Electric circuits for non-Hermitian Chern insulators
EElectric circuits for non-Hermitian Chern insulators
Motohiko Ezawa
Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan
We analyze the non-Hermitian Haldane model where the spin-orbit interaction is made non-Hermitian. TheDirac mass becomes complex. We propose to realize it by an LC circuit with operational amplifiers. A topo-logical phase transition is found to occur at a critical point where the real part of the bulk spectrum is closed.The Chern number changes its value when the real part of the mass becomes zero. In the topological phase of ananoribbon, two non-Hermitian chiral edges emerge connecting well separated conduction and valence bands.The emergence of the chiral edge states is signaled by a strong enhancement in impedance. Remarkably it ispossible to observe either the left-going or right-going chiral edge by measuring the one-point impedance. Fur-thermore, it is also possible to distinguish them by the phase of the two-point impedance. Namely, the phase ofthe impedance acquires a dynamical degree of freedom in the non-Hermitian system. Introduction:
Non-Hermitian topological systems open anew field of topological physics . They are experimentallyrealized in photonic systems , microwave resonators ,wave guides , quantum walks and cavity systems . Thewinding number and the Chern number are general-ized to non-Hermitian systems. There are some properties notshared by the Hermitian topological systems . For instance,the Hall conductance and the chiral edge conductance are not quantized in the non-Hermitian Chern insulators al-though the Chern number is quantized . The typical modelof the Chern insulator is the Haldane model, but there is so farno extension to the non-Hermitian model in literatures. Fur-thermore, although there are several studies on non-HermitianChern insulators , there are so far no reports on howto realize them physically.In this paper, we propose to realize the non-HermitianHaldane model by an electric circuit. An electric circuitis described by a circuit Laplacian. Provided it is identi-fied with a tight-binding Hamiltonian , any results obtainedbased on a tight-binding Hamiltonian may find correspondingphenomena in an electric circuit. Indeed, the SSH model ,graphene , Weyl semimetal , nodal-line semimetal ,higher-order topological phases , Chern insulators ,non-Hermitian topological phases and Majorana edgestates have been simulated by electric circuits. The edgestates are observed by measuring the impedance .We study a non-Hermitian Haldane model on the honey-comb lattice, where the spin-orbit interaction is made non-Hermitian. It is constructed by an LC circuit together withoperational amplifiers as shown in Fig.1. The spin-orbit termyields complex Dirac masses at the K and K (cid:48) points. Thechiral edge states emerge in the topological phase. They haveimaginary energy. We show a topological phase transition tooccur when the real part of the bulk-energy spectrum is closed.The Chern number is determined only by the real part of theDirac mass, while the absolute value of the bulk energy doesnot close at the transition point. Non-Hermitian chiral edgesemerge in nanoribbon geometry, which acquire pure imagi-nary energy at the high symmetry points. A prominent fea-ture is that the left-going and right-going chiral edge modesare distinguished by the phase of the two-point impedance.Furthermore, it is possible to observe only one of them by ameasurement of the one-point impedance. FIG. 1: (a) Illustration of an electric circuit realizing the non-Hermitian Haldane model, which consists of capacitors C and op-erational amplifiers acting as the spin-orbit interaction. (b) Structureof an operational amplifier circuit . (c) Each node is ground by a setof capacitor, inductor and resistor ( C A , L A , R g ) or ( C B , L B , R g ). Non-Hermitian Haldane model:
