Effect of a Zeeman field on the superconductor-ferromagnet transition in metallic grains
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Effect of a Zeeman field on the superconductor-ferromagnet transition in metallicgrains
S. Schmidt and Y. Alhassid
Center for Theoretical Physics, Sloane Physics Laboratory,Yale University, New Haven, Connecticut 06520, USA
K. Van Houcke
Universiteit Gent, Vakgroep Subatomaire en Stralingsfysica - Proeftuinstraat 86, B-9000 Gent, Belgium
We investigate the competition between pairing correlations and ferromagnetism in small metallicgrains in the presence of a Zeeman field. Our analysis is based on the universal Hamiltonian, validin the limit of large Thouless conductance. We show that the coexistence regime of superconductingand ferromagnetic correlations can be made experimentally accessible by tuning an external Zeemanfield. We compare the exact solution of the model with a mean-field theory and find that the lattercannot describe pairing correlations in the intermediate regime. We also study the occurrence ofspin jumps across the phase boundary separating the superconducting and coexistence regimes.
PACS numbers: 73.21.La, 75.75+a,74.78.Na,73.22.-f
I. INTRODUCTION
The hallmark of the BCS model of superconductiv-ity in metals is the presence of an excitation gap ∆.This gap is caused by the formation of Cooper pairs de-scribing correlated electron pairs in time-reversed states.Thus, pairing correlations in superconductors tend tominimize the total spin of the electron system. Ferro-magnetic correlations, on the other hand, prefer to max-imize the total spin and form a macroscopic magneticmoment. Early work predicted a state in whichboth pairing and ferromagnetic order are present, if fer-romagnetism is caused by localized paramagnetic impu-rities. The experimental observation that both statesof matter can coexist in heavy fermion systems andhigh-Tc superconductors came as a surprise and ledto the search for new theoretical models to describe thiscoexistence. A BCS-like model of s -wave pairing com-bined with a simple Stoner-like model of ferromagnetismwas used to derive such an intermediate state within amean-field approximation. However, it was argued thatsuch a state is unstable in the bulk.
Furthermore,it was shown that a proper Hartree-Fock mean-field the-ory of the model does not support coexistence of s -wavesuperconductivity and ferromagnetism. A similar model of BCS-like pairing and exchange in-teraction was shown to be valid in small metallic grainsin the mesoscopic regime for a Thouless energy E T that islarge compared with the single-particle mean-level spac-ing δ . In such a finite-size system, a partiallypaired state with finite spin polarization exists within anarrow parameter regime. Since this coexistence regimeis relatively small, it would be difficult to observe it ex-perimentally. It has been suggested that the probabilityof spin polarization in the presence of pairing correla-tion may be enhanced by mesoscopic fluctuations or byan asymmetric spin-dependent bandwidth of the single-particle spectrum. Here we study the competition between ferromagneticand pairing correlations in metallic grains in the crossoverregime from a few-electron system (∆ ≪ δ ) to the bulk(∆ ≫ δ ). We use Richardson’s solution of the BCS-like interaction and the known solution of the exchangemodel to determine the ground state of the grain. Forsufficiently small grains, there is a regime in pairing gap∆ /δ and exchange coupling J s /δ , in which the groundstate is partly paired and partly polarized. We show that,in the presence of a Zeeman field, the exchange couplingat which the crossover from a pure superconducting stateto the coexistence regime takes place decreases to valuesthat can be realized in several metals. The onset of mag-netization with increasing exchange coupling at a givenpairing gap corresponds to a spin jump ∆ S ≥
