\mathcal{H}}^4[{\mathbb{Z}}_{\text{trans}}^3\times \mathrm{SO}(3{)}_{\mathrm{int}},\mathrm{U}(1)]$. 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 $\mathcal{O}\in {\mathcal{H}}^4[{\mathbb{Z}}_{\text{trans}}^2\times \mathrm{SO}(3{)}_{\mathrm{int}},\mathrm{U}(1)]$, associated with defining an effective theory of the symmetry defects. The bulk-boundary correspondence requires $\mathcal{O}$ of the boundary and ${\mathrm{\Omega }}_{\text{weak}}$ 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 $\mathfrak{b}(T_y,T_x)$.

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 that move the $e$ particles vertically (thick black line) and $m$ particles horizontally (dashed black line) involve operators on vertical links, so are charged under the symmetry operations.

Figure 5

(a) Braiding an anyon $c_{\mathbf{0}}$ around a unit length of $\mathbf{g}$-defect branch line is equivalent to the local action of $R_{T_i}^{-1}{R}_{\mathbf{g}}^{-1}{R}_{T_i}{R}_{\mathbf{g}}$ on the anyon (shown here for $i=x$). This implies that $\mathbf{g}$-defect branch lines carry topological charge $\mathfrak{b}(T_i,\mathbf{g})$ per unit length in the $\widehat{i}$ direction. (b) The same result is obtained by comparing the results of braiding $c_{\mathbf{0}}$ around two loops, each of which encloses the $\mathbf{g}$ defect, and where one is translated relative to the other in the $\widehat{i}$ direction. This alternative derivation allows a generalization to the case where the $\mathbf{g}$-defect branch line is not localized.

Figure 6

Braiding a $\mathbf{g}$ defect around an $\widehat{i}$ dislocation ($T_i$ defect) leaves a $\mathbf{g}$-defect branch line loop encircling the dislocation. If $\mathbf{g}$-defect branch lines carry $\mathfrak{b}(T_i,\mathbf{g})$ per unit length, the branch line loop carries a total topological charge of $\overline{\mathfrak{b}(T_i,\mathbf{g})}$. The topological charge of the $\mathbf{g}$ defect will change by $\mathfrak{b}(T_i,\mathbf{g})$, which matches the translational symmetry action ${\rho }_{T_i}(a_{\mathbf{g}})=[\mathfrak{b}(T_i,\mathbf{g})\times a{]}_{\mathbf{g}}$. Here, we display a $T_x$ defect with a $\mathbf{g}$-defect loop encircling it, explicitly labeling the $\mathfrak{b}(T_x,\mathbf{g})$ topological charge per directed unit length of the $\mathbf{g}$-defect branch line [$\mathfrak{b}(T_y,\mathbf{g})$ labels are left implicit].

Figure 7

(a) In the toric code with unconventional symmetry, $Z$ defects exist at the end points of a defect branch line (indicated by the dashed line), which is produced by conjugating the spins on the vertical links on one side of the branch line by ${\sigma }^z$ for terms in the Hamiltonian that straddle the branch line. This changes $-A_v\mapsto +A_v$ for vertex terms straddling the branch line, energetically favoring an $e$ quasiparticle (indicated by solid black dots) at those vertices. (b) The configuration obtained from that in (a) by acting on the system with the translation operator $R_{T_x}$. (c) The configuration obtained from (a) by adiabatically transporting the defect in the $\widehat{x}$ direction by one lattice spacing. In this case, the topological charge of the defect must change from $I_Z$ to $e_Z$. (d) The configuration obtained from that in (c) by adiabatically transporting the defect in the $\widehat{x}$ direction by one lattice spacing. In this case, the configuration of the physical system is unchanged, but is interpreted differently. (e) The configuration obtained from that in (c) by adiabatically transporting the defect in the $\widehat{y}$ direction by one lattice spacing.

Figure 8

Physical interpretation of the contribution to the obstruction from the anyonic spin-orbit coupling from topological charge per directed unit length of defect branch lines. (a) Pair creating a $\mathbf{g}\text{−}\overline{\mathbf{g}}$ defect pair, adiabatically transporting the $\mathbf{g}$ defect counterclockwise along a path enclosing a unit cell, and then annihilating the defects generates a loop of $\mathbf{g}$-defect branch line encircling the unit cell (dashed blue line). This applies local $\mathbf{g}$ action $U_{\mathbf{g}}$ on the unit cell. Each edge of the unit cell can then be associated with a topological charge, as indicated in the figure. (b) The topological charges associated with each edge of the unit cell can be obtained by applying Wilson lines in the horizontal and vertical directions, as shown. The precise manner in which the crossing point is resolved does not affect the final result. (c) The configuration of anyon strings associated to the process $U_{\mathbf{g}}{U}_{\mathbf{h}}$. Solid lines correspond to the $U_{\mathbf{g}}$ process and dashed lines to the $U_{\mathbf{h}}$ process. (d) The configuration of anyon strings associated to the process $U_{\mathbf{h}}{U}_{\mathbf{g}}$.

Figure 9

An $\widehat{x}$ dislocation in the toric code model. When the unconventional ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry is imposed, the coefficient of the vertex term ${\sigma }_1^x{\sigma }_2^x{\sigma }_3^x$ must be set to zero to respect the $Z$ symmetry. This results in a degeneracy between topological charge values of the translational symmetry defect that differ by $\mathfrak{b}(T_x,Z)=e$.