We investigate a non-Hermitian Haldane model on the honeycomb lattice, wherethe Haldane interaction is non-Hermitian. As described later,it is realized by an electric circuit of the configuration illus-trated in Fig.1. The honeycomb lattice is a bipartite lattice,which consists of the A and B sites. In the basis of the A and B sites, the Hamiltonian is given by H = (cid:18) g ( k ) + U f ( k ) f ∗ ( k ) − g ( k ) − U (cid:19) , (1)with f ( k ) = t ( e − ik y / √ + 2 e ik y / √ cos k x , (2) g ( k ) = i √ λ r ( e ik x + e − i kx + √ ky + e − i kx −√ ky ) − λ l ( e − ik x + e i kx + √ ky + e i kx −√ ky )] , (3) a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l FIG. 2: Band structure of the non-Hermitian Chern insulator withzigzag nanoribbon geometry. (a1)–(c1) real part, and (a2)–(c2) imag-inary part of the band structure. (a) Topological phase ( U = 0 ), (b)critical point ( U = λ ) and (c) trivial phase ( U = 2 λ ). Color repre-sents how the wave-function localizes at the right (magenta) or left(cyan) edges. We have set λ = 0 . t and γ = 0 . t . where f ( k ) describes hopping, and g ( k ) describes the non-Hermitian Haldane interaction. It is reduced to the originalHaldane interaction for λ l = λ r . We have added one-sitestaggered potential ± U , which alternates between A and B sites. Let us define λ = (cid:0) λ l + λ r (cid:1) / and γ = (cid:0) λ l − λ r (cid:1) / ,where γ represents the non-Hermiticity.The energy is given by E = ± (cid:113) | f ( k ) | + ( g ( k ) + U ) ,which we show in Fig.2. Characteristic features read as fol-lows. First of all, it is complex in general. The real partRe[ E ] looks very similar to the energy of the original Hal-dane model with γ = 0 , while the imaginary part Im[ E ] haspeaks at the K and K (cid:48) points, i.e., at K ξ = (cid:0) ξ π , (cid:1) with ξ = ± . The positions of the K and K (cid:48) points do not shiftby the non-Hermitian Haldane interaction. We also show theenergy spectrum in the Re[ E ]-Im[ E ] plane in Fig.3(a1)–(e1),where the conduction and valence bands are separated alongthe line Re[ E ]=0 except for U = λ . It is called a line gap .We study physics near the Fermi level. We make the Taylorexpansion of (2) around the K and K (cid:48) points. Hereafter, let ususe k as the momentum measured from each of these points.We obtain f ( k + K ξ ) = ξk x − ik y for | k | (cid:28) , makingthe Taylor expansion. Hence, the low-energy physics near theFermi level is described by the Dirac theory, H ξ ( k ) = (cid:126) v F ( ξk x τ x + k y τ y ) + m ξ τ z , (4)where v F = √ (cid:126) t is the Fermi velocity, and m ξ = U − ξλ − iγ/ √ (5)is the Dirac masses at the K and K (cid:48) points. Non-Hermitian Chern number:
Since the conduction andvalence bands are separated by the line gap for U (cid:54) = λ , theChern number is well defined for U (cid:54) = λ , and characterizesthe phase even in the non-Hermitian theory.The non-Hermitian Chern number is defined by C = 12 π (cid:90) BZ F ( k ) d k, (6)where F ( k ) is the non-Hermitian Berry curvature F ( k ) = ∇ × A ( k ) with the non-Hermitian Berry connection A ( k ) = − i (cid:10) ψ L (cid:12)(cid:12) ∂ k (cid:12)(cid:12) ψ R (cid:11) . The integration of the Chern num-ber is taken over the Brillouin zone.We first summarize general results on the non-HermitianChern number in the two-band systems. We expand theHamiltonian by the Pauli matrices, H = (cid:80) i = x,y,z h i τ i , with h i being complex-valued functions. Its right and left eigen-values are given by (cid:12)(cid:12) ψ R (cid:11) = [2 E ( E − h z )] − / ( h x − ih y , E − h z ) T , (cid:10) ψ L (cid:12)(cid:12) = [2 E ( E − h z )] − / ( h x + ih y , E − h z ) , (7)which satisfy the biorthogonal condition, (cid:10) ψ L (cid:12)(cid:12) ψ R (cid:11) = 1 . Thenon-Hermitian Berry connection is calculated as A α = ( h x ∂k k α h y − h y ∂ k α h x ) / [ E ( h − E )] . (8)The non-Hermitian Berry curvature reads F ( k ) = 2 − E − / ε µυρ h µ ∂ k x h ν ∂ k y h ρ . (9)The Chern number (6) with this F ( k ) is understood as thePontryagin number or the wrapping