8 v(x) a (x) a x 2d V(x) a 2d 0 −V$_{0}$ FIG. 2. Left: One-dimensional model of a single isolated atom, de- scribed by equation (73). The atomic potential, v ( x ) , is modeled by the potential well of width 2 d , embedded between two infinitely- thick potential barriers of height V 0 . Right: The potential V ( x ) in a ring-shaped periodic lattice composed of N one-dimensional atoms ( N = 6 ). The lattice period is a , the ring circumference is L = Na . In the numerical calculations of Sect. V, the parameters of the ring are chosen to emulate the GaAs ring with conduction electrons sub- jected to the periodic potential of N quantum dots: the electron ef- fective mass is m = 0 . 067 m 0 , the period of the quantum-dot lattice is a = 40 nm, the ring length is L = Na , and the size and depth of the potential well (quantum dot) are 2 d = 15 nm and V 0 = 18 meV, respectively. For these parameters the potential well supports one bound state, the ground state with energy ε$_{a}$ = − 10 . 4 meV. The ground-state orbital ϕ$_{a}$ ( x ) is sketched schematically. V. THE LCAO APPROXIMATION, APPLICATION TO THE 1D LATTICE WITH RECTANGULAR POTENTIAL WELLS Φ$_{k}$( x ) = C ∞ ∑ l = −∞ e $^{ikla}$ϕ$_{a}$ ( x − la ) , (75) V ( x ) = ∞ ∑ l = −∞ v ( x − la ) , (74) ε$_{n}$ ( φ ) = N 2 ∑ j = − N 2 +1 e − ik$_{n}$ ( φ ) $^{ja}$T$_{j}$ , (72) ε$_{n}$ ( φ ) = N$_{−}$ 1 2 ∑ j = − N$_{−}$ 1 2 e − ik$_{n}$ ( φ ) $^{ja}$T$_{j}$ , (71) [ − ℏ 2 2 m d 2 dx 2 + v ( x ) ] ϕ$_{a}$ ( x ) = ϵ$_{a}$ϕ$_{a}$ ( x ) , (73) According to the definition (44), in the infinite 1D crystal the matrix element Γ$_{N}$ represents the hopping amplitude of a single hop from the Wannier state at site l to the Wannier state at site l + N . In the ring with N lattice sites the site l + N is a periodic replica of site l , which means the same site. Appearance of Γ$_{N}$ in expression (47) therefore implies, that the electron in the ring with N sites can make a single hop from the Wannier state at site l to its periodic replica at site l + N . In other words, the electron hops from site l to the same site by making one round around the ring, and Γ$_{N}$ is the hopping amplitude of the process. Note that we speak about hopping in the ring, but in terms of the Wannier states of the infinite crystal (c.f. Eq. 44). Similarly, the coefficient Γ$_{2}$$_{N}$ describes the process in which the electron hops from site l to the same site by mak- ing two rounds around the ring. It follows from relation (36) and equation a$_{j}$ = 2Γ$_{j}$ , that Γ$_{2}$$_{N}$ is much smaller than Γ$_{N}$ . Its contribution to the right hand side of Eq. (47) is therefore negligible. The meaning of the coefficients Γ$_{3}$$_{N}$ , Γ$_{4}$$_{N}$ , etc., is analogous and their contribution is negligible even more. We can summarize the result (47) as follows. The persistent current in the insulating ring with N lattice sites, derived in terms of the crystal Wannier functions, is due to the fact that the electron at the ring site l is allowed to make a single hop from site l to its periodic replicas l ± N . (The sign minus is allowed as well due to the symmetry reason.) To evaluate the persistent current (Eq. 47) for a specific crystal lattice, we need to determine the crystal Wannier func- tions of the lattice. As an example we consider the ring- shaped 1D lattice in Fig. 2. The lattice is composed of the artificial 1D atoms with a single atomic level ϵ$_{a}$ . The atomic orbital ϕ$_{a}$ ( x ) and energy ϵ$_{a}$ in the isolated atom positioned at x = 0 obey the Schrödinger equation The process in which an electron at the ring site l hops from site l to the same site by making one round around the ring may seem meaningless, because such process does not exist in the basis of the ring Wannier functions. In that basis the energy spectrum ε$_{n}$ ( φ ) is given by equation (60). As shown in appendix B, equation (60) can be rewritten for odd N as and for even N as where v ( x ) is the atomic potential modeled as a rectangular potential well centered at x = 0 (figure 2, left). In accord with the general recipe of Sect. II, assume that the ring with N atoms (figure 2, right) is periodically repeated as shown in figure 1. This repetition gives rise to the infinite 1D crystal with potential V ( x ) which is periodic with period a and ranges from x = −∞ to x = ∞ . Evidently, V ( x ) is a sum of all isolated atomic potentials, that means where T$_{−}$$_{j}$ = T ∗ j . According to the definition (57), the matrix element T$_{j}$ represents the amplitude of hopping from the ring Wannier state w$_{l}$ ( x ) to the ring Wannier state w$_{l}$ ′ ( x ) , with j = l − l ′ being the hopping length. The maximum hopping length between the ring Wannier states, allowed by equations (71) and (72), is j = ± Int( N/ 2) only, and the current in the insulating ring (Eq. 62) is due to the fact that the ring Wannier states depend on magnetic flux. Hopping from l to l ± N , l ± 2 N , etc., emerges after the ring Wannier states are expressed (see Eq. 64) through the flux-independent crystal Wannier states. This provides a different picture of the same physics. where la is the position of the l -th lattice point. The Bloch function Φ$_{k}$( x ) and Bloch energy ϵ ( k ) of the electron mov- ing in such periodic potential can be expressed analytically in the approximation of localized atomic orbitals (LCAO). The Bloch function is approximated by the LCAO ansatz [21]