Competition of non-Hermitian skin effect and topological localization of corner states observed in circuits

We first consider a LC network posing honeycomb lattice, as shown in Fig. 1. Each unit cell includes two nodes labeled by A (red) and B (blue) with reciprocal intracell coupling $C_{3}$ and non-reciprocal intercell coupling $C_{1}\pm C_{2}$. Each node is grounded with a different combination of inductance and capacitance to guarantee the same on-site admittance. All capacitance and induction parameters in our theoretical calculations are real and correspond to the values of the real electrical components. The non-reciprocal capacitance coupling is realized by using operational amplifiers designed as current-inversion negative impedance converters (INIC) (see dashed red box in Fig. 1), thus we can switch the sign of $C_{2}$ as reversing the direction of current (see Appendix A for details).

The current and voltage in the electric circuit follow Kirchhoff’s law. Considering the monochromatic frequency response, we have

$$I_{a}(\omega)=\sum_{b}J_{ab}(\omega)V_{b}(\omega),$$

(1)

with the circuit Laplacian

$$J_{ab}(\omega)=i\omega\left[-C_{ab}+\delta_{ab}\left(\sum_{n}C_{an}-\frac{1}{\omega^{2}L_{a}}\right)\right],$$

(2)

where $a,b,n=1,2,...$ label different nodes, $I_{a}$ is the current flowing into node $a$, $V_{b}$ is the voltage on node $b$, $L_{a}$ is the grounded inductance at node $a$, and $\delta_{ab}$ represents the Kronecker delta function. The sum is taken over all nearest-neighbour nodes.

We choose the two basis vectors of the honeycomb lattice as $\mathbf{a}_{1}=\frac{\sqrt{3}}{2}\hat{x}+\frac{1}{2}\hat{y}$, $\mathbf{a}_{2}=\frac{\sqrt{3}}{2}\hat{x}-\frac{1}{2}\hat{y}$ with the unit lattice constant (see Fig. 1). We first consider the infinite circuit with PBC. Then, the circuit Laplacian in momentum space can be expressed as a $2\times 2$ matrix

$$\mathcal{J}(\omega,{\boldsymbol{k}})=-i\omega\left(\begin{array}[]{cc}h_{0}&h_{1}(\mathbf{k})\\ h_{2}(\mathbf{k})&h_{0}\\ \end{array}\right),$$

(3)

with

	$\displaystyle h_{0}=\frac{1}{\omega^{2}L}-C_{0},~{}~{}~{}C_{0}=2C_{+}+C_{3},$		(4)
	$\displaystyle h_{1}(\mathbf{k})=C_{3}+2C_{+}e^{-i\frac{\sqrt{3}}{2}k_{x}}\cos\frac{k_{y}}{2},$		(5)
	$\displaystyle h_{2}(\mathbf{k})=C_{3}+2C_{-}e^{i\frac{\sqrt{3}}{2}k_{x}}\cos\frac{k_{y}}{2},~{}~{}~{}C_{\pm}=C_{1}\pm C_{2}.$		(6)

In conventional tight-binding formalism, $h_{1(2)}(\mathbf{k})$ is usually taken as the element of a Bloch Hamiltonian and the Hermicity requires $h_{2}(\mathbf{k})=h_{1}^{*}(\mathbf{k})$. However, the non-reciprocal coupling leads to the non-Hermicity, i.e., $h_{2}(\mathbf{k})\neq h_{1}^{*}(\mathbf{k})$ in the above equations. At the resonant frequency $\omega_{0}=1/\sqrt{LC_{0}}$, the diagonal elements of circuit Laplacian vanish, i.e., $h_{0}=0$, the system then has a chiral symmetry $\sigma_{z}^{-1}\mathcal{J}\sigma_{z}=-\mathcal{J}$ with $\sigma_{z}=\text{diag}\{1,-1\}$ being the Pauli matrix. By diagonalizing Eq. (3), we obtain the following symmetric two-band spectra with respect to the zero-admittance

$$j_{\pm}(\mathbf{k})=\pm i\omega_{0}\sqrt{h_{1}(\mathbf{k})h_{2}(\mathbf{k})}.$$

(7)

The admittance spectra is purely imaginary for Hermitian system (for Hamiltonian $\mathcal{H}=i\mathcal{J}/\omega$). Due to $h_{2}(\mathbf{k})\neq h_{1}^{*}(\mathbf{k})$, the admittance spectra (7) takes complex values. However, for a finite system, the purely imaginary spectra emerge (see below), which is inconsistent with spectra (7) by assuming PBC. Next, we will address this issue.

