Filling up complex spectral regions through non-Hermitian disordered chains

Hui Jiang and Ching Hua Lee

Department of Physics,National University of Singapore,Singapore 117551,Republic of Singapore

Keywords: non-Hermitian skin effect,disordered chain,disorder localization,non-Hermitian spectra

1. Introduction

The spectra of non-Hermitian systems lie in the 2D complex plane,and can exhibit intriguing geometric and topological spectral transitions.[1–16]In particular, it is known[17–20]that if the spectrum under periodic boundary conditions(PBCs)is a loop that encloses a nonvanishing region,the spectrum of the same system under semi-infinite boundary conditions(SIBCs)will fill up the interior of this loop. This intriguing fact is due to the non-local nature of the non-Hermitian skin effect (NHSE), which has inspired numerous theoretical and experimental[21–25,25,25–35]developments and challenged various longheld paradigms in physics. The NHSE, which arises in non-Hermitian lattices with broken reciprocity, amplifies and pumps all states towards a boundary, such that the effects of boundary hoppings become non-perturbatively large.[36–68]

Intuitively, a semi-infinite 1D lattice system can have a spectrum that fills up a 2D region because its eigenstates only need to satisfy boundaries conditions on one side, and are hence free to accumulate against it with any spatial decay length. As such, an eigenstate is characterized by two continuous variables: its wavenumber and decay length,the latter which possesses no Hermitian analog. However, true semiinfinite systems can neither be numerically nor experimentally simulated,and their non-Hermitian properties have so far been mathematical curiosities.

In this work,we show how to construct finite 1D systems whose spectra nevertheless fill up the 2D interiors of their PBC spectra. This is achieved with appropriately designed disordered couplings which mathematically simulate the effects of semi-infinite boundaries,namely,the co-existence of a continuum of different decay length scales of the skin eigenstates.Notably,the density of states in the 2D complex plane can be fine-tuned towards a variety of desired profiles by tuning the disorder distribution. While it is arguably easy to fill up a 2D spectral region with the eigenenergies of many separate (uncoupled)1D chains,doing so with a single chain is nontrivial due to the non-local effects of NHSE accumulation that can propagate across very distant parts of the chain.[55,56]As such,our construction can be construed as a stochastic means to subtly control the distribution of skin decay lengths,and also the propagation of skin accumulation tendencies.

2. Exploring the complex energy plane by tuning boundary conditions

The starting point of our work is the observation that,by modifying the boundary conditions of a non-Hermitian system with unbalanced hoppings, we can access a continuum of complex energy spectra,and thus sweep across the interior of the PBC loop.[54–56,69]This is thanks to the non-Hermitian lattice’s extreme sensitivity to the boundary conditions,a phenomenon commonly known as the non-Hermitian skin effect.In the extreme limit of open boundary conditions(OBCs),the eigenenergy continuum is given by whereE(k) is the dispersion of the original HamiltonianH(k), andκcis the imaginary part of the momentum that represents the boundary localization of the former bulk eigenstates,[20–23,37–40,70,71]which now decay like～e-κcx. In general,κccan be a complicated function ofk, and is determined by the condition that ˉE(k),k ∈[0,2π)does not enclose any nonzero area.[23,24,36–38,71–76]But for the purpose of this work,κcremains as a constant.

Interpolatingκfromκcto 0,the energy spectrumE(k+iκ) will start off as the OBC energy spectrum, and continuous evolve till it coincides with the PBC spectrum atκ=0,in the process passing through the entire region enclosed by the PBC loop. Below,we describe how to physically access such intermediate values ofκby tuning the boundary hoppings.

Using the Hatano–Nelson model[14,43,69,77–79]for convenient illustration,since it only has nearest neighbor unbalancing hoppings,we have

withE=tz+t′/zandz=exp(-κ+ik). Solving this characteristic equation givesκ,as shown in Table 1,which can then be substituted into ˉE=E(k+iκ) to yield the spectrum (see Appendix A).It is seen that under OBCs,κonly depends on the relative amplitudest/t′of the left/right hoppings. However, when the boundary and bulk hoppings are not equal,κwill depend either ont/μort′/μ′, depending on the relative strength oftandt′as well as whetherμμ′ortt′is larger.

