Non-Hermitian scattering on a tight-binding lattice
NNon-Hermitian scattering on a tight-binding lattice
Phillip C. Burke, Jan Wiersig, and Masudul Haque
1, 3 Department of Theoretical Physics, Maynooth University, Maynooth, Kildare, Ireland Institut f¨ur Physik, Otto-von-Guericke-Universit¨at Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany Max-Planck Institute for the Physics of Complex Systems, Dresden, Germany (Dated: July 20, 2020)We analyze the scattering dynamics and spectrum of a quantum particle on a tight-binding latticesubject to a non-Hermitian (purely imaginary) local potential. The reflection, transmission and ab-sorption coefficients are studied as a function of the strength of this absorbing potential. The systemis found to have an exceptional point at a certain strength of the potential. Unusually, all (or nearlyall) of the spectrum pairs up into mutually coalescing eigenstate pairs at this exceptional point.At large potential strengths, the absorption coefficient decreases and the effect of the imaginarypotential is similar to that of a real potential. We quantify this similarity by utilizing properties ofa localized eigenstate.
I. INTRODUCTION
In recent years there has been a surge of interest inquantum systems that are described by non-HermitianHamiltonians. Although Hermiticity is regarded as a pos-tulate of standard quantum mechanics, non-HermitianHamiltonians are useful as effective descriptions of sys-tems where loss or gain plays an important role, such asopen quantum systems [1] and optical systems describedby wave equations formally analogous to a Schr¨odingerequation [2–4]. By now, a number of experimental plat-forms for the study of non-Hermitian quantum mechanicsare available. These include lasers or optical resonators[5–8], coupled optical waveguides [9–13], microwave res-onators [14–17] and arrays thereof [18], optical microcav-ities [2, 19, 20], optomechanical systems [21], photoniccrystals [22, 23], acoustics [24–27] atom-cavity compos-ites [28], exciton-polariton systems in semiconductor mi-crocavities [29, 30], and various other arrangements [31–36].Non-Hermitian Hamiltonians lead to various phenom-ena not present in Hermitian systems. In general, theeigenvalues of non-Hermitian Hamiltonians are complex.The left and right eigenstates of a non-Hermitian Hamil-tonian are generally not equal — we confine our discus-sion to right eigenstates. The eigenstates are in gen-eral not mutually orthogonal. This non-orthogonalitybecomes extreme at points in the parameter space re-ferred to as exceptional points [37–41]. At an excep-tional point, the eigenvalues appear to become degen-erate. However, it is not a genuine degeneracy as thecorresponding eigenvectors coalesce as well. This re-sults in our eigenstates no longer providing a basis span-ning the entire Hilbert space. The Hamiltonian ma-trix is therefore non-diagonalizable and is a defectivematrix [42, 43] at these exceptional points. The sur-viving eigenstate at an exceptional point is always chi-ral [44]; this chirality has been observed experimentally[7, 8, 16, 30, 45]. Other phenomena associated with ex-ceptional points include loss-induced transparency [9],unidirectional transmission [26, 32, 33], lasers with non- monotonic pump-dependence [5], enhanced sensing [46–48], etc. Exceptional points are also associated with thereal-to-complex spectral transition for parity-time ( PT )symmetric Hamiltonians [41].In this work, we are concerned with the non-Hermitianphysics of a quantum particle on a tight-binding lat-tice. Previous studies of non-Hermitian effects for a lat-tice particle include Anderson localization [49–52] andlocalization in quasiperiodic potentials [53, 54], invisibil-ity (reflectionless scattering) due to non-Hermitian hop-ping [55] or oscillating imaginary scatterer [56], flat-bandphysics [57], Bloch oscillations [58], PT symmetry ob-tained by combining an absorbing potential on one sitewith an emitting potential on another [59–65], etc. In ad-dition, non-Hermitian tight-binding lattices form the ba-sis of the study of non-Hermitian topological many-bodysystems, a topic of rapidly growing interest [66–68]. Afew studies have also addressed interacting many-bodysystems in non-Hermitian lattice systems [69, 70].We will consider an imaginary potential on one siteof the lattice, serving as an absorbing scattering poten-tial. This can be regarded as a lattice analog of a delta-function scattering potential in the continuum