Scale-dependent competing interactions: sign reversal of the average persistent current

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by  H. Bary-Soroker

Scale-dependent competing interactions:
sign reversal of the average persistent current
H. Bary-Soroker,1 O. Entin-Wohlman,1,∗ Y. Imry,2 and A. Aharony1,∗
1Physics Department, Ben Gurion University, Beer Sheva 84105, Israel
2Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
(Dated: October11, 2012)
The interaction-inducedorbital magnetic response of a nanoscale system, modeled bythepersis-
2
tent current in a ring geometry, is evaluated for a system which is a superconductor in the bulk.
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The interplay of the renormalized Coulomb and Fr¨ohlich interactions is crucial. The diamagnetic
0
response of the large superconductor may become paramagnetic when the finite-size-determined
2
Thouless energy is larger than or on theorder of theDebyeenergy.

t
c
PACSnumbers: 73.23.Ra,74.25.N-,74.25.Bt
O

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1 Introduction. Renormalization,oftenaccomplishedus- An intriguing situation offered by mesoscopic super-
ingtherenormalization-groupmethod,isoneofthebasic conductors, is the possibility to observe the “running”,

] concepts in physics. It deals with the way various cou- scale-dependent relevant interaction [3]. Obviously the
l
l plingconstants(e.g.,theelectronchargeortheCoulomb renormalization has to be stopped at the scale corre-
a
h interaction) change as a function of the relevant scale sponding to the system size, which turns out for many
- for the given problem. The scale may be the resolution physical observables to be the Thouless energy–the in-
s
e at which the system is examined, determined by its size verse of the time it takes an electron to traverse the fi-
m and by the relevant energy for the process under consid- nite system–which for diffusive systems is E = D/L2
c
. eration. Often, one knows the coupling constant’s “bare (D = vFℓ/3 is the diffusion coefficient) [4]. Moreover,
t
a value” at a less interesting (e.g., a very small or a very Ec isalsotheenergyscaleformanymesoscopicphenom-
m large) scale, and what is relevant for experiments is the ena. Thus, by measuring a quantity which depends on
- value atadifferent, “physical”scale;the couplingonthe the interaction, as a function of L, one may deduce the
d
latter scale is then used for the relevant physics. scale-dependence (“running” value) of that interaction.
n
An important and particularly interesting application When that quantity is sensitive to the sign of the inter-
o
c of this idea is in the theory of superconductivity. There, action(e.g.,alinearfunction,uptocorrections,ofthein-
[ two competing interactionsexist: the repulsiveCoulomb teraction, often resulting from perturbationtheory), one

interaction, starting on the large, microscopic energy may observe a rather dramatic change of the sign of the
1
v scale–typicallythe Fermienergy/bandwidthEF,andthe interaction-dependent term in that quantity!
1 attractivephonon-inducedinteraction,operativeonlybe-
Auniqueexampleforthelatteristheorbitalmagnetic
5 low the much smaller Debye energy ω . By integrating
8 D susceptibility of a mesoscopic (or even nanometric) sys-
overthinshellsinmomentum(orenergy)space[1,2]one
2 tem. Most convenient for calculations is the persistent
finally gets the variation of any coupling g, being repul-
. current [6] flowing in a mesoscopic normal-metal ring in
0 sive or attractive, between a high-energy scale, ω , and
1 > responsetoanAharonov-Bohmfluxthroughitsopening.
a low one, ω ,
2 < The magnetic moment induced by very small fluxes due
1 g(ω ) to these currents is expected to be of the same order of
v: g(ω<)= 1+g(ω )ln>(ω /ω ) . (1) magnitudeasthatofasmalldiscmadeofthesamemate-
> > <
i rial,andhavingsimilarradiusandthickness,inresponse
X (We use ~ = c = k = 1.) Notice that a re-
B to the same flux.
