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Counterexample to the Bohigas Conjecture for Transmission Through a One-Dimensional Lattice

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Counterexample to the Bohigas Conjecture for Transmission Through a One-Dimensional Lattice Counterexample to the Bohigas Conjecture for Transmission Through a One-Dimensional Lattice Ahmed A. Elkamshishy∗ Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907 USA Chris H. Greeney Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907 USA and Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907 USA Resonances in particle transmission through a 1D finite lattice are studied in the presence of a finite number of impurities. Although this is a one-dimensional system that is classically integrable and has no chaos, studying the statistical properties of the spectrum such as the level spacing distribution and the spectral rigidity shows quantum chaos signatures. Using a dimensionless parameter that reflects the degree of state localization, we demonstrate how the transition from regularity to chaos is affected by state localization. The resonance positions are calculated using both the Wigner-Smith time-delay and a Siegert state method, which are in good agreement. Our results give evidence for the existence of quantum chaos in one dimension which is a counter-example to the Bohigas- Giannoni-Schmit conjecture. In 1984, Bohigas, Giannoni and Schmit stated the cel- BGS conjecture, and a second case where a signature of ebrated (BGS) conjecture[1] that describes the statisti- chaos arises in the quantum spectrum, which matches the cal properties of chaotic spectra. This conjecture draws GOE statistics for both the nearest-neighbor distribution a connection between quantum systems whose classical and the spectral rigidity. The chaotic behavior is shown analog is chaotic and random matrix theory (RMT). to depend on a dimensionless parameter that represents Classical chaos is a consequence of the non-linearity of the degree of state localization, thereby suggesting a con- the Newtonian equations of motion, while Schr¨odinger's nection between quantum chaos and the phenomenon of equation is linear and strictly speaking has no chaos. Anderson localization[9, 10]. While RMT formally deals Nevertheless, quantum signatures of chaos can arise and with discrete spectra only, our study discusses it in the are exhibited by the statistical properties of the quantum context of very narrow resonances, which is justifiable energy level spectra, such as the level spacing distribution because of their generally extreme narrowness. and the spectral rigidity (SR) [1, 2]. The model considered in this study is a particle moving The spectrum of random matrices was first studied by through a one-dimensional lattice with the lattice poten- Wigner in 1951 [3{5], who demonstrated the existence of tial energy modeled by a sum of delta functions, one per a few universal classes based on the symmetry imposed lattice site. Thus the Hamiltonian is given by: on such matrices. In this paper, the two classes consid- ered are the gaussian orthogonal ensemble (GOE) and N=2 the Poisson distribution. P 2 X The claimed connection between chaos and RMT is the H = + αnδ(x − na): (1) 2m essence of the BGS conjecture, which relates the spectral n=−N=2 properties of quantum systems whose classical Hamilto- nians are irregular (chaotic) to the GOE class. The BGS Here N + 1 is the total number of lattice sites, and αn conjecture is formally stated as follows: Spectra of time- is the strength of the n-th delta function. In the case of reversal-invariant systems whose classical analogs are K- a perfectly clean periodic but finite lattice that we con- arXiv:2011.10631v1 [nlin.CD] 20 Nov 2020 systems[6] show the same fluctuation properties as pre- sider here, all αn are the same and all are negative, but dicted by GOE [1]. On the other hand, the spectrum of we will keep the notation general for now because part a classically regular system follows Poissonian behavior of our analysis will be an exploration of the effect of im- [1, 7, 8]. purities. The reason behind choosing the delta function The goal of the present study is to explore a one- potential is the fact that an attractive delta function po- dimensional quantum system whose classical analog is tential admits one bound state, so one can imagine the regular. Our analysis demonstrates that such systems system as having one atomic state around each atomic have quantum chaos signatures. Moreover, two limiting site, which allows us to treat both the bound states and cases are discussed: the regular case that agrees with the scattering dynamics. Modeling the lattice by consider- ing one atomic state at each site and treating the effect of site-to-site tunneling as an effective hopping parame- ter has been studied for years and is referred to as the ∗ [email protected] tight-binding approximation [9, 10]. One of our goals y [email protected] is to compare the exact solution with the results of the 2 tight-binding model. Our results show limitations of this particle. To obtain both r, and t, the solution inside the approximation that become relevant in the context of lattice has to be obtained. 