which ispurely imaginary. An imaginary scattering potential islinked to measurement [71–73], and is thus related toquantum first-passage time problems and the quantumZeno effect [74–81]. In analogy to the quantum Zeno ef-fect, it is expected that an imaginary potential will havesuppressed absorption when the strength of the poten-tial is large. This suggests that the absorption might benon-monotonic as a function of the strength of the dis-sipative potential. In this work, we explicitly show non-monotonic dependence of the amount of absorption onthe potential strength, in the context of a simple latticemodel. The Hamiltonian is H = − J L − (cid:88) j =1 (cid:16) | j (cid:105) (cid:104) j + 1 | + | j + 1 (cid:105) (cid:104) j | (cid:17) − iγ | q (cid:105) (cid:104) q | , (1)with 1 ≤ q ≤ L . Here γ is a positive constant, so thatthe imaginary potential is absorbing. The labels for thebra’s and ket’s here are site labels: The particle lives a r X i v : . [ qu a n t - ph ] J u l Impurity site
FIG. 1. The impurity is placed at one of the central sites ofthe lattice, as shown here for L = 6. In this case, it couldequivalently be placed on the fourth instead of the third site.For odd L , there is a definite central site. on an L -site chain with open boundary conditions. Thehopping strength will henceforth be set to J = 1, i.e.,energies and times are measured in units of J and 1 /J respectively, and are therefore presented without units.Also, the spacing between sites is set to unity, so thatlengths and wavenumbers are dimensionless as well.The site q is the location of the dissipative impurity.Since we want to study reflection and transmission, itis convenient to place the particle at the center of thelattice, at either site (cid:4) L (cid:5) or (cid:4) L (cid:5) + 1 (Fig. 1).We present a study of the dynamics and eigenspec-trum of the system (1). By scattering wavepackets nu-merically off the dissipative impurity, we show how thereflection, transmission and absorption fractions dependon the strength γ of the impurity. These results are com-pared with the continuum problem, which is a variantof the standard textbook problem of quantum scatteringoff a Hermitian delta-function potential. In both casesthe absorption coefficient is found to be a non-monotonicfunction of γ , having a maximum at a point that dependson the momentum of the incident particle or wavepacket.In addition, we present the spectrum of the Hamiltonian,which shows an unusual exceptional point at γ = 2 atwhich all (or nearly all, depending on L ) of the eigenval-ues pair up. The absorption coefficient is non-monotonicand has a maximum near, but not necessary at, the ex-ceptional point. At large γ , the absorption is vanishinglysmall, and the system behaves as if the impurity were areal potential V . In particular the system has a (anti-)bound eigenstate, which allows us to draw a correspon-dence between values of γ and V . The localized eigen-state is purely a lattice phenomenon with no analogue inthe continuum.In Section II we present the scattering results and com-parisons with the continuum case. Section III discussesthe spectrum and exceptional points. In Section IV weinvestigate the system at large γ values, and draw a com-parison between real and imaginary potentials via theirbound states. In Section V we present some discussionand concluding remarks. The Appendixes present somefurther details on the eigenvalues and eigenstates. II. SCATTERING AT AN ABSORBINGPOTENTIAL — REFLECTION, TRANSMISSION,ABSORPTION
In this section, we examine the scattering of a quantumparticle by the dissipative impurity. To make a compar-
Continuum0 2 . . . γ ? γ M ag n i t ud e RTA
FIG. 2. Continuum scattering. The reflection, transmissionand absorption probabilities ( R , T , A ), plotted against thestrength γ of the dissipative delta-potential. Here k = π/ (cid:126) = 1, and m = 0 . ison with the corresponding continuum system, we firstwork out the results for the continuum system in II A,before turning back to our lattice problem in II B. A. Continuum scattering by imaginarydelta-potential
