Abstract
Bloch’s theorem was a major milestone that established the principle of bandgaps in crystals. Although it was once believed that bandgaps could form only under conditions of periodicity and longrange correlations for Bloch’s theorem, this restriction was disproven by the discoveries of amorphous media and quasicrystals. While network and liquid models have been suggested for the interpretation of Blochlike waves in disordered media, these approaches based on searching for random networks with bandgaps have failed in the deterministic creation of bandgaps. Here we reveal a deterministic pathway to bandgaps in randomwalk potentials by applying the notion of supersymmetry to the wave equation. Inspired by isospectrality, we follow a methodology in contrast to previous methods: we transform order into disorder while preserving bandgaps. Our approach enables the formation of bandgaps in extremely disordered potentials analogous to Brownian motion, and also allows the tuning of correlations while maintaining identical bandgaps, thereby creating a family of potentials with ‘Blochlike eigenstates’.
Introduction
The isospectral problem posed via the question ‘Can one hear the shape of a drum?’^{1} introduced many fundamental issues regarding the nature of eigenvalues (sound) with respect to potentials (the shapes of drums). Following the demonstration presented in ref. 2, it was shown that it is not possible to hear the shape of a drum because of the existence of different drums (potentials) that produce identical sounds (eigenvalues), namely, isospectral potentials. Although the isospectral problem has deepened our understanding of eigenstates with respect to potentials and raised similar questions in other physical domains^{3}, it has also resulted in various interesting applications such as the detection of quantum phases^{4} and the modelling of anyons^{5}.
The field of supersymmetry^{6} (SUSY) shares various characteristics with the isospectral problem. SUSY, which describes the relationship between bosons and fermions, has been treated as a promising postulate in theoretical particle physics that may complete the standard model^{6}. Although the experimental demonstration of this postulate has encountered serious difficulties and controversy, the concept of SUSY and its basis of elegant mathematical relations have given rise to remarkable opportunities in many other fields, for example, SUSY quantum mechanics^{7} and topological modes^{8}. Recently, techniques from SUSY quantum mechanics have been utilized in the field of optics, thereby enabling novel applications in phase matching and isospectral scattering^{9,10,11,12}, complex potentials with real spectra^{13} and complex Talbot imaging^{14}.
In this paper, we propose a supersymmetric path for the generation of Blochlike waves and bandgaps without the use of Bloch’s theorem^{15}. In contrast to approaches based on an iterative search for random networks^{16,17,18,19} with bandgaps, a deterministic route towards bandgap creation in the case of disordered potentials is achieved based on the fundamental wave equation. This result not only demonstrates that longrange correlation is a sufficient but not a necessary condition for Blochlike waves^{16,17,18,19} but also enables the design of randomwalk potentials with bandgaps. Such designs can facilitate the creation of a family of potentials with ‘Blochlike eigenstates’: identical bandgaps and tuneable longrange correlations, even extending to conditions of extreme disorder analogous to Brownian motion. We demonstrate that the counterintuitive phenomenon of ‘strongly correlated wave behaviours in weakly correlated potentials’ originates from the ordered modulation of potentials based on spatial information regarding the ground state, which is the nature of SUSY. We also show that our approach for Blochlike waves can be extended to multidimensional potentials under a certain condition, allowing highly anisotropic control of disorder.
Results
Relation between eigenstates and potential correlations
To employ the supersymmetric technique^{7,9}, we investigate waves governed by the onedimensional (1D) Schrodingerlike equation, which is applicable to a particle in nonrelativistic quantum mechanics or to a transverse electric mode in optics. Without a loss of generality, we adopt conventional optics notations for the eigenvalue equation H_{o}ψ=γψ, where the Hamiltonian operator H_{o} is
k=i_{x} is the wavevector operator, k_{0} is the freespace wavevector, V_{o}(x)=–[n(x)]^{2} is the optical potential, n(x) is the refractive index profile, ψ is the transverse field profile and n_{eff} is the effective modal index for the eigenvalue . Two independent methods are applied to equation (1) for verification, the Finite Difference Method^{20} (FDM) and the Fourier Grid Hamiltonian^{21} (FGH) method, whereby both yield identical results for the determination of bound states (see the Methods section).
