Magnetic moments in biased quantum circuits

Michele Cini,1,2 Enrico Perfetto,1 and Gianluca Stefanucci1,2,3

1Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Rome, Italy2Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, Via E. Fermi 40, 00044 Frascati, Italy

3European Theoretical Spectroscopy Facility (ETSF)�Received 23 November 2009; revised manuscript received 16 February 2010; published 9 April 2010�

We consider a quantum ring connected to leads and the current which is excited by biasing the circuit in theabsence of external magnetic field. The magnetic moment Mring that arises in this way depends on the currentdistribution inside the ring. We perform a thought experiment in which Mring is determined by measuring thetorque due to an infinitesimally small probe magnetic field. This leads to a definition Mring, which is given bythe derivative of the grand-canonical energy of the quantum ring with respect to an external magnetic flux inthe zero flux limit. We develop the many-body formalism by Green’s-function techniques and carry on illus-trative model calculations. The resulting theory predicts that at small bias the current in the ring is alwayslaminar, that is, the magnetic moment vanishes in linear-response theory. The approach most naturally lendsitself to include induction effects by a self-consistent procedure.

DOI: 10.1103/PhysRevB.81.165202 PACS number�s�: 73.23.Ad, 72.10.Bg

I. INTRODUCTION

There is widespread interest in systems containing meso-scopic and nanoscopic rings where quantum coherence ef-fects are important. The persistent currents1 in mesoscopicmetallic rings2 have been investigated in depth. Recentprogress in connecting aromatic molecules to metallic leadsbrought the ringlike topology into the nanoworld.3 The quan-tum behavior of electrons and fundamental paradigms suchas Aharonov-Bohm oscillations,4,5 quantum interference pat-tern of the total current,6–8 and of the ring bond currents9–12

have received considerable attention.Many state-of-the-art computations of ring magnetic mo-

ments produced by the current excited by the bias have al-ready been reported by several authors.7,13–16 The establishedquantum transport theory17–22 yields the currents flowing oneach bond as quantum averages of the current operators. Thispaper deals with the current distribution inside multiply con-nected circuits excited by the bias and their magnetic effectsbut we focus on the physical interpretation of those resultsand on the relation to an actual magnetic measurement per-formed on the ring. Indeed we argue that different experi-mental setups will measure markedly different quantities;therefore, we aim to an operational definition suitable tonanorings of molecular dimensions, when one must considerthe possibility that the measurement perturb the system.

Below for the sake of definiteness we specialize the dis-cussion to circuits containing one tight-binding ring con-nected to biased leads. The continuum counterpart deserves asimilar discussion. We consider a system which consists of aleft �L� and right �R� biased leads connected to a polygonalring with N sites, see Fig. 1, described by the Hamiltonian23

H = Hring + HL + HR + HT + Hbias. �1�

Here, for the quantum ring we use the tight-binding Hamil-tonian

Hring = �m�n=1

N

hmncm† cn + H.c. �2�

with hopping integrals hmn=0 if m and n are not nearest

neighbors. Further, in Eq. �1� H� represents the unperturbed

�=L,R lead while HT stands for the tunneling Hamiltonianconnecting the leads to the ring. The occupation of the sys-tem in equilibrium is fixed by the chemical potential �. Thesystem is driven out of equilibrium by the bias term

Hbias = ULNL + URNR �3�

with N� the total number operator for electrons in lead �.Eventually it reaches a steady state which is17 a Slater deter-minant of current-carrying eigenstates of Eq. �1� with left-going/right-going states populated up to the electrochemicalpotential �+UR/L of the right/left lead. This interplay ofequilibrium and out-of-equilibrium terms is also familiar inthe partition-free time-dependent formulation19 where thecontribution of the current-carrying states is weighted by theFermi functions which assign the occupations before the ex-ternal potential is applied.

It is intuitively clear that the current pattern must be asuperposition of a circulating current which produces thering moment and a laminar one which relates to the interac-tion of the bias with the external circuit. How to calculate themagnetic moment and hence the circulating current is themain contribution of this work.

The classical definition of the ring magnetic momentMring is24

Mring =1

2N�

m�n=1

N

Imn�r�m ∧ r�n� , �4�

where r�m is the position of the m site of the ring, R is theradius of the ring, and Imn is the current flowing along them-n bond. For example, in the interesting special case whenthe ring is a regular N-sided polygon, we may write,

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Mring =R2

2sin

2�

N�

m�n=1

N

Imn =S

N�

m�n=1

N

Imn, �5�

where S= 12R2N sin2�

N is the area of the ring. In order toevaluate Mring we need to know the currents Imn in the in-ternal wires. In quantum mechanics18,23

Jmn = − i�hmncm† cn − hnmcn

†cm� �6�

is the electron current operator between sites m and n con-nected by a bond with hopping integral hmn. Such interpre-tation naturally stems from the continuity equation

d

dtnm = �

n

Jmn �7�

in which the rate of change in the density nm on site m isseen as the sum of the currents flowing from site m to allconnected sites n. �In a similar way one obtains the familiarformula for the current density in a continuum system.� We

shall call Jmn the bond current operator since it depends onthe operators straddling a bond.

For an isolated ring, in the absence of a bias, a current canbe excited by a threading magnetic flux; all the sides of thepolygon have a common value Imn� I, and the classical re-sult �5� we reduces to the elementary formula Misolated=SI.In state-of-the-art calculations for wired rings7,13–16 the mag-netic moment is calculated by setting in Eq. �4� Imn=Jmn

��Jmn�, where the average is taken over the many-body cur-rent carrying states. Next, we show that such a prescription isquestionable.

A. Need for a proper definition of the vortex current

In classical circuits the continuity equation is encoded inKirchhoff’s law, which determines the current together withOhm’s law. Indeed, since the electric field is irrotational, ifthe bonds have equal conductance Ohm’s law requires thatthe current be irrotational as well as divergenceless; in anycase the current is completely specified. On the other hand,in a macroscopic ring connected to leads, one can measurethe current in each wire by using an amperometer or byexploring the magnetic field around each branch of the cir-cuit and performing the line integral. To sum up, the currentin each wire can be independently measured and is deter-mined by Kirchhoff’s and Ohm’s laws.

This does not apply to ballistic transport in the quantumregime. For circuits containing loops the continuity equationalone cannot uniquely determine the currents Imn. The cur-rent in Eq. �6� solves Eq. �7� but it is not the unique solution.

One is free to add a vortex field Jmn→Jmn+V for m�n andtherefore there is no reason that the physical current flowingthrough the bond m-n is the same as the expectation value of

Jmn. In the continuum case this corresponds to the possibilityof adding a vortex field V� , with �� ·V� =0, to the current den-sity. Moreover the current cannot be taken to be defined at agiven bond in quantum systems where one cannot tell if theelectron goes through the upper or lower branch of the ring.

B. Peierls prescription and symmetric phases

In the tight-binding description an external magnetic flux� piercing the ring is included using the Peierls prescription,i.e.,

hmn → hmne�2�i/�0��R� m

R� n A� dr� = hmnei��mn/c�, �8�

where �0= hce is the flux quantum, which becomes �0=2�c,

where c is the speed of light, in atomic units �=1, e=1; theline integral of the vector potential A� goes from site m to siten; the phases are related to the flux � threaded by the ring bythe relation

�m�n=1

N

�mn =� A� dr� = � . �9�

For an isolated ring, one can regain the classical resultMisolated=SI by setting

Misolated = cS d

d��Hring�

�=0�10�

and using the Hellmann-Feynman theorem; this gives

Misolated=SI with I=J��Jmn�. Let us try to extend this pro-cedure to the wired rings. The formula in Eq. �4� with Imn=Jmn can be obtained as the derivative of the total energy ofthe system with respect to � at �=0 provided that we adoptthe symmetric phase choice �mn=� /N for m�n. Indeed, theHellmann-Feynman theorem now yields,

Mring � cS d

d��H�

�=0

= cS �m�n=1

N d

d�hmnei��/Nc�cm

† cn + H.c.��=0

=S

N�

m�n=1

N

Jmn, �11�

and S= 12R2N sin2�

N is the ring area. This agrees with Eq. �5�.Should we feel reassured after this confirmation of the

classical formula? The answer is definitely no since this is anad hoc derivation based on contrived assumptions. The mag-netic moment Mring should only depend on the magnetic flux� or from the flux derivative, not on the specific choice ofthe phase factors �mn. We illustrate this point by consideringfor simplicity the triangular ring �N=3� with sites a, b, and cconnected to one-dimensional leads through sites a and b,see Fig. 2. The formula �11� with �ab=� and �bc=�ca=0

yields Mring=SJab. On the other hand, the choice �ab=�bc

FIG. 1. �Color online� Sketch of a ring with N=7 sites in contactwith a L and R lead. The site m of the ring is positioned in r�m. Thering is connected to lead L via site a and to lead R via site b.

