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FIG. S2. (a) The schematic of the anomalous scattering process. (b) The path in momentum space preserves the component along θ direction and transverses the incident wave ki, which is denoted by |z| = 1. (c) The Bloch spectrum of H(k) in Eq.(S7) as k transverses the path in (b). (d) the flow of zeros z in Eq.(S19) when the reference energy changes from E1 to E2, here only the zeros in the red annular region are shown.
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In this paper, we propose the concept called dynamical degeneracy splitting to characterize the anisotropic decay behaviors in non-Hermitian systems. We show that when the system has dynamical degeneracy splitting, it will exhibit (i) anomalous scattering in the bulk and (ii) the non-Hermitian skin effect under the open boundary condition of a gene...
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Context 1
... wave as multiband (q > 1) cases, but in a more compact form. In this case, the codimension-1 scattering potential in Eq.(S16) becomes an impurity line lying on the r ⊥ = 0 and along the θ-direction, which is denoted as L θ . For the left (right) side of the impurity line L θ , the corresponding region is denoted by r ⊥ < 0 (r ⊥ > 0), as shown in Fig. S2(a). We define z := e ik ⊥ , then the scattered wave ...Context 2
... loss of generality, we assume the incident plane wave k i = (k θ i , k ⊥ i ) with energy E ν (k i ) comes from r ⊥ < 0 region as show in Fig. S2(a). Correspondingly, the transmitted wave is in the r ⊥ > 0 region and the reflected wave is in the r ⊥ < 0 region. From Eq.(S20) one can see that when |r ⊥ | → ∞, the transmitted wave is dominated by the maximum z max in ∈ {z in } (having the maximal amplitude |z max in |), and the reflected wave is mainly controlled by the minimum z min ...Context 3
... for the scattering process shown in Fig. (S2)(a), the momentum k θ along the impurity line L θ is preserved. When we give the momentum k i of the incident wave, the path in k space that is perpendicular to the θ direction is determined. This path traverses the momentum k i , as shown in Fig. (S2)(b). Here we use the Hamiltonian in Eq.(S7) as an example, and choose the incident wave ...Context 4
... for the scattering process shown in Fig. (S2)(a), the momentum k θ along the impurity line L θ is preserved. When we give the momentum k i of the incident wave, the path in k space that is perpendicular to the θ direction is determined. This path traverses the momentum k i , as shown in Fig. (S2)(b). Here we use the Hamiltonian in Eq.(S7) as an example, and choose the incident wave to be k i = (k x i , k y i ) = (π/2, 0) with the energy . Now we can define the Bloch spectral winding number regarding the reference energy E 0 + iη [6,7] ...Context 5
... N zeros refers to the number of zeros of g(E 0 + iη, k θ i , z) inside |z| = 1 and N poles is the order of the pole at the origin. The spectral winding number is ill-defined when η = 0. As shown in Fig. S2(c), the η is tuned from η < 0 to η > 0 (correspondingly, the reference energy goes from E 1 → E 0 → E 2 ...Context 6
... that we always choose the rightward incident wave as shown in Fig. S2(a). Correspondingly, as k increases, the Bloch spectrum always passes through E 0 from right to left in the complex energy plane, as shown in Fig. S2(c). Therefore, the spectral winding number always increases by 1 when the reference energy E 0 + iη runs from the region below E 0 to the region above E 0 , namely w(E 0 + 0 + ) − w(E 0 + 0 ...Context 7
... that we always choose the rightward incident wave as shown in Fig. S2(a). Correspondingly, as k increases, the Bloch spectrum always passes through E 0 from right to left in the complex energy plane, as shown in Fig. S2(c). Therefore, the spectral winding number always increases by 1 when the reference energy E 0 + iη runs from the region below E 0 to the region above E 0 , namely w(E 0 + 0 + ) − w(E 0 + 0 − ) = 1. For example, in Fig. S2(c), w(E 1 ) = −1 and w(E 2 ) = 0. Meanwhile, in this process N poles is always invariant, thus the number of zeros ...Context 8