To examine the relationship between wave eigenspectra and the correlations of potentials, three types of randomwalk potentials are analysed: crystals, quasicrystals and disordered potentials, which are generated by adjusting the refractive index profile. Figure 1a represents a 1D binary Fibonacci quasicrystal (the sixth generation with an inflation number, or sequence length, of N=8, substituting A→B and B→BA for each generation using A as the seed), where each element is defined by the gap between the highindex regions: A (or B) for a wider (or narrower) gap. The crystal and the disordered potential are generated using the same definition of elements, while the crystal has an alternating sequence (BABABA…), and the disordered potential has equal probabilities of A and B for each element (that is, it is a Bernoulli random sequence^{22} with probability P=0.5). To quantify the correlation, the Hurst exponent^{23,24} H is introduced (Fig. 1b; see the Methods section). As N increases, both the crystal and the quasicrystal have H values that approach 0 (that is, they exhibit ‘ballistic behaviour’ with strong negative correlations^{24}), in stark contrast to the Bernoulli random potential, which has H∼0.4 (close to ideal Brownian motion, with H=0.5).
Figure 1c–h illustrates the stationary eigenstates for each potential, which are calculated using the FDM and the FGH method. Consistent with previous studies^{16,17,18,19,25,26,27,28,29}, Blochlike waves with wide bandgaps are obtained for the ordered potentials of the crystal (Fig. 1c) and the quasicrystal (Fig. 1d), and the Blochlike nature becomes more apparent with increasing N (Fig. 1f,g). By contrast, no bandgap is observed for the Bernoulli random potential, which lacks any correlations (Fig. 1e), especially for larger N (Fig. 1h); this lack of correlation originates from the broken coherence of this case, which hinders the destructive interference that is necessary for the formation of bandgaps. It should also be noted that many eigenstates are localized within this random potential, exhibiting a phenomenon that is widely known as Anderson localization^{29}.
Supersymmetric transformation for quasiisospectral design
In light of the results in Fig. 1, we now consider the following question: ‘Is it possible to design nearly uncorrelated (or Brownian) potentials with H∼0.5 while preserving the original bandgaps?’. To answer this question positively, we exploit the SUSY transformation to achieve quasiisospectral potentials^{7,9}. In equation (1), it is possible to decompose the Hamiltonian operator as follows: H_{o}–γ_{0}=NM, where N=−ik/k_{0}+W(x), M=ik/k_{0}+W(x), W(x) is the superpotential that satisfies the Riccati equation W(x)^{2}–i[kW(x)]/k_{0}+γ_{0}=V_{o}(x) and γ_{0} is the groundstate eigenvalue of H_{o}ψ_{0}=γ_{0}ψ_{0}. Then, the inversion of the N and M operators yields the SUSY Hamiltonian H_{s} with the SUSY partner potential V_{s}(x): H_{s}=MN+γ_{0}=k^{2}/k_{0}+V_{s}(x). From the original equation H_{o}ψ=γψ, the relation H_{s}·(Mψ)=γ·(Mψ) is obtained, thus proving isospectrality with γ and the transformed eigenstates of Mψ^{7,9}. For the later discussion of twodimensional (2D) potentials, it is noted that the isospectrality between H_{o} and H_{s} can also be expressed in terms of the intertwining relation MH_{o}=H_{s}M=MNM (MH_{o}ψ=H_{s}(Mψ)=γ(Mψ) when H_{o}ψ=γψ), where the operator M is the intertwining operator^{30,31}.
