Non-Hermitian topological phases and exceptional lines in topolectrical circuits
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Non-Hermitian topological phases and exceptional lines in topolectrical circuits
S M Rafi-Ul-Islam, ∗ Zhuo Bin Siu, † and Mansoor B.A. Jalil ‡ Department of Electrical and Computer Engineering, National University of Singapore, Singapore
We propose a scheme to realize various non-Hermitian topological phases in a topolectrical (TE)circuit network consisting of resistors, inductors, and capacitors. These phases are characterized bytopologically protected exceptional points and lines. The positive and negative resistive couplings R g in the circuit provide loss and gain factors which break the Hermiticity of the circuit Laplacian.By controlling R g , the exceptional lines of the circuit can be modulated, e.g., from open curvesto closed ellipses in the Brillouin zone. In practice, the topology of the exceptional lines can bedetected by the impedance spectra of the circuit. We also considered finite TE systems with openboundary conditions, the admittance spectrum of which exhibits highly tunable zero-admittancestates demarcated by boundary points (BPs). The phase diagram of the system shows topologicalphases which are characterized by the number of their BPs. The transition between different phasescan be controlled by varying the circuit parameters and tracked via impedance readout betweenthe terminal nodes. Our TE model offers an accessible and tunable means of realizing differenttopological phases in a non-Hermitian framework, and characterizing them based on their boundarypoint and exceptional line configurations. A. Introduction
There is growing interest in studying topological states in various platforms such as topological insulators , coldatoms , photonics systems , superconductors , and optical lattices due to their extraordinary properties suchas topologically protected edge states and unconventional transport characteristics . Such topological states havebeen studied in Hermitian and lossless systems , where the eigenenergies are always real. However, Hermitiansystems do not exhibit many interesting phenomena such as exceptional points , the skin effect , biorthogonalbulk polarization , and wave amplification and attenuation . In the pursuit of more exotic characteristics intopological phases, researchers have shifted attention from Hermitian to non-Hermitian systems . In contrast toHermitian systems, non-Hermitian systems in general exhibit complex eigenvalues unless the system obeys somespecific symmetries such as the PT symmetry where P and T are the parity and time reversal operations, respectively.One iconic feature of non-Hermitian systems is the existence of exceptional points, where two or more eigenvectorscoalesce and the Hamiltonian becomes nondiagonalizable. This feature leads to many novel transport phenomena, suchas unidirectional transparency , unconventional reflectivity , and super sensitivity . Generally, the exchange ofenergy or particles between lattice points and the surrounding environment induces non-Hermiticity in the system.One way to induce non-Hermiticity is to add imaginary onsite potentials at different sublattices that represent gainor loss in the system depending on the sign of the potentials. In addition, asymmetric sublattice couplings mayalso induce non-Hermiticity in the system Hamiltonian . However, realizing non-Hermitian systems in condensedmatter , acoustic metamaterials , and optical structures is in practice difficult because of the limited control oversublattice couplings, instability of the complex eigenspectra , and limitations in experimental accessibility. In pursuitof alternative platforms to overcome the aforementioned experimental limitations, topolectrical (TE) circuits have emerged as an ideal platform to not only realize non-Hermitian systems, but also to investigate many emergingphenomena such as Chern insulators , the quantum spin Hall effect , higher-order topological insulators ,topological corner modes , Klein tunneling and perfect reflection . Appropriately designed TE circuitscan emulate the topological properties of materials and offer unparalleled degrees of tunability and experimentalflexibility through the conceptual shift from conventional