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Bulk-boundary correspondence in point-gap topological phases

Daichi Nakamura, Takumi Bessho, Masatoshi Sato

Abstract

A striking feature of non-Hermitian systems is the presence of two different types of topology. One generalizes Hermitian topological phases, and the other is intrinsic to non-Hermitian systems, which are called line-gap topology and point-gap topology, respectively. Whereas the bulk-boundary correspondence is a fundamental principle in the former topology, its role in the latter has not been clear yet. This Letter establishes the bulk-boundary correspondence in the point-gap topology in non-Hermitian systems. After revealing the requirement for point-gap topology in the open boundary conditions, we clarify that the bulk point-gap topology in open boundary conditions can be different from that in periodic boundary conditions. On the basis of real space topological invariants and the $K$-theory, we give a complete classification of the open boundary point-gap topology with symmetry and show that the nontrivial open boundary topology results in robust and exotic surface states.

Bulk-boundary correspondence in point-gap topological phases

Abstract

A striking feature of non-Hermitian systems is the presence of two different types of topology. One generalizes Hermitian topological phases, and the other is intrinsic to non-Hermitian systems, which are called line-gap topology and point-gap topology, respectively. Whereas the bulk-boundary correspondence is a fundamental principle in the former topology, its role in the latter has not been clear yet. This Letter establishes the bulk-boundary correspondence in the point-gap topology in non-Hermitian systems. After revealing the requirement for point-gap topology in the open boundary conditions, we clarify that the bulk point-gap topology in open boundary conditions can be different from that in periodic boundary conditions. On the basis of real space topological invariants and the -theory, we give a complete classification of the open boundary point-gap topology with symmetry and show that the nontrivial open boundary topology results in robust and exotic surface states.
Paper Structure (44 sections, 3 theorems, 117 equations, 8 figures, 17 tables)

This paper contains 44 sections, 3 theorems, 117 equations, 8 figures, 17 tables.

Table of Contents

  1. From PBC to yOBC in Eq. (8)
  2. Proof of the BBC for point-gap topological phases in AZ$^\dagger$ symmetry classes
  3. class A
  4. class AIII
  5. class $\text{AI}^{\dag}$
  6. class $\text{DIII}^{\dag}$
  7. class $\text{AII}^{\dag}$
  8. class $\text{CII}^{\dag}$
  9. class $\text{C}^{\dag}$
  10. class $\text{CI}^{\dag}$
  11. Bulk-boundary correspondence in 2D class AIII
  12. The BBC for point-gap topological phases in 38-fold symmetry classes
  13. Classes without SLS
  14. class $\text{BDI}$
  15. class $\text{D}$
  16. ...and 29 more sections

Key Result

Lemma 1

Suppose $h_\pm$ is diagonalizable and $[h_{+},h_-]=0$, and let $|\phi_n\rangle$ be an eigenvector diagonalizing $h_+$ and $h_-$ simultaneously, Then, we obtain the eigenvectors and eigenenergies of $H$ from those of $h_{\pm}$, where $c_n^\pm$ is a constant.

Figures (8)

