Mirror skin effect and its electric circuit simulation

We analyze impacts of crystalline symmetry on the non-Hermitian skin effects. Focusing on mirror symmetry, we propose a novel type of skin effects, a mirror skin effect, which results in significant dependence of energy spectrum on the boundary condition only for the mirror invariant line in the two-dimensional Brillouin zone. This effect arises from the topological properties characterized by a mirror winding number. We further reveal that the mirror skin effect can be observed for an electric circuit composed of negative impedance converters with current inversion where switching the boundary condition significantly changes the admittance eigenvalues only along the mirror invariant lines. Furthermore, we demonstrate that extensive localization of the eigenstates for each mirror sector result in an anomalous voltage response.

The above progresses for Hermitian and non-Hermitian systems lead us to the following issue; understanding impacts of crystalline symmetry on non-Hermitian topological properties which is crucial because a variety of non-Hermitian topological phenomena are expected as is the case for Hermitian systems. In particular, it is expected that the interplay between crystalline symmetry and non-Hermiticity may result in a novel type of skin effects. In spite of the significance of the above issue, there are few works addressing effects of crystalline symmetry on non-Hermitian skin effects.
Therefore, in this letter, we analyze effects of mirror symmetry on non-Hermitian skin effects, shedding new light on the interplay between crystalline symmetry and non-Hermitian topology. Our analysis discovers a novel type of skin effects, a mirror skin effect which results in significant dependence of energy spectrum on the boundary condition only along mirror invariant lines in the two-dimensional Brillouin zone. We also elucidate that a mirror winding number characterizes this skin effect. We verify the mirror skin effect by numerically diagonalizing a tight-binding model with the mirror winding number taking one. Furthermore, by making use of the ubiquity of the topological phenomenon, we theoretically suggest that the mirror skin effect can be observed for an electric circuit composed of negative impedance converters with current inversion (see Fig. 1). In this system, switching the boundary conditions drastically changes the impedance for the mirror invariant lines, which serves as a distinct evidence of the mirror skin effect for the electric circuit.
Theory of mirror skin effect.-Let us first elucidate that the topology protected by mirror symmetry induces a novel skin effect.
For comparison, we start with a brief review of the ordinary skin effect for symmetry class A. Consider a twodimensional system under the periodic boundary condition for the x-direction which can be regarded as a set The blue (cyan) symbols represent negative impedance converters with current inversion connecting intra-layer (inter-layer) nodes, respectively. We denotes upper (lower) layer as A (B), respectively. The black symbols denotes capacitors. Dots represent nodes. This system preserves the mirror symmetry whose mirror plane is illustrated with the vertical yellow plane.
of one-dimensional systems aligned along the x-direction in the momentum space. When the winding number ν tot (k x ) takes a finite value for the subsystem with a given momentum k x , the skin effect occurs; the energy spectrum significantly changes by switching the boundary condition for the y-direction, [i.e., the periodic boundary condition (PBC) to the open boundary condition (OBC)]. This relation can be understood with topological deformation; each subsystem for given momentum k x is topologically deformed into the Hatano-Nelson model showing the skin effect. The above fact indicates that the ordinary skin effect of class A is induced by the winding number ν tot (k x ). (For later use, we call it total winding number.) In contrast to the ordinary skin effect mentioned above, the mirror skin effect elucidated below occurs even when the total winding number is zero for arbitrary momenta k x . In the rest of this paper, we assume ν tot (k x ) = 0 unless otherwise stated. Firstly, we note that the presence of mirror symmetry results in an additional topological invariant. Consider the Hamiltonian which is invariant under applying the mirror operator M x ; where P x flips the momentum k : . U m is an unitary matrix satisfying U 2 m = 1. Along the mirror invariant line specified by k * x , the Hamiltonian can be block-diagonalized for the plus and the minus sectors of the operator M x . Thus, besides the total winding number ν tot , the following mirror winding number can be defined Here, ν ± (k * x ) denotes the winding number computed with the block-diagonalized Hamiltonian H ± (k * x , k y ) for each sector where E pg is the reference energy for the point gap. We note that the total winding number is computed with for the mirror invariant lines. The mirror winding number taking a nontrivial value results in a skin effect; in spite of ν tot = 0, the energy eigenvalues significantly depend on the boundary condition for the mirror invariant line in the Brillouin zone. We call this skin effect mirror skin effect because the mirror symmetry protects the topological properties.