materials system to artificial electrical networks. Thefreedom in design and control over lattice couplings allow us to investigate electronic structures beyond the limitationsof condensed matter systems. Moreover, TE circuit networks are not constrained by the physical dimension ordistance between two lattice (nodal) points but are described solely by the mutual connectivity of the circuit nodes.Besides, the freedom of choice in the connections at each node and long-range hopping make TE systems easier tofabricate compared to real material systems. Therefore, it is also possible to design an equivalent circuit networkin lower dimensions that resembles the characteristics of higher-dimensional circuits . Unlike real material systems,an infinite real system can be mimicked by finite-sized circuit networks in TE circuits. Therefore, TE networksenable us to design a non-Hemitian system in a RLC circuit network with better measurement accessibility as allthe characteristic variables such as the admittance bandstructure and the density of states can be evaluated throughelectrical measurements (e.g. impedance, voltage, and current readings). In this paper, we investigate the exceptionallines, i.e. loci of exceptional points (EP), in a non-Hermitian TE system consisting of electrical components such asresistors, inductors and capacitors as a function of the non-Hermitian parameter (i.e. resistance). We show that byintroducing non-Hermiticity to the circuit Laplacian through the insertion of positive or negative resistances betweenthe voltage nodes and the ground to induce imaginary onsite potentials in the lattice sites and tuning the non-Hermitian parameters appropriately through using by using appropriate resistance values, the loci of the EPs may beeasily switched to take the form of either a line or closed curves such as ellipses in the Brillouin zone. We show that aunique property of the TE platform over earlier works , viz. the impedance spectrum, is that the exceptional linescan be detected and easily tracked by measuring the impedance spectrum of the circuit. We further investigate finitesystems with open boundary conditions along one direction, and find that these finite systems possess a rich phasediagram with different phases possessing up to four pairs of BPs, depending on the circuit parameters. We also showthat edge states in Hermitian or pure LC circuits, become hybridized with the bulk modes in non-Hermitian RLCcircuits. In summary, our TE model provides an experimentally accessible means to investigate the phase diagramand various topological phases of non-Hermitian systems. B. Theoretical Model
We consider a two-dimensional TE circuit, shown in Fig. 1, which is composed of capacitors, inductors, positiveresistors (loss elements) and negative resistors (gain elements). The circuit has a unit cell consisting of two sublatticenodes A and B . Each node is connected by a capacitor of ± C and C y to its nearest neighbour in the x and y -direction,respectively, and a parallel combination of a common capacitor C and inductor L to the ground. The inductance L canbe varied to modulate the resonance condition. To explore the effects of loss and gain in the TE network, we consideran imaginary onsite potential iR g on the A nodes and − iR g on the B nodes. This onsite potential can be realizedby connecting resistors of resistance r a ( − r a ) between each A ( B ) node and the ground. The onsite potential arerelated to the resistors by R g = 1 /ωr a , where ω denotes the frequency of the driving alternating current. The negativeimaginary onsite potential at the B -type nodes can be obtained by using op-amp-based negative resistance converterswith current inversion (INRC) (see Supplemental section 1 for details) Alternatively, similar gain and loss terms canobtained in our TE model via using only two unequal positive resistances connected to ground for each type of nodesinstead of using negative resistance converters, which yields the mathematically similar Hamiltonain as Eq. 1 exceptfor a global shift of the imaginary part of all the admittance eigenenergies (see Supplementary Note 1 for details).The advantage of using grounding resistors with different values of (positive) resistance is the dynamic stability ofthe circuit because the absence of op-amps avoids the possibility that the voltage profiles may get over-amplified bythe negative resistance. The Laplacian of the TE circuit over the x - y plane can be expressed as C y - C C r a - r a R R xy (c)(a)(b) LC = r a + - v + v - - V dd + V dd A LC = - r a BP P QQ
