content_image/428970.jpg content_image/428980.jpg content_image/428971.jpg content_image/428973.jpg content_image/428974.jpg content_image/428978.jpg content_image/428969.jpg content_image/428972.jpg content_image/428975.jpg content_image/428977.jpg content_image/428979.jpg content_image/428976.jpg
Since T$_{l}$ ′ ,l depends only on the difference between l and l $^{′}$, we can write 

<img src='content_image/428970.jpg'>

Inverting Eq. (59) we get 

By means of Eq. (60), the ground state energy of the fully filled valence band is readily obtained in the form 

Finally, the persistent current at full-filling reads 

One can see from Eq. (57) that the matrix element T$_{0}$ rep- resents the on-site energy of the Wannier state w$_{l}$ ( x ) . Since w$_{l}$ ( x ) depends on magnetic flux, so does T$_{0}$ . This however means that the current (62) is in general not zero. We have thus arrived at the same conclusion as in the preceding sec- tion, where the persistent current in the insulating ring (Eq. 35) was derived from the Bloch states and general recipe. 

## C. Ring Wannier functions in the basis of crystal Wannier functions 

The result (47), or equivalently the result (35), can be ob- tained from the result (62), if the ring Wannier functions are expanded over the basis of the crystal Wannier functions. To show this we proceed as follows. 

The Bloch functions of the ring are given by Eq. (38) and normalized as ∫ Na 0 ϕ ∗ $_{k}$( x ) ϕ$_{k}$ ( x ) dx = 1 , where k = k$_{n}$ ( φ ) [see Eq. (48)]. The Bloch functions φ$_{k}$ ( x ) coincide with ϕ$_{k}$ ( x ) except for the normalization constant. As they are nor- malized by means of Eq. (40), in the ring we have the relation 

We set for φ$_{k}$ ( x ) in Eq. (63) the definition (43). Then we set Eq. (63) into the right hand side of Eq. (49). We obtain 

<img src='content_image/428980.jpg'>

where k = k$_{n}$ ( φ ) and summation over k means the summa- tion over n = 0 , 1 , . . . , N − 1 . Using substitution l ′ = Nr + s , 

<img src='content_image/428971.jpg'>

<img src='content_image/428973.jpg'>

<img src='content_image/428974.jpg'>

<img src='content_image/428978.jpg'>

where s = 0 , 1 , . . . ( N − 1) and r = 0 , ± 1 , ± 2 , · · · ± ∞ , we further obtain 

<img src='content_image/428969.jpg'>

Eventually 

<img src='content_image/428972.jpg'>

Equation (66) expresses the ring Wannier function at the ring site l , w$_{l}$ ( x ) , through the crystal Wannier functions W$_{l}$ ′ ( x ) . For φ = 0 equation (66) coincides with equation (5.9) in Kohn’s paper [17]. 

Setting Eq. (66) into Eq. (57) we obtain 

<img src='content_image/428975.jpg'>

By means of the matrix elements Γ$_{j}$ , equation (67) can be written as 

For j = 0 equation (68) gives 

<img src='content_image/428977.jpg'>

We substitute Eq. (69) into the right hand side of Eq. (62). We obtain the persistent current at full filling in the form 

<img src='content_image/428979.jpg'>

or equivalently in the form (47) which coincides with the re- sult (35) for a$_{j}$ = 2Γ$_{j}$ . 

In the rest of this section we discuss the physical meaning of the coefficients Γ$_{N}$ , Γ$_{2}$$_{N}$ , etc., appearing in expression (47). Since a$_{j}$ = 2Γ$_{j}$ , the meaning of the coefficients a$_{N}$ , a$_{2}$$_{N}$ , etc., is the same and does not need an extra discussion. 

<img src='content_image/428976.jpg'>

