Abstract
Critical systems represent physical boundaries between different phases of matter and have been intensely studied for their universality and rich physics. Yet, with the rise of nonHermitian studies, fundamental concepts underpinning critical systems  like band gaps and locality  are increasingly called into question. This work uncovers a new class of criticality where eigenenergies and eigenstates of nonHermitian lattice systems jump discontinuously across a critical point in the thermodynamic limit, unlike established critical scenarios with spectrum remaining continuous across a transition. Such critical behavior, dubbed the “critical nonHermitian skin effect”, arises whenever subsystems with dissimilar nonreciprocal accumulations are coupled, however weakly. This indicates, as elaborated with the generalized Brillouin zone approach, that the thermodynamic and zerocoupling limits are not exchangeable, and that even a large system can be qualitatively different from its thermodynamic limit. Examples with anomalous scaling behavior are presented as manifestations of the critical nonHermitian skin effect in finitesize systems. More spectacularly, topological ingap modes can even be induced by changing the system size. We provide an explicit proposal for detecting the critical nonHermitian skin effect in an RLC circuit setup, which also directly carries over to established setups in nonHermitian optics and mechanics.
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Introduction
Lying at the boundary between distinct phases, critical systems exhibit a wide range of interesting universal properties from divergent susceptibilities to anomalous scaling behavior. They have broad ramifications in conformal and statistical field theory^{1,2,3,4}, Schramm–Loewner evolution^{5,6}, entanglement entropy (EE)^{7,8,9,10,11,12,13,14}, and many other contexts. Recently, concepts crucial to criticalities—like band gaps and localization—have been challenged by studies of nonHermitian systems^{15,16} exhibiting exceptional points^{17,18,19,20,21,22,23,24,25,26,27} or the nonHermitian skin effect (NHSE), which are characterized by enigmatic bulkboundary correspondence (BBC) violations, robustdirected amplifications, discontinuous Berry curvature, and anomalous transport behavior^{28,29,30,31,32,33,34,35,36,37,38,39,40}.
We uncover here a class of criticality, dubbed the “critical nonHermitian skin effect (CNHSE)”, where the eigenenergies and eigenstates in the thermodynamic limit “jump” between different skin solutions discontinuously across the critical point. This is distinct from previously known phase transitions (Hermitian and nonHermitian) (Fig. 1), where the eigenenergy spectrum can be continuously interpolated across the two bordering phases. A CNHSE transition, by contrast, is characterized by a discontinuous jump between two different complex spectra along with two different sets of eigenstates. As elaborated below, this behavior appears generically whenever systems of dissimilar NHSE localization lengths are coupled, no matter how weakly. Importantly, at experimentally accessible finite system sizes, the jump smooths out into an interpolation between the two phases in a strongly sizedependent manner, such that the system may exhibit qualitatively different properties, i.e., real vs. complex spectrum or presence/absence of topological modes at different system sizes. Being strongly affected by minute perturbations around the critical point, such behavior may prove useful in sensing applications^{41,42}.
Results
Hints of the critical nonHermitian skin effect from the general Brillouin zone
In nonHermitian systems with unbalanced gain and loss, the spectra under periodic boundary conditions (PBCs) and openboundary conditions (OBCs) can be very different^{28,29,31,43,44,45}. Indeed, under OBC, eigenstates due to NHSE can exponentially localize at a boundary, in contrast to Bloch states under PBCs. This also explains the possible violation of the BBC, taken for granted in Hermitian settings.