  • Figure 1: The energy spectra of ETI in Eq. (\ref{['eq:ETI']}). (a) The full PBC spectrum (gray). The point gap is open around $E=0$, with the nontrivial 3D winding number $+1$. (b) The zOBC spectrum (blue) and the full PBC spectrum (gray) in comparison. No NHSE occurs but surface states appear in the region where the 3D winding number takes the nontrivial value. The system size is $L_x=L_y=100$ and $L_z=30$. (c) The yOBC spectrum (blue) and the full PBC spectrum (gray) in comparison. NHSEs occur, and in-gap skin modes (orange) appear. The system size in the $y$ direction is $L_y=30$ and the momentum resolutions in both $k_{x,z}$-directions are taken as $\Delta k_{x,z} = 2\pi/100$.
  • Figure 2: The energy spectra (top) and the 3D winding numbers (bottom) of the model in Eq. (\ref{['eq:ETI2']}) under different boundary conditions. The magenta lines in the top figures represent $E=-0.5+i{\rm Im}E$ with $-4<{\rm Im}E<4$. (a)(top) The bulk spectra under the full PBC (gray) and the zOBC (turquoise). The zOBC spectrum is calculated by using the non-Bloch theory in Ref.YM-19. The point-gapless region under the zOBC is wider than that under the full PBCs. (bottom) The 3D winding number under the full PBC along the magenta line in the top figure. The gray shadings represent point-gapless regions. (b)(top) The energy spectrum under the zOBC. The blue modes are surface states which are calculated with the system size $L_x=L_y=100,L_z=20$. Surface states appear even in the regions where $w_{3}|_\text{fullPBC}$ is ill-defined (inside the red circle). (bottom) The real-space 3D winding number under the zOBC along the magenta line in the top figure. The turquoise segments represent point-gapless regions.
  • Figure 3: The changes of the spectra of ETI in Eq. (\ref{['eq:ETI']}) from under the full PBC to the yOBC. $L_y$ is the same as that in Fig. \ref{['fig:ETI']}(c) but the momentum resolution around $(k_x,k_z)=(0,0)$ is much finer. The orange modes represent the modes with $k_x=k_z=0$. (a) The full PBC spectrum. A point gap is open in the region containing $E=0$ with the nontrivial 3D winding number $+1$. The modes at $k_x=k_z=0$ form a loop and have a nontrivial 1D winding number in each eigensector of $\sigma_z=\pm 1$. (b) The spectrum under a boundary condition between the full PBC and the yOBC. Here the hopping amplitude between the $y=1$ sites and the $y=L_y$ sites is $10^{-6}$. The point-gapped region with the non-zero 3D winding number shrinks. (c) The yOBC spectrum. The region including $E=0$ is completely closed by the in-gap skin effect of the modes at $k_x=k_z=0$. Comparing with Fig. \ref{['fig:ETI']}(c), we can see that the point-gapped region is completely collapsed by the modes near $k_x=k_z=0$.
  • Figure S1: The yOBC spectrum of the model in Eq. (\ref{['eq:ETI2-SM']}). The system size in the $y$ direction is $L_y=21$ and the momentum resolution in the $k_{x,z}$-directions is taken as $\Delta k_{x,z} = 2\pi/100$. No surface states appear, and instead, the skin modes appear.
  • Figure S2: The changes of the spectrum (top) and the real-space 3D winding number (bottom) of the model in Eq. (\ref{['eq:ETI2-SM']}) from under the full PBC to the yOBC. $L_y$ is the same as that in Fig. \ref{['fig:in-gap_skin_mode-SM']}, but the momentum resolution around $(k_x,k_z)=(0,0)$ is much finer in the energy spectrum. The orange circles in the top figures represent the modes with $(k_x,k_z)=(0,0)$. In the bottom figures, the 3D winding numbers are calculated at $E=0+i{\rm Im}E$ with $-4<{\rm Im}E<4$. The purple shaded regions in the bottom figures correspond to point-gapless regions in the top figures. (a) The full PBC. A point gap is open in the region containing $E=0$ with the nontrivial 3D winding number $-1$. The orange modes at $(k_x,k_z)=(0,0)$ form a loop and have a nontrivial 1D winding number in each eigensector of $\sigma_z=\pm 1$. (b) A boundary condition between the full PBC and the yOBC. The hopping amplitude between the $y=1$ sites and the $y=L_y$ sites is $0.00005$. The point gapped region containing $E=0$ shrinks, but the 3D winding number is $-1$ in the point gapped region. (c) The yOBC. The point-gapped region including $E=0$ is completely closed by the in-gap skin effect with the orange modes near $(k_x,k_z)=(0,0)$. The in-gap skin effect makes the 3D winding number ill-defined.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Corollary 1