Complex scattering potentials in the continuum havebeen considered generally in the literature [72, 73, 82].We are specifically interested in the case of an imaginarypotential of delta-function shape, which is the analog ofthe single-site potential on a lattice.In the continuum, the wavefunction ψ ( x ) satisfies thetime-independent Schr¨odinger Equation: − (cid:126) m d ψ ( x ) dx + V ( x ) ψ ( x ) = Eψ ( x ) . (2)(We will eventually set (cid:126) = 1 but retain it for now.) Wetake V ( x ) to be a negative imaginary delta potential: V ( x ) = − iγδ ( x ).Solving the scattering problem is a variation of thestandard textbook scattering problem with a real delta-function potential [83]. We take the wavefunction to beof the form e ikx + re − ikx on the left half-line ( x < te ikx on the right half-line ( x > k >
0. We then use the appropri-ate (dis)continuity conditions at x = 0 to solve for thereflection and transmission amplitudes ( r , t ). This yields r = −
11 + k (cid:126) mγ , t = 11 + mγk (cid:126) . (3)Using (3) we can obtain the reflection, transmission, andnow also absorption probability as functions of the pa-rameter γ : R = | r | , T = | t | , A = 1 − R − T . (4)We see in Fig. 2 that R = T for a particular value of γ , and that A is maximized by some value of γ . Usingequations (3) and (4), we find that these points are bothequal to γ (cid:63) = k (cid:126) m . (5)These expressions depend on (cid:126) and the mass m . Weset (cid:126) = 1. To facilitate comparison with the lattice sit-uation, we choose m = 1 / (cid:126) k / m ) on the continuum matches the low-energy part of the cosine dispersion ( − k ) on thelattice without impurity. Thus r = − γγ + 2 k , t = 2 kγ + 2 k , γ (cid:63) = 2 k . (6) B. Lattice
We now turn to the lattice problem. Through numer-ical time evolution we will calculate the reflection andtransmission fractions, R and T , and obtain the absorp-tion fraction using A = 1 − R − T .We initialize our particle as a (discrete version of) aGaussian wavepacket, localized around the site j andcarrying lattice momentum k : | ψ (0) (cid:105) = (cid:88) j ψ j (0) | j (cid:105) = N − (cid:88) j e − ( j − j σ e ikj | j (cid:105) (7)where N is a normalization constant. A positive k en-sures that the wavepacket will propagate rightwards ini-tially. The position j is chosen such that the wavepacketstarts on the left side of the lattice, and does not ini-tially overlap significantly with either the lattice edgesor the impurity. The width σ is chosen to be sig-nificantly larger than 1, but significantly smaller than L/
2. The wavepacket is evolved using the Hamiltonian: | ψ ( t ) (cid:105) = e − iHt | ψ (0) (cid:105) . Expressing the wavefunction attime t in the site basis, | ψ ( t ) (cid:105) = (cid:80) j ψ j ( t ) | j (cid:105) , the coeffi-cients ψ j ( t ) provide the occupancies, | ψ j ( t ) | .Fig. 3 shows the evolution of a wavepacket for threedifferent values of γ , initially localized near the left endof a 500-site lattice. After the particle is incident on theimpurity, we see different portions being reflected andtransmitted. Choosing a time after the collision has oc-curred, such that the reflected and transmitted packetsare well-separated from the impurity, one can define thecoefficients based on the wavefunction coefficients at thistime. The reflected (transmitted) fraction is the weightto the left (right) of the impurity. Denoting the impurity T i m e A 0 . .
20 250 500Site09001800 T i m e B 0 . .
20 250 500Site09001800 T i m e C 0 . . FIG. 3. Wavepacket evolution illustrated by a density plot ofsite occupancies | ψ j | . Here L = 500, σ = 40, k = π/ A : γ = 0 . B : γ = 2 - Shows roughly similar amounts of thewavepacket being reflected/transmitted. C : γ = 10 - Showsless of the wavepacket being transmitted than reflected. Lattice0 2 4 600 . γ ? γ M ag n i t ud e RTA
FIG. 4. Reflection, transmission and absorption probabilitiescalcuated using wavepacket evolution on the lattice. (
R, T, A plotted against γ .) Here L = 500, σ = 40, k = π/ site as q , R = q (cid:88) j =1 | ψ j | , T = L (cid:88) j = q +1 | ψ j | , A = 1 − R − T . (8)Fig. 4 shows the results of calculating the coefficients fora lattice with 500 sites, with the impurity at site 250,for a range of values for γ . The coefficients are extractedfrom time evolution with a σ = 40 wavepacket. The ob-servation of weights on the left and right parts of the lat-tice is performed at a time well after the wavepacket hasscattered off the impurity, but well before either the re-flected or the transmitted wavepacket reaches one of theboundaries. For Fig. 4, this time was t = 160. For othervalues of k (Fig. 5), the times are different as the speed ofthe wavepacket depends on k . We have checked that thedependence on σ is negligible provided 1 (cid:28) σ (cid:28) L/ k = π/
2, for which the dispersion of the wavepacket isleast severe.
