The Aharonov–Bohm effect in quadratic Gauss quantum rings

We outline a theory for the Aharonov–Bohm effect in a novel type of quantum ring, namely the quadratic Gauss quantum ring, here for brevity called the Gauss ring. A Gauss ring is a one-dimensional nonperiodic lattice with sites at z = z j = j n d with j ∈ {0, 1, 2, ⋯ , s + 1} ( s∈W ), n ≥ 2 ( n∈N ), and d the underlying period wrapped into a ring-shaped geometry. Here we discuss the electronic spectrum of quadratic Gauss rings (n = 2, s ∈ {0, 1, 2, ⋯ , 5, 6, 7}), and show that the spectrum can be described using theoretical tools recently devised to provide an understanding of the Gauss chain [D S Citrin (2023) Phys. Rev. B 107, 235 144]. We then consider the effect on the electronic states of a magnetic flux Φ threading the Gauss ring and show that the effect of the flux is to select specific electronic states from a periodic chain whose supercells are finite-length Gauss chains thus relating the states in the Gauss quantum ring with those in a periodic Gauss chain.


Introduction
A quantum ring is a quasi-one-dimensional material fashioned into a ring-shaped geometry.Such structures are of interest in part due to the Aharonov-Bohm (A-B) effect [1], where it was shown that the energy of an electron in a ring threaded by magnetic flux Φ is affected by the flux, even if there is no flux through portions of the structure in which there is nonzero probability density to find the electron.The A-B effect leads to persistent currents [2] and oscillation of the energy with Φ [3,4].The A-B effect has been observed in numerous experiments, notably in metal and semiconductor quantum rings [5][6][7][8][9][10][11][12][13][14][15][16].Subsequently, evidence of the A-B effect was seen in optical experiments and also led to interest in the excitonic A-B effect [17][18][19][20][21][22]. Many-body effects, not discussed in the present work, are reviewed in [23].
In a quantum ring the electron wavefunctions must be single-valued, i.e. in going 2π in angle f around the ring, the value of the wavefunction must match that where one starts.(Our discussion leaves out the Zeeman splitting associated with the electron spin's interaction with the magnetic field.This effect is additive in the absence of significant spin-orbit interaction and is smaller in typical cases than energy shifts due to the A-B effect.)In the absence of any additional potential in the ring, this gives one-dimensional eigenstates y f = f e im ( ) with Î m  of energy at zero magnetic flux ) where m * is the electron effective mass and R is the quantum-ring radius.(The presence of a potential will result in energy eigenstates mixing various values of m.)In the A-B effect in a quantum ring, the energy eigenstates preserve the form ψ(f), but modify the energy eigenvalues to ) where j = Φ/Φ 0 and Φ 0 = hc/e is the quantum of magnetic flux.These states contribute to the persistent currents I = ∂E m (Φ)/∂Φ of the single-particle states.Our aim in this study is to engineer a potential around a quantum ring to attain a degree of control over the energy eigenvalues and persistent currents based on a one-dimensional quasiperiodic lattice.
In recent work we have studied quadratic Gauss chains (GC) [24][25][26][27][28] in which lattice sites are located on the z axis at z = z j = j n d with Î n j , , n = 2, and d the underlying lattice constant.The name Gauss is attached to these chains due to the central rôle played in the theory by Gauss sums [24,26,28,29].Quasiperiodic lattices have been of great interest for a number of reasons.First, such structures in some cases exhibit properties intermediate between periodic and random lattices.As such, quasiperiodic lattices may show exotic behavior with regard to their spectra and localized versus delocalized states, scaling on physical properties with chain length, in the process shedding light on the nature of these effects in more familiar periodic and random lattices.GCs in particular have properties distinct from other quasiperiodic chains heretofore investigated [30][31][32][33].
Reference [25] characterized the delocalized-state spectrum in closed form; it was found to be singularcontinuous with candidate delocalized states at freespace wavevectors k, i.e. between quadratic sites, such that p kd is a rational number.The delocailized-state spectrum has been shown to consist of a hierarchy of minibands and gaps.For absolute value of the onsite potential parameter λ exceeding a k-dependent value, most states become localized, but some continue as delocalized for arbitrarily large |λ| [25].This behavior is due to a hidden symmetry of the GC and can be seen from the transfer matrix, which is periodic with its period depending on wavevector k, even though the GC itself is nonperiodic [26].That is, the transfer matrix for the GC is surprisingly periodic with a k-dependent period where k is the free-electron wavevector between scattering sites of the GC [25].Moreover, the structure factor and delocalized-state spectrum of the GC is related to theta functions and Gauss sums, indicating its inherent number-theoretic interest [24,26,28].
Here we study a quadratic GC wrapped upon itself into a quantum ring as in figure 1, calling it a Gauss ring (GR), and explore the flux-dependent properties of the electronic states.Our aim is to establish a theoretical approach to the A-B effect in such structures.We consider a purely one-dimensional GR of negligible width and assume that Φ entirely threads the GR.The GR is formed from a GC in which the j = 0 and = ¢ = + j s s 1 sites coïncide.Thus, the GR is composed of ¢ = + s s 1 sites, with j ä {0, 1, 2, L , s − 2, s − 1, s}.(It will be convenient at times to employ ¢ s rather than s.)This is equivalent to considering an ¢ = + s s 1 ( )-site GC (recalling we start with the j = 0 site) with onsite Dirac δ-function potentials at z j = j 2 d subject to Born-von Karman boundary conditions within a Kronig-Penney model.By way of approaching the problem of the GR, we will first consider the electronic structure of an infinite chain composed of a periodic sequence of ¢ = + s s 1 ( )-site GCs, i.e. a superlattice (SL) whose supercell is a ¢ = + s s 1 ( )-site GC.We call this a GC SL.We then proceed to the GR.It is shown that the electronic states in the GR are a subset of those in the GC SL where the states are selected in effect by Φ; 2πj plays the rôle of ¢ qL s where is the supercell length in the GC SL or the circumference of the GR and ÿq is the quasimomentum in the GC SL.There have been a few investigations of chains of coupled quantum rings where the properties from ring to ring are modulated by a Fibonacci sequence [34,35].We are unaware of any previous work in which the potential itself in circumnavigating a quantum ring is quasiperiodic.In our case, GRs will be seen to exhibit highly complex spectra due to the underlying GC structure, and such complexity sets in for rather low GC order.