The celebrated GBZ formalism aims to restore the BBC through a complex momentum deformation^{29,30,31,36,37,38}. Rigorously applicable for bounded but infinitely large systems, it has however been an open question whether the GBZ can still accurately describe finitesize systems. The GBZ of a momentumspace Hamiltonian H(z), z = e^{ik} can be derived from its characteristic Laurent polynomial (energy eigenequation)
where E is the eigenenergy. While the ordinary BZ is given by the span of allowed real quasimomenta k, the GBZ is defined by the complex analytically continued momentum k → k + iκ(k), with the NHSE inverse decay length \(\kappa (k)=\mathrm{log}\, z\) determined by the smallest complex deformation z → e^{ik}e^{−κ(k)} such that f(z, E) possesses a pair of zeros z_{μ}, z_{ν} satisfying ∣z_{μ}∣ = ∣z_{ν}∣ for the same E^{29,31,38}. Due to the double degeneracy of states with equal asymptotic decay rate at these E, there exist a pair of eigenstates ψ_{μ}, ψ_{ν} that can superpose to satisfy OBCs, i.e., zero net amplitude at both boundaries. As such, provided that the characteristic polynomial is not reducible, the OBC spectrum in the thermodynamic limit (denoted as E_{∞}) can be obtained from the PBC spectrum via E(e^{ik}) → E(e^{ik}e^{−κ(k)}), apart from isolated topological modes. Thus it is often claimed that the BBC is “restored” in the GBZ defined by k → k + iκ(k) or, at the operator level, with the surrogate Hamiltonian H(e^{ik}) → H(e^{ik}e^{−κ(k)})^{38}. In general, different E (energy band) solutions can admit different functional forms of κ(k), leading to banddependent GBZs that have recently also been described with the auxiliary GBZ formalism^{37}. Since e^{ik}e^{−κ(k)} is generically nonanalytic, it represents effectively nonlocal hopping terms^{38}. As such, the GBZ description challenges the very notion of locality, which is central to critical systems, by effectively “unraveling” the realspace eigenstate accumulation through replacing local hoppings with effectively nonlocal ones.
Due to the robustness of the NHSE, eigenspectra predicted from the GBZ typically are approached rapidly by the exact numerically obtained OBC spectra even for small system sizes (\({\mathcal{O}}(1{0}^{1})\) sites). In principle, the convergence should be exact in the thermodynamic limit, but in practical computations, floatingpoint errors ϵ_{0} are continuously amplified as they propagate across the system. We hence expect accurate numerical spectra only when \(L\,<\mathrm{log}\,({\epsilon }_{0})/\max (\kappa )\), a condition always checked to be satisfied here to ensure that physical phenomena presented below are not due to numerical errors common in computations with nonreciprocal systems. However, the numerical agreement in eigenspectra between finitesize systems and the GBZ predictions fails spectacularly near a critical point where f(z, E) changes from being reducible to irreducible. To understand the significance of this algebraic property of reducibility, consider a set of coupled irreducible subsystems described by the characteristic polynomial
where f_{i}(z, E) is the characteristic polynomial of the ith subsystem, and f_{0} is a constant that represents the simplest possible form for the subsystem coupling. When f_{0} = 0, f(z, E) completely factorizes into irreducible polynomials, as expected from a Hamiltonian H(z) that blockdiagonalizes into irreducible sectors associated with the individual f_{i}(z, E)’s. In particular, the OBC spectrum of this completely decoupled scenario is derived from the independent κ_{i}(k)’s of each subsystem, each determined by z_{μ}, z_{ν} from the same subsystem.
Yet, a nonzero coupling f_{0}, no matter how small, can have marked physical consequences by hybridizing different sectors of f_{i} significantly. Indeed, such hybridization is inevitable in the thermodynamic limit, with OBC eigenstates formed from superpositions of eigenstates ψ_{μ}, ψ_{ν} from dissimilar subsystems, each corresponding nonBloch momenta \(i\mathrm{log}\,{z}_{\mu /\nu }\). Hence the GBZs, i.e., κ(k)’s of the coupled system, which are defined in the thermodynamic limit, are thus determined by all pairs of ∣z_{μ}∣ = ∣z_{ν}∣ not necessarily from the same subsystem. Therefore, the GBZs in the coupled case, no matter how small is f_{0}, can differ from the decoupled GBZs at f_{0} = 0. That is, the thermodynamic limit and the f_{0} → 0 limit are not exchangeable. However, since an actual finite physical system cannot possibly possess very different spectrum and band structure upon an arbitrarily small variation in its system parameter, the GBZ picture becomes inapplicable when describing finite systems (small or large) in the presence of CNHSE.