C. Comparison between Continuum and Lattice
Comparing Figures 2 and 4, we see that our latticeresults are very similar to the continuum results, exceptfor a rescaling of γ . In the continuum case, we havefound that the main feature [maximum of A ( γ ), or cross-ing point of R ( γ ) and T ( γ )] occurs at a value of γ thatis proportional to the momentum, γ (cid:63) = 2 k . One there-fore expects that in the lattice case γ (cid:63) should also de-pend on the momentum of the scattered particle. Morespecifically, since the single-particle dispersion changesas k → − k in going from the continuum to lattice,one expects from the dependence of γ (cid:63) = 2 k in the con-tinuum that the dependence might be γ (cid:63) = 2 sin k on thelattice.We can extract γ (cid:63) for various momenta by runningour numerical time evolution of wavepacket scattering forvarious momenta and identifying the maximum of A ( γ ).The results are shown in Fig. 5, comparing the contin-uum and lattice case. Indeed the momentum dependenceof the γ (cid:63) appears to be ≈ k on the lattice, with amaximum of γ (cid:63) ≈ k = π/ III. SPECTRUM AND EXCEPTIONAL POINTS
It turns out that the value γ ≈ γ . As the eigen-values are complex, the real and imaginary componentsare shown separately. We also show the 14 eigenvaluesin the complex plane, for three different values of γ , inFig. 7. For any value of γ , the real part of the eigenvaluesare generally spaced between − γ = 2,the eigenvalues coalesce in pairs. (The coalescence is vis-ible in the real parts — the imaginary parts are already π/ π/ π/ . . k γ ? Continuum, Eq . (5)Lattice, Numerics2 sin k FIG. 5. Comparing results of the value of γ for which ab-sorption is maximised in the continuum ( γ (cid:63) = 2 k ) and onthe lattice (obtained from numerical wavepacket evolution).Lattice results are obtained with L = 250 and σ = 15. Forcomparison, the function 2 sin k is plotted (dashed curve). paired up even at γ < γ = 2, we find numerically that one of the eigenstatesbecomes equal to − i times the other eigenstate.In Appendix A we show analytically that the eigen-values always group into degenerate pairs at γ = 2, foran even- L lattice with the impurity at one of the centralsites. One can also show that the corresponding eigen-states for every such pair are linearly dependent.Unlike exceptional points which separate a PT -symmetric phase from a PT -symmetry-broken phase, theeigenvalues of our system are complex on both sides ofthe exceptional point. The imaginary parts on averagehave larger magnitude near the exceptional point, andgenerally decrease as one moves away from γ = 2, withone striking exception. The exception corresponds toone of the two eigenvalues whose real part becomes zero.The imaginary part becomes large and negative as γ in-creases, and eventually becomes ≈ − γ . This eigenvaluecorresponds to a bound state localized at the dissipativeimpurity, which we will analyze in the next section.The structure of the spectrum discussed here for L =14 is true for L mod 4 = 2. For other values of L , thereare variations, which we detail in Appendix C. In par-ticular, for odd values of L , there is only a single pairof eigenvalues coalescing ( L mod 4 = 3), or none at all( L mod 4 = 1). However, even with an odd number ofsites the localized eigenstate still exists for large values -2.00.02.0 R e ( E ) γ I m ( E ) FIG. 6. Energy spectrum of the Hamiltonian (1), for L = 14,as function of the potential strength γ . Real and imaginaryparts of the eigenvalues are plotted separately. γ = 1 . γ = 2 γ = 2 . E ) I m ( E ) -2 0 2Re( E ) -2 0 2Re( E ) FIG. 7. Eigenvalues of the Hamiltonian (1), for L = 14, forthree values of γ , below, at and above the exceptional point.In each case, the L eigenvalues are plotted on the complexplane. For γ = 2, only L/ of γ . In Appendix D we also discuss the dependence ofthe location of the impurity site. IV. LARGE γ At large γ the absorption decreases, suggesting thatthe effect of the imaginary potential is similar to that ofa real potential. In this section we draw a comparisonbetween the effects of real and imaginary on-site poten-tials.In Section III we saw there was a single eigenvalue, − − − − − Site j | h j | φ i | γ = 2 . | V | = 2 . FIG. 8. Site occupancies of the localized eigenstate, for botha real ( V ) and an imaginary ( − iγ ) potential of magnitude 2.5,and L = 42 sites. The scale is log-linear. with a corresponding eigenstate, which had a purelyimaginary negative component. At large γ the eigenen-ergy approaches − iγ , for which a plausible explanationwould be that the eigenstate is localized at or aroundthe impurity site q and hence its energy is primarily de-termined by the − iγ | q (cid:105) (cid:104) q | term in the Hamiltonian (1).Indeed the corresponding eigenstate is numerically foundto be exponentially localized around the impurity site(Fig. 8).For comparison, we also consider the effect of a realpotential, i.e., the Hermitian Hamiltonian H = − J L − (cid:88) j =1 (cid:16) | j (cid:105) (cid:104) j + 1 | + | j + 1 (cid:105) (cid:104) j | (cid:17) + V | q (cid:105) (cid:104) q | (9)Here V is a real parameter which could be either positiveor negative. It is known that this Hamiltonian supportsa bound state for negative V and an anti-bound statefor positive V . (The spectrum, which is real, containsone state which separates from the band and at large | V | approaches V .) This eigenstate is exponentially localizedaround site q .In Fig. 8 we show the exponential localization of theeigenstate both for the real potential ( | V | = 2 .
5) and forthe dissipative impurity γ = 2 .