In the following, we verify that the mirror winding number results in the above significant dependence by numerically analyzing a tight-binding model. The Hamiltonian reads, where ρ i (i = 1, 2, 3) are the Pauli matrices and ρ 0 is the 2 × 2 identity matrix. The above Hamiltonian preserves the mirror symmetry with M x = ρ 2 P x . Therefore, for k * x = 0 or π, the Hamiltonian can be block-diagonalized with ρ 2 . For k * x = 0 (k * x = π), the mirror winding number , while the total winding number is zero for the arbitrary value of k x .
In Fig. 2 the energy spectrum of the Hamiltonian (4) are plotted for (t, µ, ∆) = (1, 2, 1.8) at k x = 0, π/6, π/2, π. The data denoted with blue (orange) dots represent the energy eigenvalues for the PBC (OBC) along the y-direction, respectively. Figure 2(a) indicates that the energy spectrum under the PBC form a circle enclosing the origin of the complex plane which is consistent with the relation, ν M = 1 for k x = 0. Imposing the OBC along the y-direction significantly changes the spectrum; energy eigenvalues are aligned along the real axis (i.e., ImE n ∼ 0 with n = 1, 2, · · · , dimH). This striking dependence of the energy spectrum is a signal of the skin effects. Here, we note that the mirror symmetry plays an essential role; away from the mirror invariant line, the spectra obtained for the two distinct boundary conditions coincide with each other [see Figs. 2(b) and (c)].
At k x = π, the mirror symmetry is preserved which again induces the skin effect [see Fig. 2 In association with the significant change of the energy eigenvalues, the eigenvectors shows extensive localization. Figure 3 plots amplitude of the right eigenvectors | i y |Ψ nR | 2 for k x = 0, π/6. Here, |Ψ nR denotes the right eigenvector of the Hamiltonian (4) (i.e., H|Ψ nR = |Ψ nR E n with n = 1, · · · , dimH), and i y labels the sites along the y-direction. We note that the eigenstates are extended in the bulk under the PBC along y-direction (see Sec. S1 in the supplemental material [67]). Imposing the OBC for y-direction results in extensive localization of eigenstates. Figure 3 Based on the above significant difference of energy spectra and the eigenstates, we can conclude that a nontrivial value of the mirror winding number ν M results in the mirror skin effect. We furthermore propose that this mirror skin effect can be observed for an electric circuit composed of negative impedance converters (see Fig. 1), the details of which are discussed in the rest of this paper.
Mirror skin effect in an electric circuit.-Before detailed proposal for the implementation of the circuit showing the mirror skin effect, let us briefly review how an electric circuit mimics a generic tight-binding Hamiltonian. Consider an electric circuit where the voltage V a (ω) is applied at nodes a = 1, 2, · · · with angular frequency ω. In this case, based on the Kirchhoff's law, the current I b (ω) at node b is given by Thus, the admittance matrix J ba (ω) serves as a Hamiltonian for the corresponding tight-binding model, which means that topological phenomena can also be observed for electric circuits [13]. For instance, the Su-Schrieffer-Heeger model can be realized for an electric circuit composed only with capacitors and inductors. The energy conservation of the electric circuit implies the Hermiticity of the matrix J ba (ω) up to the global phase factor i. Now, let us discuss how to experimentally verify the mirror skin effect for electric circuits. In order to implement a circuit showing the mirror skin effect, we need to reproduce non-Hermitian terms of the Hamiltonian [i.e., the second and the third term of Eq. (4)], which can be accomplished by employing the negative impedance converters [68,69]. Specifically, we propose that an electric circuit shown in Fig. 1 serves as a platform of the mirror skin effect. The corresponding admittance matrix is given by in the momentum space. I α (ω, k) and V α (ω, k) (α = A, B) denotes the Fourier transformed current and voltage, respectively [70,71]. The detailed derivation of Eq. (6) is given in Sec. S2 A of the supplemental material [67]. This model preserves the mirror symmetry with M x = ρ 1 P x . In addition, its topology is characterized by the mirror winding number taking ν M (k * x ) = −1 (k * x = 0, π) for the parameter set summarized in caption of Fig. 4. We note that the relation ν tot = 0 holds for arbitrary momentum k x . We mention here that the circuit elements of the above parameters are commercially available. Numerical data elucidating the above topological properties are shown in Sec. S2 B of the supplemental material [67].
In the following, we see that the above model shows the mirror skin effect. For the electric circuit, the skin effect can be experimentally observed by measuring the admittance eigenvalues j n with n = 1, · · · , dimJ [i.e., eigenvalues of the admittance matrix (6b)]. One can access the admittance eigenvalues by the impedance measurement [J −1 ab (ω)] [69]. When the skin effect occurs, the admittance spectrum significantly depends on the boundary condition as we have seen in Fig. 2(a). Figure 4