r pulsive/attractive interaction is “renormalized down-
a
wards/upwards” with decreasing energy scale. What Moreover, it is well known [4, 5] that the impurity
makes superconductivity possible is that at ω the re- ensemble-averaged persistent current for noninteracting
D
pulsionismuchsmallerthanitsvalueonthemicroscopic electrons is smaller by typically two ordersof magnitude
scale. Then, at ω the attraction may win and at lower thantheinteraction-dominatedterms. Thereforethelat-
D
energies the totalinteractionincreasesin absolute value, terrepresentsratherwellthewholemagneticresponseof
untilitdivergesatsomesmallscale,whichisjustthecon- the electrons and its measurement as a function of L
ventional“T ” ofthe givenmaterial. The upper panelof will amount to monitoring the renormalized interaction.
c
Fig. 1 presents this qualitative text-book picture. In This picture is borne out by our calculations. We imple-
this Letter we present quantitative results for a diffusive ment the ideas of Refs. 1 and 2 in the derivation of the
system whose size L exceeds the elastic mean-free path superconductive fluctuation-induced partition function,
ℓ=v τ (v isthe Fermivelocityandτ isthe mean-free and obtain the effect of the two competing interactions
F i F i
time). in the Cooper channel. We use this result to study the
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averagepersistentcurrentofalargeensembleofmetallic agram” showing the regions where the first harmonic of
rings in the presence of these interactions. thecurrentisparamagneticordiamagnetic,asafunction
of E /ω . Note in particular the existence of a region
c D
g where although the attraction |g | exceeds the repulsion
a
g ,thecurrentisparamagnetic, contrarytotheusualex-
r
pectation [5]. The Thouless energy can be increased by
making the ring smaller, and/or less disordered.
Ž Tc Tca ΩD EF Renormalizing the two interactions. The Hamil-
gHE L
r F Ω tonian we consider consists of a single-particle part,
H (r) = (−i∇−eA)2/(2m)−µ+u(r), and a term de-
g HΩ L 0
a D scribing local repulsion and attraction, of coupling con-
stants g ≡λ N/V and g ≡λ N/V, respectively,
r r a a
e
H= dr ψ†(r)H (r)ψ (r)
Z (cid:16)Xσ σ 0 σ
+(λr −|λa|)ψ↑†(r)ψ↓†(r)ψ↓(r)ψ↑(r) , (2)
0.2 |g |=0.1 |g |=0.15 |g |=0.2 (cid:17)
a a a
0.15
g where N is the density of states, V the system volume,
r
0.1 and ψ (ψ†) destroys (creates) an electron of spin com-
σ σ
0.05 ponent σ at r. Here, gr is the bare repulsion, µ is the
chemical potential, u(r) is the disorder potential due to
0
0 0.2 0.4 0.6 0.8 1 nonmagnetic impuritiees, and A is the vector potential,
Ec/ωD with A =Φ/L, Φ giving rise to a persistent current. As
the magnetic fields needed to produce such currents are
rather small, the Zeeman interaction will be ignored.
FIG. 1: Upper panel: the renormalized interactions, Eq. Thepartitionfunction,Z,correspondingtotheHamil-
(1), as functions of the energy, for a metal which is a su- tonian (2) is calculated by the method of Feynman
perconductor in the bulk. When only the attraction exists,
path integrals, combined with the Grassmann algebra
its absolute value is renormalized up (with ω> = ωD) un- of many-body fermionic coherent states [7]. Introducing
til it diverges at Tca (the dash-dotted curve). The repulsion the bosonic fields φ(r,τ) and ∆(r,τ) via two Hubbard-
(dashed curve) is renormalized down from its bare value at
ω> = EF. The total interaction starts at ωD from the bare Stratonovichtransformationsoneisabletointegrateover
valuega(ωD)+gr(ωD)andthenitsabsolutevalueisrenormal- the fermionic variables; the result is then expanded to
ized upuntilit divergesat the“true” transition temperature second order in the bosonic fields [8]
Tc (full curve). In a mesoscopic system the renormalization
stops at the Thouless energy and therefore the relative loca-