1, followedby successive spin increments of ∆ S = 1. The magni-tude of the initial spin jump depends on the value of ∆ /δ .Similar spin jumps were found in the crossover from a su-perconducting state to a paramagnetic state, wherethey are reminiscent of a first-order transition in the bulk.We apply a mean-field theory similar to the one used inRef. 25 and compare with the exact results. In contrastto the exact solution, we find that the mean-field ap-proximation cannot describe pairing correlations in theintermediate regime of partial spin polarization. II. MODEL
An isolated metallic grain in which the single-particledynamics are chaotic and whose dimensionless Thoulessconductance g T = E T /δ is large ( g T ≫ ˆ H = X kσ ǫ k c † kσ c kσ − G ˆ P † ˆ P − J s ˆ S + gµ B H ˆ S z . (1)Here c † kσ is the creation operator for an electron in thesingle-particle level ǫ k with either spin up ( σ = +) orspin down ( σ = − ). The one-body term in (1) describesthe kinetic energy plus confining single-particle potential.The second term on the r.h.s. of Eq. (1) is a pairing in-teraction with strength G and where P † = P i c † i + c † i − isthe pair creation operator. The third term in (1) is anexchange interaction expressed in terms of the total spinoperator ˆ S = P kσσ ′ c † kσ τ σσ ′ c kσ ′ ( τ i are Pauli matrices).The parameter J s is the exchange coupling constant (es-timated values of J s for a variety of materials were tab-ulated in Ref. 26). The inclusion of such an exchange in-teraction in quantum dots explained quantitatively themeasured peak height and peak spacing statistics. The last term on the r.h.s. of Eq. (1) describes the cou-pling of an external Zeeman field H (applied in the z direction) to the spin of the dot. Here g is the g -factor ofthe electrons in the grain (taken to be positive) and µ B is the Bohr magneton. Orbital diamagnetism can be ne-glected for small grains. The charging energy e ˆ N / C ( C is the capacitance of the grain) is a constant for agrain with a fixed number of electrons N and was omit-ted in the Hamiltonian (1).In this work, we do not consider mesoscopic effects thatoriginate in the random matrix description of the single-particle Hamiltonian. To construct a typical phase dia-gram of a single grain, we consider a generic equidistantspectrum ǫ k = kδ with − N o ≤ k ≤ N o at half filling.Thus we have N = 2 N o for an even number of electrons( p = 0) and N = 2 N o + 1 for an odd number ( p = 1). III. EXACT SOLUTION
In the absence of a pairing interaction ( G = 0), theHamiltonian (1) can be solved in closed form. The or-bital occupations ˆ n k = ˆ n k + + ˆ n k − commute with ˆ S andare good quantum numbers. The empty ( n k = 0) anddoubly occupied ( n k = 2) orbitals do not contribute tothe total spin, so the total spin of the grain is obtainedby coupling the singly occupied levels with spin 1 / S and spin projection M . For a specific set B of b singly occupied levels, the total spin ranges from S = p/ S = b/ d b ( S ) = (cid:0) bS + b/ (cid:1) − (cid:0) bS +1+ b/ (cid:1) . A complete setof eigenstates is then given by |{ n k } , γ, S, M i where γ are quantum number distinguishing between eigenstateswith the same spin. The pairing interaction can only scatter time-reversedpairs from doubly occupied to empty orbitals but doesnot affect the singly occupied levels (referred to as“blocked” levels). It is therefore