The solution W(x) is simply obtained from the Riccati equation through W(x)=[_{x}ψ_{0}(x)]/[k_{0}ψ_{0}(x)] for unbroken SUSY^{7,9}, which also provides the groundstate annihilation equation Mψ_{0}=[ik/k_{0}+W(x)]ψ_{0}=O. Because V_{s}(x)=–[n_{s}(x)]^{2} is equivalent to
the index profile n_{s}(x) after the SUSY transformation can finally be obtained as follows:
Equation (3) demonstrates that the SUSY transformation can be achieved deterministically based solely on the groundstatedependent functionality. Figure 2 illustrates an example of serial SUSY transformations applied to the 1D Fibonacci quasicrystal potential defined in Fig. 1, where the small value of N=5 is selected for clarity of presentation. For each SUSY transformation, all eigenstates of each previous potential, except for the ground state, are preserved in the transformed spatial profiles, while the shape of the designed potential becomes ‘disordered’ through ‘deterministic’ SUSY transformations.
Potentials with Blochlike states and tuneable randomness
Because the presence of deterministic order is essential for Blochlike waves and bandgaps, regardless of the presence of longrange correlations in their spatial profiles^{16,17,18,19,25,26,27,28,29}, the ‘randomlyshaped’ potentials (Fig. 2) that can be ‘deterministically’ derived by applying SUSY transformations to ordered potentials offer the possibility of combining Blochlike waves and disordered potentials. To investigate the wave behaviour associated with the SUSY transformation, we consider a largerN regime in which the wave behaviours are clearly distinguished between ordered (Fig. 1f,g) and disordered potentials (Fig. 1h). Figure 3a,b presents the results obtained after the 10th SUSY transformation for the crystal (Fig. 3a,c) and the quasicrystal (Fig. 3b,d) with N=144. Although the shapes of the SUSYtransformed potentials and the spatial information of the eigenstates in Fig. 3a,b are markedly different from those of the corresponding original potentials in Fig. 1c,d, the eigenspectrum of each potential is preserved, save for the annihilation of the 10 lowest eigenstates, which is consistent with the nature of SUSY transformations. From the SUSY transformation Mψ={ik/k_{0}+_{x}ψ_{0}(x)/[k_{0}ψ_{0}(x)]}·ψ, it is also expected that the distribution of the ground state ψ_{0}(x) with respect to the original state ψ primarily affects the effective width^{32} of the transformed eigenstate Mψ. In crystals that have highly overlapped intensity profiles between eigenstates, the effective width of ψ decreases progressively from serial SUSY transformations due to the ‘bound’ distribution of ψ_{0}(x). For a quasicrystal, the variation of the effective width showed more complex behaviour owing to its spatially separated eigenstates (see Supplementary Note 1 and Supplementary Figs 1 and 2 for the comparison between crystal and quasicrystal potentials).
The eigenspectral conservation is apparent in Fig. 3c,d, which depicts the variation in the effective modal index that occurs during the SUSY transformations (up through the 20th SUSY transformation). As shown, the eigenspectrum of each potential is maintained from the original to the 20th SUSY transformation, save for a shift in the modal number, and, therefore, the bandgaps in the remainder of the spectrum are maintained during the serial SUSY transformations (∼125 states after the 20th SUSY transformation following the loss of the 20 annihilated states). Consequently, bandgaps and Blochlike eigenstates similar to those of the original potentials are allowed in SUSYtransformed potentials with disordered shapes (Fig. 3a,b) that can be classified as neither crystals nor quasicrystals.
Figure 4a–h illustrates the shape evolutions of the crystal and quasicrystal potentials that are induced through the SUSY process, demonstrating the increase in disorder for both potentials. Again, note that the SUSYbased modulation is determined by equation (3), starting from the groundstate profile ψ_{0}(x), which is typically concentrated near the centre of the potential (Fig. 3a,b). To investigate the correlation features of SUSYtransformed potentials with bandgaps, we again consider the Hurst exponent. Figure 4i,j shows the Hurst exponents for the transformed crystal and quasicrystal potentials as functions of the number of SUSY transformations for different sequence lengths (N=34, 59, 85 and 144).