tion of Ec vs. ωD and the ratio |ga|/gr(max{Ec,ωD}) will Z/Z0 =Z D(∆,∆∗)Z D(φ,φ∗)exp[−S] , (3)
determinethesignoftheeffectiveinteractionatωD. Thefig-
e
ure is drawn for demonstration purposes with theunrealistic
parameter EF/ωD =2.5. Lower panel: the lines in the plane with the action S given by
gr −Ec/ωD where the first harmonic of the current changes
e
signfromparamagnetic(abovethecurve)todiamagnetic(be- S = βN|∆ |2/|g |+βN|φ |2/g
low), for three values of the attraction ga. Here and in all Xq,ν(cid:16) q,ν a q,ν r
figures below, T =ωD/125≃2−4K, and EF/ωD =25. e e
− ∆ +iφ ∆∗ +iφ∗ Π , (4)
q,ν q,ν q,ν q,ν q,ν
The renormalization of the interactions has to be (cid:0) (cid:1)(cid:0) (cid:1) (cid:17)
e
carried out down to an energy scale roughly given by and Z0 denoting the partition function of noninteract-
max{T,E }, where T is the temperature. We denote ing electrons (β = 1/T). Equation (3) represents the
c
by g the attraction and by g = [g−1 +ln(E /ω )]−1 partition function due to superconductive fluctuations,
a r r F D
the repulsion, both at ω , such that for |g | > g , the with the action S, a sum over the wavevectors q and
D a r
material in its bulk form is a supeerconductor. When the bosonic Matsubara frequencies ν = 2πnT, written
e
E > T then depending on the ratios E /ω and in terms of the polarization Π averaged over the impu-
c c D
|g |/g (max{E ,ω }) the resultant interaction may be rity disorder [9]. It is this object in which the different
a r c D e
repulsive or attractive, and the response may be para- renormalization of the competing interactions manifests
magnetic or diamagnetic. These two types of behavior itself.
are reflected by the sign of the average current at small In a diffusive system, the disorder-averaged polariza-
fluxes. The lower panel of Fig. 1 presents a “phase di- tionisgivenasasumoverfermionicMatsubarafrequen-
3
cies, ω =πT(2n+1), Equation (9) is our result for the superconductive
fluctuation-inducedpartitionfunction. Itisbasedonthe
∞
2πΘ[ω(ω+ν)] N 1 renormalized repulsion, Eq. (8).
Π =N = ,
q,ν |2ω+ν|+Dq2 T n+F The fluctuation-induced average persistent cur-
Xω nX=0 q,ν
e rent. The thermodynamic persistent current is ob-
F =0.5+(Dq2+|ν|)/(4πT) , (5)
q,ν tained by differentiating the free energy with respect
to the dimensionless flux, ϕ = Φ/Φ , i.e., I =
where Θ is the Heaviside function. The first of Eqs. (5) 0
(eT)/(2π)∂(lnZ)/∂ϕ where Φ = h/e is the flux quan-
is valid for |2ω+ν| ≪ τ−1 and qℓ ≪ 1. It represents a 0
i tum. In the ring geometry, the flux enters the longi-
divergentsumandthereforeacutoffisneeded. Thecutoff
tudinal component, q , of the wavevector q, q ≡ q =
energy is determined by the relevant interaction. The k k n
2π(n+2ϕ)/Lwherenisaninteger[3]. Hence(exploiting
sums in the terms of Eq. (4) involving ∆ are bounded
the Poissonsummation formula)
by ω and therefore are cut off at n = ω /(2πT)−
D max D
1; these sums are denoted ΠωD. The sum multiplying
∞
|φ|2 is cut off by EF, i.e., nmax = EF/(2πT)−1; this I = sin(4πmϕ)Im , (11)
sum is denoted by ΠEF. Therefore, there appear two mX=1
polarizations,the one cut off by ω andthe other by E
D F
where [3]
T ω (E )
NΠωq,Dν(EF) =Ψ(cid:16) D2πTF +Fq,ν(cid:17)−Ψ(cid:16)Fq,ν(cid:17) , (6) I =[(2ieT)/π] ∞ dxe2πix∂ln[a ]/∂x , (12)
m x,m,ν
Z
where Ψ is the digamma function, Xν −∞
Performing the integration over the fields φ yields with a [Eq. (10)] given in terms of F =
x,m,ν x,m,ν
0.5 + |ν|/(4πT) + (πE x2)/(m2T) [see Eq. (5)]. The
Z 1 1 c
= D(∆ ,∆∗ )exp − − integral in Eq. (12) is handled by a contour integration,
Z0 Yq,νZ q,ν q,ν h (cid:16)gr(q,ν) |ga|(cid:17) andthe signofthe resultwhich determines the magnetic
βN response, depends on a subtle balance between the con-
× −ΠωD |∆ |2 , (7)
|g |−g (q,ν) q,ν q,ν tribution of the poles and that of the zeros [3].