Anomalous finitesize scaling from CNHSE
For illustration, we turn to a minimal example of two coupled nonHermitian 1D Hatano–Nelson chains^{46} each containing only nonreciprocal (unbalanced) nearest neighbor (NN) hoppings (Fig. 2a). Its Hamiltonian reads as
with \({g}_{a}(z)={t}_{a}^{+}z+{t}_{a}^{}/z+V\) and \({g}_{b}(z)={t}_{b}^{+}z+{t}_{b}^{}/zV\), \({t}_{a/b}^{\pm }={t}_{1}\pm {\delta }_{a/b}\) being the forward/backward hopping of chains a and b. This model can be also realized with a reciprocal system with the NHSE in a certain parameter regime (Supplementary Note 1). When t_{0} = 0, the two chains are decoupled, and the characteristic polynomial is reducible as f(z, E) = [g_{a}(z) − E][g_{b}(z) − E]. Each factor f_{a/b}(z, E) = g_{a/b}(z) − E determines the skin eigensolutions of its respective chain. However, even an infinitesimal coupling t_{0} ≠ 0 generically makes f(z, E) irreducible, providing that the two chains correspond to different GBZs (see “Methods” section). Specifically, consider the simple case of \({t}_{a}^{+}={t}_{b}^{}=1\) and \({t}_{a}^{}={t}_{b}^{+}=0\). Without couplings (t_{0} = 0), the two chains under OBC respectively yields a Jordanblock Hamiltonian matrix in real space, with the spectrum given by E = ±V. Because the eigenstates of the decoupled chains are exclusively localized at the first or the last site, their GBZs collapse^{45}. By contrast, for any t_{0} ≠ 0, \(f(z,E)={E}^{2}E(z+{z}^{1})+(z+V)({z}^{1}V){t}_{0}^{2}\) is irreducible (here \({t}_{0}^{2}={f}_{0}\) from Eq. (2)), insofar as the eigenenergy roots \(E=\cos k\pm \sqrt{{t}_{0}^{2}+{(V+i\sin k)}^{2}}\) are no longer Laurent polynomials in z = e^{ik} that can be separately interpreted as de facto subsystems with local hoppings. In fact, in higher degree polynomials, an algebraic expression for z may not even exist as implied by the Abel–Ruffini theorem. Importantly, the corresponding OBC E_{∞} spectrum and the GBZ for t_{0} ≠ 0 are now qualitatively different. As derived in the Methods section, setting ∣z_{a}∣ = ∣z_{b}∣ gives OBC spectrum (in the thermodynamic limit): \({E}_{\infty }^{2}=\frac{1{\eta }^{2}}{1+{\eta }^{2}}+{V}^{2}+{t}_{0}^{2}\pm 2\sqrt{{t}_{0}^{2}{\eta }^{2}+{\eta }^{2}{t}_{0}^{2}}/(1+{\eta }^{2})\), with \(\eta \in {\mathbb{R}}\). Clearly, even one now takes the t_{0} → 0 limit, \({E}_{\infty }^{2}\) only simplifies to \({E}_{\infty }^{2}\to {V}^{2}+\frac{1\pm i\eta }{1\mp i\eta }\), which is not the abovementioned OBC spectrum of the two decoupled chains. Likewise, the t_{0} → 0 limit of the coupled GBZ, which can be shown to be the locus of \(z=\pm \sqrt{{V}^{2}+{e}^{i\theta }}V\) and \(z=1/[V\pm \sqrt{{V}^{2}+{e}^{i\theta }}]\), θ ∈ [0, 2π], has nothing in common with the collapsed GBZs of the decoupled case.