(a)[(b)]
(a) shows the admittance spectrum for k x = 0 (k x = π/6), respectively. For the momentum invariant line k x = 0, the admittance spectrum significantly changes depending on the boundary condition [see Fig. 4(a)]; the eigenvalues form a circle for the PBC, while they form a straight line for the OBC. Away from the mirror invariant line, the mirror operation is not closed for each momentum k x , which results in the absence of the skin effect [see Fig. 4(b)].
Extensive localized states, observed in Fig. 3(a), are also observed through the voltage profile with the current feed for each mirror sector. In Fig. 4(c) [(d)], we plot the voltage profile with the current feed I in = I +L (I in = I −R ), respectively. Here, I +Liyα = iI s δ iyLy (δ αA + iδ αB ), I −Liyα = iI s δ iy0 (δ αA − iδ αB ), and I s = 0.0001A. Tuning the phase of the feed current is accomplished with the setup illustrated in Fig. 4(e). In Fig. 4(c), we observe that the voltage response becomes large around the right edge i y = L y although the current is fed at the left edge i y = 0. The essentially same result can also be observed for the other mirror sector [see Fig. 4(d)].
The above anomalous response for the feed current arises from the extensively localized states. Firstly, we note that the voltage profile for inputted current can be obtained as V a = b J −1 ab I inb with the impedance matrix J −1 ab = n R an j −1 n L † nb [69,72]. Here the matrix R (L † ) denotes the set of the right (left) eigenvectors b J ab R bn = R an j n ( a L † na J ab = j n L † nb ), respectively. a and b detote the set of the labels i y and α = A, B. We note that n L † nb I +Lb = 0 when n labels the states for the minus sector. Thus, we can see that the only eigenvectors for the plus sector contribute to the voltage response. In addition, when one of the states (n = n 0 ) for the plus sector is dominant [73], the voltage is esti- Therefore, we observe that the extensive localized states induce the anomalous voltage response for I in = I +L . The anomalous response shown in Fig. 4(d) can be understood in a similar way.
The above results indicate that the mirror skin effect can be observed for the circuit shown in Fig. 1

. Specifically, it induces significant change of admittance eigenvalues [Figs. 4(a) and(b)] and the anomalous voltage response [Figs. 4(c) and(d)] both of which can be observed in experiments.
Summary.-In this letter, we have analyzed interplay between mirror symmetry and skin effects, shedding new light on crystalline symmetry and non-Hermitian topology. Our analysis has clarified a novel type of skin effects, a mirror skin effect which results in the significant dependence both of the energy spectrum and the states on the boundary condition only along mirror invariant lines in the two-dimensional Brillouin zone. The topological characterization of this skin effect can be done with the mirror winding number. The mirror skin effect has been verified by numerically diagonalizing a tightbinding Hamiltonian with the mirror winding number taking one. Furthermore, we have proposed how to implement the electric circuit for the experimental observation of the mirror skin effect. In this system, switching the boundary condition significantly changes the admittance eigenvalues, which serves as a distinct evidence of the mirror skin effect.

Supplemental Materials:
Mirror skin effect and its electric circuit simulation S1.

AMPLITUDE OF THE RIGHT EIGENSTATES OF EQ. (4)
In Fig. 3(a), we have seen that the eigenstates |Ψ nR n = 1, 2, · · · are extensively localized around the boundary. Here, we show that the boundary condition is essential for the above extensive localization. Figure S1 plots the amplitude of the eigenstates under the PBC along the y-direction. Figure S1(a) [(b)] shows data for k x = 0 (k x = π/6), respectively. In these figures, we can see that the states are delocalized for the PBC.  We start with explaining how the negative impedance converters with current inversion induces the non-Hermitian terms. The element shown in Fig. S2 responses as [68,69] where the vectors I in I out T and V in V out T represent the current and the voltage illustrated in Fig. S2. C 1 denotes the capacitance. This can be seen as follows. We can tune the current and voltage as shown in Fig. S2. In this case, based on the Kirchhoff's law, we obtain where R a and C a represent the resistance and the capacitance, respectively. V a and I a denote the current and the voltage as illustrated in Fig. S2, respectively. ω denotes the angular frequency. Solving these equations yields Eq. (S1).
Here, we note that current I in has the opposite sign; the ordinary capacitor responses as with capacitance C 0 . The above additional sign plays an essential role in the non-Hermitian hopping. Now, let us consider the circuit illustrated in Fig. 1. Connecting intra-layer nodes by the negative impedance converters with current inversion [Eq. (S1)] yields where I αRi (V αRi ) denotes the current and the voltage of the node specified with α and R i , respectively. R i is the position vector at site i. α = A, B specifies the layer. e µ (µ = x, y) denotes the unit vector for each direction.
Applying the Fourier transformation, we obtain with Therefore, we can see that connecting intra-layer nodes by the elements defined in Eq. (S1) yields the third term of Eq. (6b). In a similar way, we can see that connecting inter-layer nodes with the elements (S1) yields the fourth term of Eq. (6b); in the real-space, we obtain which yields Concerning the other circuit elements illustrated in Fig. 1, we obtain which yields I αk = −2iωC 0 (cos k x + cos k y − 2) + (iωL 0 ) −1 V αk . (S10) Summing up the contributions [Eqs. (S5), (S8), and (S10)], we end up with the admittance matrix (6b).