We design a finite rhombus lattice with $N$ nodes as shown in Fig. 2 (a), with all nodes being labeled. The circuit Laplacian in real space is an $N\times N$ matrix. It reads

$$J_{N}=-i\omega\left(\begin{array}[]{ccccccc}h_{0}&C_{3}&0&C_{+}&0&0&\cdots\\ C_{3}&h_{0}&0&0&C_{-}&0&\cdots\\ 0&0&h_{0}&C_{3}&0&0&\cdots\\ C_{-}&0&C_{3}&h_{0}&0&0&\cdots\\ 0&C_{+}&0&0&h_{0}&C_{3}&\cdots\\ 0&0&0&0&C_{3}&h_{0}&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\ \end{array}\right)_{N\times N},$$

(8)

with $N=70$ in the following calculations and experiments. The eigenequation of Eq. (8) can be expressed as

$$J_{N}\psi_{n}=j_{n}\psi_{n},~{}~{}~{}n=1,2,...N,$$

(9)

where $j_{n}$ and $\psi_{n}$ are the eigen-admittances and eigenfunctions, respectively. To facilitate the analysis, we define a nonreciprocal parameter $\gamma=C_{2}/C_{1}~{}(0\leq\gamma<1)$. The reciprocal case corresponds to $\gamma=0$.

We first evaluate the admittance spectrum $j_{n}$ as a function of $1/\gamma$ at resonant frequency shown in Fig. 2(b), with fixed capacitors $C_{2}=0.2$ nF, $C_{3}=1.2$ nF, and inductance $L=75~{}\mu$H. A representative example of admittance spectrum for $C_{1}=0.47$ nF is plotted in Fig. 2(c), in which the two corner states are marked by red circles. We find that eigen-admittances for OBC are purely imaginary, which does not match the complex spectra (7) obtained by PBC.

To gain more insight, we re-write the admittance (8) as $J_{N}=-i\omega H$. The eigenequation for $H$ is $H\psi_{n}=E_{n}\psi_{n}$, which has the same eigenfunction of $J_{N}$, with the eigenvalue $E_{n}=j_{n}/(-i\omega)$. Since $j_{n}$ is purely imaginary, $E_{n}$ is purely real. Then, we take a similarity transformation to the Hamiltonian

$$S^{-1}HS=\bar{H},$$

(10)

with $S=\text{diag}\{1,1,r^{-1},r^{-1},r,r...\}$. Here, we choose $r\!=\!\sqrt{C_{+}/C_{-}}~{}(r\geq 1)$, to ensure the transformed Hamiltonian $\bar{H}$ being Hermitian (see Appendix B for details). Without loss of generality, we chosse $C_{-}>0~{}(C_{1}>C_{2})$ to garantee the Hermicity of $\bar{H}$. Since the similarity transformation does not change the eigenvalues, i.e., $\bar{H}\bar{\psi}_{n}=E_{n}\bar{\psi}_{n}$ with $\bar{\psi}_{n}=S^{-1}\psi_{n}$, $E_{n}$ is therefore real which explains the purely imaginary eigenvalue $j_{n}=-i\omega E_{n}$ for OBC. We thus call $H$ a pseudo-Hermitian Hamiltonian.

Next, by assuming the PBC, we express $\bar{H}$ in momentum space as

$$\bar{\mathcal{H}}(\mathbf{k})=\left(\begin{array}[]{cc}0&q^{*}(\mathbf{k})\\ q(\mathbf{k})&0\\ \end{array}\right),$$

(11)

with

$$q(\mathbf{k})=C_{3}+2\sqrt{C_{+}C_{-}}e^{i\frac{\sqrt{3}}{2}k_{x}}\cos\frac{k_{y}}{2}.$$

(12)

It is noted that Eq. (11) is equivalent to replacing $k_{x}$ by $k_{x}-i\frac{2}{\sqrt{3}}\ln r$ in non-Bloch Hamiltonian elements $h_{1(2)}(\mathbf{k})$ in (3). As an example, by taking the range of $k_{x}$ in $[-\frac{2\pi}{\sqrt{3}},\frac{2\pi}{\sqrt{3}}]$ [from $M^{\prime}\to\Gamma\to M$ in the original BZ, see Fig. 2 (d)], we introduce a new phase factor $\beta=\exp[i\frac{\sqrt{3}}{2}(k_{x}-i\frac{2}{\sqrt{3}}\ln r)]=re^{i\frac{\sqrt{3}}{2}k_{x}}$ shown by the red circle in Fig. 2(e), which is the GBZ. We observe a radius expansion compared to the original factor $e^{i\frac{\sqrt{3}}{2}k_{x}}$ (blue dashed unit circle). One should notice that circuit implementation does not depend on the relative orientation and distance between nodes, because the hopping strength here is independent of the real distance in the circuit, unlike the case in solids. The symmetry determines the type of circuit lattice. In this regard, the factor $\sqrt{3}/2$ of $k_{x}$ is not important for the key results.

Moreover, we plot both the eigenvalues of Eq. (3) in original BZ and those of Eq. (11) in GBZ in Fig. 2(f). We find that the eigenvalues are complex in the original BZ and purely imaginary in GBZ. The latter one thus repairs the bulk-boundary correspondence in a finite system with OBC. The concept of GBZ [29] and pseudo-Hermitian Hamiltonian develops the non-Bloch theory which paves the way to the study of non-Hermitian topology.

