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

We consider (2+1)-dimensional 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-Schultz Mattis 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, exotic bulk excitations, i.e., topological order, is necessarily present even though the enlarged unit cell contains integer spin. In the second case, topological order can be avoided but then a symmetry protected topological (SPT) phase is necessarily realized. The resulting SPTs display a dyonic character in that they associate spin/charge with symmetry flux, allowing the flux in the unit cell to screen the fractional spin/charge on the sites. We provide an explicit formula that encapsulates this physics, which identifies a specific set of allowed SPT phases. This also provides a route to constructing models of SPT states by decorating dimer models of Mott insulators, which should be useful in their physical realization.

Figure 1

Degrees of freedom in the decorated BFG model (1). The top-layer Ising d.o.f. ${\sigma }_I$ (${\sigma }_I^z=\pm 1$) lives on the honeycomb lattice and the bottom-layer spin d.o.f. $S_i$ ($S_i^z=\pm \frac{1}{2}$) lives on the kagome lattice. The Ising coupling signs $s_{IJ}=+1$ on red bonds, and $s_{IJ}=-1$ on blue bonds. The thick red bonds represent the “$y$-odd zigzag chains” used in Eq. (18). A local area enclosed by the loop $L$ (thick gray loop) satisfying the low-energy constraint (three spin down per hexagon) is shown, together with the corresponding electric field lines (thick green lines).

Figure 2

(a) Schematic phase diagram of the decorated BFG model by tuning $\lambda$. We have already fixed $J_z\gg J_{\perp },h$. In the limit $\lambda \to 0$, the Ising layer is decoupled and the ground state is just that of the original BFG model with $Z_2$ topological order. This is an SET state with spinon carrying $S^z=\frac{1}{2}$. When $\lambda$ is tuned to be within the parameter regime where $J_z\gg \lambda \gg J_{\perp },h$ and $\frac{h^2}{{\lambda }^2}\gg \frac{J_{\perp }}{J_z}$, we have an SPT state with Ising defect carrying $S^z=\frac{1}{2}$ as discussed in the main text. There is a possible direct phase transition triggered by the condensation of Ising-odd visons at some intermediate ${\lambda }_c$. (b) A schematic view of vison condensation. The honeycomb lattice where Ising d.o.f. lives is shown and the spin d.o.f. lives in the bond center. Two visons are created at the ends $I,J$ of the string operator along $C$ (blue string): ${\sigma }_I^z{\sigma }_J^z{\prod }_{k\in C}2S_k^z$ with $S_k^z$ runs over all the black dot shown in the graph. Alternatively, we can view the string operator as the product of bond variable $S_i^z{\sigma }_H^z{\sigma }_K^z$ along $C$. Due to the constraint $2S_i^z\left(s_{HK}{\sigma }_H^z{\sigma }_K^z\right)=1$, the string operator has a well-defined eigenvalue (product of $s_{HK}$'s along the blue bonds) acting in the low-energy subspace, indicating that the visons are condensed and the $Z_2$ topological order is confined. Note that the condensed visons in the present case are dressed by local ${\sigma }^z$ operator and hence carry the quantum number of the Ising symmetry, resulting in an SPT state.

Figure 3

A pair of Ising defects (only one is shown) is created at the end points of the branch cut (dashed black line) after modifying the original Hamiltonian $H$ in Eq. (1) into $H^{\prime }$. The sign $s_{IJ}$ is flipped in $H^{\prime }$ along the branch cut comparing with the original model (red bond: $s_{IJ}=+1$, blue bond: $s_{IJ}=-1$). For any loop $L$ enclosing the Ising symmetry defect as the gray loop shown here, the product $\prod s_{IJ}$ around the loop flips sign comparing with the original model. Due to the low-energy constraint $2S_i^z\left(s_{IJ}{\sigma }_I^z{\sigma }_J^z\right)=1$, the total $S^z$ around the loop $L$ is changed by an odd integer comparing to the case without an Ising defect. The result is that Ising defect is topologically bound with a half-integer spin. See the discussion in the main text.

Figure 4

The Ising d.o.f. $\sigma$ live on the honeycomb lattice and the spin d.o.f. $\mathbf{\tau }$ live on the triangular lattice. The Ising coupling signs $s_{IJ}=+1$ on red bonds, and $s_{IJ}=-1$ on blue vertical bonds. The thick red bonds represent the “$y$-odd zigzag chains” used in Eq. (29).

Figure 5

Illustration of adiabatically separating a pair of $g$ defect/antidefect along the $x$ direction with $g^3=I$. For simplicity, one may imagine Hamiltonian to host nearest-neighbor (NN) terms. Along the $x$ direction, due to the magnetic translation symmetry (20), the NN interactions on the vertical bonds have a three-unit-cell periodicity (solid, dashed, and dotted bonds). While the $g$ defect crosses the entanglement cut at $x_0+\frac{1}{2}$, the Hamiltonian along the branch cut (dashed gray line) is effectively translated along $x$ direction by one unit cell. After separating such pairs of defects for every row, the final Hamiltonian is related to the original Hamiltonian by $T_x^{\mathrm{orig}}$.

Figure 6

The decomposition of global IGG into plaquette IGG. $\lambda$'s from different plaquettes commutes with each other, and the action of any two $\lambda$'s in the same plaquette leaves the tensor invariant.