... as k increases, the Bloch spectrum always passes through E 0 from right to left in the complex energy plane, as shown in Fig. S2(c). Therefore, the spectral winding number always increases by 1 when the reference energy E 0 + iη runs from the region below E 0 to the region above E 0 , namely w(E 0 + 0 + ) − w(E 0 + 0 − ) = 1. For example, in Fig. S2(c), w(E 1 ) = −1 and w(E 2 ) = 0. Meanwhile, in this process N poles is always invariant, thus the number of zeros inside |z| = 1 curve N zeros increases +1. Equivalently, there is one pole in Eq.(S20) that moves into |z| = 1 curve as the reference energy changes from E 1 to E 2 , as shown in Fig. S2(d). Therefore, for the rightward ...Context 9
... 0 + 0 + ) − w(E 0 + 0 − ) = 1. For example, in Fig. S2(c), w(E 1 ) = −1 and w(E 2 ) = 0. Meanwhile, in this process N poles is always invariant, thus the number of zeros inside |z| = 1 curve N zeros increases +1. Equivalently, there is one pole in Eq.(S20) that moves into |z| = 1 curve as the reference energy changes from E 1 to E 2 , as shown in Fig. S2(d). Therefore, for the rightward incident wave, there is always one pole approaches to |z| = 1 curve from inside when η → 0 + . Finally, we can obtain only the two cases of scattered wave as discussed in the main ...Context 10
... wave as multiband (q > 1) cases, but in a more compact form. In this case, the codimension-1 scattering potential in Eq.(S16) becomes an impurity line lying on the r ⊥ = 0 and along the θ-direction, which is denoted as L θ . For the left (right) side of the impurity line L θ , the corresponding region is denoted by r ⊥ < 0 (r ⊥ > 0), as shown in Fig. S2(a). We define z := e ik ⊥ , then the scattered wave ...Context 11
... loss of generality, we assume the incident plane wave k i = (k θ i , k ⊥ i ) with energy E ν (k i ) comes from r ⊥ < 0 region as show in Fig. S2(a). Correspondingly, the transmitted wave is in the r ⊥ > 0 region and the reflected wave is in the r ⊥ < 0 region. From Eq.(S20) one can see that when |r ⊥ | → ∞, the transmitted wave is dominated by the maximum z max in ∈ {z in } (having the maximal amplitude |z max in |), and the reflected wave is mainly controlled by the minimum z min ...Context 12
... for the scattering process shown in Fig. (S2)(a), the momentum k θ along the impurity line L θ is preserved. When we give the momentum k i of the incident wave, the path in k space that is perpendicular to the θ direction is determined. This path traverses the momentum k i , as shown in Fig. (S2)(b). Here we use the Hamiltonian in Eq.(S7) as an example, and choose the incident wave ...Context 13
... for the scattering process shown in Fig. (S2)(a), the momentum k θ along the impurity line L θ is preserved. When we give the momentum k i of the incident wave, the path in k space that is perpendicular to the θ direction is determined. This path traverses the momentum k i , as shown in Fig. (S2)(b). Here we use the Hamiltonian in Eq.(S7) as an example, and choose the incident wave to be k i = (k x i , k y i ) = (π/2, 0) with the energy . Now we can define the Bloch spectral winding number regarding the reference energy E 0 + iη [6,7] ...Context 14
... N zeros refers to the number of zeros of g(E 0 + iη, k θ i , z) inside |z| = 1 and N poles is the order of the pole at the origin. The spectral winding number is ill-defined when η = 0. As shown in Fig. S2(c), the η is tuned from η < 0 to η > 0 (correspondingly, the reference energy goes from E 1 → E 0 → E 2 ...Context 15
... that we always choose the rightward incident wave as shown in Fig. S2(a). Correspondingly, as k increases, the Bloch spectrum always passes through E 0 from right to left in the complex energy plane, as shown in Fig. S2(c). Therefore, the spectral winding number always increases by 1 when the reference energy E 0 + iη runs from the region below E 0 to the region above E 0 , namely w(E 0 + 0 + ) − w(E 0 + 0 ...Context 16
