$PT$ Symmetry's Real Topology

Abstract

Symmetry-protected topological phases have been a central theme in condensed matter physics and beyond over the past two decades. Most efforts have focused on topological classifications of physical systems under given symmetries, while the intrinsic topology of the symmetries themselves has received much less attention. Here, we show that, in generic non-interacting spinless crystals, the spacetime inversion symmetry $PT$ naturally carries a real vector-bundle structure whose topology is characterized by Stiefel--Whitney (SW) classes. In contrast to previous work, where SW classes were used to describe the topology of real valence bundles protected by $PT$, we identify SW classes associated to the $PT$ symmetry itself. These symmetry SW classes can endow the \emph{total} real bundle of a $PT$-symmetric band structure with nontrivial topology, overturning the common assumption that the total bundle is always trivial. As a consequence, valence and conduction bands can exhibit asymmetric SW classes, in sharp contrast to the usual symmetric scenario. We further demonstrate that the symmetry SW classes provide a refined distinction between atomic insulator phases. Our results underscore the importance of treating crystal symmetries as topological objects in their own right, rather than focusing solely on the topology of energy bands.

Figures (1)

• Figure 1: a Schematic of the 1D lattice model with nearest-neighbor hopping amplitude $t$. Sites of different colors represent sublattices with opposite onsite energies $\pm \epsilon$. This setup illustrates the evolution between distinct atomic insulator phases separated by a critical point. b Structure of the 2D semimetal on a square lattice. Gray (red) bonds denote positive (negative) hopping amplitudes. c The Brillouin zone featuring two nodal points (red dots). Each point carries a topological charge $(w_{1}^{+},w_{1}^{-})=(1,1)$ defined on an enclosing circle. d Bulk band structure of $\mathcal{H}(\mathbf{k})$ in Eq. \ref{['eq:Hsquare']} along high-symmetry lines. e The Kagome lattice with primitive vectors $\mathbf{a}_1, \mathbf{a}_2$. All nearest-neighbor hoppings are $t$, and $\epsilon_i$ denotes the onsite energy of the $i$-th site. f Bulk band structure for the model in e along the path $\Gamma-K-M-\Gamma$. g Wilson loop spectra for the conduction bands of Eq. \ref{['eq:kagomeH']}, calculated along circles $C_{k_2}$ ($k_2$ fixed) and parameterized by $k_1$, confirming $w_2^{+}=1$. h The 3D cubic lattice model with four sites per unit cell. Bond colors follow the convention in b. i Brillouin zone exhibiting two nodal loops (red circles). Each loop carries a charge $(w_2^{+},w_2^{-})=(1,1)$ defined on an enclosing sphere. j Bulk band structure for the model in h along the path $\Gamma-Y-S-R-\Gamma$.