Class II Sublattice Symmetry in Topological Phases

• Class II sublattice symmetry is a non-local, glide-reflection-based operation in bipartite lattices that combines a π momentum translation with energy inversion.
• It enforces unique zero-mode constraints, yielding specific band crossing rules (e.g., 4n+2 for odd and 4n for even orbitals) and alters conventional topological invariants.
• Concrete lattice models such as the Hofstadter semimetal and dimerized Hofstadter insulator illustrate its role in defining Z2 indices and shaping edge state structures.

Class II sublattice symmetry refers to a distinct manifestation of sublattice symmetry in bipartite lattice systems where the periodicity of the primitive unit cell is incommensurate with the periodicity of the sublattice labeling. In contrast to the conventional (Class I/AIII) chiral symmetry, Class II sublattice symmetry is realized via a non-local transformation in momentum space: a glide reflection that combines momentum translation by $\pi$ and energy inversion. This altered symmetry operation leads to fundamentally different physical consequences for topological phases, classification tables, and zero-mode constraints.

1. Primitive-Cell–Sublattice Mismatch and the Definition of Class II Symmetry

For a bipartite lattice model, standard sublattice (chiral) symmetry of Class I (AZ class AIII) presupposes that primitive lattice translations preserve sublattice labeling. The Hamiltonian in this case has the form: $$H(k) = \begin{pmatrix} 0 & Q(k) \ Q^\dagger(k) & 0 \end{pmatrix}$$ with the sublattice operator: $$\Gamma_{AIII} = \tau_z\otimes\mathbb{1}_N,\qquad\{\Gamma_{AIII}, H(k)\} = 0.$$ This symmetry is internal and local, corresponding to $\Gamma_{AIII} H(k) \Gamma_{AIII}^{-1} = -H(k)$.

In Class II, sublattice symmetry arises when the primitive translation $L_x$ exchanges sublattices, i.e., $\{L_x, S\} = 0$ for $S$ the sublattice-sign operator. The $A$/$B$ period is doubled relative to the cell period. Consequently, in $k$-space: $S e^{ik_x} S^{-1} = -e^{ik_x} = e^{i(k_x+\pi)},$ so the symmetry generator is

$\Gamma_{II}(k_x) := U_S\,\mathcal L^x_\pi,$

where the action induces

$$U_S\,H(k_x,\bar k)\,U_S^\dagger = -H(k_x+\pi,\bar k),$$

expressed compactly as

$$\Gamma_{II}(k) H(k) \Gamma_{II}(k)^{-1} = -H(k+\pi\,\hat x).$$

This symmetry operation is a glide reflection: it combines translation in $k$-space by $\pi$ and energy inversion.

2. Glide Reflection: Band Structure and Zero-Mode Constraints

The physical implication is that any eigenstate at momentum $k$ and energy $E$ is mapped to momentum $k+\pi$ and energy $-E$: $E(k+\pi) = -E(k),$ with the $(k,E)$-space dispersion invariant under the glide operation $(k,E)\mapsto(k+\pi,-E)$. This yields strict constraints on zero-modes:

• For an unpaired single band $E_0(k)$, $E_0(k+\pi)=-E_0(k)$, so zero crossings per $2\pi$ period must be even, but not a multiple of 4: generically $4n+2$.
• For a pair of bands $\{E_a, E_{\bar a}\}$ related by $E_a(k+\pi)=-E_{\bar a}(k)$, gapped crossings occur in multiples of $4n$.
• For an odd number of orbitals per primitive cell, zero-mode count is $4n+2$; for even, it is $4n$.

3. Topological Classification: Strong Invariants and Dimension Parity Switching

Class II symmetry alters the topological classification table relative to AIII:

Dim ($d$)	1	2	3	4	5	6	7	8
Class I (AIII)	$\mathbb{Z}$	0	$\mathbb{Z}$	0	$\mathbb{Z}$	0	$\mathbb{Z}$	0
Class II (glide)	0	$\mathbb{Z}_2$	0	$\mathbb{Z}_2$	0	$\mathbb{Z}_2$	0	$\mathbb{Z}_2$

Thus, the nontrivial strong invariants in Class II occur in even dimensions and are $\mathbb{Z}_2$-valued, in contrast to odd-dimensional $\mathbb{Z}$ (winding) invariants in AIII. Even/odd dimension roles are swapped and winding invariants vanish in Class II.

4. Explicit Model Implementations and $\mathbb{Z}_2$ Invariants

Several concrete lattice models illustrate the consequences of Class II symmetry:

Hofstadter Semimetal (Undimerized)

With Landau gauge, flux $\Phi=2\pi p/q$ and primitive cell of $q$ sites: $$H = \sum_{x,y}[t\,c^\dagger_{x+1,y} c_{x,y} + J\,e^{i2\pi p x/q} c^\dagger_{x,y+1} c_{x,y} + \text{h.c.}]$$ Translation $x \rightarrow x+1$ exchanges sublattice, satisfying $$\Gamma_{II}(k_x) H(k_x) \Gamma_{II}^{-1} = -H(k_x+\pi)$$. For odd $q$ there are $4n+2$ zero-modes, for even, $4n$.

Dimerized Hofstadter Insulator ($\Phi=\pi/2$)

Vertical hoppings alternate: $J_1, J_2$. The glide symmetry persists: $$U_S\,\mathcal H^{(p)}(k_x,k_y)\,U_S^\dagger = -\mathcal H^{(p)}(k_x+\pi,k_y)$$ At half-filling, a full gap opens at zero energy. The $d=2$ $\mathbb{Z}_2$ invariant is

$$\nu = \frac{1}{2\pi} \int_{\text{half-torus}} dk_x dk_y\, f_{-}(k_x,k_y) + \frac{1}{\pi} \gamma^y_{-}(k_x=-\pi) \bmod 2,$$

with $f_{-}$ the valence-band Berry curvature and $\gamma^y_{-}$ Berry phase. Numerically, $\nu = 1$ when $J_2 > J_1$, yielding zero-energy edge states that obey $(k,E)\mapsto(k+\pi,-E)$ glide symmetry.

5. Generalization: Stiefel Manifolds, Flat Bands, and Class II in CSNES

For chiral symmetry with $N_A \neq N_B$:

• Sublattice operator $S = \text{diag}(1_M,-1_N)$; Hamiltonian takes off-diagonal form $$H(k) = \begin{pmatrix}0 & Q(k) \ Q(k)^\dagger & 0\end{pmatrix}$$ with $Q$ generically rank $M$.
• Flat bands (zero modes) number $N-M$ at $E=0$.

Classifying spaces become real Stiefel manifolds $V_M(\mathbb{R}^N)$ for “Class II” if $\mathcal{PT}^2=+1$. Homotopy groups for the real case exhibit $\mathbb{Z}_2$ and even-Chern $(2\mathbb{Z})$ invariants, consistent with the $\mathbb{Z}_2$ structure for Class II symmetry.

6. Consequences and Physical Interpretation

Class II sublattice symmetry is not an internal “chiral” symmetry but a non-local symmetry that enforces a spatial glide in the Brillouin zone and energy inversion. This leads to modified zero-mode counting rules in semimetals, unique edge state structures in topological insulators, and a reorganization of the periodic table of topological phases, with $\mathbb{Z}_2$ indices as the primary invariants in even dimensions and the complete absence of strong winding invariants in odd dimensions (Xiao et al., 17 Apr 2024). Concrete lattice realizations include both ungapped and gapped configurations, making Class II symmetry a powerful framework for engineering and diagnosing topological matter with spatially entangled symmetries.