Figure 1. a) Schematic of the two-dimensional TE lattice in the x - y plane. The blue and magenta circles represent sublattices A and B respectively. The alternating sublattice sites A and B are connected to each other in the x -direction by an inductor − C and capacitor C for the intracell and intercell connections, respectively (the dashed rectangle delineates a unit cell). Alongthe y -direction, nodes of different sublattices are connected diagonally by a capacitor C y . b) Schematic circuit of a negativeresistance converter, which introduces a π -phase difference and therefore converts a loss resistive term r a to a gain term − r a .The combination of two resistors having the same resistance R along with an ideal operational amplifier with supply voltages+ V dd and − V dd results in current inversion and hence acts as a negative resistance converter. c) Grounding mechanism ofthe TE circuit. All nodes are connected to ground via a parallel combination of a common capacitor ( C ) and inductor ( L ).Furthermore, in parallel to these, a positive resistor r a (loss term) and negative resistor − r a (gain term) is connected to groundfrom the A and B nodes, respectively. The negative resistor − r a is implemented by means of INRC depicted in (b). H D ( k x , k y ) = − ( C (1 + cos k x ) + 2 C y cos k y ) σ x − C sin k x σ y + iR g σ z , (1)where the σ i s denote the Pauli matrices in the sublattice space. In the absence of resistances, Eq. 1 exhibitsboth the chiral symmetry, i.e., C H D ( k x , k y ) C − = − H D ( k x , k y ) and inversion symmetry, i.e., I n H D ( k x , k y ) I − n = H D ( − k x , − k y ), where the chirality and inversion operators are C = σ z and I n = σ x , respectively. However, a finite R g would break the inversion symmetry of the Laplacian in Eq. 1 although the parity time ( PT ) symmetry as definedby PT H D ( k x , k y ) T − P − = H ∗ D ( k x , k y ), would still be preserved (here, P = σ x is the parity operator and T iscomplex conjugation). Therefore, the Laplacian in Eq. 1 can still have real eigenvalues despite its non-Hermiticity .More specifically, a non-zero R g in Eq. 1 corresponds to the insertion of alternating iR g and − iR g terms on thediagonal of the Laplacian, which preserves the commutation with the PT operator . In contrast to the Hermitiancase, PT symmetry in the non-Hermitian TE system eigenmodes can be broken depending on the model parameters,i.e., the eigenmodes of Eq. 1 are not necessarily the eigenstates of the PT operator even when the Laplacian itselfrespects PT symmetry. In this situation, complex admittance spectra emerge with exceptional points where both thehole- and particle-like admittance bands coalesce. Therefore, the complex admittance dispersion for the circuit modeltake the form of E D ( k x , k y ) = q C (1 + cos k x ) + 4 C y cos k y + 4 C C y cos k y (1 + cos k x ) − R g , (2)where the ± refers to the two admittance bands, respectively. By tuning the circuit parameters, we can obtaindifferent numbers of real solutions for ~k for E D = 0 in Eq. (2), which translates into different number of exceptionalpoints in the Brillouin zone (BZ). The exceptional points occur at w ex = ( k x , k y ) = ( π, ± arccos( R g / C y )) and w ex = ( π, ∓ π ± arccos( R g / C y )). To illustrate the effect of non-Hermitian gain or loss, we plot the admittance (a) (b) (c) ky ky ky ky ky ky I m ( E D ) [ m Ω - ] I m ( E D ) [ m Ω - ] R e ( E D ) [ m Ω - ] R e ( E D ) [ m Ω - ] A b s ( E D ) [ m Ω - ] A b s ( E D ) [ m Ω - ] ky A b s ( E D ) [ m Ω - ] ky R e ( E D ) [ m Ω - ] ky I m ( E D ) [ m Ω - ] Figure 2. Absolute value, real part, and imaginary part of the complex admittance as a function of k y with the parameters C = 0 .
78 mF, C y = 0 .
39 mF, and k x = π . We consider three representative values of the grounding resistance, i.e., (a) R g = 0mΩ − Hz − , (b) R g = 0 . − Hz − , and (c) R g = 0 .
78 mΩ − Hz − . Note that case (c) corresponds to the critical value ofthe non-Hermitian parameter R g = 2 C y = 0 .