This paradoxical singular behavior of GBZs leads to anomalous scaling behavior in finitesize systems that are more relevant to experimental setups. The discontinuous critical transition illustrated above becomes a smooth crossover between the different OBC E_{∞} solutions. As the size N of a coupled system is varied, its physical OBC spectrum interpolates between the decoupled and coupled OBC E_{∞} solutions. As illustrated in Fig. 2b for the twochain model Eq. (3) at small coupling t_{0} = 0.01 (with t_{1} = 0.75 and δ_{a} = −δ_{b} = 0.25 for welldefined skin modes), the OBC spectrum (black dots) changes markedly from N = 10 to 80 unit cells. For small N = 10, the spectrum approximates the OBC E_{∞} (green) for t_{0} = 0 (which lies on the real line), with the associated GBZs given by two perfect circles in the complex plane (Fig. 2c). At large N = 80, the spectrum converges toward the true OBC E_{∞} (red curve) with nonzero coupling, where the associated respective GBZs of the two bands (also shown in Fig. 2c) are much different from the two circles as decoupled GBZs. Indeed, the eigenstates for N = 10 are almost entirely decoupled across the two chains, while those for N = 80 are maximally coupled/decoupled depending on whether they approach the red/green E_{∞} curves. In the intermediate N = 20 case, the OBC spectrum lies far between the two E_{∞}’s, and cannot be characterized by their GBZs. The sizedependent behavior of the OBC spectrum is further elaborated through a spectralflow study^{31} in Supplementary Note 2.
Let us now explain the aboveobserved marked sizedependent spectra via the competition between dissimilarly accumulated skin modes and the couplings across them. The general conditions for such are unveiled in the “Methods” section. In our model (Eq. 3), the inverse decay lengths in chains a, b are given by \({\kappa }_{a/b}=\frac{1}{2}\mathrm{log}\,({t}_{a/b}^{+}/{t}_{a/b}^{})\), which will be dissimilar as long as δ_{a} ≠ δ_{b}. After performing a similarity transformation that rescales each site j by a factor of \({e}^{j{\kappa }_{b}}\), chain b becomes reciprocal with \({\kappa }_{b}^{\prime}=0\) while chain a has a rescaled inverse decay length \({\kappa }_{a}^{\prime}={\kappa }_{a}{\kappa }_{b}\). If \({\kappa }_{a}^{\prime}\,\ne\,0\), chain a always possesses exponentially growing skin modes scaling like \(\,{e}^{{\kappa }_{a}^{\prime}N}\) at one end. As such, the coupling t_{0}, even if being extremely small, still affects the spectrum and eigenstates markedly as the system size N increases, as further elaborated in the “Methods” section.
Scalefree exponential wavefunctions
A hallmark of conventional critical systems is scalefree powerlaw behavior, particularly in the wavefunctions. Interestingly, such scalefree behavior can also be found in the exponentially decaying wavefunctions, i.e., skin modes. Shown in Fig. 3a are the profiles of the slowest decaying eigenstates ψ(x) of H_{2chain} at different system sizes N = 20, 40, 60, and 80, with the horizontal axis normalized by N. These featured eigenstates belong to the top of the central black ring in Fig. 2b, with their distance from the coupled OBC E_{∞} ring (red) decreasing as ~N^{−1}. Unlike usual exponentially decaying wavefunctions with fixed spatial decay length, here ∣ψ(x)∣ ~ e^{−κx} with κ ~ N^{−1} (Fig. 3b), such that the overall profile ψ(x) has no fixed length scale. Such unique scalefree eigenmodes result from the slow critical migration of the eigenstates between E_{∞} solutions (Fig. 2a, inset).
