Topological states in the Kronig-Penney model with arbitrary scattering potentials

We use an exact solution to the fundamental finite Kronig-Penney model with arbitrary positions and strengths of scattering sites to show that this iconic model can possess topologically non-trivial properties. By using free parameters of the system as extra dimensions we demonstrate the appearance of topologically protected edge states as well as the emergence of a Hofstadter butterfly-like quasimomentum spectrum, even in the case of small numbers of scattering sites. We investigate the behaviour of the system in the weak and strong scattering regimes and observe drastically different shapes of the quasimomentum spectrum.


I. INTRODUCTION
The Kronig-Penney (KP) model is one of the fundamental models of solid state physics and has since its inception [1] received significant attention. It combines predictive power with accessibility and has, in fact, become a standard model that is taught in almost all solid state classes for undergraduate students. Despite its underlying simplicity that neglects interactions between the particles, it is particularly well suited to describe the behaviour of electrons in metals [2][3][4][5][6][7]. More recently an experimental realization of the KP potential for ultracold atoms in optical lattices was proposed [8] and demonstrated [9] .
One important aspect of the success of the KP model lies in its flexibility. It allows to describe impurities or disorder in an easy and straightforward manner by assuming the scattering potentials to be located at nonperiodic positions or having random strengths [10,11]. Here we present an analytical solution for the finite KP model, when all scatterers are placed at arbitrary positions and have arbitrary strengths. This can be used for the exact treatment of effects that were only explored numerically before, such as localisation [12][13][14] or the existence of topologically non-trivial states [15,16]. As an example we use the explicit solution in order to investigate appearance of the edge states and Hofstadter butterfly-like features in a finite, continuous system.
To obtain the single-particle solutions of the arbitrary finite KP model we use the coordinate Bethe ansatz approach. This method of solving one-dimensional quantum many-body problems was first described by Hans Bethe in 1931 [17], and has since then been successfully applied to a large number of problems in lower dimensions [18][19][20][21][22].
Our manuscript is organised as follows. In Sec. II we outline the solution to the arbitrary finite KP model using the Bethe ansatz. For this we first define the problem in Sec. II A and then derive and present the explicit expressions for the Bethe equation and the eigenfunctions, * irina.reshodko@oist.jp in Secs. II B and II C, respectively. In Sec. III we use these solutions to demonstrate the appearance of edge states in the finite KP model with a lattice shift as an extra dimension. Finally, in Sec. IV, we explore the emergence of a Hofstadter butterfly-like energy spectrum in the amplitude-modulated KP model.

A. Model
We consider a one-dimensional system consisting of an infinite potential box of size L, in which M point-like scatterers of arbitrary strengths h = (h 1 , . . . , h M ) are placed at arbitrary positions y = (y 1 , . . . , y M ) with y n ∈ [− L 2 , L 2 ] and y n < y m for n < m (see the schematic in Fig. 1), i.e., (1) We show below that this potential cannot, in general, be solved with the Bethe ansatz for a system of point-like interacting bosons. However, the non-interacting and the infinitely strongly interacting (Tonks-Girardeau) limit can be solved, the latter by making use of the Bose-Fermi mapping theorem [23,24]. For both it is necessary to consider only the single-particle Hamiltonian and the corresponding Schrödinger equation The essence of the coordinate Bethe ansatz approach is that the eigenstates of any system can be represented as a superposition of plane waves with different quasimomenta for each particle [25]. By taking into account all possible scattering events one can construct a set of consistency equations, called the Bethe equations, and only those quasi-momenta which satisfy the Bethe equations are allowed in the system. Once the quasimomenta are determined, the energy of the system is given by a simple sum of their squares. A necessary condition for a system to be integrable, however, is that it satisfies the Yang-Baxter relations [26,27]. These stem from the requirement that all of the three-body scattering processes in the system can be decomposed into a series of twobody scattering events whose order does not matter.
Unfortunately, for larger numbers of particles and barriers, or for a non-symmetric placement of a single barrier, the Yang-Baxter relation cannot be satisfied in the regime of finite interactions. This can be seen straightforwardly by considering two particles with different quasimomenta located in the same region between two scatterers. The three-body scattering events occur when both particles hit the same barrier at the same time. While these events can in principle be decomposed into the two particles scattering between themselves, and each particle individually scattering with the barrier, the order in which the particles scatter against the barrier matters. This is because the second particle to scatter on the barrier will be subject to a different dynamical evolution depending on the quasi-momentum of the other particle. Consequently, the Yang-Baxter relations cannot be fulfilled and the model cannot be solved analytically with the Bethe ansatz for finite interaction strengths.
It is worth noting, however, that the interacting case was recently studied for a specific example by Liu and Zhang [28], who considered one scatterer at the centre of the infinite box (M = 1, y 1 = 0). They showed that this system can be partially solved for two particles and arbitrary scattering strengths, by finding the eigenstates for which the Yang-Baxter relations are satisfied.
Let us also mention that a different approach to the single-particle problem was recently proposed by Sroczyńska et al. [29]. In this work the authors use a Green's function approach to solve the problem of a single-particle moving in an arbitrary trapping potential which has regularized delta scatterers superposed. The solution presented here will coincide with the onedimensional solution of Ref. [29] after substitution of the Green's function for the infinite square well.