Theory
We begin by describing an electron in a chain; below we impose appropriate boundary conditions to results in the GC SL or the GR.Let the Hamiltonian be with m * the effective mass and λ the onsite potential strength (λ < 0 attractive).We do not account for the curvature of the ring or the transverse confinement, and treat the problem purely in one dimension.Therefore, we consider the GC SL or GR to be described by a one-dimensional Kronig-Penney model with Dirac δ-function potentials.Hereon, we take d = 1 and call the distance measured along the GC or around the GR circumference z, rather than employ the angle f in the latter case.To minimize the number of parameters, we define the normalized Hamiltonian  ( ) will later be imposed on the GR.The electronic states are conveniently described using a transfer-matrix approach.To proceed, we will need to know how a wavefunction is affected by crossing a site of the GC or GR.Consider the wavefunction at fixed energy = E k 2 ¯between sites z j−1 < z < z j .Because between sites the potential vanishes, the wavefunction must )-site QC that will be formed into an ¢ s -site GR by identifying the j = 0 with the = ¢ j s site. (The actual number of sites in the GC shown is be that of a free particle,  as yet undetermined.Given that ψ is continuous, we integrate the Schrödinger equation from z j − ò to z j + ò in the limit ò → 0, to give the conditions

A e B e
A e B e a , 3 We can recast this in matrix form as the transfer matrix across site Î j .The transfer matrix from just after the site at z j to just before z j+1 is Consequently, the transfer matrix ) .Consider now forming a chain that is a periodic sequence of ¢ = + s s 1 ( )-site GCs, i.e. a GC SL.Its transfer matrix is thus LT s T s T s T s L .Now, Q j and M are unimodular, and thus so is T s and indeed so is T s raised to any .For the ¢ = + s s 1 ( )-site GR, however, y y = ¢ L 0 s ( ) ( ), whence we require in that case x s = 1.
Before considering the GR, it will be helpful to understand an important point about the hidden symmetry of the infinite GC.One can show [25] that for |x s | 1 we necessarily have p = ¢ kd r s with Î s  and Î r  for candidate delocalized states in the infinite GC.In the present study, we first consider the GC SL.
We now focus on the A-B effect [1] of GRs in the presence of a magnetic flux Φ threading the GR.In this case, the Hamiltonian of equation ( 1) is replaced with ) is the circumference of the ¢ = + s s 1 ( )-site GR, and j = Φ/Φ 0 with Φ the magnetic flux and Φ 0 = hc/e the flux quantum.For z j < z < z j+1 , the potential term in H does not appear and we have In this region, the (positive-energy) eigenstates of H can be written where A and B are coefficients to be determined.It is easily verified that the expression above for ψ(z) is indeed an eigenfunction of H with eigenvalue or = E k 2 ¯.Note that ψ(z) depends explicitly on Φ, but also implicitly on Φ as the allowed values of k depend on Φ.The energy eigenvalues E ¯likewise depend on Φ.
In the context of the transfer-matrix approach, let  s be the transfer matrix around the ¢ = + s s 1 ( )-site GR in the presence of Φ.To obtain  s from T s , note that we need to replace Q j by ) .We therefore have for the transfer matrix around the GR in the presence of Φ, Born-von Karman boundary conditions hold when the energy eigenvalues E ¯-or equivalently k-satisfy One finds by inspection of equation ( 12) that the spectrum is periodic in Φ with period Φ 0 .
The electronic states of the GR are a subset of those of the GC SL.Compare equation (8) for the GC SL with equation (13) for the GR, respectively: where we have slightly rearranged equation (15) for the GR using equation (12).These are twisted boundary conditions.Clearly, the solutions of equation ( 15) are a subset of those of equation (14) where j in essence specifies the subset selected.In other words, varying Φ in the GR enables one to selectively explore electronic states in the GC SL.In effect, 2πj in the GR plays the rôle of ¢ qL s in the GC SL with q the crystal momentum.

Results