Anomalous correlations and entanglement entropy
The OBC spectra can be gapped for certain system sizes where EE obeys an area law scaling, and then become gapless at other sizes where the EE scaling is replaced by a logarithmic dependence on system size^{47}. This indicates that the CNHSE can lead to an unusual scaling behavior of the EE. Consider for instance the OBC H_{2chain} (Eq. 3) with parameters chosen to gap out the OBC spectrum at small system sizes N. With all \({\rm{Re}}[E]\,<\,0\) states occupied by spinlessfree Fermions, the realspace entanglement entropy S (blue curve in Fig. 3b) exhibits a crossover from the decoupled gapped regime at N ≤ 5 where it remains a constant due to the sizeindependent area of boundaries (two ends), to the gapless regime N > 20 where it approaches the \(\frac{1}{3}\mathrm{log}\,N\) behavior of a gapless system (yellow line). In generic CNHSE scenarios with multiple competing OBC E_{∞} loci, S can scale differently at different system size regimes, choices of fillings, and entanglement cuts, challenging the notion of single welldefined scaling behavior. As further shown in Supplementary Note 3, the twoFermion correlator 〈ψ(1)ψ(x)〉 characterizing the EE also crossovers from rapid exponential decay at small N to 1/x powerlaw decay at large N. Remarkably, the probability of finding another Fermion nearby generally increases drastically when the system is enlarged (with filling fraction maintained).
Sizedependent topological modes
Topological modes are usually associated with bulk invariants in the thermodynamic limit, with finitesize effects having a diminishing role in the face of topological robustness. This intuition is not necessarily true in nonHermitian systems, as hinted from ref. ^{34}, where an infinitesimal instability can cause a Z_{2} topological transition in the thermodynamic limit^{34}. Remarkably, the CNHSE here can cause topological edge modes to appear only at certain system size regimes. Consider replacing the nonreciprocal intrachain couplings of our H_{2chain} model with interchain couplings with nonreciprocity ±δ_{ab} between adjacent unit cells (Fig. 4a), as described by the following CNHSE Su–Schrieffer–Heeger (SSH) model^{48}:
where h_{y}(z) = iδ_{ab}(z + 1/z), h_{z}(z) = V + δ_{−}(z − 1/z), and h_{0}(z) = t_{1}(z + 1/z) + δ_{+}(z − 1/z), with δ_{±} = (δ_{a} ± δ_{b})/2. H_{CNHSESSH} is so named because interestingly, at δ_{−} = δ_{ab}, it can be transformed via a basis rotation σ_{z} → σ_{x} into an extended SSH model with nonreciprocal intercell couplings given by ±2δ_{−} and a uniform nonreciprocal nextnearest neighbor hopping given by t_{1} ± δ_{+} (Supplementary Note 4), which is known to possess a Ztype topologically nontrivial phase.
When δ_{ab} = 0, the system is decoupled into two Hatano–Nelson chains, which must be topologically trivial. The OBC spectrum E_{∞} in the decoupled case and the associated inverse decay length κ are shown in Fig. 4b, c (green curves), with positive/negative κ corresponding to skin modes accumulating population at opposite boundaries. Also shown in Fig. 4b, c (red curves) are E_{∞} in the coupled case and the corresponding κ for the hybridized skin modes. With small N = 20 unit cells in Fig. 4b, the finitesize OBC spectrum (gray dots) qualitatively agrees with the decoupled E_{∞} (green), with a realvalued gap at E = 0 along the \({\rm{Im}}[E]=0\) axis (inset). Upon the size increase to N = 30 and then to N = 40, such a gap first closes on the complex plane and then develops into a point gap with two zeroenergy degenerate modes lying in its center. The Ztype topological origin of such ingap modes is also carefully verified in Supplementary Note 5. The gap closure and then the emergence of ingap topological modes resemble the typical behavior of a topological phase transition. Yet, here it is an intriguing sizeinduced effect. Further, the emergence of ingap modes only requires exponentially weaker interchain coupling (i.e., smaller δ_{ab}/δ_{a}) for larger N, as shown in the “phase” diagram shown in Fig. 4d.