number of h .For the K and K (cid:48) points ( ξ = ± ) we explicitly find F ξ ( k ) = − ξv m ξ + U )[( m ξ + U ) + v k ] − / . (10)As in the case of the Hermitian system , the integration of k is elongated in (6) from to ∞ since the Berry curvature hasstrong peaks at the K and K (cid:48) points. The Chern number is C ξ = ( m ξ + U ) / (cid:18) (cid:113) ( m ξ + U ) (cid:19) , (11)which is the same as in the Hermitian system although m ξ now takes a complex value. We find C ξ = − ξ/ forRe[ m ξ + U ] > and C ξ = ξ/ for Re[ m ξ + U ] < , whichare independent of the non-Hermiticity γ since γ is the imag-inary part of m ξ as in (5). As a result, the topological phasediagram is the same as in the Hermitian model. The system istopological for | λ | > | U | and trivial for | λ | < | U | . Non-Hermitian chiral edge states:
According to the bulk-edge correspondence in nanoribbon geometry, the topologicalphase is characterized by the emergence of the left-going andright-going chiral edge states which cross each other at E =0 . An intuitive reason is that the topological number cannotchange its value without gap closing. We ask how it is affectedby the non-Hermitian Haldane term.We show the energy spectra in the Re[ E ]-Im[ E ] plane inFig.3(a2)–(e2). There are two edge states connecting the va-lence and conduction bands in the topological phase. Onthe other hand, the edge states attach themselves to the samebands in the trivial phase.We may understand the structure as follows. In the Hal-dane model ( γ = 0 ), it is well known that the two chiral edgesalong the two edges cross at E = 0 , where k = π , charac-tering the topological phase ( m < λ ). We ask how the cross-ing point moves as the non-Hermiticity γ is introduced. Byperturbation theory in γ , the behavior of E edge ( k ) at k = π FIG. 3: Energy spectrum of the non-Hermitian Chern insulator in Re E -Im E plane (a1)–(e1) with bulk and (a2)–(e2) with zigzag nanoribbongeometry. (a1), (a2) and (b1), (b2) for topological phase ( C = 1 ) with U = 0 and U = 0 . λ , respectively; (c1), (c2) for critical point with U = λ ; (d1), (d2) and (e1), (e2) for trivial phase ( C = 0 ) with U = 1 . λ and U = 2 λ , respectively. Magenta (cyan) color indicates the edgestates localized at the right (left) edge, while black color indicates the bulk states. The edge states connect the conduction and valence bandsin the topological phase, while they attach themselves to the same bands in the trivial phase. Parameters λ and γ are the same as in Fig.2. is obtained analytically as E edge ( π ) = − iλγ for the A sitesand E edge ( π ) = 2 iλγ for the B sites. Consequently, the cross-ing points at k = π become pure imaginary for γ (cid:54) = 0 , as inFig.3(a2). The bulk-band structure changes suddenly at thecritical point ( m = λ ), resulting in the switching of the edgestates between the topological and trivial phases. Electric circuit realization:
The Haldane model is con-structed as in Fig.1. First, a capacitor C is placed on eachneighboring link of the honeycomb lattice. Second, we bridgea pair of the same type of sites by an operational amplifier,which acts as a negative impedance converter with current in-version and act as the spin-orbit interaction . Then, A and B sites are grounded by a set of capacitor, inductor and resistor( C A , L A , R g ) and ( C B , L B , R g ), respectively.Let us first review the operational amplifier when it op-erates in a negative feedback configuration. The current en-tering the operational amplifier from the bottom is given by I in = ( V j − V op ) /R a , while the current leaving from the topis I out = ( V j − V i ) /R c . We set an infinite impedance of theoperational amplifier, where any current cannot flow into theoperational amplifier. Then, the output current is given by I out = ( V op − V j ) /R b . As a result, we find (cid:18) I in I out (cid:19) = 1 R c (cid:18) − ν ν − (cid:19) (cid:18) V j V i (cid:19) , (12)with ν = R b /R a , where R a , R b and R c are