Finding suitable nonlocal order parameters that distinguish various symmetry-protected topological (SPT) phases is an important subject in view of experimental and numerical detection of SPT phases. By ``simulating'' the generating manifold of cobordism group in the operator formalism, the authors here propose nonlocal operations as diagnoses for SPT phases protected by point group symmetries. The nonlocal operations involve ``partial point group transformations'', which are obtained by point group transformations restricted to a spatial subregion on a given ground-state wave function. Through analytical and numerical calculations, the authors show that the complex $U(1)$ phase of the ground state expectation value of such partial point group transformations may serve as an order parameter for those SPT phases. The examples in the paper include the ${Z}_{8}$ and ${Z}_{16}$ invariants of topological superconductors protected by inversion symmetry in $(1+1)$ and $(3+1)$ dimensions, respectively, as well as the lens space topological invariants in $(2+1)$ -dimensional fermionic topological phases.Finding suitable nonlocal order parameters that distinguish various symmetry-protected topological (SPT) phases is an important subject in view of experimental and numerical detection of SPT phases. By ``simulating'' the generating manifold of cobordism group in the operator formalism, the authors here propose nonlocal operations as diagnoses for SPT phases protected by point group symmetries. The nonlocal operations involve ``partial point group transformations'', which are obtained by point group transformations restricted to a spatial subregion on a given ground-state wave function. Through analytical and numerical calculations, the authors show that the complex $U(1)$ phase of the ground state expectation value of such partial point group transformations may serve as an order parameter for those SPT phases. The examples in the paper include the ${Z}_{8}$ and ${Z}_{16}$ invariants of topological superconductors protected by inversion symmetry in $(1+1)$ and $(3+1)$ dimensions, respectively, as well as the lens space topological invariants in $(2+1)$-dimensional fermionic topological phases.

arXiv:1609.05970Physical review. B./Physical review. BMay 25, 201773 citations

Towards a Notion of Symmetry for Topological Phases

Matthew Titsworth, Tobias Hagge

Bulletin of the American Physical SocietyOct 26, 20120 citations

Symmetry-resolved entanglement of C 2 -symmetric topological insulators

Kyle Monkman, Jesko Sirker

For a many-body system of arbitrary dimension, we consider fermionic ground states of noninteracting Hamiltonians invariant under a ${C}_{2}$ cyclic group. The absolute difference $\mathrm{\ensuremath{\Delta}}$ between the number of occupied symmetric and antisymmetric single-particle states is a topological invariant in this case. We prove lower bounds on the configurational and the number entropy based on this invariant. Entanglement entropies are particularly important for identifying topological phases protected by spatial symmetries because they lack gapless boundary modes in general. In topological crystalline insulators, the topological invariant $\mathrm{\ensuremath{\Delta}}$ and the entropy bounds can be directly determined from high-symmetry points in the Brillouin zone.For a many-body system of arbitrary dimension, we consider fermionic ground states of noninteracting Hamiltonians invariant under a ${C}_{2}$ cyclic group. The absolute difference $\mathrm{\ensuremath{\Delta}}$ between the number of occupied symmetric and antisymmetric single-particle states is a topological invariant in this case. We prove lower bounds on the configurational and the number entropy based on this invariant. Entanglement entropies are particularly important for identifying topological phases protected by spatial symmetries because they lack gapless boundary modes in general. In topological crystalline insulators, the topological invariant $\mathrm{\ensuremath{\Delta}}$ and the entropy bounds can be directly determined from high-symmetry points in the Brillouin zone.

arXiv:2210.07406Physical review. B./Physical review. BMar 3, 20236 citations

Topological invariants and localized wave functions for some topological phases

晋林

Mar 23, 20170 citations

Complete Crystalline Topological Invariants from Partial Rotations in ( 2 + 1 ) D Invertible Fermionic States and Hofstadter’s Butterfly

Yuxuan Zhang, Naren Manjunath, Ryohei Kobayashi +1 more

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.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.

arXiv:2303.16919Physical Review LettersOct 23, 202317 citations

U(1)

phase of the ground state expectation value of such partial point group transformations may serve as an order parameter for those SPT phases. The examples in the paper include the

{Z}_{8}

and

{Z}_{16}

invariants of topological superconductors protected by inversion symmetry in

(1+1)

and

(3+1)

dimensions, respectively, as well as the lens space topological invariants in

(2+1)

-dimensional fermionic topological phases.

For a many-body system of arbitrary dimension, we consider fermionic ground states of noninteracting Hamiltonians invariant under a

{C}_{2}

cyclic group. The absolute difference $\mathrm{\ensuremath{\Delta}}$ between the number of occupied symmetric and antisymmetric single-particle states is a topological invariant in this case. We prove lower bounds on the configurational and the number entropy based on this invariant. Entanglement entropies are particularly important for identifying topological phases protected by spatial symmetries because they lack gapless boundary modes in general. In topological crystalline insulators, the topological invariant $\mathrm{\ensuremath{\Delta}}$ and the entropy bounds can be directly determined from high-symmetry points in the Brillouin zone.

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.