Proposal for circuit demonstration
The CNHSE is most simply realized when the two subsystems have equal and opposite κ values, since the system is then net reciprocal. Consider the RLC circuit as illustrated in Fig. 5. It is governed by Kirchhoff’s law I = JV, where I, V are the input currents and potentials at nodes 1A, 1B, 2A, 2B, ... and J is the circuit Laplacian given, at AC frequency ω = (LC)^{−1/2}, by
where \(\Delta (k)={r}^{1}+2{R}^{1}2({R}^{1}+\omega C)\cos k\). The second line was obtained via a unitary basis transformation \({\sigma }_{y}\to {\tilde{\sigma }}_{y}=U{\sigma }_{y}{U}^{1}={\sigma }_{z}\) that transforms the circuit Laplacian into a form similar to Eq. (3) (with V = 0), which is susceptible to the CNHSE. In this rotated basis, we evidently have two effective chains coupled by −r^{−1}, each with unbalanced gain/loss couplings that give rise to equal and opposite NHSE. Note that RLC components are all reciprocal and cannot realize the nonreciprocal effective chains individually. However, with the basis transformation given above, the two effective chains become entangled in a way such that they are net reciprocal and hence easy to realize with RLC components, as illustrated in Fig. 5.
One can experimentally demonstrate the CNHSE by building copies of the circuit with different numbers of unit cells N (or alternatively by adjusting its length with appropriately placed switches), and mapping their Laplacian (admittance) spectra via established approaches^{49,50,51,52}. For instance, one can systematically connect a current source I to each node α, one node at a time (the current exits through the ground), and measure the resultant electrical potentials V_{β,α} at each node β. The spectrum of J is given by the inverse of the eigenvalues of the matrix V_{β,α}/I. In the presence of the CNHSE, the spectral plots for different N should qualitatively resemble that in Fig. 2b, since Eq. (5) is of the form of Eq. (3). Due to the robustness of the skin effect, component uncertainties in an actual experiment should minimally affect the resultant spectrum, as verified by simulation results presented in Supplementary Note 6. In particular, the circuit Laplacian spectra, which manifest the CNHSE, are almost undisturbed by uncertainty tolerances of up to 20%.
Discussion
In mathematical terms, the CNHSE arises when the energy eigenequation exhibits an algebraic singularity that leads to inequivalent auxiliary GBZs across the transition. The CNHSE heralds a whole class of discontinuous critical phase transitions with rich anomalous scaling behavior, challenging traditional associations of criticality with scalefree behavior. Even a vanishingly small coupling between dissimilar skin modes can be consequential as the system size increases. This insight is much relevant to sensing and switching applications. Beyond our twochain models, there are other scenarios that can engineer coupling between subsystems of dissimilar NHSE length scales and hence yield CNHSE (e.g., see “Methods” section for a discussion of general twoband models). In particular, we anticipate fruitful investigations in various experimentally feasible settings such as electric circuits^{53,54,55,56}, cold atom systems^{57,58}, photonic quantum walks^{59}, and metamaterials^{41,60}, all of which are investigated with finitesize systems and hence highly relevant to the CNHSE.
Methods
Discontinuous transition of GBZ in twochain models
The discontinuous transition induced by an infinitesimal transverse coupling in the thermodynamic limit, and also the crossover in a finite system, exist only when the two decoupled chains have different κ of their OBC skin solutions. To see this, we consider a general twochain model described by Hamiltonian
where g_{a,b}(z) only contains terms with nonzero order of z. When decoupled, the two chains correspond to the polynomials g_{a,b}(z) + V_{a,b}, respectively, and possess the same κ solutions when and only when g_{b}(z) = cg_{a}(z), with c a nonzero coefficient. When a nonzero transverse coupling t_{0} is introduced, the characteristic polynomial of the twochain system can always be written in the form of
where A, B are two coefficients determined by other parameters. Therefore for two chains with the same κ solutions, a transverse coupling t_{0} only modifies the energy offset between them, without inducing a transition of skin solutions.
Nevertheless, the above factorization does not hold when the coupling term t_{0} is zdependent, corresponding to interchain couplings between different unit cells. Under this condition, P_{c}(z) cannot be factorized into two subpolynomials of g_{a}(z) and g_{b}(z) = cg_{a}(z), meaning that the skin solution is changed for the system.