In this section we start by considering the GC SL composed of supercells consisting of ¢ = + s s 1 ( )-site GCs, i.e. of period ¢ L s .In this case, the delocalized states are determined by |x s | 1.In an infinite QC, the candidate delocalized states (they can become localized for |λ| above a k-dependent value) have kd/π a rational number [25].As shown in [26], this is because for p = ¢ kd r s , we can write the transfer matrix for the infinite GC as a product of repeated factors of T s .For the GC SL with each period consisting of an ¢ = + s s 1 ( )-site GC considered here, this restriction on k is lifted; however, we shall see a vestige of the effect related to the infinite GC just mentioned.
Figure 2 shows the values of kd for delocalized states for various values of λ in the GC SL with each period ¢ L s an ¢ = + s s 1 ( )-site GC.Recall, k is a surrogate for normalized energy E ¯via = k E; ¯we do not show states for < E 0 ¯(kd imaginary) for λ < 0. Results are shown for s ä {0, 1, 2, L ,5, 6, 7}, i.e.
{ }in each case, the first panel is for negative values of λ while the second for positive λ.For a given λ, the horizontal black lines show allowed values of kd for delocalized states, i.e. bands, while the white areas correspond to localized states, i.e. bandgaps.Of course bands and bandgaps also exist for values of λ between those shown.The case s = 0 in figure 2(a) and (b) treats a periodic lattice whose unit cell consists of a single quadratic site with a freespace interval of length d, and thus the lattice constant is L 1 = d.We see a lower (upper) bandedge at kd = mπ with Î m  for λ < 0 (λ > 0).For s 1, bandgaps originate at p = ¢ kd m s near λ = 0.For λ < 0 (λ > 0) there are two types of lower (upper) bandedges.The positions of those at kd = 2mπ or kd = mπ for s even or odd are independent of λ, while the other lower (upper) bandedges trend toward lower (higher) kd as λ decreases (increases).As s increases, the bandwidths and bandgaps become narrower in kd.(Note, simultaneous with an increase in s is an increase in the GC SL period ¢ L s .)Also, as bands narrow for higher s, some bands narrow to the point that they are no longer clearly visible in the plots; nonetheless, they persist as |λ| increases.As one proceeds from bandgap to bandgap, the bandedge states alternate such that y y =  ¢ L 0 s ( ) ( ).This will mean that only half the bandedge states seen in figure 2  as expected.Many of the trends noted concerning figure 2 apply here as well.We also note that as s increases, the plots of the λ-dependent spectra resembles the stack structures computed in [25].By s = 7, a stack structure emerges.The λ-independent bandedges in the case of figure 2 here translate into GR states whose energies are independent of λ.
Why the λ-independence of some GR states?Consider states varying as y = z k z sin ( ) or kz cos around the GR where z is the coördinate around the GR circumference.Clearly, at λ = 0, all states apart from kd = 0 are two-fold degenerate.Increasing |λ| from zero breaks this degeneracy.Those states whose energies E ¯and hence kd are independent of λ vary as kz sin with kd = 2mπ or kd = mπ for s even or odd.In these cases, ψ(z) has nodes at the locations of the quadratic sites, and thus E ¯is independent of λ.The kz cos -like states for these values of kd in a perturbation picture sample the quadratic sites, and thus their energies depend on λ.
Finally, we consider the magnetic-flux dependence of the energy eigenvalues.In figure 4 are shown the GR states for a ¢ = + = s s 1 2GR for various values of j = Φ/Φ 0 .The value ¢ = s 2 is merely chosen to illustrate the general properties discussed below.The energy eigenvalues are obtained by solving equation (13).Note that the energy eigenvalues are periodic in j with period 1 as expected.Starting at j = 0 in figures 4(a) and (b), as we saw as in figure 3(c) and (d) which are identical with figures 4(a) and (b), that the values of kd are equally spaced at λ = 0 as is well known.The presence of the quadratic sites perturbs the energy eigenvalues, showing up as nonequispaced kd as |λ| increases from 0. As j increases from 0, the degeneracies at λ = 0 are broken, with the degeneracy at λ = 0 eventually restored at j = 0.5, though the kd values for λ ≠ 0 differ for j = 0 and 0.5.States whose energies are λ-independent at j = 0 no longer enjoy this property.Further increasing j again leads to breaking the λ = 0 degeneracy, which is again restored at j = 1, which is identical with the case j = 0.In addition, the energy eigenvalues are even in j; noting the periodicity in j, compare, say, figures 4(c) and (o) or  ( ) in the presence of magnetic flux Φ threading the GR.

