Exact solution of non-Hermitian systems with generalized boundary conditions: size-dependent boundary effect and fragility of skin effect
Cui-Xian Guo, Chun-Hui Liu, Xiao-Ming Zhao, Yanxia Liu, Shu Chen
EExact solution of non-Hermitian systems with generalized boundary conditions:size-dependent boundary effect and fragility of skin effect
Cui-Xian Guo, Chun-Hui Liu,
1, 2
Xiao-Ming Zhao, Yanxia Liu, and Shu Chen
1, 2, 3, ∗ Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China
Systems with non-Hermitian skin effects are very sensitive to the imposed boundary conditionsand lattice size, and thus an important question is whether non-Hermitian skin effects can survivewhen deviating from the open boundary condition. To unveil the origin of boundary sensitivity,we present exact solutions for one-dimensional non-Hermitian models with generalized boundaryconditions and study rigorously the interplay effect of lattice size and boundary terms. Besides theopen boundary condition, we identify the existence of non-Hermitian skin effect when one of theboundary hopping terms vanishes. Apart from this critical line on the boundary parameter space, wefind that the skin effect is fragile under any tiny boundary perturbation in the thermodynamic limit,although it can survive in a finite size system. Moreover, we demonstrate that the non-HermitianSu-Schreieffer-Heeger model exhibits a new phase diagram in the boundary critical line, which isdifferent from either open or periodical boundary case.
Introduction.-
It is well known that the spectrumof a periodic crystal can be characterized by the Blochwave vector and the periodic boundary condition (PBC)is usually taken for the convenience of calculating theband structure [1]. If the system size is large enough,the bulk spectrum is stable against boundary perturba-tions even though the translation invariance of the sys-tem is broken [2–4]. This constitutes the foundation forunderstanding why the bulk energy levels of a large sys-tem with open boundary condition (OBC) can be repro-duced from the Bloch band calculation. However, sucha paradigm is challenged in some non-Hermitian systems[5–9], for which the wave functions in large systems withOBC accumulate on the boundary accompanying with aremarkably different eigenvalue spectrum from the peri-odic system [10–16]. This phenomenon is coined as thenon-Hermitian skin effect (NHSE) [9] and recently at-tracted intensive studies [17–33].The NHSE suggests that the change of boundary con-dition may induce dramatic change of bulk propertiesof non-Hermitian systems [9–16, 34–40]. Size-dependentNHSEs are also observed in some coupled non-Hermitianchains [41, 42] and non-reciprocal chains with impurity[43]. These studies indicate that both boundaries and lat-tice size play an important role in these boundary sen-sitive effects. Although the spectral flow from PBC toOBC is studied by introducing an imaginary flux [13, 44],it is still elusive to get a quantitative understanding ofthe sharp change of spectrum and wave functions of skinmodes under tiny boundary perturbations. A more chal-lenging task is to count quantitatively the interplay effectof system size and boundary perturbations and unveil theintrinsic reason behind the boundary sensitive effects. Asnumerical methods for boundary sensitive problems are ∗ Corresponding author: [email protected] time consuming and sometimes unreliable due to the exis-tence numerical errors and calculation precision [45, 46],exact solutions are highly desirable for analytically ex-ploring the size-dependent boundary effect.In this letter, we present exact solutions of non-Hermitian models with non-reciprocal hopping undergeneralized boundary conditions, which enable us to ex-plore rigorously the interplay effect of lattice size andboundary perturbations. Our analytical results show ex-plicitly how the lattice size and boundary terms affect thesolutions of eigen equations. Particularly, we find the ex-istence of NHSE in a critical line on the boundary param-eter space, including the OBC as a special case. Apartfrom the critical line, the NHSE is unstable against anytiny boundary perturbations in the thermodynamic limitand thus is fragile, although it may survive in a finite sizesystem. Moreover, we find that the two-band system canexhibit a new phase diagram in the critical line, whichis different from either PBC or OBC case, but is a com-bination of the two cases. Our work demonstrates novelphenomena induced by the boundary terms from the per-spective of exact solution and provides a firm ground forunderstanding boundary sensitivity phenomena in non-Hermitian systems.
