Complete Crystalline Topological Invariants from Partial Rotations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:math> Invertible Fermionic States and Hofstadter’s Butterfly
Yuxuan Zhang, Naren Manjunath, Ryohei Kobayashi, Maissam Barkeshli
Abstract
The theory of topological phases of matter predicts invariants protected only by crystalline symmetry, yet it has been unclear how to extract these from microscopic calculations in general. Here, we show how to extract a set of many-body invariants ${{\mathrm{\ensuremath{\Theta}}}_{o}^{\ifmmode\pm\else\textpm\fi{}}}$, where $o$ is a high symmetry point, from partial rotations in $(2+1)\mathrm{D}$ invertible fermionic states. Our results apply in the presence of magnetic field and Chern number $C\ensuremath{\ne}0$, in contrast to previous work. ${{\mathrm{\ensuremath{\Theta}}}_{o}^{\ifmmode\pm\else\textpm\fi{}}}$ together with $C$, chiral central charge ${c}_{\ensuremath{-}}$, and filling $\ensuremath{\nu}$ provide a complete many-body characterization of the topological state with symmetry group $G=U(1){\ifmmode\times\else\texttimes\fi{}}_{\ensuremath{\phi}}[{\mathbb{Z}}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{M}]$. Moreover, all these many-body invariants can be obtained from a single bulk ground state, without inserting additional defects. We perform numerical computations on the square lattice Hofstadter model. Remarkably, these match calculations from conformal and topological field theory, where $G$-crossed modular $S$, $T$ matrices of symmetry defects play a crucial role. Our results provide additional colorings of Hofstadter's butterfly, extending recently discovered colorings by the discrete shift and quantized charge polarization.