78 mΩ − Hz − , beyond which the admittance spectrum becomes purely imaginary.All the exceptional points are represented by open red circles. spectrum as a function of wavevector k y and fix k x = π for three representative values of R g . In the absence of gainor loss (i.e., R g = 0), the admittance spectrum becomes purely real with two Dirac points or exceptional points (seeFig. 2a). Because the Laplacian in Eq. 1 obeys chiral symmetry, for a given k y , the admittance eigenvalues alwayscome in pairs with equal magnitude but opposite signs. For non-zero values of R g , the admittance dispersion becomescomplex and the band-touching degeneracy points split into pairs of band-touching exceptional points. For instance,in Fig. 2b, the splitting of the degeneracy points is most evident in the admittance plots in the left-most and middlecolumns. Here, the degeneracy point with zero admittance at k y ≈ − . k y ≈ − . k y ≈ − . k y ) with PT -symmetrical eigenmodes or purely imaginary (in the complementary range of k y ), in which case the eigenmodesbreak the PT symmetry. The boundaries between the real and imaginary admittances are defined by the EPs, whereall the eigenmodes coalesce at the eigenvalue of zero. Two of the four EPs are located at k y = ∓ π ± arccos( R g / C y )while the other two are at k y = ± arccos( R g / C y ). As the magnitude of R g increases, the range of k y corresponding tothe real (imaginary) part of the admittance spectrum shrinks (expands). At some critical value given by R c = 2 C y , thewhole spectrum becomes purely imaginary with a thre EPs (see Fig. 2c). When R g exceeds R c , the two admittancebands will become gapped and no EP exists in the Brillouin zone (not shown in Fig. 2). In this case, the admittanceeigenmodes break the PT symmetry for the entire range of k y in the BZ. As can be seen from Fig. 2c, the real part ofthe admittance spectrum vanishes at the critical resistance R c . In summary, we can obtain a variable number of EPsdepending on the R g parameter, i.e., two, four, three, and zero EPs for R g = 0, R g < C y , R g = 2 C y , and R g > C y ,respectively.To obtain the exceptional lines (the loci of the exceptional points), we use the equation for the degeneracy pointsof the admittance spectrum: ( C (1 + cos k x ) + 2 C y cos k y ) + ( C sin k x ) = R g . (3)Eq. 3 governs the loci of the exceptional points in the k x - k y plane. A finite R g will transform a single pair of band-touching points into exceptional or nodal lines on the k z = 0 plane characterized by Eq. 3. On these exceptional lines,both the real and imaginary parts of the eigenvalues vanish. The top panels of Fig. 3a and b show the exceptionallines for the parameter sets in Fig. 2b and c , respectively. The exceptional points shown in Fig. 2b and c thencorrespond to the k y cross sections of the exceptional lines in Fig. 3 at k x = π . In addition to the above analysis of (a) (b) -2 0 2-3-2-10123 -2 0 2-3-2-10123 k y k y k y k y kx kx kx kx log [ Ê ] -2 0 2 -3-2-10123 -3-2-101234 -2 0 2-3-2-10123 -3-2-10123 log [ Ê ] Figure 3. The top panels show the loci of the exceptional points in 3D non-Hermitian systems, i.e., systems with finite resistivecouplings, at C = 0 .
78 mF, C y = 0 .
39 mF, k z = π/
2, and a. R g = 0 . R g = 0 .
78 mF. The lower panels show thecorresponding impedance spectra for the corresponding three-dimensional systems with the same values of C , C y and R g asthe two-dimensional systems. (Note that the quantities plotted are the base 10 logarithms of the impedances in Ohms.) the exceptional lines based on the admittance spectrum, an alternative visual representation of the exceptional linescan also be obtained from the impedance spectrum of the TE circuit. In general, the impedance between any twoarbitrary nodes p and q in the circuit can be measured by connecting an external current source providing a fixedcurrent I pq to the two nodes and measuring the resulting voltages at the two nodes V p and V q . The impedance in