5. At these values, theeigenstate is more strongly localized (has smaller local-ization length) for the case of the real potential, Eq. (9).Approximating the occupancies at site j by the form ∝ e ( j − q ) /α , where q is the impurity position, one canextract the localization length α . By extracting α forthe localized eigenstate for various values of γ in the caseof our non-Hermitian system (1), and for various val-ues of V in the case of the system (9), we can assign toeach γ > V , for which the same localizationlength is obtained. Results of this calculation are shownin Fig. 9, for a system with L = 42 sites. This quantifiesthe idea that, at large γ , an absorbing impurity behaveslike a real-valued impurity. γ | V | γ = | V | FIG. 9. A correspondence between the parameters of the realand imaginary potentials, using the localization length of thebound states, for L = 42. For values of γ <
2, there is no bound state. For γ slightly larger than 2, the localization length correspondsto the bound state of a very weak real potential (verysmall | V | ). As γ grows, the corresponding | V | increasesand asymptotically approaches | V | = γ . In other words,the effect of an absorbing impurity of large strength γ (cid:29) γ . It is well-known that the negative real delta-potential has a singlebound (localized) state. However, neither the positivereal potential, nor the imaginary potential, have boundstates. (If one assumes that there is a bound state forsome potential, λδ ( x ), one finds that λ must have a realcomponent, which is negative.) Hence no quantitativecorrespondence can be drawn in terms of the localizationlength, as we have done for the lattice.The existence of a strongly localized eigenstate pro-vides a simple ‘spectral’ interpretation of the suppres-sion of absorption at large γ that we have presented inSubsection II B. For large γ , the localized eigenstate hasnear-zero overlap with the incident wavepacket, becausein the initial state the wavepacket is far from the im-purity site. Thus, the wavepacket is ‘shielded’ from theimpurity, because its dynamics is confined to the sub-space of all the other eigenstates which have near-zeroweight at the impurity site. Therefore the wavepacketundergoes almost no absorption. Curiously, for the sup-pression of absorption in the continuum case (SubsectionII A), the same interpretation cannot be used, as there isno localized eigenstate in that case.In Appendix B we show site occupancy profiles for asample of some of the eigenstates. Other than the spe-cial (localized) eigenstate, the other eigenstates resemble those for a real potential — the eigenstates at the bottomand top of the band have few nodes, while those near thecenter of the band have many nodes. V. DISCUSSION AND CONTEXT
We have studied the scattering dynamics and the spec-trum of a tight-binding single-particle system with a non-Hermitian absorbing impurity at one site, focusing on thecase where the impurity is near the center of the lattice.Setups loosely similar to ours have been explored ina few other recent works. In Ref. [56], scattering off alocalized lattice impurity is studied, in the case wherethe strength and phase of the impurity are oscillating.Scattering was studied using Gaussian wavepackets, asin the present work. For certain parameters, the os-cillatory non-Hermitian impurity was reported to allowperfect transmission (‘Floquet invisibility’). In Ref. [87],the lattice impurity was placed at the lattice edge andthe role of the non-orthogonality of the eigenstates onthe non-unitary time evolution was explored. In ad-dition, some related issues have been discussed in thecontext of PT -symmetric lattice systems formed by hav-ing imaginary potentials on multiple sites [59–65]. Thespectrum of lattices with two impurities has been stud-ied in Refs. [60, 65]. Ref. [59] reported an eigenstatewhich is localized on the two impurity sites — this maybe considered a PT -symmetric version of the localizedeigenstate we have studied. Refs. [60, 61] have madecomparisons between the non-Hermitian system and cor-responding Hermitian system, as we have done. Afterthe appearance of our preprint, our single-particle non-Hermitian Hamiltonian has appeared in Ref. [88] as aneffective Hamiltonian. Experimentally , lattice systems with localized losseshave been studied in several contexts. In the setup ofRef. [89, 90], a Bose-Einstein condensate is realized in aone dimensional optical lattice, with engineered losses ona single site acting as a local dissipative potential. Con-necting single-particle results such as ours to many-bosonphysics in such a setup remains an interesting challengefor future work.A realization more similar to the single particle tight-binding system considered in this work is that with pho-tonic lattice