The figures show that, for successive applications of SUSY transformations, the Hurst exponents of the crystal and quasicrystal potentials (H=0∼0.1) increase and saturate at H∼0.8. For example, at N=144, the negative correlations (H<0.5) of the crystal and quasicrystal potentials (H=0∼0.1) become completely uncorrelated, with H=0.51 after the 10th SUSY transformation; the correlations are even weaker than that of the Bernoulli random potential (H=0.35∼0.48, Figs 1b and 4i,j) and approach the uncorrelated Brownian limit of H=0.5. After the 10th SUSY transformation, the correlation begins to increase again into the positivecorrelation regime (H⩾0.5, with longlasting, that is, persistent, potential shapes), thereby exhibiting a transition between negative and positive correlations in the potentials. This transition from an ‘antipersistent’ to ‘persistent’ shape originates from the smoothing of the original potential caused by the slowly varying term in equation (3), which is derived from the nodeless groundstate wavefunction ψ_{0}(x).
We note that this ψ_{0}(x)dependent modulation shows a dependence on the size (or sequence length N) of the potentials; for a potential with a large size, ψ_{0}(x) varies weakly over a wide range, thus decreasing the relative strength of the SUSYinduced modulation (Fig. 4i,j). Thereby, the number of SUSY transformations required for extreme randomness (H∼0.5) increases with the size of the potential (Fig. 4i,j, (S_{B}, N)=(4, 34), (6, 55), (8, 89) and (10, 144), where S_{B} is the required SUSY transformations for H∼0.5). Eventually, the SUSY transformation to periodic potentials of infinite size n(x)=n(x+Λ) preserves the periodicity because the SUSY transformation with the Bloch ground state ψ_{0}(x)=ψ_{0}(x+Λ) will repeatedly result in periodic potentials n_{s}(x)=n_{s}(x+Λ).
These results reveal that the application of SUSY transformations to ordered (crystal or quasicrystal) potentials allows for remarkable control of the extent of the disorder while preserving Blochlike waves and bandgaps. Therefore, a family of potentials with ‘Blochlike eigenstates’, for its members have identical bandgaps but tuneable disorders, can be constructed through the successive application of SUSY transformations to each ordered potential, with a range of disorder spanning almost the entire regime of Hurst exponents indicating negative and positive correlations (0≤H≤0.8), including the extremely uncorrelated Brownian limit of H∼0.5. As an extension, in Supplementary Note 2, we also provide the design strategy of randomwalk discrete optical systems (composed of waveguides or resonators) that deliver Blochlike bandgaps, starting from the firstorder approximation of Maxwell’s equations, that is, coupled mode theory^{33,34}.
Extension of SUSY transformations to 2D potentials
In stark contrast to the case of 1D potentials, which exclusively satisfy a 1:1 correspondence between their shape and ground state^{7}, it is more challenging to achieve isospectrality in multidimensional potentials. Although studies have shown the vectorform SUSY decomposition of multidimensional Hamiltonians^{35,36,37}, such an approach, which is analogous to the Moutard transformation^{38}, cannot guarantee isospectrality. This approach only generates a pair of scalar Hamiltonians with eigenspectra that, in general, do not overlap but together compose the eigenspectrum of the other vectorform Hamiltonian^{35,36,37}. Here we employ an alternative route^{30,31,39} starting from the intertwining relation MH_{o}=H_{s}M to implement a class of multidimensional isospectral potentials.
Without the loss of generality, we consider the 2D Schrodingerlike equations with the Hamiltonian of and its SUSY partner Hamiltonian . To satisfy the intertwining relation MH_{o}=H_{s}M, the ansatz for the intertwining operator M can be introduced^{30,31}, similarly to the 1D case:
where M_{o}, M_{x} and M_{y} are arbitrary functions of x and y. From equation (4), the intertwining relation MH_{o}=H_{s}M can be expressed in terms of operator commutators as follows:
where V_{d} is the modification of the potential through the SUSY transformation: V_{d}(x,y)=V_{s}(x,y)–V_{o}(x,y). Although here we focus on the 2D example, it is noted that equation (4) can be generalized to Ndimensional problems^{30,31} as M=M_{o}(x_{1}, x_{2}, …, x_{N})+M_{i}(x_{1}, x_{2}, …, x_{N})·_{i} while maintaining equation (5).