sufficient to diagonalizethe reduced BCS Hamiltonian P kσ ǫ k c † kσ c kσ − GP † P us-ing the single-particle subspace U of empty and doublyoccupied levels. This problem was solved by Richard-son. The eigenenergies are given by E m = m X µ =1 E µ , (2) where E µ are parameters that characterize the eigenstateand m = ( N − b ) / E µ are found by solving the set of m couplednon-linear equations1 G + 2 m X ν =1 ν = µ E ν − E µ = X i ∈U ǫ i − E µ ( µ = 1 , . . . , m ) . (3)To each set of m doubly occupied levels at G = 0, there isa unique solution for Richardson’s parameters at G = 0.We note that in the general case Richardson’s equationsdepend on the seniority quantum numbers of the levels(the seniority is the number of electrons not coupled tospin zero). In our case, the levels are doubly degenerateand the seniority of a doubly occupied level is zero.The eigenstates constructed from the subset U ofempty and doubly occupied levels have spin zero, sothe total spin of the grain is determined by the spin-1/2 coupling of the singly occupied levels in B . Theeigenstates of the full Hamiltonian (1) are then given by |B , { E µ } , γ, S, M i with energies of E = E m + X k ∈B ǫ k − J s S ( S + 1) + gµ B HM . (4)In this work, we focus on the ground state of the grainas a function of the interaction couplings G and J s . Tothat end, we find the lowest energy E ( S ) in (4) for agiven spin S and then minimize with respect to S . Theenergy E ( S ) is found by choosing a set B of b = 2 S singly occupied levels that are placed closest to the Fermienergy. We then populate these b levels with spin-downelectrons, resulting in a good-spin state with spin S andspin projection M = − S . For a given set B , we solvedRichardson’s equations using the method of Ref. 30.The physical parameter desc ribing the pairing Hamil-tonian is ∆ /δ , where ∆ is the bulk pairing gap and δ the single-particle mean-level spacing. The low-energyspectrum of the grain (for J s = H = 0) is determined bythe value of this parameter. We can truncate the totalnumber of levels from N o to N r < N o , and renormalize G such that the low-energy spectrum of the grain remainsapproximately the same. For a picketfence spectrum, therenormalized coupling constant is given by G r δ = 1arcsinh (cid:16) N r +1 / /δ (cid:17) . (5)Strictly speaking, this holds in the absence of an ex-change interaction. However, since the exchange interac-tion affects only the blocked levels, we expect the renor-malization (5) to hold as long as the number of blockedlevels is small compared with the total number of levels inthe band. The quality of this approximation depends onthe choice for N r and was discussed in detail in Ref. 31. IV. MEAN-FIELD APPROXIMATION
We compare the findings based on the exact solutionwith a mean-field theory. The mean-field approach isbased on a trial wave function of the form | ψ S i = Y k ∈B c † k − Y j ∈U (cid:16) u ( S ) j + v ( S ) j c † j + c † j − (cid:17) | i (6)with the normalization condition ( u ( S ) j ) + ( v ( S ) j ) = 1.The wave function ψ S has b = 2 S singly occupied levelswith spin-down electrons (set B ) chosen to be closest tothe Fermi energy, and is of the BCS form within theremaining set of levels U . The lowest state with spin S is found by minimizing the expectation value h ψ S | ( ˆ H − µ ˆ N ) | ψ S i with respect to the variational parameters v ( S ) j .Here µ is a chemical potential ensuring that the averagenumber of particles is N .The mean-field energy at fixed spin S is given by E mf ( S ) = 2 X k ∈U ǫ k (cid:16) v ( S ) k (cid:17) − ∆ S G + X k ∈B ǫ k − J s S ( S + 1) − gµ B HS , (7)where (cid:16) v ( S ) k (cid:17) = 12 − ǫ k − µ q ( ǫ k − µ ) + ∆ S , (8)and ∆ S is a spin-dependent pairing gap. The gap param-eter and chemical potential are determined by solving thegap equation together with the particle number equation2 G = X k ∈U q ( ǫ k − µ ) + ∆ S , (9a) N = 2 X k ∈U (cid:16) v ( S ) k (cid:17) + b . (9b)For an equidistant spectrum, the chemical potential canbe determined by symmetry considerations and is givenby µ = − (1 − p ) δ/ N o δ ≫ ∆ S .Here, we used the same approximations as in Ref. 25and neglected a term in the energy E mf ( S ) which is pro-portional to ( v ( S ) j ) . The result (7) is in agreement withthe leading term of an expansion in the inverse number ofelectrons 1 /N . . Comparing (7) and (4) with M = − S ,we see that the exchange and Zeeman terms are treatedexactly in this mean-field approximation. The ground-state spin in the mean-field approximation is found byminimizing E mf ( S ) in (7) with respect to S . V. GROUND-STATE PHASE DIAGRAM
The ground-state spin of the grain is determined bythe competition between various terms in the universal ∆/δ J s / δ (a) (b) FMFMSC SC
21 12 S=0S=0 0
SC−FM
FIG. 1: Ground-state phase diagram in the J s /δ –∆ /δ planefor an even number of electrons ( N = 50). Left panel: ex-act results. Right panel: mean-field approximation (see text).The phase diagrams show a superconducting (SC) phase anda ferromagnetic (FM) phase. The exact phase diagram alsoexhibits an intermediate regime (SC-FM) in which the groundstate is partially polarized but still has pairing correlations.The intermediate regime in the mean-field phase diagram de-scribes a state that is partially polarized state but does notinclude pairing correlations. In particular, the dashed lineseparates an S = 0 SC phase from an S = 0 phase with nopairing correlations. The numbers shown in the intermediateregime are the spin values in the corresponding sectors. Hamiltonian. The one-body part (kinetic plus confiningone-body potential) and pairing interaction favor mini-mal spin S = p/ /δ, J s /δ and gµ B H/δ . Using the exact solution,we find three different phases: a superconducting phasewhere the number of pairs is maximal and S = p/
2, aferromagnetic phase where the system is fully polarized S = N/ S < N/
2, in which b = 2 S electrons reside in singly occupied levels closest to theFermi energy and the remaining electrons are paired togive spin zero.The phase diagram in the ∆ /δ – J s /δ plane of a grainwith even number of electrons and in the absence of Zee-man field ( H = 0) is presented in Fig. 1(a). For weakpairing, the superconducting and ferromagnetic phasesare separated by an intermediate regime. The boundariesof this intermediate regime are described by two criticalvalues J (1) s and J (2) s of the exchange interaction that arefunction of ∆ /δ . The critical value J (1) s /δ is a monotoni-cally increasing function of ∆ /δ , i.e., a stronger exchangeinteraction is required to polarize a grain with strongerpairing correlations. However, J (2) s /δ is almost insensi-tive to ∆ /δ . The intermediate regime shrinks at larger∆ /δ and eventually disappears above ∆ /δ ∼
3. Forstronger pairing correlations, the superconducting phasemakes a direct transition to the ferromagnetic phase. Inthis regime (not shown in Fig. 1), the phase boundaryexhibits a strong dependence on the bandwidth N o .For comparison, we show the mean-field results inFig. 1(b). We observe that the mean-field results arequalitatively different from the exact solution. The re-gion to the right of the thick solid line and dashed linedescribes an superconducting phase with ∆ = 0. How-ever, there is no superconducting solution (i.e., ∆ = 0)for ∆ /δ ≤ .