B. Bethe equations
In the following we outline the exact solution of eq. (2), with the potential given by expression (1), using the Bethe ansatz. We start by considering solutions for each of the regions between the scatterers, determined by the free-particle Schrödinger equation We construct an ansatz for the full solution of eq. (3) composed of piecewise plane waves with quasimomentum k as being the Heaviside step function. Each term in the sum corresponds to a region between two scatterers or a scatterer and the adjacent walls, x ∈ D n = (y n−1 , y n ), and we have set y 0 = −L/2 and y M +1 = L/2. This ansatz has to satisfy the boundary conditions imposed by the scatterers and walls, which are of the form The Bethe ansatz approach now consists of constructing equations for the quasi-momenta k, from which the energies follow as E =h 2 k 2 /2m. To do so, we will first construct expressions for all coefficients A Starting with the boundary conditions at the walls, we first substitute the ansatz in eq. (4) into eq. (5), and obtain expressions for the first and the last elements of the reflection matrix of the form Next, the continuity and scattering conditions given in eqs. (6) and (7) at the j-th barrier lead to By considering eqs. (11) and (12) for j = n, n − 1 and taking into account that A j , we find the recursive form for the reflection matrix elements for the inner regions D n , n = 2 . . . M . Together with eqs. (9) and (10), these expressions correspond to the two ways of inverting the sign of the quasi-momentum k by reflecting the particle at the left or the right wall. Thus, the two expressions for each region have to be equivalent, yielding the M + 1 Bethe equations that define the allowed quasi-momenta of the system. Next we prove that all these Bethe equations are equivalent. For this we represent the process of reflecting a particle as a sequence of scattering events at each barrier, denoted by the elements of the scattering matrix and reflections against the left and right wall, denoted by R It is easy to see that by multiplying both sides of the equation by the inverse scattering matrices in the appropriate sequence, one can reconstruct similar equations for all other regions. Consequently, we only need one Bethe equation for the single variable k, and we choose the one that assumes the particle to be in the rightmost region, as it has the simplest form given by By unwrapping the this recursive expression, we can then construct the Bethe equation for any given values of the system parameters, which can be algebraically simplified to The sum is over all ordered sets of n scatterer indices, i.e., (p 1 , . . . , p n ), and we have also defined y p0 = −L/2 and y pn+1 = L/2. The Bethe equation constructed in this way is an algebraic transcendental equation, whose roots can generally only be found via numerical methods or analytical methods for small number of roots [30].

C. The wavefunction
From eqs. (11) and (12) we can also obtain an explicit recursive expression for the elements of the scattering matrix of the form n and the reflection matrix.
We therefore have everything to express all coefficients of the ansatz wavefunction in eq. (4) in terms of A The remaining coefficient A 1 is in principle fixed by normalization of the wavefunction. The explicit form of the coefficients is then given by

III. EDGE STATES
Since we now have access to all single-particle states of the system, let us look at states that appear in situations that are not perfectly periodic. One example are states that are located inside the band gaps of the system. These topologically protected states are known as edge states, and the most famous consequence of their appearance in two dimensions is the quantum Hall effect [31][32][33]. In fact, even though topological effects generally require higher dimensions, it has been shown that in certain one-dimensional systems additional degrees of freedom can be used as a virtual second dimension -a superspace [34] Two examples are one-dimensional Fermi-Hubbard-type system [16] (where the superspace is realized via modulated on-site energies) and continuous cosine lattices [35] (where the superspace is represented by a lattice shift). These systems have recently been numerically investigated, showing the existence of edge states. Due to the generality of the exact solution presented above, the arbitrary finite KP model is a perfect candidate to study the appearance of edge states and other topologically non-trivial states. It has the flexibility to represent real world systems such as crystals made from multiple atomic species, impurities in the spatial periodicity with respect to position and scattering strength, and effects stemming from finite size of the system.
Below we investigate the appearance of edge states for two paradigmatic structures: a uniform lattice and a superlattice. In both cases the superspace is realised by a relative shift of the lattice of scatterers with respect to the box (see Fig. 2).