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=0 and �ca=� corresponds to the same flux � but yields

Mring=SJca. Different choices give different results, whichdisagree from the result in Eq. �11� corresponding to thesymmetric phase choice. In Sec. II C below, we show that thedifferent choices are indeed not gauge equivalent for an in-finite system because there is a subtle point about represent-ing the external circuit by infinite wires.

It is evident that the above difficulties constitute the warn-ing light of a physical problem. The solution is physical andwe must ask how the magnetic moment could be measured.

II. THOUGHT EXPERIMENTS

The approach introduced originally by Stern and Gerlachto measure magnetic moments uses a weak external probefield and a mechanical force measurement. They measuredthe force acting on the Ag atoms by means of the deflectionof the spin-polarized atomic beams. In the case of a wiredring we could think about a direct mechanical experimentperformed by an atomic force microscope. We develop agedanken experiment as an idealized version of an actualexperiment that one should make in order to give an opera-tional meaning to Mring. Let us now better specify whichforce �or torque� one should measure.

While the system is biased, and the current flows, weswitch a very weak probe magnetic field of strength B at anangle � with respect to the normal of the ring. The field mustbe weak because we want to know Mring caused by thecurrent, without the modification that a large field could pro-duce. The probe field produces forces on the ring, and on thewhole circuit, that one can measure.

A. Torque acting on the circuit

We denote by � the magnetic flux through the ring and by������ the current carrying state of the system after all tran-sient effects have disappeared. The state ������ is thereforean eigenstate of the Hamiltonian in the presence of the mag-

netic flux which we call H���.The total energy of the infinite system diverges but one

could consider

E��� = ������H��������� − ���0��H�0����0�� , �12�

which may be interpreted as the total energy change inducedby the field.

The flux depends on the direction of the field and theangular derivatives of E��� define a mechanical torque act-ing on the system. Let us assume for convenience that thefield couples exclusively to the bonds within the ring andtherefore the magnetic perturbation is localized. This can bearranged if Hring depends on the field but the rest of H is field

independent. Even so, the field in the ring changes the wave

function everywhere in the circuit and hence ������H���− Hring��������� does depend on the flux. Physically, thismeans that the field-free external circuit experiences atorque. If the whole circuit acts as a rigid body and we mea-sure a torque acting on it, we are actually measuring themagnetic moment of the whole system, not the one of thering. The use of the total energy to calculate the torque ex-perienced by the ring is therefore ambiguous. To measureMring we must arrange a different experiment.

B. Local measurement

We define Mring for the connected ring by a magneticmeasurement to be performed in situ on the ring itself. Thismeans that we perturb the system by inserting a small fluxthrough the ring and measure the resulting torque on the ringitself, e.g., by an atomic force microscope. Although ours isa thought experiment, it is also a suggestion of a real onesince the recent developments of technology can make ourproposal a practical one. We shall see that the Hamiltoniancontains enough information to compute Mring since thecoupling to a probe field �which is eventually set to zero� viathe Peierls prescription encodes all the necessary informa-tion. A local experiment requires a measurement of theforces acting on the ring and the result can be interpreted interms of the ring magnetic moment Mring. The ring energy is

Ering��� = ������Hring��� − �Nring������ , �13�

where Nring is the number operator of the ring and � is theequilibrium chemical potential as discussed in Sec. I. Thusone gets the magnetic moment Mring directly by measuring

Ering��� − Ering�0� = − MringB cos � = − Mring�

S, �14�

where S is the ring surface. For isolated rings, Nring is a

conserved operator, ������Nring������ is flux independent,

and therefore one can safely discard the �Nring term. Forconnected rings, the term which references the energy to thechemical potential is needed to ensure the invariance of thetheory under a uniform energy shift. Otherwise any shift in

the energy origin Hring→ Hring+ENring would give a flux-

dependent contribution proportional to ������Nring������and hence it would change the magnetic moment, which isabsurd.

