Physica A 200 (1993) 519-524

North-Holland

SDZ: 0378-4371(93)E0142-2

Persistent currents in interacting lattice models

Michael Abraham and Richard Berkovits The Jack and Pearl Resnick Institute for Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Can 52900, Israel

We present the results of an exact calculation of the averaged persistent current and its root mean square value in interacting disordered 1D rings. While the averaged persistent current

exhibits a variety of interesting behaviors depending on the disorder and the strength of

electron-electron interactions, the r.m.s. value depends very weakly on the strength of

interaction. In general we observe that no dramatical increase of the averaged current or its

r.m.s. value as a result of interactions is possible for these systems.

Recently, the interest in persistent currents of quasi-1D disordered rings has arisen due to two experimental measurements performed on those systems. One of the experiments performed by the Bell-Labs group [l] on lo7 quasi-1D disordered rings measured the averaged amplitude of the current to be of order 10-2ev,lL (where ur is the Fermi velocity, and L is the ring’s circumference). On the other hand, a second experiment by the IBM group [2] on three single rings measured the current to be of order eu,lL.

Theory for non-interacting electrons in an ordered 1D ring threaded by a magnetic flux predicts a persistent current periodic in the magnetic flux of amplitude eu,lL [3-81. For a disordered ring it is theoretically predicted that the averaged current amplitude will drop as a function of disorder. These results are also correct for a multi-channeled quasi-1D ring. The magnitude of the theoretically calculated current for the above experimental realizations is of order 10-3eu,lL. The r.m.s. value of the current is calculated to be of the same order as the experimentally measured averaged amplitude of the current

[4,6,71. The large difference between the measured current amplitude in those two

experiments and theory, as well as the two orders of magnitude difference between the experiments themselves which cannot be explained as a result of fluctuations in current amplitude since they largely exceed the calculated r.m.s. value of the current, has recently led to many theoretical attempts to explain this discrepancy. Most of the published works try to explain this discrepancy as the result of the fact that previous theoretical work on persistent currents ignored the role of electron-electron interactions in these systems [9-131. The

03784371/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

520 M. Abraham, R. Berkovits I Persistent currents in interacting lattice models

main motivation for considering the role of interactions in the disordered case is the hope that electron-electron interactions will tend to homogenize the system, thus offsetting the reduction in current due to disorder, resulting in the increase of the averaged current [9,10,12]. Other works deal with the influence of interactions on the r.m.s. value of the amplitude, which when some classes of diagrams are included seems to increase [11,13]. Nevertheless, no consensus on the exact influence of interactions in both cases has yet been reached.

In a previous publication [14] we presented an exact study of the influence of interactions on a 1D single band spinless fermion model on a small lattice. Our results for the behavior of the averaged current for different values of interactions and disorder are summed up in a schematic way in fig. 1. It can be seen that interactions play a different role for different regimes of parameter space. There is a line of maximum enhancement for strong disorders and medium strength of interaction. Even for maximum enhancement the value of current is very far from the ordered non-interacting value of eu,lL. For strong interactions the current is suppressed by the Mott-Hubbard transition. Maxi- mum suppression of the current by interactions is achieved along a line corresponding to weak disorder and strong interactions. Returning to the general question of the influence of interactions on the averaged current, it can

aB

I b

S weak interactions

Fig. 1. A schematic drawing of the main features of the averaged current dependence on disorder and interactions. The localization length 5, and Bohr radius aB which is inversely proportional to

the interaction are defined in the text.

M. Abraham, R. Berkovits I Persistent currents in interacting lattice models 521

be seen that although a complex behavior of the current as a function of interaction is observed, none of the changes will increase the averaged current to the value ev,lL. The reason is that for values of interaction which are big enough to cancel out effects of disorder the current will decrease because of the Mott-Hubbard transition.

The Mott-Hubbard transition is the result of the absence of Galilean invariance in the interacting lattice model resulting in a metal-insulator transition [15]. For systems which are Galilean invariant, i.e. continuous models with no underlying periodic potential, the transition does not occur for clean systems, and for the disordered case strong interactions can increase the averaged current to a value of order ev,/3L [16]. Therefore, the underlying periodic potential of the metal plays an important role in determining the influence of interactions on the persistent current.