A. Equidistant scatterers of equal heights
We first consider a set of equidistant scatterers of equal height h n = h > 0 for n = 1, .., M and introduce a shift in the barrier positions ∆ ∈ [−1, 1] with respect to the walls of the box To simplify the presentation of our results, we will use natural units, m =h = 1, from now on. The spectrum of the quasimomenta as a function of ∆ is shown in Fig. 3(a), where one can clearly see the appearance of gap states between the bands, even in this case with a rather small number of barriers. We also show the probability density of the first two edge states as a function of ∆, demonstrating localization of the two wavefunctions ( Fig. 3(b,c)). The strongest localisation of the wavefunction is achieved for k values in the middle of the band gap (∆ = ± 1 2 ), and the position within the box depends on the slope of the quasimomentum function: positive and negative slopes correspond to localisation at the right or the left edge, respectively. This is in agreement with the results of a recent study in a continuous cosine potential [35].

B. Equidistant scatterers of non-uniform heights
While superlattice-type structures are more complicated, they can still be treated straightforwardly using the above solution. Here we consider again a system of equidistant scatterers (cf. Eq. (27)), but with alternating heights h = {0.4, 1.4}. As expected, the quasimomentum spectrum becomes more complicated with additional gaps appearing (see Fig. 4(a)), which are due to the existence of two different sub-lattices [36]. At the same time edge states are still present and one can conclude that their appearance is robust against variations in the strengths of the scatterers. In fact, edge states also survive in a model with a set of periodically placed scatterers of random heights, an example of which is shown in Fig. 5.

IV. HOFSTADTER BUTTERFLY AND COCOON SPECTRA
Another characterstic topological effect is the appearance of a fractal pattern in the energy spectrum of a system [37]. This was first predicted by Hofstadter for electrons on an infinite 2D lattice in the presence of a magnetic field, where the particles experience a phase shift φ due to the magnetic field after a full loop over a lattice plaquette. Such an energy spectrum has since then become known as a Hofstadter butterfly due to its distinct shape. For finite systems, however, the fractal nature of the energy spectrum is known to be lost [38][39][40], but the overall shape of the butterfly is preserved, with states appearing in the bandgaps. In one-dimensional systems similar effects can be observed when using a superspace [16,41], and here we will investigate the Hofstadter butterfly-like quasi-momentum spectrum of the arbitrary finite KP model as it emerges with increasing numbers of scatterers.
The model we are considering consists of equidistant barriers at positions y n = −L/2 + anL, with a = 1/(M + 1). The heights of the scatterers are modulated The scattering potential is periodic in φ (with period φ 0 ≡ (M + 1)/2), and φ therefore plays the role of the flux from the original Hofstadter study. Note that our model describes a continuous system, whereas the original Hofstadter argument was made for a system in the tight-binding approximation [37].
We will first study the case of all positive scatterers, and fix the heights of the scatterers to vary between h min = 0.1 and h min = 1.5. We then find the quasimomentum spectrum for φ ∈ [0, φ 0 ] and show it in Fig. 6(a) for M = 17 scatterers. One can see that the spectrum is symmetric around k = 0 and splits into bands, whose width depends on the minimum scatterer strength h min . Each of these bands has a shape that resembles a Hofstadter butterfly, but this shape becomes less pronounced in higher bands due to the finite height of the scatterers. The gradual emergence of the butterflylike shape with increasing the number of scatterers can be seen in Fig. 7. One can also see that in each band the state with the largest absolute value of quasimomentum is fully flat and corresponds to a delocalized state with k l flat = π(M + 1)l/L, where l = ±1, ±2, ... is the band index. This state has nodes exactly where the scatterers are located and its energy is therefore not affected by them. Let us finally study the case which also includes negative values for the scatterers' strengths. We limit ourselves to weak negative scatterers to avoid the presence of bound states.
The quasimomentum spectrum for M = 17 scatterers with minimum and maximum strengths h min = −0.5 and h max = 0.5 is shown in Fig. 6(b). The two spectra in Fig. 6(a) and (b) are very similar, especially for k values close to the band-edge, which confirms again that the properties of the systems are mostly determined by the position distribution of the scatterers rather than their changes in strength. While the butterfly structure in this weakly-scattering, finite-sized system has not yet fully developed, one can see a prominent cocoon-shaped feature appearing around k = 0 in the case where scatterers have negative as well as positive strengths.

V. CONCLUSIONS
We have used the coordinate Bethe ansatz to derive an analytical solution of the finite Kronig-Penney model with delta scatterers of arbitrary heights positioned at arbitrary points within a box. The concise form of these solutions allows to treat many problems that were only accessible numerically until now in an exact way and is likely to give insight into many problems in solid state physics, such as impurities, finite systems, or disordered systems. As an example, we have used the solution in a topologically non-trivial system and shown the existence of edge states in the continuous finite Kronig-Penney model. We have also demonstrated the appearance of a Hofstadter butterfly-like quasimomentum spectrum with modulated scatterer heights, as well as the presence of a cocoon-shaped feature in the spectrum in the case when the scatterers can be both positive and negative. The solution we present can be readily applied to studies of localization in various distributions of the barrier heights and positions, in solid state and in optical lattices.