Table 1. Approximate analytic forms of κ of the model Eq. (1) in various regimes, for large L. Here, κc =1/2log(t/t′), the value of κ where the spectrum collapses into the OBC spectrum that encloses zero area. With appropriate tuning of boundary hoppings μ,μ′,we will be able to obtain any κ that lies between the PBC and OBC cases,i.e.,[0,κc].

The upshot is that if we take the boundary hoppingsμ,μ′to be random numbers from(0,t)and(0,t′),the energy spectrum will fall within the PBC spectral loop and, after multiple random trials,the combined energy spectra will fill up the loop,as shown in Fig.1(b). So far,this is not very surprising,since we are using an ensemble of 1D systems to fill up a 2D region. In the following sessions, we shall demonstrate how we can instead construct a single 1D system whose eigenenergies fill up the interior of a spectral loop.

Fig.1. Filling up a spectral ellipse with a large number of separate Hatano–Nelson chains with random boundary hoppings. Panels(a1)and(a2)show the random distributions of μ,μ′ boundary hoppings, while (b) shows the combined energy spectra E (red crosses) from 40 different random trials,which fill up the interior of the elliptical PBC loop of the model given by Eq.(2)with parameters t =2,t′=1,L=50. From Table 1,only the distribution P(μ)affects the filling,which here forms“bands”of approximately equal density due to the chosen step-like distribution of P(μ). The energies from an illustrative(μ,μ′)pair is indicated in light blue.

3. Filling 2D spectral region with a 1D non-Hermitian lattice

Previously, we filled the 2D spectral region enclosed by the PBC loop via an ensemble of 1D chains. However, to do the same with a single 1D chain is a nontrivial feat, even for one constructed by concatenating many 1D chains. This is because the NHSE relentlessly pump all states towards one direction,including across the concatenated chains,thereby fundamentally modifying their individual nature. Specifically,we expect different behavior from boundary hoppings that close up the individual 1D chains, compared to those that connect many 1D chains into one long periodic chain.

To construct a bona fide 1D lattice whose spectrum does fill up a 2D spectral region, we instead consider a modelH[Fig.2(a)]of the form whereαis the chain index, andnlabels the sites in theα-th chain, whose eigenstates|ψα〉are localized at boundaries ofαchain — skin states. The second termHbcontains all the random couplings between adjacent chainsαandα+1, and takes the form

andΞα,Ξ′αareM×Mrandom matrices whose elements represent their random couplings between the chains. Together,they connect the chains into a long PBC loop via 2NM2random couplings. Even for the weak random coupling, the eigenstates are localized states rather than skin states, which can be importance weighted combination of|ψα〉. And the system with localized eigenstates becomes insensitive to the boundary conditions. As elaborated later,the coupling lengthM ∈[1,L] profoundly affects the filling, as demonstrated in Fig.2(c)and subsequent figures.

We choose the random couplings matrix elementsξeiφofΞα,Ξ′αfrom a random ensemble with amplitudesξGaussian distributed with mean 0 and varianceσ,and phaseφuniformly distributed[Fig.2(a)].Taking sizeNL ?1 of system to avoid random uncertainties,we set the results as single disorder realization.Since the spectrum changes dramatically when the inter-chain couplings are small, analogous to the boundary couplings discussed in the previous section, the coupling amplitude varianceσsignificantly affects the filling behavior[Fig.2(b)].

To quantify how completely and evenly the spectral loop region is filled by the eigenenergies, we introduce two metrics: Cr, the coverage rate and Pr, the participation rate. To define the coverage rate(Cr),we divide the interior region of the PBC loop(spectrum ofHαunder PBCs)intoNparts(for a largeN);then we count the numberN′of parts which contain one or more eigenvalues ofHEq. (4) within itself. The ratio

is the coverage rate Cr. The larger the Cr, the more complete is the energy filling;a small Cr indicates that the filling occurs very inhomogeneously. This definition of Cr remains meaningful even when the PBC loop is irregular and it is hard to directly see how well it is filled by the eigenenergies. Next,we also define the participation rate(Pr),which represents the fraction of eigenenergies ofHthat are within the PBC loop ofHα,i.e.,

where num(E)is the number of eigenenergies within the PBC loop, andNLis the total number of sites in our lattice model Eq.(4). A high participation rate indicates that few eigenenergies are outside the PBC loop.