Hatano-Nelson model with generalized bound-ary conditions.-
We start with the Hatano-Nelson (HN)model [47, 48] with generalized boundary condition de-scribed by H = N − (cid:88) n =1 (cid:104) t L c † n c n +1 + t R c † n +1 c n (cid:105) + δ R c † c N + δ L c † N c (1)where N is the number of lattice sites, δ L , δ R ∈ R deter-mines the generalized boundary conditions, and t L , t R ∈ R are imbalanced hopping amplitudes which can be pa-rameterized as t L = te − g and t R = te g with real t and g .This is the minimal model which can display nontrivialsize-dependent boundary effect. a r X i v : . [ qu a n t - ph ] F e b FIG. 1: (A-C) IPR on the parameter space of δ L /t L and δ R /t R for HN model with N = 10 , ,
80, respectively. (a-f) Energyspectrum (red circles and dots) corresponding dots ’a-f’ in (A-C), respectively. The analytical results (red circles) are in exactagreement with the numerical results (red dots). The green and blue line represents energy spectrum corresponding OBC caseand PBC case in the thermodynamic limit, respectively. Common parameters: t L = 1 , t R = 0 . The corresponding eigenvalue equation can be writtenas H | Ψ (cid:105) = E | Ψ (cid:105) , where | Ψ (cid:105) = ( ψ , ψ , · · · , ψ N ) T . Theabove eigenvalue equation consists of a series of equa-tions, including bulk equations as follows t R ψ s − Eψ s +1 + t L ψ s +2 = 0 (2)with s = 1 , , · · · , N −
2, and the boundary equationsgiven by − Eψ + t L ψ + δ R ψ N = 0 and δ L ψ + t R ψ N − − Eψ N = 0. By comparing the above two equations withEq.(2), they are equivalent to the following boundaryconditions t R ψ = δ R ψ N , (3) δ L ψ = t L ψ N +1 . (4)Due to spatial translational property from bulk equa-tions, we set the ansatz of wave function Ψ i which satis-fies the bulk equations Eq.(2) as followsΨ i = ( z i , z i , z i , · · · , z N − i , z Ni ) T . (5)By inserting Eq.(5) into the bulk equation Eq.(2), weobtain the expression of eigenvalue in terms of z i : E = t R z i + t L z i . (6)For a given E , there are two solutions z i ( z , z ), and thusthey should fulfill the following constraint condition: z z = t R t L . (7)Therefore, the superposition of two linearly indepen-dent solutions is also the solution of Eq.(2) correspond-ing the same eigenvalue, i.e., Ψ = c Ψ + c Ψ = ( ψ , ψ , · · · , ψ N ) T , where ψ n = (cid:80) i =1 ( c i z ni ) = c z n + c z n with n = 1 , , · · · , N .To solve the eigen equation, the general ansatz ofwave function should satisfy the boundary conditions.By inserting the expression of Ψ into Eqs.(3) and (4),the boundary equations transforms into H B ( c , c ) T = 0with H B = (cid:18) t R − δ R z N t R − δ R z N z (cid:0) δ L − t L z N (cid:1) z (cid:0) δ L − t L z N (cid:1) (cid:19) . The condition for the existence of nontrivial solutions for( c , c ) is determined by det[ H B ] = 0, which gives rise tothe general solution:( z N +11 − z N +12 ) − δ R δ L t L ( z N − − z N − ) − (cid:34) δ L t L + δ R t R (cid:18) t R t L (cid:19) N (cid:35) ( z − z ) = 0 . (8)Eq.(8) and Eq.(7) together determine the solution of z and z exactly. The solutions of z and z give the finite-size generalized Brillouin zone , which may be different fordifferent lattice size. According to the constraint condi-tion of Eq.(7), we can always set the solution as z = re iθ , z = re − iθ (9)with r = (cid:113) t R t L = e g , then Eq.(8) becomessin[( N + 1) θ ] − η sin[( N − θ ] − η sin[ θ ] = 0 , (10)where η = δ R δ L t R t L and η = δ L t L r − N + δ R t R r N = δ L t L e − gN + δ R t R e gN . The corresponding eigenvalue is given by E = 2 √ t R t L cos θ. (11)The solutions θ of Eq.