systems, such as those in Refs. [91, 92]. Inthis setup, photonic lattices are realized using femtosec-ond laser writing to inscribe waveguide arrays with ap-propriate index profiles in fused silica. The physics ofphotons in such an architecture can be well-describedby a tight-binding model, with an additional spatial di-rection taking the role of time. This setup, or its vari-ants, has been used to demonstrate a number of paradig-matic tight-binding phenomena, including Bloch oscilla-tions [93] and Anderson localization [94, 95]. Both one-dimensional and two-dimensional lattices have been real-ized, and lossy sites and other types of non-Hermiticityhave been explored [12, 92, 96]. It is possible to cre-ate localized excitations (wavepackets) and observe theirpropagation [95, 97]. Thus, studies of scattering off lossysites should be possible in such a setup.Another possible experimental setting for observingscattering off non-Hermitian potentials in a tight-bindinglattice might be microwave realizations using coupled di-electric resonators, such as that discussed in [18]. Thissetup is well approximated by a nearest-neighbour tight-binding Hamiltonian. The resonance frequency of an iso-lated resonator, and the coupling strength between tworesonators (due to the evanescent electromagnetic field),correspond to the on-site energy and to the hopping term,respectively. A controllable on-site loss is created by plac-ing an absorbing material on a particular resonator.In the present work, by explicit time evolution start-ing from initial states which are momentum-carryingwavepackets, we found the reflection, transmission andabsorption coefficients ( R , T , A ) as a function of the im-purity strength γ and of the incident momentum k . Theabsorption was shown to first increase and then decreaseas the strength γ is increased. It can be argued thatthis non-monotonic behavior is related to the quantumZeno effect. The experimental non-monotonic behaviorof Ref. [34] can be interpreted in the same light. We havedemonstrated and analyzed the effect in a simple latticesetting. We have also compared with the scattering of asingle particle in a continuum from an absorptive delta-potential.We have also presented the spectrum of the non-Hermitian system. The system we focus on — even num-ber of sites, impurity at one of the central sites — has anunusual exceptional point structure. At the same valueof γ , all the eigenstates of the systems coalesce in pairs.This is not a higher-order of exceptional point [84–86],rather, it is a collection of many second-order coales-cences at the same point in parameter space. At larger γ ,the spectrum contains one localized eigenstate. This isanother way in which a strong absorptive impurity actslike a real-valued impurity potential. This feature is par-ticular to the lattice as there are no bound states in thecorresponding continuum problem. The eigenvalue cor-responding to the localized eigenstate has a purely imag-inary value.Our work opens up several avenues of research. Wehave explored scattering dynamics. A detailed study ofother types of dynamics remains to be done, not onlyfor tight-binding lattices, but also for continuum parti-cles subjected to localized absorbers. Extending suchdynamical considerations to nonlinear cases [76, 90, 98]also deserves further exploration. The spectral part ofthe present study provides motivation for a more thor-ough investigation of the spectrum of relatively simplenon-Hermitian models. The structure we have found —many pairs coalescing at the same point — suggests thatnon-Hermitian spectra may hold more surprises not yetknown in the literature. ACKNOWLEDGMENTS
We thank S. Nulty (Maynooth) for analytically demon-strating the degeneracy at γ = 2 (Appendix A). PCBthanks Maynooth University (National University of Ire-land, Maynooth) for funding provided via the John &Pat Hume Scholarship. Appendix A: Analytical expressions for spectrum
In the main text, we have shown numerically that theeigenvalues of our system coalesce in pairs at γ = 2, foreven L , when the impurity site q is one of the centralsites, i.e., when q = L/ q = ( L/
2) + 1. In this Ap-pendix, we analyze the eigenvalues analytically. We ex-press the characteristic polynomial (whose roots are theeigenvalues) in a form which allows us to predict, first,that all the eigenvalues pair up when q is one of the cen-tral sites, and second, that this is a multiple exceptionalpoint because each eigenstate pair is linearly dependent.The characteristic polynomial is treated in Section A 1and the case of q = L/ q = L/