Fig. 2. Filling of spectral region from a single long chain with disordered couplings. (a) Structure of our 1D chain model Eqs.(4–6), which consists of N Hatano–Nelson chain segments(green and yellow)that are randomly coupled to adjacent chains via their first and last M sites, as given by matrices Ξ and Ξ′. The hopping amplitudes ξ ∈(0,σ) are Gaussian random distributed,and phases φ are uniformly distributed. (b)The dependence of the coverage rate(Cr)Eq.(8)and participation rate(Pr)Eq.(9)on the hopping amplitude variance σ,for different M.While Cr is generally insensitive to σ,it improves significantly with M. Pr remains almost complete at 1 for σ ≤0.1,beyond which it decreases. Panels(c1)–(c4)show the energy spectrum at different σ and M combinations as indicated in(b);as M increases,the interior eigenvalues within the spectral loop proliferate,finally resulting in a filled spectral interior. Excessive σ, however, causes the filling to exceed the PBC loop boundaries. (d)The best filling spectrum from(b),with representative eigenstates shown in(e). Generally,more localized states occur deeper in the loop interior. EOBC and EPBC refer to the eigenenergies of base model Hα Eq.(5).

From Fig.2(b),it is evident that the longer the rangeMof the inter-chain random couplings, the better the coverage Cr.This is because short-ranged inter-chain couplings inΞα,Ξ′αonly couple sites close to the end of the chains,and the entire lattice is still akin to a long PBC chain with somewhat complicated couplings. Indeed, at smallM, the spectrum ofHis still a well-defined loop[Fig.2(c1)],which by definition covers the PBC interior region very poorly. AsMincreases,more distant sites between adjacent chains are randomly coupled,and some eigenenergies start to appear in the interior of the spectral loop ofH[Fig. 2(c2)]. They almost always appear in its interior because disorder generically breaks translation invariance,thereby allowing for localized skin mode accumulation.These localized modes have larger effectiveκby virtue of their shorter decay lengths,and hence tend towards the interior of the spectral loop. This is further examined in the next section;here we mention that intuitively,we expect these random coupling-induced localized skin modes energies to be closer to the real line because the net effect of many random couplings is to prevent any state from being amplified or attenuated too many times consecutively. Finally, the coverage Cr reaches its maximum[Fig.2(b)]whenM=L,i.e.,when all sites in our lattice Eqs.(4)–(6)are randomly coupled. In this limit,all vestiges of a PBC spectral loop are gone,and we observe a continuous eigenstate density within the PBC spectral loop ofHα.

The filling of the 2D spectral region can be further optimized by tweaking the probability distribution of the individual inter-chain hopping strengthsξeiφ. We shall keep the phaseφas being uniformly distributed,and just vary the varianceσof the Gaussian-distributed amplitudeξ(with zero mean). As shown in Fig.2(c), the cloud of eigenenergies becomes larger asσgrows,since stronger random couplings invariably perturb their“trapped”localized skin modes energies more strongly. We obtain the best filling withM=L, and when the eigenenergy cloud just fills the PBC spectral loop ofHαwithout going out of it. This occurs at a critical value ofσ=σc,where the participation ratio Pr just starts to decrease from 1[Fig.2(b)].

So far, we have only used the Hatano–Nelson model Eq. (5) as the base modelHα. As a model with a simple elliptical PBC loop,it is an appropriate paradigmatic model for disorder spectral filling. However, our approach also works for generic models with nontrivial spectral winding loops.

For instance,let us add to the disorder couplingHbEq.(6)a different base Hamiltonian given by[40]

which has hoppingsasites to the left with amplitudet, and hoppingsbsites to the right with amplitudet′, as illustrated in Fig. 3(a). TheseHαformNidentical chains with open boundary conditions, connected to each other by disordered couplings just as before Eqs.(4)and(6).

In the same way, we can also get the best filling conditions. The larger the Cr, the better the spectral region filling,in Fig. 3(c). There have the same results with the previous Hatano–Nelson model,the value ofMis closer toL,the better PBC loop-filled energy spectrum. Similarly, the model with the best loop-filled energy spectrum can be obtained(M=L),and the optimal distribution of random couplingsξexp(iφ)for Fig.3(d)can be obtained by considering the coverage rate(Cr)and participation rate(Pr).