(10) may take real or complex de-pending on the values of η and η . In the presence ofboth nonzero boundary terms, i.e., with fixed δ L,R (cid:54) = 0, η always increases exponentially with N [49], and thusthe solutions are very sensitive to even a tiny bound-ary perturbation when N is large enough. Such a size-enhancing boundary sensitivity has no correspondence inthe Hermitian limit with r = 1 ( g = 0).The OBC corresponds to the special case with δ R = δ L = 0, for which we have η = η = 0 and Eq.(10) has Nreal solutions given by θ = mπN +1 ( m = 1 , , · · · , N ), whichexcludes θ = 0 , π due to z (cid:54) = z . The correspondingeigenvalues are real, and the eigenstates are given byΨ = (cid:0) r sin[ θ ] , r sin[2 θ ] , · · · , r N sin[ N θ ] (cid:1) T . (12)For cases with either δ R = 0 ( δ L (cid:54) = 0) or δ L = 0 ( δ R (cid:54) =0), we have η = 0 and η = δ L t L r − N or η = δ R t R r N .As long as | η | <
1, Eq.(10) has N real solutions, andthe corresponding eigenvalues given by Eq.(11) are real.Particularly, in the thermodynamic limit we have | η | → δ R = 0 (a fixed δ L ) and r > | t R | > | t L | )or δ L = 0 (a fixed δ R ) and r < | t R | < | t L | ), and thesolutions θ = mπN +1 are identical to the OBC case. Theanalytical results indicate clearly that in these cases thesystem exhibits NHSE as all wavefunctions accumulateeither on the left ( r <
1) or right ( r >
1) edge in thelarge size limit.Now we consider the general case with nonzero δ L and δ R . In the region of 0 < δ R t R , δ L t L <
1, we have 0 < η < η >
0. When η < η , Eq.(10) has N real solu-tions. When η > N + 1 − η ( N −
1) which always holdstrue in the large N limit, Eq.(10) has no real solutions but N complex solutions, and the corresponding eigenvaluesare complex. In this case, we have | z | (cid:54) = | z | . In the ther-modynamic limit, we have | z / | → | z / | → t R t L ,suggesting that the spectrum approaches to the periodicspectrum [50].To give a concrete example, we display the energyspectra and averaged inverse participation ratio (IPR)in Fig.1 for the case of t R /t L < ≤ δ R t R , δ L t L ≤
1. The averaged IPR is de-fined as IPR = N (cid:80) Ns =1 IPR s = N (cid:80) Ns =1 (cid:80) n |(cid:104) n | Ψ s (cid:105)| ( (cid:104) Ψ s | Ψ s (cid:105) ) , where Ψ s is the s -th right eigenstate Ψ of H . WhileIPR ∼ N approaches zero in large N limit for homoge-neously distributed eigenstates, a finite IPR gives signa-ture of NHSE. As shown in Fig.1 (A) for N = 10, theeigenstates in the yellow region are similar to the OBCcase, and the corresponding eigenvalues are real as dis-played in Fig.1 (a)-(d). When we increase the size N,the yellow region becomes narrow. The eigenvalues withthe same parameters as in Fig.1 (d) become complex asdisplayed in Fig.1 (e) and (f) for N = 20 and 80, respec-tively. Particularly, for N = 80, we see that the spectrumalmost completely overlaps with the PBC spectrum, andthe blue region almost spreads over the whole parameterspace except a very narrow region near the axis of δ L = 0,which is consistent with our analytic prediction. N | z | =0.01=1=100 n || =0.01 =100 n || =0.01 =100 -1 -0.5 0 0.5 1 Re(z) -1-0.500.51 I m ( z ) N=10 N=20 N=100 BZ -2 -1 0 1 2
Re(z) -2-1012 I m ( z ) N=10 N=20 N=100 BZ (a) (b)(c)(d) (e)
FIG. 2: (a) | z | as a function of lattice size N for HN modelwith µ = 0 . , , N = 10 and N = 100 independent of t R , t L , respectively. (d,e) The finite-size generalized Brillouin zones z with µ = 0 . µ = 100 for different size N = 10 , ,