28. Furthermore, in each of the partially po-larized regions with spin 0 < S < N/
2, the correspond-ing pairing gap vanishes ∆ S = 0 and there are no pairingcorrelations present. While solutions with ∆ S = 0 exist,they occur for values of ∆ /δ for which a higher spin statewith no pairing correlations has lower energy (because ofthe exchange interaction). For example, a solution with∆ = 0 exists only for ∆ /δ > . . However, at thisstrength of pairing correlation we observe a direct tran-sition from S = 0 to S = 4 with ∆ S = 0 as the energyof the lowest S = 4 state with no pairing correlationsis lower than the paired S = 1 state. As a result, theboundaries which separate different spin phases are flat,e.g., independent of the pairing strength. In fact, in themean-field approximation the ground-state wave functionis a Slater determinant through the whole intermediateregime of partial spin polarization. Thus no coexistenceof pairing and spin polarization is observed within themean-field approach.In contrast, the exact solution shows that pairing cor-relations are present as long as the system is not fullypolarized. This can be seen in the shift of the spin tran-sition lines to higher values of the exchange interactionstrength as the pairing gap ∆ /δ is increased. Thus, theexact solution predicts a regime in which pairing correla-tions and spin polarization coexist. In the following, weonly discuss results obtained from the exact solution.More detailed phase diagrams for H = 0 are shown inthe top row of Fig. 2 for both grains with even [panel (a)]and odd [panel (b)] number of electrons. For weak pairingwe observe an odd-even effect (in number of electrons).In particular, the critical value J (1) s is larger for the oddgrain, even though the presence of a blocked level in theodd superconducting phase weakens pairing correlationsin the odd grain. This is because increasing the spin from1 / / gµ B H/δ = 2 . J (1) s for a givenpairing gap smaller. Second, at given exchange strength J s /δ , it increases the critical value of ∆ /δ at which par-tial spin polarization is destroyed. Both effects togetherincrease the size of the intermediate regime in the ∆ /δ – J s /δ plane. ∆/δ J s / δ FMSCFM FM SCFM SCSC−FM(c)(a) (b)(d)
S=0
SC 3/212312 S=0 S=1/2S=1/23/25/27/2
FIG. 2: Phase diagrams in the J s /δ –∆ /δ plane at a fixedZeeman field gµ B H = 0 (top panels) and gµ B H/δ = 2 . VI. SPIN JUMPS
As we increase the exchange coupling constant J s /δ atfixed ∆ /δ and Zeeman field, the spin increases by discretesteps from its minimal value S = p/ S = N/
2. In the absence of pairing (∆ = 0),the transition from spin S to spin S + 1 occurs for anexchange coupling of J s /δ = (2 S + 1) − gµ B H/δ S + 2 at ∆ = 0 . (10)The corresponding ∆ = 0 phase diagrams in the gµ B H/δ –∆ /δ plane are shown in Fig. 3(a) and 3(b) foreven and odd grains, respectively. In particular, thephase boundaries are given by J (1) s = δ ( p + 1) / ( p + 2) − gµ B H/ ( p + 2) and J (2) s = δ ( N − /N − gµ B H/N . Theground-state spin increases as a function of J s in steps of∆ S = 1. An interesting qualitative change in the pres-ence of pairing correlations is the possibility of a spinjump ∆ S >
1. For ∆ /δ < .
6, the ground-state spinstill increases in steps of ∆ S = 1 versus J s . However, for0 . < ∆ /δ < .
8, the ground-state spin jumps from 0 to2 within the range 0 . < J s /δ < .
9. The size of thefirst-step spin jump gets larger with increasing ∆ /δ . Allsubsequent steps are of size one [see Fig. 2(a)].A similar effect was observed when superconductivitybreaks down due to the presence of a large external Zee-man field. The experimental findings were qualitativelyexplained using the mean-field theory we discussed pre-viously (but without the inclusion of an exchange inter-action). It was concluded in Ref. 25 that the first-orderphase transition from a superconductor to a paramagnet,observed in thin films, is “softened” in metallic grains. g µ Β Η/δ J s / δ FMSC SC−FM SC(a)(c) FMFM (b)(d)S=1/2 3/2 5/2 7/2S=1/2 3/2 5/221S=0 1 2 3S=0 2 4 6 2 4 6
FIG. 3: Phase diagrams in the J s /δ – gµ B H/δ plane at fixed∆ /δ = 0 (top row) and ∆ /δ = 2 (bottom row) for an evengrain (left column) and for an odd grain (right column). Num-bers denote the spin in each sector. Here we have shown that spin jumps also occur in thepresence of exchange correlations. In the absence to anexternal Zeeman field, these spin jumps are predicted tooccur at exchange coupling values J s /δ > .
