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where V ( x ) is the infinite periodic potential with period L , obtained by the periodic repetition of the ring potential V ( x ) . Equation (9) has the well known Bloch solution 

where the function u$_{k}$ ( x ) fulfills the periodic condition 

and k is the electron wave vector from the interval ( −∞ , ∞ ) . Clearly, the wave function (5) and Bloch solution (10) coin- cide for k = 2 π L φ φ $_{0}$. Let us discuss this coincidence in detail. 

Consider first the ring with zero magnetic flux. To obtain the wave functions in such ring, it suffices to take the Bloch function (10) and to restrict it by the periodic condition 

which is the condition (7) for φ = 0 . Due to the condition (12), the wave vector k becomes discrete: 

Thus, in the ring with zero magnetic flux and specified poten- tial V ( x ) , the eigen-function ϕ$_{n}$ ( x ) and eigen-energy ε$_{n}$ can be calculated simply by setting k = 2 π L n into the Bloch so- lutions ϕ$_{k}$ ( x ) a ε ( k ) , calculated for the same potential V ( x ) repeated with period L from x = −∞ to x = ∞ . This recipe can be generalized to nonzero magnetic flux as follows. 

Arbitrary magnetic flux φ can be written in the form 

where φ , is the reduced flux from the range < − φ 0 2 , φ 0 2 ) or alternatively from < 0 , φ$_{0}$ ) , and n is one of the values n = 0 , ± 1 , ± 2 , . . . . Setting (14) into (5) one can write (5) in the form 

where the function ψ , n,φ , ( x ) ≡ exp ( i 2 π L nx ) ψ$_{φ}$ ( x ) obeys the periodic condition ψ , n,φ , ( x ) = ψ , n,φ , ( x + L . The boundary condition (7) now reads 

Similarly, in the Bloch function theory it is customary to express the wave vector k by means of the relation 

where k ′ is the reduced wave vector from the first Brillouin zone and the integer n plays the role of the energy band num- ber. Using (17) we can write the Bloch function (10) as 

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where the function u ′ n,k ′ ( x ) ≡ exp ( i 2 π L nx ) u$_{k}$ ( x ) fulfills the periodic condition u ′ n,k ′ ( x ) = u ′ n,k ′ x + L . It is then easy to verify that 

The equations (15) and (16) coincide with the equations (18) and (19), respectively, if 

or alternatively, if 

One can thus formulate the following general recipe. In the ring with a known potential V ( x ) , the eigen-function ϕ , n,φ , ( x ) ≡ ϕ$_{φ}$ ( x ) and eigen-energy ε$_{n}$ ( φ $^{′}$) ≡ ε ( φ ) can be calculated by setting (20) or (21) into the Bloch solutions ϕ$_{n,k}$ , ( x ) ≡ ϕ$_{k}$ ( x ) a ε$_{n}$ ( k $^{,}$) ≡ ε ( k ) , calculated for the same potential V ( x ) repeated with period L from x = −∞ to x = ∞ . The general recipe holds for any potential V ( x ) obeying the 

cyclic condition (8). It therefore holds also for any potential which additionally obeys the periodic condition 

where a = L/N and N is the (integer) number of periods a within the period L . This type of potential will be con- sidered in the rest of the paper. Figure 1 illustrates how an infinite 1D crystal is created from a 1D ring with lattice pe- riod a and length L = Na . Such crystal is still described by Schrödinger equation (9), except that now V ( x ) = V ( x + a ) and consequently u$_{k}$ ( x ) = u$_{k}$ ( x + a ) . However, the condition u$_{k}$ ( x ) = u$_{k}$ ( x + L ) holds as well because V ( x ) = V ( x + L ) . We can rederive the general recipe briefly, if we compare u$_{k}$ ( x ) = u$_{k}$ ( x + L ) with ψ$_{φ}$ ( x ) = ψ$_{φ}$ ( x + L ) and ϕ$_{k}$ ( x ) = exp( ikx ) u$_{k}$ ( x ) with ϕ$_{φ}$ ( x ) = exp ( i 2 π L φ φ $_{0}$x ) ψ$_{φ}$ ( x ) . We see that ϕ$_{φ}$ ( x ) coincides with ϕ$_{k}$ ( x ) for k = 2 π L φ φ $_{0}$. 

It remains to express the persistent current. The Bloch elec- tron in state ( n, k $^{′}$) moves with velocity v$_{n}$ ( k $^{′}$) = 1 ℏ ∂ε$_{n}$ ( k $^{′}$) ∂k ′ . We set into the Bloch electron velocity the formula (20). We obtain the expression v$_{n}$ ( φ $^{′}$) = L e ∂ε$_{n}$ ( φ $^{′}$) ∂φ ′ , which is the elec- tron velocity in state ( n, φ $^{′}$) in the ring. The current carried by the electron in state ( n, φ $^{′}$) reads 

The total persistent current circulating in the ring is 

where we sum (at zero temperature) over all occupied states n = 0 , ± 1 , ± 2 , . . . up to the Fermi level. In the following text we use the formulas (23) and (24) with symbol φ ′ changed to φ , where φ ∈ < − φ 0 2 , φ 0 2 ) or alternatively φ ∈ < 0 , φ$_{0}$ ) . 

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