Note that ������Nring������ depends on �, and thereforeEqs. �13� and �14� predict that the term proportional to �contributes to Mring. Physically, this must be expected be-cause the insertion of a flux can attract charge of either signin the wired ring and this effect must be accounted for.

Since forces are easier to measure than energies, we pro-pose measuring Mring as

FIG. 2. Sketch of a ring with three sites in contact with twoleads. The sites of the ring are labeled with Roman letters.

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Mring = −SdEring

d�

�=0. �15�

We have stressed that no magnetic field is needed since thecurrent is excited by the bias, however in the presence of flux� the obvious extension is

Mring = −SdEring

d�

�

. �16�

The magnetic moment is due to a current

Iring = −c

SMring �17�

that will be referred to below as the circulating current.Physically one expects that at least the flux due to self-induction should be introduced, and since this flux is propor-tional to the current in the ring, flux and current should befound self-consistently. However we defer this developmentto future work. For isolated rings, one can take �=0 and getback the formula �10� which represents the persistent currentresponse of an isolated ring by an external magnetic field.

C. Ring local gauge invariance

There is another strong reason why the local definition inEq. �15� using Hring is suitable for the ring moment while theglobal definition �with H instead of Hring� is not. Both defi-nitions are expressed in terms of observables and are obvi-ously gauge invariant. The magnetic moment Mring in Eq.�15�, however, enjoys a more subtle invariance, that allowsus to choose the Peierls phases within the ring as we pleaseprovided Eq. �9� is fulfilled, without keeping into accountwhat happens at large distances, in the rest of the system. Itis a stronger condition than the standard gauge invariance.

Consider the patterns in Fig. 3. In a1, a2, and a3 the ringis connected to wires with open-boundary condition, in b1,b2, and b3 the external circuit is considered. An arrow along

the m-n bond means that �R� m

R� n A� dr�=�, no arrow means that

the hopping integral is real; ai and bi have the same choiceof phases. Two configurations are equivalent �i.e., connectedby a gauge transformation� if the flux threaded in any closedpath is the same. We shall use the symbol � to denote thisequivalence. By inspection, a1�a2�a3 and b2�b3 but theexternal circuit is threaded by a flux � in b1 and by a flux 0in b2 and b3. Physically, one expects that the ring has awell-defined magnetic moment but in the case of a ring con-nected to leads there are different gauge-inequivalent ways toinsert the flux. The different choices imply threading flux inthe external circuit or not, and yield different total energies inEq. �12�, and hence different magnetic moments. It is neces-sary to know the geometry of the field and circuit in order toevaluate the total energy given in Eq. �12�. This confirms ourinterpretation that Eq. �12� does not bring unambiguous in-formation to extract the ring magnetic moment.

On the contrary the invariance of Eq. �13� follows from

the fact that the operator Hring���−�Nring is a local operatorand hence its average only depends on the projection ontothe ring of the single particle states �k� forming the Slater

determinant ������. The effects of any flux concatenatedwith the circuit is removed in this way. For example, thesingle-electron wave functions which make up the many-body wave functions undergo a phase change in the triparound the external circuit in b1 and not in b2; however thesolution at a given energy can be taken to be locally the samein the ring region, despite the change in a distant bond.

D. Magnetic moment of the ring

Equation �15� for Mring with Ering in Eq. �13� is the cen-tral result of this work. It does not suffer from any of theproblems of the standard approach. The ring current, whichis not uniquely determined by the continuity equation, is nowfixed by the dynamics. It represents the forces acting on thering acting as a rigid body hinged on the circuit and is inde-pendent of all interactions of the field far from the ring.Below we develop a many-body theory by Green’s-functiontechnique to calculate Mring in Eq. �15� and explore itsphysical contents.