1. General location, q We want to find the eigenvalues of the L × L matrix[ H q ] jk = − δ j,k +1 − δ j +1 ,k − iγδ jq δ jk . (A1)Here 1 ≤ q ≤ L . The characteristic polynomial of thismatrix up to a minus sign is the determinant of the tridi-agonal matrix λ . . . . . . . . . . . . . . . λ . . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . ... ... ... 1 λ + iγ . . . . . . ... ... ... ... . . . . . . . . . . . . ... ... ... ... ... 1 λ λ . (A2)Now the determinants of tridiagonal matrices satisfy arecurrence relation. If P n is the determinant of the n × n matrix with elements A ij = b i δ i,j +1 + c j δ i +1 ,j + a i δ ij , (A3)then P n = a n P n − − c n − b n − P n − . (A4)This recurrence relation can be verified by determinantexpansion and appears in numerous sources, e.g., is men-tioned in Section 8.4 of Ref. [42]. The characteristic poly-nomial of H (A1), i.e., the determinant of the matrix(A2), therefore satisfies P n = λP n − − P n − , if n (cid:54) = qP q = ( λ + iγ ) P q − − P q − , if q > P = 1 P = λ + iγδ q . (A5)A standard method of solving such linear recurrence re-lations is to use the Z transform. Ignoring the secondline in Eq. (A5), i.e., ignoring the impurity, we can getan expression for P n in terms of P , P and λ . Defining F ( z ) = Z { P n } and using a shift theorem, we get z F ( z ) − z P − zP = λ ( zF ( z ) − zP ) − F ( z ) . (A6)After solving for F ( z ) and decomposing into partial frac-tions, one can take the inverse Z transform, yielding P n = P √ λ − (cid:2) ( x + ) n +1 − ( x − ) n +1 (cid:3) + P − λP √ λ − x + ) n − ( x − ) n ] (A7)where x ± ( λ ) = (cid:2) λ ± √ λ − (cid:3) .Defining: K n ( λ ) := √ λ − (cid:2) ( x + ) n +1 − ( x − ) n +1 (cid:3) for n ≥
00 for n < P n = P K n + ( P − λP ) K n − . (A9)Since we have derived this ignoring the impurity,Eqs. (A7), (A9) are valid either for q = 1, in which case P = λ + iγ , or for values of n less than q .For q = 1, we have P = 1 and P = λ + iγ so that P n = K n + iγK n − , and therefore: P L = K L + iγK L − for q = 1 . (A10)We now turn to q >
1. For n < q , Eqs. (A7) and (A9)are valid directly with P = 1 and P = λ , i.e., with P − λP = 0, so that P n = K n for q > n < q. (A11)We have expressions for P n up to n = q −
1, but wewant P L and L ≥ q . To go beyond q , we define a new se-quence of functions Q n ( λ ), satisfying the same recurrencerelation as P n (A5), except with new initial conditions: Q = P q − and Q = P q = ( λ + iγ ) P q − − P q − . Thuswe need to solve Q n = λQ q − − Q q − , Q = K q − ,Q = ( λ + iγ ) K q − − K q − . (A12) Now we have already solved the same recurrence relationfor P n , using the Z transform. The solution is Q n = Q K n + ( Q − λQ ) K n − . Therefore Q n = K q − K n + ( iγK q − − K q − ) K n − . (A13)Noting that Q n ( λ ) = P n + q − ( λ ), the determinant of thefull matrix can be found as P L ( λ ) = Q L − q +1 ( λ ). Thus P L ( λ ) = K q − K L − q +1 + ( iγK q − − K q − ) K L − q . (A14)We now introduce a slight change of notation: We referto this polynomial as P L,q . In other words, the charac-teristic polynomial of the Hamiltonian matrix of a latticeof size L and having the impurity at position q will becalled P L,q . Note that Eq. (A14) reduces to Eq. (A10)for q = 1; thus P L,q = K q − K L − q +1 + ( iγK q − − K q − ) K L − q (A15)for all positions of the impurity, 1 ≤ q ≤ L .By binomial-expanding ( x ± ) n +1 , one can show that P L,q ( − λ ∗ ) = ( − L P L,q ( λ ) ∗ . (A16)This shows that the zeros of P L,q (eigenvalues of H ) aresymmetric by reflection through the imaginary axis inthe complex plane, since if λ = a + ib is a zero then − λ ∗ = − a + ib is also a zero. This symmetry is obviousfrom the spectra shown in Fig. 7.
2. Impurity at center
We now turn to the case we have focused on in thispaper: when L is even and q = L/ q = L + 1. In thiscase, P L, L = K L − K L +1 + ( iγK L − − K L − ) K L = K L − ( λK L − K L − ) + ( iγK L − − K L − ) K L = − ( K L − ) + K L (cid:16) λK L − − K L − (cid:17) + iγK L − K L = ( K L ) − ( K L − ) + iγK L − K L . Now precisely when γ = 2, this can be written as P L,L/ = (cid:16) K L + iK L − (cid:17) . (A17)This means that every root of the polynomial is a zeroof order at least 2, i.e., the eigenspectrum is doubly de-generate at γ = 2. We have thus analytically derived themost prominent feature of the spectrum presented in themain text.We now argue that, for a tridiagonal system such hasours, a coalescence of eigenvalues implies a coalescence ofeigenstates, i.e., that the eigenstates corresponding to theequal eigenvalues are always linearly dependent. Con-sider some eigenvalue λ and corresponding eigenvector X = ( x , x , . . . , x L ) T . Due to the form of the matrix,all the components x i can be written as a function of λ and the terms on the diagonals, times the first compo-nent x . If we have any two eigenvectors with the sameeigenvalue λ , the functions in the eigenvectors are thesame functions, and hence the eigenvectors only differ inthe choice of x , i.e., they are linearly dependent. Thus,if there is a degeneracy at some point, the eigenvectorsare linearly dependent, and hence we have an exceptionalpoint. Appendix B: Eigenstates