the resistances inthe operational amplifier circuit: See Fig.1(b).With the AC voltage V ( t ) = V (0) e iωt applied, the Kirch-hoff’s current law reads I a ( ω ) = (cid:80) b J ab ( ω ) V b ( ω ) ,where I a is the current between the site a and the ground,while V a is the voltage at the site a . The circuit Laplaciancorresponding to the circuit in Fig.1(a) is given by J ( ω ) = [3 iωC + iωC A iωC B − iωL A − iωL B + 1 R g ] I − (cid:18) g J ( k ) + U J f J ( k ) f ∗ J ( k ) − g J ( k ) − U J (cid:19) . (13) with f J ( k ) = iωC (cid:18) e − iak y / √ + 2 e iak y / √ cos ak x (cid:19) − iωL ,g J ( k ) = − √ R b R a R c (cid:18) e iak x + e − ia kx + √ ky + e − ia kx −√ ky (cid:19) − R c (cid:18) e − iak x + e ia kx + √ ky + e ia kx −√ ky (cid:19) ] ,U = iω C A − C B ) − iω (cid:0) L − A − L − B (cid:1) . (14)Since J is a × matrix, we may equate it with the Hamilto-nian (1). It follows that J = iωH + 1 /R g , provided t = − C, λ r = R b / ( ωR a R c ) , λ l = 1 /ωR c ,U = −
12 ( C A − C B ) + 12 ω (cid:0) L − A − L − B (cid:1) . (15)The resistors in the operational amplifier circuit are tuned tobe ν = 1 , or R a = R b , in the literature so that the systembecomes Hermitian. However, the system is non-Hermitianfor a general value of the resistors, leading to ν (cid:54) = 1 , which isthe main theme of this paper. It is noted that the topologicalphase transition is induced solely controlling the capacitors C A , C B or the inductors L A , L B connected to the ground. Wehave also added resistors R g between nodes and the ground,whose role is shift the imaginary part of the energy. Admittance spectrum and impedance peaks:
The ad-mittance spectrum consists of the eigenvalues of the cir-cuit Laplacian (13). Indeed, it is possible to ob-serve experimentally the admittance band structure in a di-rect measurement . It corresponds to the band structure incondensed-matter physics.Two-point impedance is given by Z ab = V a − V b I ab = G aa + G bb − G ab − G ba , where G is the green function definedby the inverse of the Laplacian J , G ≡ J − . In the case of theHermitian system, it diverges at the frequency where one ofthe eigenvalues of the Laplacian is zero J n = 0 . Namely, thepeaks of impedance are well described by the zeros of the ad-mittance. On the other hand, in the case of the non-Hermitiansystem, it takes a maximum value at the frequency satisfying FIG. 4: (a1)–(c1) Admittance curves in the k - ω plane given by(17), along which the impedance peaks appear. (a2)–(c2) Realpart of the two-point impedance Z AB , which is quite similar tothe impedance peaks in (a1)–(c1). (a3)–(c3) Imaginary part of thetwo-point impedance Z AB . (a4)–(c4) Real part of the one-pointimpedance G A at the edge at the A sites. (a5)–(c5) Real part ofthe one-point impedance G B at the edge at the B sites. (a) U = 0 ,(b) U = λ and (c) U = 2 λ . The red (green) regions indicate posi-tive (negative) values of the impedance. We have set λ = 0 . t and γ = 0 . t . Re[ J n ] = 0 . After the diagonalization, the circuit Laplacianyields J n ( ω ) /iω = − (2 ω L A ) − − (2 ω L B ) − + 3 C − ε n ( ω ) + 1 /iωR g , (16)where ε n is the eigenvalue of the corresponding Hamiltonian.It is pure imaginary for the Hermitian system but becomescomplex for the non-Hermitian system. Solving Re J n ( ω ) =0 , we obtain ω R ( ε n ) = 1 / (cid:112) ( − Re [ ε n ] + 3 C ) / (1 / L A + 1 / L B ) , (17)at which the impedance has a peak. The eigenvalue ε n should be solved numerically. The impedance resonance becomesweaker when the eigenvalue ε n has an imaginary part. How-ever, by tuning the resistors R g , we can decrease the imagi-nary part to enhance the impedance peak.We consider a nanoribbon, where ω R ( ε n ) is a functionof the momentum k x . We show it as a function of k x inFig.4(a1)–(c1). The impedance has peaks at these frequen-cies. The impedance-peak structure in Fig.4(a1)–(c1) havesome feature common to the band structure in Fig.2(a1)–(c1).The chiral edge states cross