thecircuit is then given by Z pq = V p − V q I pq = X i | ψ i,p − ψ i,q | λ i , (4)where ψ i,a and λ i are the voltage at node a and the eigenvalue of the i th eigenmode of the (finite-width) Laplacian.One of the key characteristics of Eq. 4 is that the impedance diverges (increases to a large value) in the vicinity ofthe zero-admittance modes ( λ i = 0 ) for non-zero eigen-mode voltages ψ i,p and ψ i,q . Therefore, the locus of the highimpedance readout can be used to mark out the exceptional lines in momentum space. The lower panels of Fig. 3aand b depict the corresponding impedance spectra for the parameter sets in Fig. 2b and c respectively. The plottedimpedance is that across the two nodes in a unit cell (i.e. with nodes p and q chosen to be terminal points at eitherend of the circuit). (For the case of C = 2 C y = R g plotted in Fig. 3b, the k y = ± π lines are also exceptional lines.)For both resistive values, the locus of high impedance readouts coincides exactly with the exceptional lines in theadmittance spectrum (compare upper and lower panels of Fig. 3). This suggests the possible electrical detection ofexceptional lines in the TE system via impedance measurements. C. Zero-admittance states in finite system
To gain further insight into the zero-admittance states of a non-Hermitian system, we will study a 2D dissipativeTE system described by Eq. 1 , which is finite along the x -direction, i.e., having open boundary conditions alongthat direction, but is infinite in the y -direction. Before investigating the properties of the finite system, we will firstexplain some properties of the infinite -sized 2D system. For this system, Kirchoff’s current law at the A and B nodesat resonance can be written as − EV Ax,y = − C V Bx,y + C V Bx − ,y − C y ( V Bx,y +1 + V Bx,y − ) + iR g V Ax,y , (5)and − EV Bx,y = − C V Ax,y + C V Ax +1 ,y − C y ( V Ax,y +1 + V Ax,y − ) − iR g V Bx,y . (6)By substituting the ansatz V x,y = λe ik x + ik y in Eqs. 5 and 6, we obtain( E − iR g σ z ) | λ i = ( C (1 + χ x ( σ x − iσ y ) + χ − x ( σ x + iσ y )) + 2 C cos k y σ x ) | λ i (7)where χ x = e ik x and λ = ( λ A , λ B ) T . For a given E and k y , χ x can be solved from Eq. 7 as χ x = − ( t + C − p ) ± √ ∆ tC , (8)where t = C + 2 C y cos k y and p = E + R g , and∆ ≡ ( t + C − p ) − (2 tC ) . (9)When ∆ < χ x ;(∆ < = − ( t + C − p ) ± i p | ∆ | tC . (10)It can be readily seen that | χ x ;(∆ < | = 1, and because χ x ≡ exp( ik x ), this indicates that the corresponding valuesof k x would be real when ∆ <
0. In this case, k x = arg( χ x ;(∆ < ) = ± arctan( p ( − ∆ ) / ( t + C − p )). On theother hand, when ∆ > χ x ;(∆ > is real, and, in general, | χ x ;(∆ > | 6 = 1. This indicates that the correspondingvalues of k x would be imaginary. At the boundary between the two cases, we have∆ = 0 ⇒ ( t + C − p ) = (2 tC ) ⇒ χ x ;(∆ =0) = − p (2 tC ) tC = − sign(2 tC ) , so that the corresponding k x = 0 , π depending on the sign of 2 tC . When ∆ <
0, the ± i p | ∆ | terms in Eq. 10result in a finite separation along the k x axis between two points on the same equal admittance contours (EACs) fora given k y value (see Fig. 4). The two values of k x meet when ∆ = 0. Figure 4a depicts the case where χ x = +1when ∆ = 0. Here, for a given k y , k x on the EAC becomes single-valued at k x = 0. Figure 4b depicts the case where χ x = − = 0, and here k x becomes single-valued at k x = π . For both cases, the values of k y where ∆ = 0mark the boundaries for the existence of real k x . We now consider the nanoribbon geometry with a finite width alongthe x -direction. We show in the Supplementary Materials that for the nanoribbon geometry with non-zero | R g | , thezero-admittance exceptional points do not occur within the bulk energy gaps, but in the bulk bands where real values -2 0 2-3-2-10123 -2 0 2 kx kx k y Figure 4. The equal admittance contours (EACs) at E = 0 for (left) with parameters C = 0 .
78 mF, C y = 0 . R g = 1 . − Hz − , and (right) with parameters C = 0 .
39 mF, C y = 0 .