GBZ solutions E_{∞} for the twochain model
For analytic tractability, we consider the case of Eq. 3 of the main text with \({t}_{a}^{+}={t}_{b}^{}=1\) and \({t}_{a}^{}={t}_{b}^{+}=0\) (i.e., t_{1} = δ_{a} = −δ_{b} = 0.5), but nonzero b and V. We obtain
with the characteristic polynomial given by
To find the GBZ solutions E_{∞} for comparison with the actual OBC solutions, we solve for roots ∣z_{+}∣ = ∣z_{−}∣ of f(z, E) = 0 (with \(\Sigma ={E}^{2}{V}^{2}{t}_{0}^{2}+1\)):
For ∣z_{+}∣ = ∣z_{−}∣ to hold, the square root quantity must differ from Σ by a complex argument of π/2^{38} i.e.,
where \(\eta \in {\mathbb{R}}\). Simplifying, we obtain \(\Sigma =\frac{2}{1+{\eta }^{2}}\left(1\pm \sqrt{{t}_{0}^{2}+{\eta }^{2}({t}_{0}^{2}1)}\right)\) or, in terms of \({E}^{2}\to {E}_{\infty }^{2}\),
as in the main text, with η tracing out a oneparameter continuous spectrum. The GBZ can be numerically obtained by substituting Eq. (12) into the expression for z_{±} in Eq. (10) with E = E_{∞}. From that, we obtain two momentum values \({k}_{\pm }={\rm{Re}}[i\mathrm{log}\,{z}_{\pm }]\) with \(\kappa ({k}_{+})=\kappa ({k}_{})=\mathrm{log}\, {z}_{+} =\mathrm{log}\, {z}_{}\) as the inverse length scales. Note however that because of the proximity to the t_{0} = 0 critical point, this value of κ(k_{±}) is significantly different from the actual inverse OBC skin depth for a large range of finite system sizes.
Dissimilar skin modes in general twoband models
In a more general picture, the CNHSE and the sizedependent variation may exist when different parts of the system have dissimilar skin accumulation of eigenmodes. In the twochain model, we mainly consider regimes with small interchain couplings, thus the two energy bands (overlapped or connected in most cases) with dissimilar skin modes are mostly given by one of the two chains respectively. To unveil the condition of having dissimilar skin modes in a general twoband system, we consider an arbitrary twoband system described by a nonBloch Hamiltonian \(H(z)={h}_{0}(z){\mathbb{I}}+{\sum }_{n = 1,2,3}{h}_{n}(z){\sigma }_{n}\), with z = e^{ik}e^{−κ(k)}, and κ(k) a complex deformation of momentum k describing the NHSE. Its characteristic polynomial is given by
with \(P(z)={\sum }_{n = 1,2,3}{h}_{n}^{2}(z)\). NHSE can be described by a GBZ where the solutions of f(z, E) = 0 satisfy E_{α}(z_{μ}) = E_{α}(z_{ν}) with ∣z_{μ}∣ = ∣z_{ν}∣ and α = ± the band index, and \(\kappa (k)=\mathrm{log}\, z\) gives the inverse decay length. Conventionally, NHSE is studied mostly for a system with only nonzero h_{0}(z) (i.e., a oneband model) or P(z) (e.g., the nonreciprocal SSH model), where the zeros of f(z, E) lead to E_{±} = h_{0}(z) and \({E}_{\pm }^{2}=P(z)\), respectively. In either case, we can see that the two bands of E_{±} must have the same inverse skin localization depth κ(k), as E_{α}(z_{μ}) = E_{α}(z_{ν}) must be satisfied for α = ± with the same z_{μ,ν}. To have dissimilar skin modes for the two bands, h_{0}(z) and P(z) must both be nonvanishing, and possess different skin solutions. That is, although h_{0}(z_{μ}) = h_{0}(z_{ν}) and \(P({z}_{\mu ^{\prime} })={h}_{0}({z}_{\nu ^{\prime} })\) can still be satisfied with ∣z_{μ}∣ = ∣z_{ν}∣ and \( {z}_{\mu ^{\prime} } = {z}_{\nu ^{\prime} }\), we cannot have \({z}_{\mu }=z^{\prime}\) and \({z}_{\nu }=z^{\prime}\) at the same time, otherwise the same κ(k) can be obtained for the two bands.