87. Such ex-change coupling values are significantly larger than thevalues for most metals (see Fig. 9 in Ref. 26). More-over, the exchange is an intrinsic material property andis difficult to tune experimentally.The regime of spin jumps can be tuned to lower andmore typical values of J s by applying an external Zeemanfield. We have already seen in Figs. 2(c) and 2(d) that arelatively weak Zeeman field increases the size of the in-termediate regime. It also means that spin jumps can beobserved at smaller values of the exchange strength thatare accessible to experiments. This is demonstrated inFig. 3(c) and 3(d) where phase diagrams in the gµ B H/δ –∆ /δ plane are shown for a given pairing gap of ∆ /δ = 2.For example, a Zeeman field of gµ B H/δ ≈ → J s /δ ≈ .
55 at ∆ /δ = 2 [see Fig. 3(c)] as com-pared to J s = 0 . δ at ∆ = 0 . δ without Zeeman field[see Fig. 2(a)].The idea of a Zeeman-field tuning of the values of ex-change coupling where spin jumps occur is best illus-trated in Fig. 4, where spin staircase functions are shownversus J s /δ . In the presence of pairing correlations andthe absence of Zeeman field, the ground-state spin stair-case is shifted to the right and compressed as ∆ /δ in-creases [see Fig. 4(a)], reflecting the fact that the inter-mediate region shrinks [see Fig. 2(a) and 2(b)]. For aneven grain with ∆ /δ = 0 .
7, a spin jump of ∆ S = 2 setsin at J s ≈ . δ , while for ∆ /δ = 0 .
9, a spin jump of∆ S = 3 occurs at J s ≈ . δ . For a finite Zeeman fieldof gµ B H/δ = 2, the spin staircase functions that exhibit J s / δ S (b)(a) FIG. 4: Ground-state spin versus exchange coupling J s /δ foran even grain at a fixed Zeeman field strength gµ B H = 0(top panel) and gµ B H/δ = 2 (bottom panel). Solid linescorrespond to a grain with no pairing correlations ∆ /δ = 0.The dashed lines describe staircase functions with a spin jumpof ∆ S = 2 for ∆ /δ = 0 . /δ = 2 (bottom).The dotted lines correspond to staircases with a spin jump of∆ S = 3 for ∆ /δ = 0 . /δ = 2 . similar spin jumps are shifted to smaller values of the ex-change strength but larger values of the pairing gap [seeFig. 4(b)]. Spin jumps of ∆ S = 2 (∆ S = 3) occur at J s /δ = 0 .
55 ( J s /δ = 0 .
64) and ∆ /δ = 2 (∆ /δ = 2 . /δ atwhich spin jumps occur as well as the size of these jumpsincrease at larger values of gµ B H/δ . The ratio ∆ /δ (forthe given metal) can be made larger by studying a largergrain, hence reducing the mean level spacing δ .As an example, niobium has an exchange interactionstrength of J s /δ ≈ . S = p/
2) at all values of ∆ /δ [seeFig. 2(a) and 2(b)]. At a Zeeman field gµ B H/δ = 1,the ground-state spin of niobium changes from 0 to 1at ∆ /δ = 0 .
66. However, at gµ B H/δ = 2 .
6, a spinjump of ∆ S = 2 occurs from 0 to 2 at ∆ /δ = 2 . /δ , we canroughly estimate the corresponding critical size of themetallic grain given the bulk gap value ∆ = 3 .
05 meVand Fermi momentum k F = 11 . − of niobium. In aFermi gas model, the mean-level spacing is related to thevolume of the grain by δ = 2 π ~ / ( mk F V ). Assuminga hemispheric grain with radius r , we have the relation r Nb ≈ . Nb /δ ) / . Thus, the Hamiltonian (1)with an equidistant spectrum predicts a 0 → r ≈ .
35 nm and Zee-man field of gµ B H = 4 .
62 meV, and a 0 → r ≈ .
48 nm and a Zeeman field of gµ B H = 3 .
69 meV.