... that we always choose the rightward incident wave as shown in Fig. S2(a). Correspondingly, as k increases, the Bloch spectrum always passes through E 0 from right to left in the complex energy plane, as shown in Fig. S2(c). Therefore, the spectral winding number always increases by 1 when the reference energy E 0 + iη runs from the region below E 0 to the region above E 0 , namely w(E 0 + 0 + ) − w(E 0 + 0 − ) = 1. For example, in Fig. S2(c), w(E 1 ) = −1 and w(E 2 ) = 0. Meanwhile, in this process N poles is always invariant, thus the number of zeros ...Context 17
... as k increases, the Bloch spectrum always passes through E 0 from right to left in the complex energy plane, as shown in Fig. S2(c). Therefore, the spectral winding number always increases by 1 when the reference energy E 0 + iη runs from the region below E 0 to the region above E 0 , namely w(E 0 + 0 + ) − w(E 0 + 0 − ) = 1. For example, in Fig. S2(c), w(E 1 ) = −1 and w(E 2 ) = 0. Meanwhile, in this process N poles is always invariant, thus the number of zeros inside |z| = 1 curve N zeros increases +1. Equivalently, there is one pole in Eq.(S20) that moves into |z| = 1 curve as the reference energy changes from E 1 to E 2 , as shown in Fig. S2(d). Therefore, for the rightward ...Context 18
... 0 + 0 + ) − w(E 0 + 0 − ) = 1. For example, in Fig. S2(c), w(E 1 ) = −1 and w(E 2 ) = 0. Meanwhile, in this process N poles is always invariant, thus the number of zeros inside |z| = 1 curve N zeros increases +1. Equivalently, there is one pole in Eq.(S20) that moves into |z| = 1 curve as the reference energy changes from E 1 to E 2 , as shown in Fig. S2(d). Therefore, for the rightward incident wave, there is always one pole approaches to |z| = 1 curve from inside when η → 0 + . Finally, we can obtain only the two cases of scattered wave as discussed in the main ...Context 19
... wave as multiband (q > 1) cases, but in a more compact form. In this case, the codimension-1 scattering potential in Eq.(S23) becomes an impurity line lying on the r ⊥ = 0 and along the θ-direction, which is denoted as L θ . For the left (right) side of the impurity line L θ , the corresponding region is denoted by r ⊥ < 0 (r ⊥ > 0), as shown in Fig. S4(a). We define z := e ik ⊥ , then the scattered wave ...Context 20
... loss of generality, we assume the incident plane wave k i = (k θ i , k ⊥ i ) with energy E ν (k i ) comes from r ⊥ < 0 region as show in Fig. S4(a). Correspondingly, the transmitted wave is in the r ⊥ > 0 region and the reflected wave is in the r ⊥ < 0 region. From Eq.(S27) one can see that when |r ⊥ | → ∞, the transmitted wave is dominated by the maximum z max in ∈ {z in } (having the maximal amplitude |z max in |), and the reflected wave is mainly controlled by the minimum z min ...Context 21
... for the scattering process shown in Fig. (S4)(a), the momentum k θ along the impurity line L θ is preserved. When we give the momentum k i of the incident wave, the path in k space that is perpendicular to the θ direction is determined. This path traverses the momentum k i , as shown in Fig. (S4)(b). Here we use the Hamiltonian in Eq.(S7) as an example, and choose the incident wave ...Context 22
... for the scattering process shown in Fig. (S4)(a), the momentum k θ along the impurity line L θ is preserved. When we give the momentum k i of the incident wave, the path in k space that is perpendicular to the θ direction is determined. This path traverses the momentum k i , as shown in Fig. (S4)(b). Here we use the Hamiltonian in Eq.(S7) as an example, and choose the incident wave to be k i = (k x i , k y i ) = (π/2, 0) with the energy E 0 = 3/2. As k transverses the path in Fig. (S4)(b), a loop-shape spectrum on the complex energy plane can be obtained as plotted in Fig. (S4)(c). Note that the incident wave has the rightward ...Context 23