100 independent of t R , t L , respectively. The curve formed by black dots repre-sents Brillouin zone for PBC case. While the finite system may exhibit NHSE in someboundary parameter regions, our analytical and numeri-cal results indicate clearly that in the large size limit theNHSE can only survive on two critical lines, i.e., δ L = 0for t R t L < δ R = 0 for t R t L >
1. The NHSE is fragileagainst the general boundary perturbation, and the sys-tem with nonzero δ L and δ R exhibits similar behaviorsas the PBC system in the large size limit.Except of the general solution constructed by the com-bination of two linearly independent solutions, there ex-ists a kind of solution composed of only one of them, i.e.,we only need c (cid:54) = 0 and c = 0 (or c = 0 , c (cid:54) = 0), andthe boundary equation H B ( c , T = 0 requires that t R = δ R z N , δ L = t L z N , (13)which can be satisfied simultaneously only if t R δ R = δ L t L = µ. (14)We note that the special case µ = 1 just corresponds tothe PBC. Under the boundary condition (14), the solu-tion of z is determined by z N = µ , which gives rise to z = N √ µe i mπN , ( m = 1 , , · · · , N ). It then follows thatthe energy spectrum is given by E = ( t L N √ µ + t R N √ µ ) cos( θ ) + i ( t L N √ µ − t R N √ µ ) sin( θ )with θ = mπN , and eigenstates asΨ = (cid:16) N √ µe iθ , (cid:0) N √ µe iθ (cid:1) , · · · , (cid:0) N √ µe iθ (cid:1) N (cid:17) T . (15)While we have always | z | = 1 under the PBC case, forthe general case, | z | = N √ µ is not equal to 1. While thesystem may exhibit NHSE for a finite N , the NHSE willdisappear in the large size limit as | z | always approaches1 when N → ∞ for a fixed µ , as displayed in Fig.2 (here z = z ). Therefore, this case is similar to the PBC casein the thermodynamic limit. If we take µ = r N , we have E = 2 √ t L t R cos θ with θ = mπN . This special case is theso called modified PBC studied in Ref.[22]. Non-Hermitian Su-Schrieffer-Heeger model.-
We can also exactly solve the 1D non-Hermitian Su-Schrieffer-Heeger (SSH) model with generalized bound-ary condition, described by H = M − (cid:88) n =1 [ t L c † nA c nB + t R c † nB c nA + t R c † ( n +1) A c nB + t L c † nB c ( n +1) A ] + δ R c † A c MB + δ L c † MB c A , (16)where t L/ R and t L/ R are imbalanced hopping termbetween intracell and intercell sites, and M is the numberof cells. For PBC case, the model is easy to be solved inthe Bloch momentum space [9, 51–53], and we can obtainthe phase diagram with the phase boundaries determinedby the gap closing condition: | t R /t L | = 1 or | t L /t R | =1, as shown in Fig.3a. For simplicity, in the following weshall focus on the situation with all parameters t L/ R and t L/ R taking positive.In the same framework we can obtain the analyticalsolutions of the non-Hermitian SSH model with general-ized boundary condition [50]. From the expression of E in terms of z i , it follows that z and z fulfill the con-straint condition: z z = t R t R t L t L . (17)Similarly, the boundary equation leads to( z M +11 − z M +12 ) + χ ( z M − z M ) − χ ( z M − − z M − ) − χ ( z − z ) = 0 (18)with χ = t R t L − δ R δ L t L t L , χ = t R δ R δ L t L t L and χ = δ L t L + δ R t R (cid:16) t R t R t L t L (cid:17) M . Due to the constraint conditionof Eq.(17), we can always set the solution as the form ofEq.(9) with r = (cid:113) t R t R t L t L . Then Eq.