Figure 7

An example of $g$-defect line. The $g$-defect line is obtained by inserting $W_g$ on only one side of the virtual legs crossed by the red dashed line. The tensors close to the defect core should be revised in order to make the tensor wave function symmetric and nonvanishing. Following the usual convention, we say that the defect line always points from $g^{-1}$ defect to $g$ defect, and we always insert $W_g$ to the left when one goes forward along the line. Therefore, in the figure we can identify the right end as the $g$ defect [remember $W_g\left(d\right)=W_g{\left(u\right)}^{-1}]$. The gray area encloses a $g$ defect and we can to measure its projective representation through the action of ${\eta }^{\prime }(a,b)$ on the boundary virtual legs (see the discussion in the main text).

Figure 8

Invariance of the wave function under $U^g\left(a\right)$. In the figure we can see that ${\tilde{W}}_a=\left[{\lambda }_a\left(g\right)\right]\left(d\right)·W_a$ where the defect line crosses the boundary and ${\tilde{W}}_a=W_a$ elsewhere. Such a definition ensures that no boundary excitations are created by applying $U^g\left(a\right)$ (for the moment we do not care about what happens at the defect core). In deriving the second figure, we have used the invariance of the tensor under $W_{a}a$, the identity $W_a{W}_g^{-1}=W_g^{-1}{\xi }_a\left(g\right)W_a$, and invariance of the tensor under plaquette IGG ${\lambda }_a\left(g\right)$.

Figure 9

Measurement of projective representation carried by $g$ defect. From Eq. (D6), we know that $\tilde{\eta }(a,b)={\lambda }_a\left(g\right){·}^{W_{a}a}{\lambda }_b\left(g\right)·\eta (a,b)·{\lambda }_{ab}^{-1}$ where the boundary is crossed by the defect line and $\tilde{\eta }(a,b)=\eta (a,b)$ elsewhere. In the first equality we have used Eq. (D5). In the second equality we have used the decomposition of ${\eta }^{\prime }(a,b)$ and Eq. (C11). In the third equality we have used the tensor invariance under plaquette IGG. In the last equality we have used the identity $\lambda {\left(r\right)}^{-1}·W_g^{-1}=W_g^{-1}{·}^{W_g}\lambda {\left(r\right)}^{-1}$ and the tensor invariance under plaquette IGG.

Figure 10

The definition of phase-gauge transformation $W(\alpha (a,b))$.

Figure 11

The decomposition rule of ${}^{W_{T_x}{T}_x}W(\alpha (a,b))·W(\alpha (a,b){)}^{-1}$ (left-hand side) as a product of plaquette IGG $\lambda (\alpha (a,b))$ (right-hand side).

Figure 12

The original tensor before insertion of $g$ defect is required to be invariant under the revised symmetry operation and the revised plaquette IGG.

Figure 13

(a) The definition of the new tensor ${\tilde{T}}^{(x,y)}$ after the insertion of ${\left[W_g\left(u\right)\right]}^x$ to the upper leg of every original tensor $T^{(x,y)}$. (b) The new translation operation $W_{T_x}{T}_x$. It can be readily checked that ${\tilde{T}}^{(x,y)}$ is invariant under such translation. Note that we have $T_x=g^y{T}_x^{\mathrm{orig}}, T_y=T_y^{\mathrm{orig}}$, and $W_{T_y}=\mathbf{1}$. (c) The new onsite symmetry operation ${\tilde{W}}_{a}a$. It is shown in Fig. 14 that ${\tilde{T}}^{(x,y)}$ is invariant under such symmetry operation. (d) The new plaquette IGG for the new tensor ${\tilde{T}}^{(x,y)}$. As before, $\lambda$'s from different plaquettes commute with each other, and the action of any two $\lambda$'s in the same plaquette leaves the tensor invariant. The tensor ${\tilde{T}}^{(x,y)}$ invariance under plaquette IGGs follows trivially from Fig. 12.

Figure 14

The revised tensor $\tilde{T}(x,y)$ is invariant under the newly defined symmetry operation. The first equality comes from the commutation relation $W_a{W}_g^x=W_g^x{\xi }_a(g,-x)W_a$. In the second equality, we have used the invariance of tensor under $W_a$ as shown in Fig. 12 and the decomposition of ${\xi }_a(g,-x)$. In the third equality, we have used the identity ${\lambda }_a(g,-x)\left(l\right){=}^{W_{g}g}\left[{\lambda }_a(g,-x-1)\right]\left(l\right)·{\lambda }_a\left(g\right)\left(l\right)$. And, we have also used the invariance of tensor under plaquette IGG as in Fig. 13 in the third and fourth equalities.

Figure 15

Every site tensor carries a projective representation characterized by ${\left[{\delta }_g^{\omega }(a,b)\right]}^{-1}$. We show this by applying ${\tilde{W}}_{a}a{\tilde{W}}_{b}b{\left({\tilde{W}}_{ab}ab\right)}^{-1}$ on both the physical legs and the virtual legs of tensor ${\tilde{T}}^{(x,y)}$, which should leave the tensor invariant without generating any phase. But, from the calculation we find that the action on virtual legs will contribute a factor ${\delta }_g^{\omega }(a,b)$, therefore, the representation on the physical legs is $D\left(a\right)·D\left(b\right)={\left[{\delta }_g^{\omega }(a,b)\right]}^{-1}D\left(ab\right)$, i.e., they are projected onto the $D_2\left(a\right)$ sector with projective representation. In the calculation above, we have used Eq. (F5) in the first equality. And we have used the invariance of the tensor under the plaquette IGG defined in Fig. 13 in the second equality.