Modulated symmetries from generalized Lieb-Schultz-Mattis anomalies

Abstract

Symmetries rigidly delimit the landscape of quantum matter. Recently uncovered spatially modulated symmetries, whose actions vary with position, enable excitations with restricted mobility, while Lieb-Schultz-Mattis (LSM) type anomalies impose sharp constraints on which lattice phases are realizable. In one dimensional a spin chain, gauging procedures have linked modulated symmetry to LSM type anomaly, but a general understanding beyond 1D remains incomplete. We show that spatially modulated symmetries and their associated dipole algebras naturally emerge from gauging ordinary symmetries in the presence of generalized LSM type anomalies. We construct explicit lattice models in two and three spatial dimensions and develop complementary field theoretic descriptions in arbitrary spatial dimensions that connect LSM anomaly inflow to higher-group symmetry structures governing the modulated symmetries. Our results provide a unified, nonperturbative framework that ties together LSM constraints and spatially modulated symmetries across dimensions.

Figures (9)

• Figure 1: Visual illustration of the LSM anomaly in the 1D chain \ref{['spin1_d']}. On each site, we have a projective representation of $\mathbb{Z}_N\times \mathbb{Z}_N$, i.e. $Z_j X_j = \omega X_j Z_j$.
• Figure 2: (a) Three types of terms defined in \ref{['spin2d2']} that respecting the $0$-form dipole symmetry \ref{['algebra']}. (b) $0$-form dipole symmetry, forming dipole algebra \ref{['algebra']} which is schematically portrayed as an inverse of a triangle in the bottom.
• Figure 3: (a) Three types of terms defined in \ref{['terms']} which constitute the Hamiltonian \ref{['89']}. (b) $1$-form dipole symmetry, forming dipole algebra \ref{['dual94']} which is schematically portrayed as a triangle in the bottom.
• Figure 4: (a) The first three terms in \ref{['2toric']}. (b) Example of the dipole algebra \ref{['dipole01']} in the case of $N=2$ and $L_x$ even. Note that as opposed to previous studies which discuss dipole algebra involving the same form, we have unusual dipole algebra which involves $0$-form and $1$-form symmetry. Namely, acting a translational operator on the $0$-form symmetry yields stack of $1$-form symmetries. In the present case, the stack of $1$-form symmetries can be deformed into identity of one $1$-form symmetry, depending on whether $L_y$ is even or odd via flatness condition of the gauge field.
• Figure 5: (a) Spin coupling terms in the first line of \ref{['01spin']}. (b) (Top) Two $0$-form symmetries in \ref{['20form']}. (Bottom) Example of the $1$-form symmetry, corresponding to $U_Z^{(1),x}$ in \ref{['1_form']}. (c) Visual illustration of the LSM anomaly in our model: in each slab (the area inside the gray dashed line, we have two operators, $Z_{(\hat{x}+\frac{1}{2},1)}$, $\left(\prod_{\hat{y}=1}^{L_y}X_{\mathbf{l}_x}\right)$ which do not commute, signaling anomaly involving $0$-form and $1$-form symmetries and translational one in $x$-direction, in analogy to the 1D case (Fig. \ref{['1dlsm']}). (d) Gauss's law for gauging two 0-form symmetries \ref{['gauss3']}. (e) Gauss's law for gauging $1$-form symmetries \ref{['gauss4']}.