Non-Bloch winding number. Conventional bulk topological index evaluated from the Bloch Hamiltonian can dictate topological states in the finite system with OBC. For non-reciprocal non-Hermitian systems, one can use the transformed Bloch Hamiltonian to define topological invariants. The so-called non-Bloch winding number is given by [29]

$$W=\frac{1}{2\pi i}\oint_{q(\mathbf{k})}\frac{1}{q}dq.$$

(13)

The integral contour is along $q(\mathbf{k})$ on the complex plane. The value of $W$ depends on if the integral path encloses the singularity point at origin, as shown in Fig. 3(a). Based on the residue theorem, we obtain

	$\displaystyle W=0,~{}~{}\text{for}~{}~{}C_{3}>2\sqrt{C_{+}C_{-}}\left|\cos\frac{k_{y}}{2}\right|,$		(14)
	$\displaystyle W=1,~{}~{}\text{for}~{}~{}C_{3}<2\sqrt{C_{+}C_{-}}\left|\cos\frac{k_{y}}{2}\right|.$		(15)

The topological phase transition occurs at

$$2\left|\cos\frac{k_{y}}{2}\right|=\frac{C_{3}}{\sqrt{C_{1}^{2}-C_{2}^{2}}}.$$

(16)

As an example, we choose $k_{y}=0$, $C_{3}=1.2$ nF, and $C_{2}=0.2$ nF, so the corresponding topological phase transition happens at $C_{1}=0.62$ nF. The evolution of winding number varying with $1/\gamma$ is shown in Fig. 3(b), which is close to the phase transition point shown in the admittance structure in Fig. 2(b) (where the phase transition point approximately locates at $C_{1}=0.56$ nF ).

Non-Bloch $Z_{2}$ Berry phase. Note that both the original non-Bloch Hamiltonian in Eq. (3) and the transformed Hamiltonian Eq. (11) hold the chiral symmetry, i.e., $\sigma_{z}^{-1}\bar{\mathcal{H}}\sigma_{z}=-\bar{\mathcal{H}}$, we thus introduce the non-Bloch $Z_{2}$ Berry phase

$$\theta=\int_{l}A(\mathbf{k})\cdot d\mathbf{k},$$

(17)

where $A(\mathbf{k})=i\Phi^{\dagger}(\mathbf{k})\partial_{\mathbf{k}}\Phi(\mathbf{k})$ is the Berry connection defined though the transformed Hamiltonian (11), with the lower-band wavefunction $\Phi(\mathbf{k})$. The definition for non-Bloch $Z_{2}$ Berry phase is generally applicable to all pseudo-Hermitian system. The integral path $l$ can be chosen as $M^{\prime}\to\Gamma\to M$, i.e., $k_{y}=0,k_{x}\in[-\frac{2\pi}{\sqrt{3}},\frac{2\pi}{\sqrt{3}}]$. Because of the two-fold rotational symmetry, $\theta$ must be quantized as either $0$ or $\pi$. Figure 3(b) plots the dependence of $\theta$ as a function of $1/\gamma$ (red dots), and the phase transition point locates at $C_{1}=0.62$ nF which is identical to the result from the non-Bloch winding number.

We plot the corner states in reciprocal case $\gamma=0$ and non-reciprocal case $\gamma=0.19$ in Figs. 4(a) and 4(b), respectively. The former one shows a symmetric distribution while the latter one demonstrates that the left corner state is smeared out into the bulk due to the non-Hermitian skin effect. To explore how the corner localization competes with the skin effect, we extract the amplitude asymmetry along the central line, see Fig. 4(c). We find the wavefunction can be well described by formula $|\psi(x)/\psi(0)|=|\cos(\mu x)|\exp(-x/\lambda)$ with the fitting parameters $\mu=\{0.846,0.869,0.874,0.877\}$ and $\lambda=\{3.1,3.64,4.36,5.37\}$ for different $\gamma=\{0,0.06,0.13,0.19\}$, respectively. Here, $x$ is the coordinates value of the nodes (corresponding to nodes No. $31\sim 35$ in Fig. 2(a)) on the $x$-axis in unit of the lattice constant. The spatial oscillation (a non-zero $\mu$) weakly depends on the circuit non-reciprocity. For the reciprocal case, the topological corner state has a characteristic localization length $\lambda(0)=3.1$ (in unit of the lattice constant). From Fig. 4(d), we observe that the localization $\lambda$ grows exponentially with $\gamma$. As increasing non-reciprocity, the left corner states survives when $\gamma$ is not so large, with the fitted characteristic non-reciprocal parameter as $\gamma_{0}=1/6.3=0.16$. We therefore conclude that the non-Hermitian skin effect due to the non-reciprocal coupling tends to drag the topological corner state from the left into the bulk when $\gamma>\gamma_{0}$. The result does not change significantly with the system size (see Appendix C).

Moreover, the connection between the wave function and node voltage is given by the following formula [6, 15, 26]

$$V_{a}=Z_{a}I_{a}=I_{a}\sum_{n}\frac{\left|\psi_{n,a}\right|^{2}}{j_{n}(\omega)},$$

(18)

where $n$ runs over all 70 eigenstates, $V_{a}~{}(Z_{a})$ represents the voltage (impedance) between the node $a$ and ground. The voltage and impedance would show a resonance peak when one sweeps the frequency. In the following experiments, we measure the voltage response.