2 2 2
¯giving the Schrödinger equation y y = H E ¯¯with E ¯an eigenvalue of H ānd ψ the eigenfunction.For the GC SL composed of many ¢ = + s s 1 ( )-site GCs, we require that the Bloch theorem y in the length of the supercell and ÿq is the quasimomentum.Born-von Karman boundary conditions y y = ¢ L 0 s ( )

Figure 1 .
Figure 1.An ¢ + s 1 ()-site QC that will be formed into an ¢ s -site GR by identifying the j = 0 with the = ¢ j s site. (The actual number of sites in the GC shown is¢ + = + s s 1 2and in the GR ¢ = + s s 1.) Index  Î ¢ j s 0, 1, 2, , {} labels GC sites at z = z j = j 2 d along the ring.Due to Born-von Karman boundary conditions in the ring, the wavefunction of an electron satisfies y y =

2
Figure 1.An ¢ + s 1 ()-site QC that will be formed into an ¢ s -site GR by identifying the j = 0 with the = ¢ j s site. (The actual number of sites in the GC shown is¢ + = + s s 1 2and in the GR ¢ = + s s 1.) Index  Î ¢ j s 0,1, 2, , { } labels GC sites at z = z j = j 2 d along the ring.Due to Born-von Karman boundary conditions in the ring, the wavefunction of an electron satisfies y y = ¢ L 0 s ( ) ( )where = ¢ ¢ L s d s natural-number power.By the Bloch theorem, ÿq the quasimomentum, which in terms of transfer matrices means the eigenvalues ξ ± of T s for a given = E k 2 ¯corresponding to delocalized states satisfy ξ + ξ − = 1 or more conveniently from a computational standpoint

Figure 2 . 2
Figure 2. Delocalized states of the periodic QC with ¢ = + s s 1 sites per period = ¢ ¢ L s d s 2 as a function of kd and λ for various s.The states satisfy |x s | 1.

Tr 1 s 1 2 .
will serve as states of the GR where only y y the Bornvon Karman boundary conditions.We now turn to the ¢ s -site GR for Φ = 0 where the states satisfy the Born-von Karman boundary conditions, or in other words = T In figure3are plotted the GR states as functions of kd for various λ and s.The states shown correspond to half the bandedge states seen in figure2, namely those for which y y At λ = 0, the values of k are given by p

Figure 3 . 1 2 2 (
Figure 3. States of the GR as a function of kd and λ for various s.The states shown satisfy x s = 1 and thus obey Born-von Karman boundary conditions at z = 0 and = ¢ = + z s d s d 1 2 2 ( ) at zero magnetic flux Φ = 0.

Figure 4 .
Figure 4. States of the GR as a function of kd and λ for s = 1 and various j = Φ/Φ 0 .The states shown satisfy =  Tr 1 s