... k i of the incident wave, the path in k space that is perpendicular to the θ direction is determined. This path traverses the momentum k i , as shown in Fig. (S4)(b). Here we use the Hamiltonian in Eq.(S7) as an example, and choose the incident wave to be k i = (k x i , k y i ) = (π/2, 0) with the energy E 0 = 3/2. As k transverses the path in Fig. (S4)(b), a loop-shape spectrum on the complex energy plane can be obtained as plotted in Fig. (S4)(c). Note that the incident wave has the rightward velocity, which requires v x (k i ) > 0 defined in Eq.(S5). Now we can define the Bloch spectral winding number regarding the reference energy E 0 + iη [4,5] ...Context 24
... This path traverses the momentum k i , as shown in Fig. (S4)(b). Here we use the Hamiltonian in Eq.(S7) as an example, and choose the incident wave to be k i = (k x i , k y i ) = (π/2, 0) with the energy E 0 = 3/2. As k transverses the path in Fig. (S4)(b), a loop-shape spectrum on the complex energy plane can be obtained as plotted in Fig. (S4)(c). Note that the incident wave has the rightward velocity, which requires v x (k i ) > 0 defined in Eq.(S5). Now we can define the Bloch spectral winding number regarding the reference energy E 0 + iη [4,5] ...Context 25
... N zeros refers to the number of zeros of g(E 0 + iη, k θ i , z) inside |z| = 1 and N poles is the order of the pole at the origin. The spectral winding number is ill-defined when η = 0. As shown in Fig. S4(c), the η is tuned from η < 0 to η > 0 (correspondingly, the reference energy goes ...Context 26
... that we always choose the rightward incident wave as shown in Fig. S4(a). Correspondingly, as k increases, the Bloch spectrum always passes through E 0 from right to left in the complex energy plane, as shown in Fig. S4(c). Therefore, the spectral winding number always increases by 1 when the reference energy E 0 + iη runs from the region below E 0 to the region above E 0 , namely w(E 0 + 0 + ) − w(E 0 + 0 ...Context 27
... that we always choose the rightward incident wave as shown in Fig. S4(a). Correspondingly, as k increases, the Bloch spectrum always passes through E 0 from right to left in the complex energy plane, as shown in Fig. S4(c). Therefore, the spectral winding number always increases by 1 when the reference energy E 0 + iη runs from the region below E 0 to the region above E 0 , namely w(E 0 + 0 + ) − w(E 0 + 0 − ) = 1. For example, in Fig. S4(c), w(E 1 ) = −1 and w(E 2 ) = 0. Meanwhile, in this process N poles is always invariant, thus the number of zeros ...Context 28
... as k increases, the Bloch spectrum always passes through E 0 from right to left in the complex energy plane, as shown in Fig. S4(c). Therefore, the spectral winding number always increases by 1 when the reference energy E 0 + iη runs from the region below E 0 to the region above E 0 , namely w(E 0 + 0 + ) − w(E 0 + 0 − ) = 1. For example, in Fig. S4(c), w(E 1 ) = −1 and w(E 2 ) = 0. Meanwhile, in this process N poles is always invariant, thus the number of zeros inside |z| = 1 curve N zeros increases +1. Equivalently, there is one pole in Eq.(S27) that moves into |z| = 1 curve as the reference energy changes from E 1 to E 2 , as shown in Fig. S4(d). Therefore, for the rightward ...Context 29
... 0 + 0 + ) − w(E 0 + 0 − ) = 1. For example, in Fig. S4(c), w(E 1 ) = −1 and w(E 2 ) = 0. Meanwhile, in this process N poles is always invariant, thus the number of zeros inside |z| = 1 curve N zeros increases +1. Equivalently, there is one pole in Eq.(S27) that moves into |z| = 1 curve as the reference energy changes from E 1 to E 2 , as shown in Fig. S4(d). Therefore, for the rightward incident wave, there is always one pole approaches to |z| = 1 curve from inside when η → 0 + . Finally, we can obtain only the two cases of scattered wave as discussed in the main ...Similar publications
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