(cid:16) a r (cid:17) i
Itiscustomaryinstudiesofpersistentcurrentstocon-
wherewehaveintroducedthewavevector-andfrequency- centrateonthelowestharmonics,notablyonI ,since
m=1
dependent renormalized repulsion gr(q,ν) higher harmonics are exponentially smaller compared to
the lowest ones and are also more sensitive to inelastic
E ω
g−1(q,ν) =g−1+Ψ F +F −Ψ D +F . scattering (dephasing) and pair-breakers (e.g., magnetic
r r (cid:16)2πT q,ν(cid:17) (cid:16)2πT q,ν(cid:17) impurities) [3]. However, when T <Ec− values are con-
(8)
e sidered the relative magnitude of harmonics higher than
the first one increases,and their contributionshould not
In the limit ω ≫{Dq2,T} one may neglect F in the
D q,ν be discarded.
arguments of the digamma functions and retrieve (ex-
When only one of the two competing interactions is
ploitingthe asymptoticformofΨ)the well-knownresult
accountedfor,thenatureofthemagneticresponseisdic-
g−1 = g−1 +ln(E /ω ). In a finite ring, however, the
r r F D tated by the sign of that interaction, being diamagnetic
smallest wavevector is of the order 1/L and therefore at
low temeperatures this procedure is valid only when the for attraction and paramagnetic for repulsion [5]. When
both interactions are present the picture is changed, as
Thouless energy is smaller than the Debye energy, i.e.,
is exhibited in the lower panel of Fig. 1, obtained by
for large enough (and/or clean enough) rings. Carrying
numerically calculating I as a function of the Thou-
out the integrations over the fields ∆ in Eq. (7) yields m=1
less energy E . The sign of I depends on the ratio
c m=1
E /ω . Asthelatterincreases,itmaychangefromnega-
Z/Z = a−1 , (9) c D
0 q,ν tive to positive. Onthe other hand, when g >|g | (and
Yq,ν r a
the bulk material will not become a superconductor at
with a =a(F =F ) and a finite temperature) the current remains paramagnetic
q,ν q,ν
for all values of E /ω .
c D
1 ω As mentioned, it is when T > E that the first har-
a(F)= −Ψ F + D +Ψ(F) c
|g |−g 2πT monicrepresentsfaithfullythepersistentcurrent. Figure
a r (cid:16) (cid:17)
2depictsthefirstthreeharmonicsofthecurrent,asfunc-
g |g | 1 ω
+ r a −Ψ F + D +Ψ(F) (10) tions of the Thouless energy (both normalized by ω ).
|ga|−gr (cid:20)|ga| (cid:16) 2πT(cid:17) (cid:21) ThefigureindicatesthatthevalueofE atwhichthetDhe
c
E E ω
× Ψ F + F −ln F −Ψ F + D . signofIm is reversedincreaseswithm. Figure3 demon-
(cid:20) (cid:18) 2πT(cid:19) (cid:18)ωD(cid:19) (cid:16) 2πT(cid:17)(cid:21) strates the dependence on the strength of the repulsion.
4
0.05
m=1 m=2 m=3 traction suffices to lead to a superconducting phase in
I /eω
m D the bulk material, we find that it does not ensure a dia-
0
magneticresponseofthe mesoscopicsystem. The reason
being that in mesoscopic rings the orbital magnetic re-
−0.05 sponseowesitsveryexistencetoafiniteThoulessenergy.
