ϕ$_{n,φ}$ ( x ) = ϕ$_{k}$$_{=}$$_{k}$$_{n}$$_{(}$$_{φ}$$_{)}$ ( x ) , ε$_{n}$ ( φ ) = ε ( k = k$_{n}$ ( φ )) , k$_{n}$ ( φ ) ≡ 2 π Na ( n + φ φ$_{0}$ ) . (38) I ( φ ) = − (2 π/φ$_{0}$ ) ∞ ∑ j = N, 2 N,... ja$_{j}$ sin ( 2 π N φ φ$_{0}$ j ) , (35) | a$_{j}$ | ≤ Ce − $^{ξj}$, (36) W ( x − la ) ≡ W$_{l}$ ( x ) = a 2 π ∫ π a − π a e − $^{ikla}$Φ$_{k}$ ( x ) dk , (39) ∫ a 0 | Φ$_{k}$( x ) | 2 dx = 1 (40) ∫ ∞ −∞ Φ$_{k}$ ′ ( x ) $^{∗}$Φ$_{k}$ ( x ) dx = 2 π a δ ( k ′ − k ) . (41) ∫ ∞ −∞ W ∗ l ′ ( x ) W$_{l}$ ( x ) dx = δ$_{l,l}$ ′ . (42) Φ$_{k}$( x ) = ∑ l e $^{ikla}$W$_{l}$ ( x ) (43) Hϕ$_{k}$ ( x ) = ε ( k ) ϕ$_{k}$ ( x ) , H = − ℏ 2 2 m d 2 dx 2 + V ( x ) , V ( x ) = V ( x + a ) . (37) IV. PERSISTENT CURRENT IN INSULATING 1D RING IN THE WANNIER FUNCTIONS FORMALISM A. Wannier functions of the infinite crystal 5 where φ ∈ < 0 , φ$_{0}$ ) . The results (33) and (34) hold for the conducting ( N$_{e}$ < N ) as well as insulating ( N$_{e}$ = N ) rings. For N$_{e}$ = N the first sum on the right hand side of Eqs. (33) and (34) becomes zero and both equations take the form Second, according to the general recipe of Sect. II, the Bloch function and Bloch energy in the ring, ϕ$_{n,φ}$ ( x ) and ε$_{n}$ ( φ ) , are given by equations where φ ∈ < − φ 0 2 , φ 0 2 ) and N can be even as well as odd. Equation (35) expresses the persistent current in a 1D ring made of a band insulator with an arbitrary valence band ε$_{k}$ . Third, the Bloch vector k is assumed to be from the Brillioun zone < − π $_{a}$, π a > , or alternatively from < 0 , 2 π a > . This means that ε ( k ) is the dispersion of a single energy band and ϕ$_{k}$ ( x ) are the Bloch functions of that band. As before, the functions ϕ$_{k}$ ( x ) and ε ( k ) do not involve any band index, and ε ( k ) is supposed to represent the valence band of the crystal. Below we define the Wannier functions by restricting us to the Bloch states of that valence band. An important property of the formula (35) can be recog- nized without specifying the dispersion law ε$_{k}$ . The disper- sion ε ( k ) is a periodic function of period 2 π/a . As long as it is analytic in the whole Brillouin zone < − π $_{a}$, π a > , the Fourier coefficients a$_{j}$ in the expansion (25) obey the inequality [20] where C and ξ are positive constants. Equation (35) combined with inequality (36) shows that the persistent current in the insulating ring is in general not zero albeit it decreases with the ring length at least exponentially. In the next sections, we will demonstrate for various specific insulating rings th e exponential decrease, and we will see that the current can be of measurable size in realistic systems. We first define the Wannier functions for the crystal. Ac- cording to a standard textbook definition, the Wannier func- tion W ( x − la ) at the lattice site x = la is defined as where Φ$_{k}$( x ) is the Bloch function of the infinite crystal. We note that Φ$_{k}$( x ) is the same Bloch function as ϕ$_{k}$ ( x ) in equa- tion (37). The only difference is that Φ$_{k}$( x ) is normalized as The pioneer results (1) and (2) can be obtained from Eqs. (33) and (34) if we keep only the term j = 1 and we put a$_{1}$ ≡ − 2Γ$_{1}$ . Equations (1) and (2) predict for insulating rings ( N$_{e}$ = N ) a zero persistent current, which contradicts equa- tion (35). Origin of this contradiction is evident. Keeping only the term j = 1 means that only the nearest-neighbor- site hopping is operative. This approach, also known as tight- binding approximation, simplifies the general dispersion l aw (25) to the cosine shaped dispersion ε ( k ) ∝ cos( ka ) . How- ever, equation (35) shows that nonzero persistent current in the insulating ring of length N is (mainly) due to the Fourier coefficient a$_{N}$ which is in general not zero for a realistic dis- persion law. We will see in the next section that a$_{N}$ physically represents the hopping amplitude for a single hop from site i to its periodic replicas i ± N . while the normalization of ϕ$_{k}$ ( x ) in the infinite crystal is un- specified. In the following text, Φ$_{k}$( x ) represents the nor- malized Bloch function of the infinite crystal and the symbol ϕ$_{k}$ ( x ) with k = k$_{n}$ ( φ ) represents the Bloch function of the ring. [In the ring ϕ$_{k}$ ( x ) is normalized by equation (48).] Since the crystal is infinite, k is a continuous variable and the Bloch functions fulfill the orthogonality condition One can then easy verify the orthogonality condition for the Wannier functions, We consider again the 1D ring with lattice period a and length L = Na , sketched in Fig. 1. Simultaneously, we con- sider the infinite 1D crystal obtained by the periodic repeti tion of the 1D ring, as it is illustrated in Fig. 1. For the sake of clarity we recall a few major points. First, the Bloch func- tion ϕ$_{k}$ ( x ) and Bloch energy ε ( k ) in the infinite crystal are described by Schrödinger equation Finally, definition (39) can also be written in the inverted f orm where one sums over all lattice points of the (infinite) cryst al.