78 mF, R g = 0 . − Hz − . The regions in theBrillouin zone where ∆ > < of k x exist for E = 0 in the infinite bulk system. Due to the finite width of the nanoribbons, these zero-admittancepoints occur as quantized bulk states. The points on the k y axis that mark the threshold for the existence of the zeroadmittance states for the nanoribbon system coincide with the k y values where the solutions of χ x at E = 0 in Eq. 8are equal, i.e., when ∆ = 0. These values of k y would mark the ‘boundary points’ (BPs) of the system. The valuesof k y corresponding to the BPs are given by the solutions of the following equation: C + 2 C y cos k y + ηC = ζR g , (11)where η = ± ζ = ± η and ζ and the two possible signs of k y in Eq. (11) therefore provide up to eight real solutions for k y . The BPs come in pairs, and hence, depending on thechoice of the coupling capacitances C and C y and resistance R g , we can obtain up to a maximum of four pairs of BPs(see later discussions on phase diagram). Note that the BPs set the boundaries for the existence of exceptional points.In a nanoribbon, the exceptional points do not necessarily appear exactly at the BPs, but in between alternating pairsof BPs. To illustrate the role of the capacitive and resistive coupling parameters in determining the number of BPs,we plot the admittance spectra for finite TE circuits with N = 20 unit cells along the x -direction in Fig. 5. In theHermitian limit (i.e. R g = 0), the admittance spectrum is purely real, and each pair of quantized bands is symmetricabout the E = 0 axis. The Hermitian TE circuit can host two or four BPs depending upon the relative strength ofthe capacitive couplings. If C > C y , we only have a pair of BPs occurring at k y = ± π/
2. However, an additional pairof BPs emerges at k y = arccos( ± C /C y ) in the spectrum if C < C y (which is the case illustrated in Fig. 5a). TheHermitian system possesses chiral and time-reversal symmetries and belongs to to the BDI Altland-Zimbauer class .Treating k y as a parameter to an effectively one-dimensional model, the effective 1D model along the x direction ismathematically identical to the SSH model, for which the topological invariant is the winding number. The band-touching points B1 to B4 will then correspond to the values of k y at which the bulk band gap closes and the systemtransits between topologically trivial phases and topologically non-trivial phases. The emergence of the non-trivialstates in Fig. 5a) span over ( − π/ − arccos( C /C y )) and ( π/ C /C y )) in the k y axis. (Although thebands appear to be flat at the scale of the figure, they actually disperse weakly and touch only at isolated points.) Thetransition points between non-trivial (with edge states) and trivial regions (without edge states) are marked by theonset of high impedance states (shown in the rightmost plot of Fig. 5a). A finite non-Hermitian gain or loss term R g results in four different types of admittance regions for a given value of k y . These admittance regions can host purelyreal, purely imaginary, complex spectra with exceptional points, and complex admittances without E D = 0 states,which are labelled as I, III, IIA and IIB respectively, as shown in Figs. 5b and 5c. (For instance, the states betweenB2 and B3, and B6 and B7 in Fig. 5c have purely imaginary admittances.) Eigenstates with preserved and broken PT symmetry are present in regions I and III. Additionally, region IIA hosts exceptional points while IIB does not (seeFigs. 5b and 5c ). The difference between regions IIA and IIB can be explained via the existence and non-existenceof real solutions of χ x in Eq. 8. The transition points between any two successive regions are marked by the BPs.Moreover, the zero-admittance states in region IIA and the E D = 0 states in the other regions are characterizedby high and low impedances respectively (see impedance plots shown in the right-most column of Fig. 5b and c ).For illustration, the EPs in the region IIA are marked by crosses in Fig. 5b in both the impedance and admittanceplots. The impedance readout switches between the high and low impedance states occur at the BPs, which mark theboundaries between the IIA and other regions. Fig. 6a shows the phase diagram of the 2D non-Hermitian TE modelas a function of the resistive coupling R g and the capacitive coupling C . The different phases are characterized by OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
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I IIIA IIA IIA IIA IIA IIA IIAIIAIIAIIB IIB IIB IIB IIBIIBIIA IIA IIA IIA IIAIII III III IIII I I I I II non-trivial ** * **** *** * * * *** * ********* ky ky ky ky ky ky ky ky ky R e ( E D ) [ m Ω - ] R e ( E D ) [ m Ω - ] R e ( E D ) [ m Ω - ] I m ( E D ) [ m Ω - ] I m ( E D ) [ m Ω - ] I m ( E D ) [ m Ω - ] Z p q ( Ω ) Z p q ( Ω ) Z p q ( Ω ) Figure 5. Evolution of edge states and boundary points (BP) which mark the transition between non-zero and zero admittancesurface states in a non-Hermitian TE circuit, as the resistive and capacitive couplings are varied. We consider a finite TEcircuit with N = 20 unit cells along the x -direction. The first and second columns depict the real and imaginary parts of theadmittance spectra, while the third column plots the corresponding spectra for the impedance taken between two terminalpoints. Parameters used for Panel a: C = 0 .