III. GREEN’S-FUNCTION APPROACH

In order to compute the magnetic moment in Eq. �15� forthe system described by Hamiltonian in Eq. �1� we need tocalculate the expectation value of Ering over the stationarystate of the current carrying system in the presence of a flux�. The lesser Green’s function G�t , t�� which describessuch stationary nonequilibrium state depends on the time dif-

a2 b2

a3 b3

a1 b1

FIG. 3. �Color online� Flux patterns for a ring inserted in a wirewith open boundary conditions �a1, a2, and a3� and in a closedcircuit �b1, b2, and b3�. The arrows mark the bonds where thehopping integral is complex �see text�.

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ference t− t�. In terms of its Fourier transform G���,

Ering��� = − i� d�

2�Trring��h − ��G���� , �18�

where h has elements hmnei�mn/c and the trace is taken overthe sites of the ring. It is convenient to obtain G��� by anembedding technique with �=�L

+�R the lesser embed-

ding self-energies of the left and right leads. Thus,

G��� = GR�����L��� + �R

����GA��� . �19�

The retarded/advanced Green’s function projected onto thering is GR/A���= ��−h−�R/A����−1. Using the fluctuation-dissipation theorem for lead �=L,R one obtains for thelesser embedding self-energy ��

���=−2if��−U��Im���

R����, where f��� is the Fermi distribution function atchemical potential �. The retarded and advanced compo-nents are related as ��

R= ���A�† with �=L,R. Substituting Eq.

�19� in Eq. �18� and taking the flux derivative at �=0 weobtain for the magnetic moment

Mring = iS� d�

2�Trring� dh

d�GR�GA + �h − ��GR

� dh

d�GR� + �GA dh

d��GA�

�=0. �20�

As already noted, the first term in the square brackets yieldsa linear combination of the bond currents, combination thatdepends on the Peierls phase configuration. The indepen-dence of Mring from the phase choice is restored by adding�h−�� dG���

d� which is explicitly given by the second term inEq. �20�. We wish to emphasize that for �=0 the magneticmoment has no diamagnetic contribution.

IV. RESULTS AND DISCUSSION

We consider one-dimensional tight-binding leads as inFig. 2 with nearest-neighbor hopping tlead described by theHamiltonian

HL = tlead �j=−�

−1

�dj†dj−1 + H.c.� ,

HR = tlead�j=1

�

�dj†dj+1 + H.c.� , �21�

connected to the triangular ring via the tunneling term

HT = tlead�d−1† ca + d1

†cb� + H.c. �22�

The embedding self-energies have only one nonzero matrixelement, namely, ��L

R����ij =�ia� ja���−UL� and ��RR����ij

=�ib� jb���−UR�, where UL=−U and UR=U. The function���� can be easily calculated and reads

���� =1

2��� + i�� −�� + i�� + 2tlead

�1 +4tlead

�� + i�� − 2tlead� . �23�

Here and below all energies and currents are measured inunits of tlead. In Fig. 4 we display the magnetic moment

Mring of Eq. �20� as a function of the bias U for differentvalues of the chemical potential �=−1.5,0 ,−1.0,0.5,1.0.

We compare the results to the magnetic moment Mring ob-tained by the bond current according to Eq. �5� evaluated atImn=Jmn. The bond currents are computed using a Landauer-type formula.12 All currents vanish for bias U=2 as the rightcontinuum is lifted by 2 while the left continuum is loweredby the same amount. Since the bandwidth is 4 the bias U=2 represents the minimum value of U for which there is nomore overlap between the left and right continua. The maxi-mum of Mring versus bias U is seen in Fig. 4 to shift to lowerU with increasing �. For ��0 the maximum of Mring and

of Mring occurs at the same value of U. For �0 the maxi-mum of Mring occurs at the minimum of the bond currents,Fig. 4.

By numerical inspection we have observed that the de-rivative of Mring as function of U at zero bias is alwaysvanishing. It is interesting to dwell on the meaning of thisfinding. On physical grounds, a circulating current should belocalized and should not change the average number of elec-

trons N� in lead �=L /R. As a consequence the first-orderresponse in Mring�t� to the bias should vanish for t→�.According to the Kubo formula the response function isgiven by the average over the unbiased ground state of thecommutator between the operator of the cause at time 0 andthe one of the effect at time t. In the Hamiltonian in Eq. �1�the cause is the bias operator Hbias while the effect �in theHeisenberg picture with respect to the unbiased Hamiltonian�is Mring�t�. To first order this means that