We show some eigenstates of the system, through theiroccupancy profiles.Since the eigenvalues are complex, there is no partic-ularly natural way to order them. Here we order theeigenstates based on their real component, and then bytheir imaginary component, from smallest to largest, i.e.,1 − i comes before 1 + 2 i . Fig. 10 illustrates a selectionof the eigenstates of a system with L = 42 sites. Theyare labeled as ‘ E i ’, i.e. the eigenstate presented is thestate corresponding to the i th eigenvalue, when orderedin the described manner.Note that the eigenvectors coefficients (cid:104) j | φ (cid:105) are them-selves complex; we only show the occupancies |(cid:104) j | φ (cid:105)| and not the real and imaginary parts separately. (Here | φ (cid:105) is the eigenvector in question and j is the site index.) Appendix C: Size dependence of the spectrum
In Fig. 6 we saw coalescence of every pair of eigenvaluesat γ = 2. This was for a system with L = 14 sites, and theimpurity at site q = 7. We now outline the L -dependenceof the spectrum. The pattern is different for odd L . Foreven L , there is a difference between L values satisfying L = 4 n + 2 and those satisfying L = 4 n , where n is anon-negative integer.The case L = 14, presented in the main text, belongsto the L = 4 n + 2 sequence (6, 10, 14, 18, . . . ). InFig. 11 we show the case of L = 30, showing exactlythe same pattern: all eigenvalues pair up in a multipleexceptional point exactly at γ = 2. There are an oddnumber of pairs, and the eigenvalues with central realvalues have zero eigenvalue after the coalescence, i.e., for γ >
2. One of these two eigenvalues correspond to thelocalized eigenstate, and has imaginary part growing with γ . For even L values satisfying L = 4 n , the situation isvery similar, with one additional structure. As proved inAppendix A for even L , at exactly γ = 2, all eigenvaluespair up; this is true for both L = 4 n + 2 and L = 4 n .In addition, for L = 4 n , at a value slightly above γ =2, the two eigenvalues with real values nearest to zerocoalesce in an additional exceptional point, as seen inFig. 12 for L = 8. It is at this point, γ = γ > × − · − Site j | h j | φ i | (a) E . . . j | h j | φ i | (b) E × − · − Site j | h j | φ i | (c) E × − · − Site j | h j | φ i | (d) E FIG. 10. Occupancy profiles of a sample of the eigenstates, fora L = 42 system and γ = 2 .
5. The second shown eigenstatefrom the top is the localized eigenstate. that the localized state appears and the imaginary partof the corresponding eigenvalue separates off and startsto increase unboundedly in the negative direction. Withincreasing L in the sequence L = 4 n , the location of thenew exceptional point, γ , approaches 2.We now turn to odd L , with the impurity placed onthe central site, q = ( L + 1) /
2. For L = 4 n + 3, there0 -2.00.02.0 R e ( E ) γ I m ( E ) FIG. 11. Energy spectrum of a system with L = 30. As thisvalue is in the L = 4 n + 2 sequence, the features are the sameas those described in the main text for L = 14. -2.00.02.0 R e ( E ) γ I m ( E ) FIG. 12. Energy spectrum of a system with L = 8. For valuesof L in the sequence L = 4 n , there is an extra exceptionalpoint slightly above γ = 2. The localized eigenstate appearsbeyond this new exceptional point. -2.00.02.0 R e ( E ) γ I m ( E ) FIG. 13. Energy spectrum for L = 7. The impurity is on thecentral site, q = 4. -2.00.02.0 R e ( E ) γ I m ( E ) FIG. 14. Energy spectrum for L = 9. The impurity is on thecentral site, q = 5. is only a single exceptional point. This appears to bea third-order exceptional point, and appears at a value γ >
2. An example is shown in Fig. 13, for L = 7. Asthe system size tends to infinity, the location of the pointtends to γ →
2. There is always a single eigenvalue thathas a zero real component — the two other eigenvalueswith real parts closest to zero merge with this at theexceptional point.Finally, for L = 4 n + 1, there appears to be no ex-ceptional points; nevertheless, at large γ the eigenvalues1pair up gradually. An example is shown in Fig. 14 for L = 9. A single eigenvalue remains unpaired with zeroreal component. Although this does not merge with anyother eigenvalue, around γ ≈ γ ,indicating that the corresponding eigenstate becomes lo-calized.In summary, although there are differences in detailbetween the four cases, there is always a bound state atlarge γ , and around γ = 2 there is always some reorgani-zation of the spectrum. With increasing L , the locationof these features converge toward γ = 2. Appendix D: Effect of impurity location
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