The derivation in the Methods section (Equations 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21) starting from equation (5) demonstrates that the procedure of the 1D SUSY transformation can be applied to a 2D potential for each x and y axes independently, when the potential satisfies the condition of V_{o}(x,y)=V_{ox}(x)+V_{oy}(y). We also note that serial 2D SUSY transformations are possible because the form of V_{o}(x,y)=V_{ox}(x)+V_{oy}(y) is preserved during the transformation, consequently deriving a family of 2D quasiisospectral potentials. Figure 5 shows an example of SUSY transformations in 2D potentials, maintaining Blochlike eigenstates. Both the x and y axes crosssections of the 2D original potential V_{o}(x,y)=V_{ox}(x)+V_{oy}(y) have profiles of N=8 binary sequences (Fig. 5a), as defined in Fig. 1. Following the procedure of equations (17, 18, 19, 20, 21) in the Methods section, we apply SUSY transformations to the x and y axes separately, achieving the highly anisotropic shape of the potential as shown in Fig. 5b (the 5th x axis SUSYtransformed potential) and Fig. 5c (the 5th y axis SUSYtransformed potential). It is evident that this anisotropy can be controlled by changing the number of SUSY transformations for the x and y axes independently, and the isotropic application of SUSY transformations recovers the isotropic potential shape (Fig. 5d). Regardless of the number of SUSY transformations and their anisotropic implementations, the region of bandgaps of the original potential is always preserved (Fig. 5e). Interestingly, the annihilation by 2D SUSY transformation occurs not only in the ground state but also in all of the excited states sharing a common 1D groundstate profile (for details see the Methods section, Supplementary Note 3 and Supplementary Fig. 9). Consequently, the width of the bandgap can be slightly changed owing to the annihilation of some excited states near the bandgap.
To investigate the correlation features of 2D SUSYtransformed potentials, we quantify the angledependent degree of the correlation. Figure 5f,g shows the angledependent variation of the Hurst exponent for the anisotropic (the 5th x axis SUSYtransformed potential, Fig. 5b) and isotropic (the 5th x and y axes SUSYtransformed potential, Fig. 5d) disordered potential. Compared with the original potential (grey symbols in Fig. 5f,g, for Fig. 5a), H increases along the axis with the SUSY transformations (x axis in Fig. 5f and x and y axes in Fig. 5g). The potential is disordered at all angles, especially in the diagonal directions (±45°), owing to the projection of the SUSYinduced disordered potential shapes (45° profiles in Fig. 5b–d).
Discussion
To summarize, by employing supersymmetric transformations, we revealed a new path toward the deterministic creation of randomwalk potentials with ‘crystallike’ wave behaviours and tuneable spatial correlations, extending the frontier of disorder for Blochlike waves and identical bandgaps. Despite their weak correlations and disordered shapes, SUSYtransformed potentials retain the deterministic ‘eigenstatedependent order’ that is the origin of bandgaps, which is in contrast to the hyperuniform^{18,40,41,42} disorder of pointwise networks and deterministic aperiodic structures such as quasicrystals^{28,43} or the Thue–Morse^{44} and Rudin–Shapiro^{45} sequences. We also extend our discussion to multidimensions, achieving highly anisotropic or quasiisotropic disordered 2D potentials, while preserving bandgaps. Our results, which were obtained based on a Schrodingerlike equation, reveal a novel class of Blochwave disorder that approaches the theoretical limit of Brownian motion while maintaining wide bandgaps identical to those of existing crystals or quasicrystals in both electronics and optics. We further envisage a novel supersymmetric relation, based on the famous SUSY theory in particle physics, between ordered potentials and disordered potentials with coherent wave behaviours in solidstate physics. The extension of the SUSY transformation to nonSchrodinger equations, for example, transverse magnetic modes in electromagnetics (as investigated in the supplementary material of ref. 11), or to the approximated Hamiltonians applicable to arbitrarypolarized optical elements (Supplementary Note 2) will be of importance for future applications, for example, polarizationindependent bandgaps based on dualpolarized eigenstates^{46}.