1 1 transmission through a finite lattice. In the domain x 2 [(n − 2 )a; (n + 2 )a], Within the tight-binding approximation, two things iqx −iqx + − are assumed: First, there exists one atomic state around Ψn(x) = Ane + Bne ≡ n + n (4) each lattice site, and secondly, there is only hopping be- p tween nearest neighbors [11]. The Hamiltonian in this where q = 2mE , and m is the particle mass. After ap- ~ case is written as: plying the the wave function continuity and derivative X y X y discontinuity conditions, the solutions in any two adja- H = iai ai + ti;jai aj (2) cent regions separated by a lattice constant a are related i <i;j> by the transfer matrix [18] as follows: where i is the energy of the atomic state site i and ti;j is + iqa 1+mαn mαn ! + the tunneling amplitude from site j to site i, and the sum n+1 e iq iq n − = −mαn −iqa 1−mαn − in the second term is taken for nearest-neighbors where n+1 iq e iq n j = i ± 1. Both can be calculated from the potential in- (5) + troduced in Eq.1. If the lattice is periodic and all atomic n ≡ T (αn; q) − sites are identical, then this model can be solved analyt- n ically [12]. However, there is much interesting physics to To obtain the full S-matrix, both transmission (reflec- study when impurities are placed in the lattice, such as tion) amplitudes should also be obtained in the case of the transport across the lattice. In that case, the period- scattering by particles incident from the right, denoting icity is broken and there is no general analytical solution. them as t0(q), and r0(q). In one dimension, the S-matrix However, the spectrum can be obtained by writing the has the form [19, 20]: Hamiltonian in a matrix form and diagonalizing it [10]. The tight-binding approximation is useful in many t r0 S = (6) cases, especially for the N ! 1 limiting case. Most r t0 of the physics of bound states can be studied within this simple approximation. However, tight-binding has limi- tations, such as the fact that a set of N negative delta The distinctions between the periodic and the disor- functions do not necessarily support N bound states. dered cases become clear from plots of the Wigner-Smith dSy Even two delta functions in one dimension do not neces- time-delay [13], Q = iS dE , and the phase shifts [14].The sarily have two bound states, which can cause difficulties trace of Q gives the total time-delay. Fig.1 shows that whenever finite lattices are considered. Secondly, within there are no resonances in the exactly periodic case with tight-binding, one can only study bound states. So, all no impurities, and only a simple, regular oscillation of the physics of scattering is missing, such as transmission the total time-delay as a function of collision energy. On resonances. the contrary, when impurities are present, there are many The approach used in this paper is to consider a fi- narrow resonances. Associated with each resonance is a nite lattice and treat it as a finite range potential. Then peak in the time-delay and a clear rise in the sum of the scattering (S) matrix is obtained for a particle in- the eigenphaseshifts by π radians as functions of energy. cident from −∞ representing a scattering from the left, Studying the statistical properties of all the resonances and for a particle incident from +1 representing scat- in the first band shows chaos signatures in the system, tering from the right, and all desired observables calcu- as is demonstrated next. lated. The resonance positions and widths can be calcu- The resonances can also be calculated using a differ- lated from either the Wigner-Smith time-delay [13{15] or ent method: The Siegert state [17] boundary conditions by imposing outgoing-wave Siegert state boundary con- allows only outgoing waves, and they take the form: ditions [16, 17]. A numerical solution is obtained for ( −iqx −Na lattices with different values of the lattice size and the Ae if x < 2 (x) = iqx Na (7) lattice constant. A main goal here is to study the real Be if x > 2 part of the resonance energy level distribution in the first energy band. In all calculations, atomic units are used, With this boundary condition, the Hamiltonian is non- Hermitian and the spectrum is complex. Each eigen- i.e. with ~ = a0 = me = Eh = 1. The solution to the time-independent Schr¨odinger value can be written as Ej = E0j − iΓj=2, where E0j th equation for the Hamiltonian introduced in Eq.1 has the is the position of the j resonance, and Γj is the width following form for particles incident from the left: [17, 21].
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