�−�

t

�Mring���,ULNL + URNR�d� = 0, �24�

where Mring is the operator of the magnetic moment and

N��t� is in the Heisenberg picture with respect to the unbi-ased Hamiltonian. This can be true for any UL ,UR only if

0 0.5 1 1.5 2U

-0.06

-0.04

-0.02

0

0.02

0.04

0 0.5 1 1.5 2U

-0.05

0

0.05

0.1

Mring

Mring

0 0.5 1 1.5 2U

-0.06

-0.04

-0.02

0

0.02

0.04

0 0.5 1 1.5 2U

-0.02

0

0.02

0.04µ = −1 µ = 0.5

µ = 1 µ = 1.5

∼

FIG. 4. �Color online� Plot of Mring �black solid line� and

Mring= 13 �Jab+Jbc+Jca� �red, dashed line� as obtained in Eq. �5�

with Imn=Jmn. All magnetic moments are in units of Stlead /c.

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�−�

t

�Mring���,NL�d� = 0 �25�

and

�−�

t

�Mring���,NR�d� = 0. �26�

The above relations imply that the circulating current doesnot carry charge from one wire to the other and should be atleast quadratic in the applied bias. By contrast, the standardtheory of Eq. �4� with Imn=Jmn usually predicts in general alinear response. On the other hand, the laminar current is notcoupled to the local magnetic field.

We have checked numerically that at small U Mring isgenerally linear in U, while Mring�U3, see Fig. 3. The ab-sence of the quadratic term in Fig. 3 is due to the high sym-metry of the system, in which we have chosen UL=−UR=−U. Indeed letting the system lie on the xy plane, with thewires along the x axis, the reflection �x ,y�→ �−x ,y� is

equivalent to U→−U and hence both Mring and Mring areodd function of U. We have numerically verified that forUL�−UR the magnetic moment has a quadratic contributionat small bias.

We note that there are special values of � �e.g., �=1� at

which the estimate Mring starts with vanishing U derivativeat small U.12 In this case Iring coincides with the bond currentalong the c-a bond for all values of U. Since Iring yields thecirculating current, we conclude that all the laminar currentgoes along the a-b bond.

In a similar way one can compute Mring for rings with Nsites, arms of different length and different hopping as wellas on-site energy parameters. We have verified that Mring=0 for symmetrically connected rings, as physically ex-pected.

V. CONCLUSIONS

The definition of the magnetic moment of a quantum ringconnected to bias leads must comply with the way it is mea-

sured. One can measure the magnetic field produced by thering, e.g., by means of a superconducting quantum interfer-ence device �SQUID�, or the response of the ring to an ex-ternal field. However the SQUID is at least a mesoscopicobject, and has not been used so far to measure the magneticmoment of molecular-sized objects; in addition the externalcircuit also produces a magnetic field. We argue that a localmechanical measurement �of a force or a torque� by anatomic force microscope would be ideally suitable for ananoscopic ring. By thought experiments we propose Mringin Eq. �14� which depends on the interaction energy with anexternal field and does not involve the bond currents �as we

call the averages �Jmn� of the current operator with m ,n in aninternal bond of the ring�. Explicit calculations show thatMring is, in general, quite unlike a linear combination of thebond currents, except for special situations when the conduc-tance of the ring vanishes.

Remarkably we have found that the circulating currentgenerating the magnetic moment has an at least quadraticrather than linear response at small bias. Such circulatingcurrent is localized inside the ring and does not contribute tothe conduction while the linear-response current is alwayslaminar and contributes to the overall Lorentz force acting onthe circuit.

Since the effects that we predict are qualitatively impor-tant, we hope that they can be directly compared with experi-ment by measuring the ring magnetic moments. We point outthat for more complex systems with several rings the indi-vidual magnetic moment of each ring makes sense if one canthread the field through and measure the mechanical force ona single ring. Otherwise one should model the experimentaccording to the detailed way it is performed. However, morecomplex systems are outside the scope of the present paper.

Finally we expect that the present theory and its exten-sions to circuits with several loops should pave the way toinclude induction and self-induction effects in quantumtransport theory. In the extended theory, even in the absenceof an external magnetic field one will need to include a flux�=LIring, where L is the self-induction coefficient, by a self-consistent procedure.

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