(18) becomessin[( M + 1) θ ] + η sin[ M θ ] − η sin[( M − θ ] = η sin[ θ ] , (19)where η = t R t L − δ R δ L √ t R t R t L t L , η = δ R δ L t R t L and η = δ L t L r − M + δ R t R r M . The corresponding eigenvalue can beexpressed as E = ± (cid:113) √ t R t R t L t L cos θ + t R t L + t R t L , FIG. 3: Phase diagram for non-Hermitian SSH model (a)PBC case; (b) OBC case; (c) case of δ L = 0 , δ R (cid:54) = 0; (d) caseof δ R = 0 , δ L (cid:54) = 0. The phase boundaries are denoted by bluelines, and bound states exist in the shadow region. The bulkstates in the blue region and orange region are located at theleft and right edge, respectively. There is no NHSE in thewhite region. where θ is the solution of Eq.(19) and may take real orcomplex depending on the values of η , η and η .When δ R = δ L = 0, i.e., the OBC case, we have η = η = 0 and η = α with α = (cid:113) t R t L t R t L . WhileEq.(19) has M real solutions corresponding to bulk stateswhen α < α c , it has M − θ = π + iϕ )corresponding to edge states when α > α c . In the ther-modynamic limit, α c = 1 + M →
1, and thus the bound-ary of topological phase transition is given by α = 1,i.e., t R t L = t R t L as shown in Fig.3b. In the topo-logical phase, the solution ϕ for edge states is given by ϕ = log (cid:104) α (cid:16) − α α M +1) (cid:17)(cid:105) and the energy correspond-ing edge states becomes E e = ± (cid:113) − √ t R t R t L t L cosh( ϕ ) + t L t R + t L t R . In the thermodynamic limit, ϕ → log( α ) and E e → δ L = 0 and δ R (cid:54) = 0, we have η = α , η = 0 and η = δ R t R r M . In the thermodynamic limitwe have η → t R /t L < t L /t R ( r < θ are identical to the OBC case. Onthe other hand, when t R /t L > t L /t R ( r > η → ∞ for a finite δ R /t R in the thermodynamiclimit. It follows that Eq.(19) has no real solutions but M complex solutions, and we have | z | → | z | → t R t R t L t L for bulk states as M → ∞ . In this case, thespectrum in the thermodynamic limit approaches to thespectrum of system with PBC. Since the spectra in theregions of r < r > r < r > δ R = 0 and δ L (cid:54) = 0, we have η = δ L t L r − M . In the large M limit, η → r > η → ∞ for r <
1. Similarly, we can get thephase diagram as shown Fig.3d. We note that the phaseboundaries in Fig.3c and Fig.3d are determined by thegap closing conditions. In the shadow regions, there existin-gap bound states. Also there are only left or right skinstates in Fig.3c or Fig.3d, in contrast to the OBC case.In the presence of finite δ L and δ R , Eq.(19) does notsupport real solutions in the large M limit, and the spec-trum shall approach the spectrum of the PBC case. Sim-ilar to the HN model, in the thermodynamic limit theNHSE is unstable to the perturbation with both δ L (cid:54) = 0and δ R (cid:54) = 0, while it may exist in the finite size system.Consequently, the phase boundaries for the general casewith finite δ L and δ R is identical to the PBC case. Conclusions.-
We exactly solved the non-HermitianHN model and SSH model with generalized boundaryconditions and predicted the existence of NHSE beyond the OBC when one of the boundary hopping terms isabsent. Apart from this critical line on the boundary pa-rameter space, the NHSE is unstable under tiny bound-ary perturbations and vanishes in the thermodynamiclimit, whereas it may exist in a finite size system. Basedon the analytical results, we uncovered the origin of size-dependent NHSE and gave quantitative description ofthe interplay effect of boundary hopping terms and latticesize. We also applied our analytical results to explore thephase diagram of non-Hermitian SSH model under differ-ent boundary conditions and identified a novel phase di-agram in the critical boundary line. Our exact solutionsalso provide an analytical method to determine finite-sizegeneralized Brillouin zones.
Acknowledgments
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