Methods
Details of the FDM and FGH method
The FDM utilizes the approximation of the secondderivative operator in the discrete form^{20}, and the FGH method, as a spectral method, uses a planewave basis with operatorbased expressions in a spatial domain^{21}. In both methods, the Hamiltonian matrices are Hermitian because of the realvalued potentials, thus enabling the use of Cholesky decomposition to solve the eigenvalue problem. To ensure an accurate SUSY process, Rayleigh quotient iteration is also applied to obtain the groundstate wavefunction. The boundary effect is minimized through the use of a buffer region (n=1.5) of sufficient length (30 μm=20λ_{0}) on each side. Deep subwavelength grids (Δ=20 nm=λ_{0}/75) are also used for the discretization of both the 1D and 2D potentials.
Calculation of the Hurst exponent
First, the discretized refractive index n_{p} (p=1, 2, …, N) is obtained at x_{p}=x_{left}+(p−1)·Δ, where x_{left} is the left boundary of the potential, which is of length L=(N−1)·Δ. Partial sequences X_{q} of n_{p} for different length scales d are then defined (2≤d≤N and 1≤q≤d). For the meanadjusted sequence Y_{q}=X_{q}–m, where m is the mean of X_{q}, we define the cumulative deviate series Z_{r} as
The range of cumulative deviation is defined as R(d)=max(Z_{1}, Z_{2}, …, Z_{d})–min(Z_{1}, Z_{2}, …, Z_{d}). Using the s.d. S(d) of Y_{q}, we can now apply the power law to the rescaled range R(d)/S(d) as follows:
This yields log(E[R(d)/S(d)])=H·log(d)+c_{1}, where E is the expectation value and c_{0} and c_{1} are constants. H is then obtained through linear polynomial fitting: H=0.5 for Brownian motion, 0≤H<0.5 for longterm negative correlations with switching behaviours, and 0.5<H≤1 for longterm positive correlations such that the sign of the signal is persistent.
The condition for 2D isospectral potentials
By assigning M=M_{o}+M_{d} (Equation 4) to the intertwining relation MH_{o}=H_{s}M with explicit forms of H_{o} and H_{s}, it becomes . Thus, we obtain equation (5); , where V_{d}=V_{s}–V_{o}. Each commutator in equation (5) is also expressed as
It is noted that the higherorder (⩾2) derivatives in equation (5) originate from the third and fourth terms in the righthand side of equation (8). Comparing equation (8) with equations (9, 10), all of the higherorder derivatives should be removed to satisfy equation (5). This then directly leads to the preconditions M_{x} and M_{y}; _{x}M_{x}=0, _{y}M_{y}=0 and _{x}M_{y}+_{y}M_{x}=0, which hold only for M_{x}(x,y)=M_{x}(y)=a_{x}–by and M_{y}(x,y)=M_{y}(x)=a_{y}+bx where a_{x}, a_{y} and b are arbitrary constants.