the ω = 0 line at k = π when U = 0 in Fig.4(a1). The crossing point moves towards the K point as U increases and reaches it when U = λ as inFig.4(b1). It is the signal of the topological phase. It disap-pears for U > as in Fig.4(c1), indicating that the system isin the trivial phase. Thus the topological phase transition in-duced by the potential U is clearly observed in the change ofthe impedance peak.We calculate the impedance with the use of Green functionsas a function of k x and ω , and show it in Fig.4(a2)–(c5), wherethe impedance strength is represented by darkness. Note thatthe momentum dependent impedance is an experimentally de-tectable quantity by using a Fourier transformation alongthe nanoribbon direction, Z αβ ( k x , y, ω ) = (cid:88) ρ Z αβ ( x ρ , y ρ , ω ) exp [ − ix ρ k x ] , (18)where ( x ρ , y ρ ) is the Bravais vector.The impedance is pure imaginary in the Hermitian model.On the other hand, there emerges the real part of theimpedance in the non-Hermitian model, since the energy be-comes complex. It is to be emphasized that the impedanceis a complex object whose real and imaginary parts are sep-arately observable in electronics by measuring the magnitudeand the phase shift of the impedance. Let us focus on the two-point impedance Re Z AB between the two outermost edges ofa nanoribbon. It is found that the chiral edge modes becomeprominent as in Fig.4(a2) and (a3). This is because the imag-inary part of the energy is enhanced at the K and K (cid:48) points.Furthermore, it is a characteristic feature of the non-Hermitiansystem that the left-going and right-going chiral edges are dif-ferentiated by the sign of Re Z AB as in Fig.4(a2), where thesign corresponds to red or green. Equivalently, they are dis-tinguished by the phase of the impedance. This is because theimaginary part of the energy is opposite between them.The one-point impedance is defined by Z a ≡ V a /I a = G aa . It is the inverse of the resistance between the point a andthe ground. The one-point impedance Z A at the outermostedge site A is shown in Fig.4(a4)–(c4), where only the left-going chiral edge has a strong peak. It is because there is onlythe left-going chiral edge at the A edge. On the contrary, thereis a strong resonance only for the right-going chiral edge whenwe measure Z B as in Fig.4(a5)–(c5). This selective detectionof the chiral edge is possible only in electric circuits. Discussion:
Topological monolayer systems such as sil-icene, germanene and stanene provides us with rich topolog-ical phase transitions . However, their experimental realiza-tions are yet to be made. In addition, it is hard to observe theedge states of the two dimensional topological insulators byARPES since the intensity is very weak. On the other hand,almost all of these physics are realizable by employing elec-tric circuits. In particular, topological phase transitions aredetectable by measuring the change of impedance. Further-more, it is possible to observe the left-going and right-goingedge states separately by measuring the one-point impedance. The author is very grateful to N. Nagaosa for helpful discus-sions on the subject. This work is supported by the Grants-in-Aid for Scientific Research from MEXT KAKENHI (GrantsNo. JP17K05490, No. JP15H05854 and No. JP18H03676).This work is also supported by CREST, JST (JPMJCR16F1and JPMJCR1874). C. M. Bender and S. Boettcher, Phys. Rev. Lett. , 5243 (1998). C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. ,270401 (2002). S. Malzard, C. Poli, H. Schomerus, Phys. Rev. Lett. , 200402(2015). V. V. Konotop, J. Yang, and D. A. Zezyulin, Rev. Mod. Phys. ,035002 (2016). T. Rakovszky, J. K. Asboth, and A. Alberti, Phys. Rev. B ,201407(R) (2017). R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani,S. Rotter and D. N. Christodoulides, Nat. Physics , 11 (2018). B. Zhu, R. Lu and S. Chen, Phys. Rev. A , 062102 (2014). S. Yao and Z. Wang, Phys. Rev. Lett.
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