0 0.2 0.4 0.6 0.8 1 Therefore the latter, when large enough can cause the
E /ω response to be paramagnetic, albeit the strength of the
c D
attractive interaction. As the Thouless energy may be
controlledexperimentally,onemayhopethatthepredic-
FIG. 2: Im(Ec) for m = 1, 2, and 3 (see the legend), with tion made in this paper will be put to an experimental
|ga|=0.2 and gr =0.1. test. Relatively large values of persistent currents may
beachievedinmolecularsystems,andsmalldiscsareex-
pected to behave similarly at small fluxes. Finally we
As it increases the Ec/ωD−range in which the response remark that there may already be experimental indica-
is diamagnetic diminishes, and so does also the maximal tions to the validity of our prediction. Reich et al. [10]
value of the diamagnetic current. In the paramagnetic- foundthatthinenoughgoldfilmsareparamagnetic. This
response regime the amplitude of the current increases may well be due to a small grain structure.
with E [4]. Figure 4 shows the flux dependence of the
c
current, for two distinct values of g , and displays the
r WeareindebtedtoY.Oregforinstructivediscussions.
possibilityforasignreversalofthe magneticresponseat
AA and OEW acknowledge the support of the Albert
lowflux. OnenotesinFig. 2thatatE /ω =0.76I
c D m=1 Einstein Minerva Center for Theoretical Physics, Weiz-
is small compared to I ; the dashed curve in Fig. 4
m=2,3 mann Institute of Science. This work was supported by
reflects the dominance of the higher harmonics over the
the BMBF within the DIP program, BSF, ISF and its
first one for relatively small g . For a higher value (solid
r Converging Technologies Program.
line there) the first harmonic is dominant.

0.2
g=0.05 g=0.1 g=0.15
r r r
∗
0.1 Also at Tel AvivUniversity,Tel Aviv 69978, Israel
I /eω [1] P. Morel and P. W. Anderson, Phys. Rev. 125, 1263
m=1 D
0 (1962); N. N. Bogoliubov, V. V. Tolmachev, and D. V.
Shirkov, A New Method in the Theory of Superconduc-
−0.1 tivity (Consultants Bureau, Inc., New York,1959).

0 0.2 0.4 0.6 0.8 1 [2] P.G. deGennes, Superconductivity of Metals and Alloys
E /ω (Addison-Wesley Publishing Co., 1989).
c D
[3] H.Bary-Soroker,O.Entin-Wohlman,andY.Imry,Phys.
FIG.3: I(Ec)forthreevaluesoftherepulsionand|ga|=0.2. Rev.Lett.101,057001 (2008); Phys.Rev.B80, 024509
(2009) and references therein.
[4] Y. Imry, Introduction to Mesoscopic Physics, 2nd ed.
(Oxford UniversityPress, Oxford, 2002).
0.2 [5] V.AmbegaokarandU.Eckern,Phys.Rev.Lett.65,381
I/eω (1990); Europhys.Lett. 13, 733 (1990).
D
0 [6] L. Gunther and Y. Imry, Solid State Commun. 7, 1391
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(1970)[Sov.Phys.JETP31,1172(1970)]andreferences
−0.2 therein; M. Bu¨ttiker, Y. Imry, and R. Landauer, Phys.
−0.25 0 ϕ 0.25 Lett. 96A, 365 (1983).
[7] A.AltlandandB.Simons,Condensed Matter FieldThe-
FIG.4: Thecurrentasafunctionofϕ. Solid(dashed)curve ory (Cambridge University Press, Cambridge, 2006).
isforgr =0.17(gr =0.1),with|ga|=0.2andEc/ωD =0.76. [8] Thisexpansionisvalidforhighenoughtemperatures,see
A.Aharony,O.Entin-Wohlman,H.Bary-Soroker,andY.
Imry,Lithuanian Journal of Physics 52, 81 (2012).
Discussion. We have obtained the wavevector- and
[9] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski,
frequency-dependent renormalized repulsion, Eq. (8). Methods of Quantum Field Theory in Statistical Physics
The two cutoffs on the bare attraction and repulsion, (Prentice-Hall, Englewood Cliffs, NJ, 1963).
ω and E , respectively, conspire to determine the sign [10] S. Reich, G. Leitus, and Y. Feldman, Appl. Phys. Lett.
D F
of the magnetic response. Whereas an overall total at- 88, 222502 (2006) and references therein.