By applying M_{x}(y), M_{y}(x), and equations (8, 9, 10) to equation (5), we achieve two linear and one nonlinear equations for three unknowns M_{o}, V_{o} and V_{d} as
As a particular solution, we consider the case of b=0 for simplicity. In this case, from equations (11, 12), M_{o} and V_{d} are determined in the form of M_{o}=f(ρ) and V_{d}=_{ρ}f(ρ), where ρ=k_{0}^{2}·(a_{x}x+a_{y}y)/2 is the transformed coordinate and f is an arbitrary function of ρ. By substituting M_{o} and V_{d}, equation (13) then becomes
which reveals the proper form of the 2D potential V_{o} for SUSY transformations
where ξ=k_{0}^{2}·(c_{x}x+c_{y}y)/2 is the transformed coordinate perpendicular to ρ, with a_{x}·c_{x}+a_{y}·c_{y}=0. The supersymmetric potential V_{s}=V_{o}+V_{d} then becomes
Using equations (15, 16), now we can implement the procedure of serial 2D SUSY transformations. First, because of equation (15), V_{o} should have the form of V_{o}(ρ,ξ)=V_{oρ}(ρ)+V_{oξ}(ξ) for two Cartesian axes of ρ and ξ. In this case, the corresponding f(ρ) is obtained by solving the following Riccati equation:
Its particular solution is listed as (ref. 47); where ϕ_{0}(ρ) is the nodeless ground state with the eigenvalue γ_{oρ} in the corresponding 1D Schrodingerlike equation
With the obtained f(ρ), we finally achieve the SUSYtransformed potential along the ρaxis satisfying the isospectrality, V_{s}(ρ,ξ)=V_{o}(ρ,ξ)+_{ρ}f(ρ), or
Equivalently, the SUSY transformation along the ξ axis is
where φ_{0}(ξ) is the nodeless ground state with the eigenvalue γ_{oξ} in the following equation:
Note that, after the SUSY transformation for the ρ or ξ axes, V_{s}(ρ,ξ) still preserves the form of V_{s}(ρ,ξ)=V_{sρ}(ρ)+V_{sξ}(ξ) (Equations 19, 20), which is the necessary condition for the SUSY transformation of 2D potentials. Therefore, serial SUSY transformations can be applied to 2D arbitrary potentials of the form V_{o}(ρ,ξ)=V_{oρ}(ρ)+V_{oξ}(ξ), and the level of SUSY transformations can be controlled independently for each axis, allowing highly anisotropic potential profiles. In addition, by assigning nonzero b, the allowed potential of V_{o}(x,y) can be extended to nonseparated forms^{30,31}.
The eigenstate annihilation in 2D SUSY transformations
In stark contrast to the groundstate annihilation in 1D SUSY transformations, the annihilation by 2D SUSY transformations is not restricted to the ground state. For simplicity, consider the case of and a_{y}=0 for ρ=x and ξ=y without any loss of generality. The Hamiltonian, which can be SUSY transformed, is then expressed as for the eigenvalue equation H_{o}ψ=γψ. For the x axis SUSY transformation, the following equation should be satisfied to annihilate the SUSYtransformed eigenstate because Mψ=O:
Note that equation (22) is satisfied when ψ(x,y)=ϕ_{0}(x)·φ(y), allowing the separation of variables in the 2D eigenvalue equation H_{o}ψ=γψ as
It is noted that the first brace has the fixed constant of γ_{ox}, the groundstate eigenvalue of the 1D Schrodingerlike equation with the potential V_{ox}(x). Meanwhile, because the second brace can be any eigenvalues of the solution φ(y) in the 1D Schrodingerlike equation with the potential V_{oy}(y), it is clear that the annihilation by 2D SUSY transformation occurs not only in the ground state but also in all of the excited states sharing ϕ_{0}(x). The detailed illustration of this result is shown in Supplementary Note 3 and Supplementary Fig. 9.
Additional information
How to cite this article: Yu, S. et al. Blochlike waves in randomwalk potentials based on supersymmetry. Nat. Commun. 6:8269 doi: 10.1038/ncomms9269 (2015).
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Acknowledgements
We thank M.A. Miri for their discussions on supersymmetric optics, especially with regard to groundstate annihilation and S. Torquato for the encouragement of our results and the introduction of hyperuniformity. This work was supported by the National Research Foundation of Korea through the Global Frontier Program (GFP) NRF2014M3A6B3063708, the Global Research Laboratory (GRL) Program K20815000003, and the Brain Korea 21 Plus Project in 2015, which are all funded by the Ministry of Science, ICT & Future Planning of the Korean government.
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S.Y. and N.P. conceived of the presented idea. S.Y. developed the theory and performed the computations. X.P. and J.H. verified the analytical methods. N.P. encouraged S.Y. to investigate supersymmetry and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.
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Supplementary Figures 19, Supplementary Note 13 and Supplementary References (PDF 1181 kb)
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Yu, S., Piao, X., Hong, J. et al. Blochlike waves in randomwalk potentials based on supersymmetry. Nat Commun 6, 8269 (2015). https://doi.org/10.1038/ncomms9269
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