Fig.3.Filling of spectral region from a single long heterogeneous chain with disordered couplings. (a)Base lattice structure of our model Eq.(10)with next-nearest neighbor hoppings; random hoppings linking them (shown in Fig.2)having amplitudes ξ ∈(0,σ)that are Gaussian random distributed,and phases φ that are uniformly distributed. (b)Division of the irregularly shaped PBC loop into large N parts for the computation of Cr. (c)Dependence of Cr[Eq.(8)]and Pr[Eq.(9)]on the hopping amplitude variance σ,for different M. While Cr increases only slightly with σ,it improves significantly with M. Pr remains almost complete at 1 for σ ≤0.15,beyond which it decreases. Panels(c1)–(c4)show the spectra at different σ and M combinations as indicated in(c);as M or σ increases,interior eigenvalues within the spectral loop proliferate,finally resulting in a filled spectral interior. (d)The best filling spectrum from(c),with representative eigenstates shown in(e). Generally,more localized states occur deeper in the loop interior. EOBC and EPBC refer to the eigenenergies of base model Hα Eq.(10).

4. Spectral filling and skin localization

A major inspiration for our approach has been the semiinfinite boundary condition, which is consistent with the coexistence of a continuum of eigenstate decay lengths (skin depths).[19–21,23,36–39,41,42,45,71,80]Within our approach, the filling of the spectral loop is indeed intimately related to the realization of a continuum of spatial localization lengths. To quantify localization, we recall the definition of the inverse participation ratio(IPR)[15,81–85]

with|Ψ〉= ∑α,nψα,n|α,n〉a chosen eigenstate of the full HamiltonianH. If a state is perfectly localized on only one site,the IPR takes the maximal value 1. In contrast, if a state is uniformly spread overNLstates,IPR=(NL)-1→0.

We next consider what can be reasonable expectation of the extent of localization for states within the spectral loop.States in a disordered system are invariably randomly shaped,but it is conceivable that they inherit the localization length of the clean background systemHα,stochastically speaking. For a particular state with energyE,its expected(clean)skin depthκ-1is given through

wherekis an unimportant wavenumber andEαis the energy of the 1-component modelHα. On the PBC loop, the clean system harbors Bloch states,andκ=0. Generally,states further from the PBC loop correspond to largerκ,and should be more localized.

In Fig.4,we indeed observe a correlation between weaker expected localizationκand weaker numerically determined localization (smaller IPR) of the actual eigenstates. Conversely, for eigenenergies closer toEOBC, whereκ →κc, the eigenstates tend to be most localized(large IPR,red). This is also consistently observed for the base modelHαEq.(10)with asymmetric hopping distances,albeit with slightly weaker correlation.

Fig.4. Locality(IPR)vs. inverse skin depth κ. (a)IPR Eq.(11)and(b)κ Eq. (12) of each eigenstate of our disordered chain Eqs. (4)–(6) at parameters of best filling (σ =0.1,M =20). Generally, small κ (green), which correspond to delocalized Bloch states in the base model Eq. (5), indeed correlate with relatively delocalized states in the disordered chain(low IPR,green). Panels (c) and (d) show the same plots with the base Hamiltonian described by Eq.(10),with similar conclusions.

5. Discussion

In this work, we have devised a way to construct disordered 1D non-Hermitian chains Eqs.(4)–(6)exhibiting eigenspectra that fill up the interiors of 2D regions in the complex energy plane.The filling extent and density can be adjusted by tuning the probability distributions of the random couplings,and effectively simulates the effects of SIBCs,which are physically unattainable.

It is interesting to compare our mechanism with that of random matrices, i.e., the Gaussian unitary, orthogonal and symplectic ensembles, etc, which can also produce evenly spaced eigenvalues within circular regions in the complex plane, akin to electrons in a quantum Hall fluid.[86–94]What is markedly different is the extent of non-locality required: In our setup [Fig. 2(a)], random inter-segment couplings extend across at mostMsites,and in the thermodynamic limit of largeN,the entire chain can still be considered as a long PBC chain withNnearest-neighbor coupled unit cells,each having a fixedL ≥Mnumber of components. However,in classical random matrix ensembles,the random elements represent all possible couplings, which in this context can be as far asNL/2 number of sites. Furthermore,our approach can easily be generalized to fill up arbitrarily-shaped regions by choosing the base HamiltonianHαwith a similarly shaped PBC loop.

