content_image/397470.jpg content_image/397483.jpg content_image/397472.jpg content_image/397474.jpg content_image/397475.jpg content_image/397481.jpg content_image/397477.jpg content_image/397478.jpg content_image/397482.jpg content_image/397479.jpg content_image/397469.jpg content_image/397471.jpg content_image/397473.jpg content_image/397476.jpg content_image/397480.jpg
We define the matrix elements Γ$_{j}$ by equations 

<img src='content_image/397470.jpg'>

where l, l ′ = 0 , ± 1 , ± 2 , · · · ± ∞ and it is taken into account that Γ$_{l}$ ′ ,l depends only on the difference j = l − l $^{′}$. The Bloch energy ϵ ( k ) can be written as ϵ ( k ) = 〈 Φ$_{k}$ | ˆ H | Φ$_{k}$ 〉 / 〈 Φ$_{k}$ | Φ$_{k}$ 〉 . If we set for Φ$_{k}$ equation (43), we easy find 

Comparison of Eq. (45) with Eq. (25) shows that a$_{j}$ = 2Γ$_{j}$ . Obviously, 

which is an alternative form of equation (44). 

Finally, setting a$_{j}$ = 2Γ$_{j}$ into the result (35) we get 

Equation (47) together with Eq. (44) express the persistent current in the insulating ring through the Wannier functions of the constituting infinite crystal. We defer discussion of th ese expressions to subsection C, where equation (47) is derived from the Wannier functions of the ring. 

Let us define the Wannier functions of the ring. In the ring with N lattice sites and a fixed value of magnetic flux only the discrete Bloch vectors k$_{n}$ ( φ ) are allowed, where k$_{n}$ (0) ∈ < − π $_{a}$, π a > or alternatively k$_{n}$ (0) ∈ < 0 , 2 π a > . In both cases there are N allowed values of n (specified below), which means that the originally continuous band ε ( k ) consists of N discrete levels ε$_{n}$ ( φ ) . The eigen-functions of these lev- els represent a complete set of N Bloch functions obeying the orthogonality condition 

where k = k$_{n}$ ( φ ) . Consequently, in the 1D ring with N lattice sites the Wannier function at site x = la can be defined as 

<img src='content_image/397483.jpg'>

where k = k$_{n}$ ( φ ) and summation over k means the summa- tion over the N allowed values of n . Definition (49) coincides with definition (3.7) in Kohn’s paper [17], except that Kohn assumes k = k$_{n}$ (0) . 

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## B. Wannier functions of the ring 

<img src='content_image/397481.jpg'>

In the following calculations we choose the Brillouin zone k$_{n}$ (0) ∈ < 0 , 2 π a > , for which the allowed n are simply 

without regards to the parity of N and sign of φ . 

Using Eqs. (48) and (49) we can easy verify the orthogo- nality condition 

Finally, definition (49) can also be written in the inverted f orm 

In this paper the Wannier functions w$_{l}$ ( x ) are often called the ring Wannier functions while the Wannier functions W$_{l}$ ( x ) are called the crystal Wannier functions. The ring Wannier func- tions are magnetic-flux dependent, the crystal Wannier func - tions are not. 

We note that the results obtained in the rest of this section can also be obtained by choosing k$_{n}$ (0) ∈ < − π $_{a}$, π a > . In that case the allowed values of n are 

for odd N , while for even N one has 

<img src='content_image/397477.jpg'>

and 

<img src='content_image/397478.jpg'>

Evidently, it is more straightforward to work with Eq. (50). 

Now we express the persistent current at full filling in terms of the ring Wannier functions. We write ε$_{n}$ ( φ ) in the form 

We need to rewrite the right hand side of Eq. (56) in terms of the matrix elements 

where l, l ′ = 0 , 1 , . . . N − 1 . We set into Eq. (57) the equation (49). We find 

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<img src='content_image/397479.jpg'>

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<img src='content_image/397471.jpg'>

<img src='content_image/397473.jpg'>

<img src='content_image/397476.jpg'>

<img src='content_image/397480.jpg'>