The results for the r.m.s. value of the current described briefly in ref. [14] will now be presented in detail. The disordered 1D ring system is represented by the following Hamiltonian:

H = C EiU :Ui + (V(B) C uf+,ai + h.c’) + Hi”, 9

I I

where l i is the on-site energy which is chosen randomly between -W/2 and W/2. V(6) = exp(i8) V, where V is a constant hopping matrix element. Thus we are considering on-site diagonal disorder. The phase 8 = 2+JL&, where s is the lattice constant, L is the length of the ring, 4 the magnetic flux, and $0 = hcle is the quantum flux unit (h is Plank’s constant, c is the speed of light, and e is the electronic charge). In the remainder of the paper we shall take & = 1. a t (a,) are the fermionic creation (destruction) operators.

The interactions between the electrons are represented by L&. We have chosen to consider a long range Coulomb interaction of the form

where K represents the average positive jellium which equals the average electron density. Of course this form of interaction is not exact for small rings, but the exact form of interaction does not change our main results. We shall define Ed = e2/s in order to simplify further discussions.

We have exactly calculated the ground state energy of the many-body Hamiltonian given in eq. (1) by numerical diagonalization. Details are given in ref. [14]. Once the lowest eigenvalue is obtained for different r$, the persistent current may be calculated using the usual definition

522 M. Abraham, R. Berkovits 1 Persistent currents in interacting lattice models

Z(4) = --c -$ E(4) , (3)

where E(4) is the lowest eigenvalue. We are interested in calculating the r.m.s. value of the current for the whole

parameter space of V, W, and E,. Obviously only two relevant energy scales appear in this problem, and as explained in ref. [14] the most convenient are W/V, and l ,lV. In terms of length scales they correspond to 5, the localization length, which follows two asymptotical forms: 5 -slln(W/V) for strong disorder (W Z+ 27rV), and 5 - s(V/ W)” for weak disorder (W % 2nV) [17], and the Bohr radius a, = (~V/E,)S.

We start by presenting the results for the r.m.s. value of the current (Z)* in the limit of no interactions (a, = 00). For the strongly localized case ,.$ < L it is expected that the r.m.s. value will have the following behavior [4]:

(1’) = C,(Z)* expWS) , (4)

where C, is a constant. In fig. 2 we present our numerical results by filled squares and compare them to eq. (4), represented by the line where C, = 0.15. It can be seen that for 6 < L, which corresponds to ln( s/s) < 2, there is an excellent fit.

Fig. 2. The averaged r.m.s. value of the current as a function of the localization length 5 for

aB = m. The filled squares describe the numerical results, while the line corresponds to eq. (4).

M. Abraham, R. Berkovits I Persistent currents in interacting lattice models 523

It is of course important to check the influence of interactions on the r.m.s. value in order to see if any large increases in the r.m.s. value, which is needed in order to explain the discrepancies between the two experiments [1,2], is possible. In fig. 3 the normalized r.m.s. value of the current as a function of disorder and strength of interactions is presented. It can definitely be stated that interactions play an insignificant role in this case, i.e. (Z’(t, au))/ (Z(~,a,))2-(Z2(~,a~=~))l(Z(~,aB=~))2. Therefore, not only does the r.m.s. value of the current not increase dramatically, there is hardly any influence of interactions on it at all.

In conclusion, we calculated exactly the persistent current of a small interacting disordered 1D ring. We found that depending on the disorder the interactions can increase or decrease the current. No dramatical increase in the averaged current or its r.m.s. value is possible. This is the result of the periodicity of the underlying lattice structure of the model. Therefore, for half-filled single-band 1D spinless models it is clear that interactions cannot explain the results of the single ring experiments [2].

We would like to thank B.L. Altshuler, Y. Avishai, and H.A. Weidenmuller for useful discussions. RB would like to thank the Alon Foundation for financial support.

Fig. 3. The averaged r.m.s. value of the current as a function of disorder and interactions. It can

he seen that interactions have very little influence on the r.m.s. value.

524 M. Abraham, R. Berkovits I Persistent currents in interacting lattice models

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