From a more general viewpoint, our constructive approach offers an avenue for stochastically “augmenting” the dimensionality of a 1D system,such that it possesses characteristics normally associated with 2D systems,such as 2D density of states. The probability distributions associated with the random couplings provide additional degrees of freedom that may ultimately emulate extra dimensions. Finally, we mention that our models can readily be physically implemented in media that admit long-ranged couplings, such as classical electrical circuits[13,16,21,25–27,32,34,35,95–107]and quantum computers,[10,108–119]and with some adaptation even Rydberg atom lattices with long-ranged interactions.[117–129]Since the non-local couplings only need to be randomly distributed according to certain loosely defined distributions, our approach is intrinsically tolerant to significant levels of noise.

Appendix A: Detailed analysis of sensitivity to boundary conditions

We study how the simplest illustrative non-Hermitian model Eq.(2)

Combining the above results, we obtain Table 1 in the main text.

By comparing with numerically obtained spectra,Fig.A1 verifies the correctness of the approximate analytical results obtained above and given in Table 1.

Now,if the boundary hoppingsμandμ′were to be random numbers from(0,t)and(0,t′),the PBC energy spectrum would be ellipses of all different aspect ratios, and after multiple random trials, the combined energy spectrum would fill up the PBC loop, as shown in Fig. A2(a2). Note that this filling is sensitive to the direction of skin mode accumulation,so ift ＞t′, only the case in Fig. A2(a2) and not that of Fig. A2(b2) will occupy the interior of the PBC loop. To maintain an approximately uniform filling density, we have concocted a step-like distribution given by

Fig. A1. Near-perfect agreement of analytically approximated spectra (black) with numerical spectra (red) for the 4 cases discussed above: μμ′ ?tt′ (a1), (a2); μμ′ ?tt′ (b1), (b2); μ /=0,μ′ =0 (c1), (c2); μ =0,μ′ /=0 (d1), (d2). The red crosses represent numerical results of Hamiltonian Eq. (A1), and the black points show the results (analytical approximate solution) from Table 1. Other parameters are t =2, t′ =1, N =100. Since t ＞t′, an exponentially small part of the state can feel μ′ =0 (μ =0), and the system still behaves like it is under PBCs (OBCs), as in (c) and (d).Qualitatively similar conclusions apply to(a)and(b).

Fig.A2. Combined energy spectrum(a2),(b2)of Hamiltonian Eq.(A1)with 2000 random trials of μ,μ′ picked from distributions P as given in(a1),(b1)when t′＜t and μμ′＜tt′,neither E nor z has anything to do with μ′. Other parameters are t=2,t′=1,N=50.

wherexrepresents eitherμorμ′.

Appendix B: Effect of relative strengths of random couplings

We now study the effect of tuning the overall strength of the random couplings inHbby considering the parametrization

We recover the results of the main text whenλ=1, and that of the clean non-Hermitian chain whenλ=0. In the latter limit,κmust correlate poorly with the IPR, since the system is essentially that of the OBC clean system,with only a single inverse skin depth. Note that withλ,the best fillingσamplitude is also rescaled by a factor ofλ.

From Fig.B1, the correlation is poor for smallλ, as expected,since the system is not too different from an OBC system with small amounts of disorder. This is evident in the“flattening”of the spectrum in the complex energy plane. The correlation as well as the filling improves asλincreases.distancesM.

Fig.B1. IPR vs. κ of our model under the rescaling λ of the random couplings. Eigenenergies are colored by the IPR Eq.(11)(a1)–(c1)or κ Eq.(12)(a2)–(c2). Panels(a3)–(c3)show the correlation between IPR and κ. The correlation,although imperfect,is best at larger disorder strengths such as λ ≈0.6.

Fig.B2. IPR vs. κ of our model(λ =1)for different maximal random coupling distances M. Eigenenergies are colored by the IPR Eq.(11)(a1)–(c1)or κ Eq.(12)(a2)–(c2).Panels(a3)–(c3)show the correlation between IPR and κ.The correlation is best at smaller disorder coupling

In general, it is also found from Fig. B2 that the correlation improves asMdecreases. This is not surprising, since smallerMimplies fewer random couplings,thereby increasing the reliability ofκfrom the base HamiltonianHαas a measure of the locality for the entire HamiltonianH.That said,it is still with maximalM=Lthat we obtain the best fillings.