Competition between skin localization and interchain coupling
As mentioned in the main text, if two coupling chains have inverse NHSE decay lengths (nonHermitian localization length scales) κ_{a}, κ_{b}, a change of basis will bring their coupling to be effective between a chain with no NHSE, and another with an effective skin depth κ_{a} − κ_{b}. Since that entails exponentially growing skin modes scaling like \({e}^{({\kappa }_{a}{\kappa }_{b})N}\) at one end, we expect the effect of even an infinitesimally small interchain coupling t_{0} to scale exponentially with N, and eventually change the OBC spectrum substantially.
Consider increasing the interchain coupling t_{0} in our twochain model (Eq. 3 of main text) from zero. At sufficiently small t_{0}, we have two practically independent OBC Hatano–Nelson chains with real spectra. Their infinitesimal coupling only shifts their eigenenergies slightly along the real line. But at a critical t_{0} = t_{c}, the OBC spectrum is rendered complex as one or more pairs of eigenenergies coalesce and repel along in the imaginary direction. Shown in Fig. 6a is the inverse exponential scaling of the critical t_{0} = t_{c} with N. We observe that \({t}_{\mathrm{c}}^{2}{e}^{({\kappa }_{a}{\kappa }_{b})N} \sim {\mathcal{O}}(1)\), in agreement with the intuitive expectation that t_{c} should scale inverse exponentially with N because the effect of t_{0} scales exponentially with N. Yet, the fact that \({t}_{\mathrm{c}}^{2} \sim {e}^{({\kappa }_{a}{\kappa }_{b})N}\) signifies that the CNHSE is fundamentally a nonperturbative effect since it differs from \({t}_{\mathrm{c}} \sim {e}^{({\kappa }_{a}{\kappa }_{b})N}\) as expected from firstorder perturbation theory with left and right eigenstates that are oppositely exponentially localized spatially.
The scaling behavior of \({e}^{({\kappa }_{a}{\kappa }_{b})N}\) also suggests that increasing N has similar consequences as increasing the nonreciprocity in the system, the strength of which is reflected by the absolute value of (κ_{a} − κ_{b}). Therefore it is also expected that the CNHSE shall emerge when we enhance the nonreciprocity but fix N. In Fig. 6b, we show the inverse exponential scaling of the critical t_{0} = t_{c} with κ_{a} − κ_{b}, where the inverse NHSE decay lengths are given by
for the two decoupled chains. The scaling behavior versus κ_{a} − κ_{b} further confirms that \({t}_{\mathrm{c}}^{2} \sim {e}^{({\kappa }_{a}{\kappa }_{b})N}\).
Data availability
Raw numerical data from the plots presented are available from the authors upon request.
Code availability
Though not essential to the central conclusions of this work, computer codes for generating our figures are available from L.L. and C.H.L. upon reasonable request.
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Acknowledgements
We thank Nobuyuki Okuma and Zhesen Yang for helpful discussions. J.G. acknowledges support from Singapore NRF Grant No. NRFNRFI201704 (WBS No. R144000378281).
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L.L. and C.H.L. contributed equally to this work. L.L. carried out preliminary studies and all authors participated in the discussions. S.M. helped to improve the design of lattice models. C.H.L. refined this project extensively. L.L. and C.H.L. carried out additional theoretical and computational studies. All authors discussed the results and participated in the writing of the manuscript. J.G. supervised the project and finalized the manuscript.
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Li, L., Lee, C.H., Mu, S. et al. Critical nonHermitian skin effect. Nat Commun 11, 5491 (2020). https://doi.org/10.1038/s41467020189174
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DOI: https://doi.org/10.1038/s41467020189174
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