39 mF, C y = 0 .
78 mF, and R g = 0 mF; Panel b: C = 0 .
78 mF, C y = 0 .
39 mF,and R g = 0 . C = 0 .
39 mF, C y = 0 .
78 mF, and R g = 0 . k y regions between the BPs are denoted as I, III, IIA and IIBregions for ease of reference in the text. (a) (b)(c) Rg C -6-4-20246 R e ( E D ) [ m Ω - ] -2-4-3-2 0 1 2 3 -2-4 2 4 00 2-4-2024 R e ( E D ) [ m Ω - ] k y x | V | | V | Figure 6. a) Phase diagram of a 2D non-Hermitian TE model showing the effect of the capacitive coupling C and the gain/lossparameter R g on the number of BPs, and hence the number of zero-admittance regions in the admittance spectrum. Weconsider a finite number of unit cells along x -direction, and set C y = 1 . N x = 20unit cell-wide nanoribbons at b. C = 2 mF and R g = 0 mF and c. C = 1 mF and R g = 1 . k y value, as indicated bythe circles in the dispersion relations. different numbers of BPs and hence different numbers of zero-admittance regions. These phases are separated fromone another by phase boundaries that are defined by the existence and number of real solutions in Eq. 11. Note thatBPs always occur and annihilate in pairs and they connect two admittance bands together. To study the effect ofnon-Hermitian term R g on the zero-admittance states, we consider a 2D TE system with open-boundary conditionsin the x -direction. In the Hermitian condition, i.e., R g = 0, the system hosts zero-energy modes in which the squareof the voltage amplitudes are localized at its edges, as shown in top panel of Fig. 6b. This corresponds to the usualnon-trivial SSH edge state. However, in the non-Hermitian case, i.e., with the introduction of a finite R g , we wouldinstead obtain bulk states at E = 0 rather than edge states (see Fig. 6c). This is indicated by the square of thevoltage amplitude being no longer localized at an edge. D. Conclusion
In conclusion, we proposed a topolectrical (TE) circuit model with resistive elements which provide loss and gainfactors that break the Hermiticity of the circuit to model and realize various non-Hermitian topological phases. Byvarying the resistive elements, the loci of the exceptional points (or exceptional lines) of the circuit can be modulated.IWe showed that the topology of the exceptional lines in the Brillouin zone can be traced by the impedance spectra ofthe circuit. Additionally, we studied a finite TE system where open boundary conditions apply in one of the dimensions.In these finite circuits, we demonstrated the tunability of both the number of exceptional points corresponding to zero-admittance states, as well as that of boundary points (BPs) which delineate the circuit parameter range where theseexceptional points exist. The regions separated by the BPs are characterized by high and low values of impedancediffering by several orders of magnitude, which are detectable in a practical circuit. We also derived a phase diagram ofthe finite TE system which delineates different between topological phases that are characterized by different numberof BP pairs (up to a maximum of four). The edge state character of zero-admittance states of Hermitian LC circuitsare transformed into exceptional points which are hybridized with the bulk modes in the non-Hermitian RLC circuits.In summary, we have proposed a tunable electrical framework consisting of RLC circuit networks as a means to realizedifferent topological phases of non-Hermitian systems, and characterize them based on their impedance output, aswell as their BP and exceptional line configurations.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corre-sponding author upon reasonable request.
Code availability
The computer codes used in the current study are accessible from the corresponding author upon reasonable request.
Acknowledgement
This work is supported by the Ministry of Education (MOE) Tier-II grant MOE2018-T2-2-117 (NUS Grant Nos.R-263-000-E45-112/R-398-000-092-112), MOE Tier-I FRC grant (NUS Grant No. R-263-000-D66-114), and otherMOE grants (NUS Grant Nos. C-261-000-207-532, and C-261-000-777-532).
Author contributions
S.M.R-U-I, Z.B.S and M.B.A.J initiated the primary idea. S.M.R-U-I and Z.B.S contributed to formulating theanalytical model and developing the code, under the kind supervision of M.B.A.J. All the authors contributed to thedata analysis and the writing of the manuscript.0
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