VII. CONCLUSION
We have shown that there exists a small region in theground-state phase diagram of a small metallic grain inwhich pairing correlations and ferromagnetism coexist.This coexistence regime becomes larger (in the J s /δ –∆ /δ plane) and therefore more accessible to experiments inthe presence of a finite Zeeman field. In particular, wepropose that for a given exchange constant (determinedby the material used), spin jumps can be observed by tun-ing a Zeeman field. We have also shown that a quantita-tive study of the intermediate regime requires the use ofthe exact solution. Furthermore, the mean-field approx-imation is qualitatively different in that it does not pre-dict any pairing correlations in the intermediate regime of partial spin polarization.In this work, we have ignored mesoscopic fluctuationsand focused on a grain with an equidistant single-particlespectrum. It would be interesting to study how meso-scopic fluctuations affect the boundaries of the interme-diate phase and the size of spin jumps.We thank L. Fang, S. Girvin, S. Rombouts, S. Rot-ter, R. Shankar and A.D. Stone for useful discussions.K. Van Houcke acknowledges financial support of theFund for Scientific Research - Flanders (Belgium), andthe hospitality of the Center for Theoretical Physics atYale University where this work was completed. Thiswork was supported in part by the U.S. DOE grant No.DE-FG-0291-ER-40608. A. A. Abrikosov and L. P. Gorkov, Zh. Eksp. Teor. Fiz. , 1781 (1960)[Sov. Phys. JETP , 1243 (1961)]. A. M. Clogston, Phys. Rev. Lett. , 266 (1962). B. S. Chandrasekhar, Appl. Phys. Lett. , 7 (1962). P. Fulde and R. A. Ferrell, Phys. Rev. , A550 (1964). A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. , 1136 (1964)[Sov. Phys. JETP , 762 (1965)]. S. S. Saxena et al. , Nature (London) , 587 (2000). C. Pfleiderer et al. , Nature (London) , 58 (2001). D. Aoki et al. , Nature (London) , 613 (2001). J. Tallon et al. , IEEE Trans. Appl. Supercon. , 1696(1999). C. Bernhard et al. , Phys. Rev B , 14099 (1999). N. I. Karchev, K. B. Blagoev, K. S. Bedell, and P. B.Littlewood, Phys. Rev. Lett. , 846 (2001). Y. Zhou, J. Li, and C. Gong, Phys. Rev. Lett. , 069701-1(2003). R. Shen, Z. M. Zheng, and D. Y. Xing, Phys. Rev. Lett. , 069702-1 (2003). R. Shen, Z. M. Zheng, S. Liu, and D. Y. Xing, Phys. Rev.B , 024514 (2003). Y. N. Joglekar, A. H. MacDonald, Phys. Rev. Lett. ,199705-1 (2004). K. B. Blagoev, K. S. Bedell, and P. B. Littlewood, Phys.Rev. Lett. , 199706-1 (2004). I. L. Kurland, I. L. Aleiner, and B. L. Altshuler, Phys.Rev. B , 14886 (2000). I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys.Rep. , 309 (2002). G. Murhty and R. Shankar, Phys. Rev. Lett. , 066801(2003). Z. Ying, M. Couco, C. Noce, and H. Zhou, Phys. Rev. B , 012503 (2006). G. Falci, R. Fazio, and A. Mastellone, Phys. Rev. B ,132501 (2003). R. W. Richardson, Phys. Rev. Lett. , 277 (1963); R. W.Richardson, and N.Sherman, Nucl. Phys. , 221 (1964);R. W.Richardson, Phys. Rev. , 792 (1967). Y. Alhassid and T. Rupp, Phys. Rev. Lett. , 056801(2003). J. von Delft and D. C. Ralph, Physics Reports , 661-173 (2001). F. Braun, J. von Delft, D. C. Ralph, and M. Tinkham,Phys. Rev. Lett. , 921 (1997). D. A. Gorokhov, and P. W. Brouwer, Phys. Rev. B ,155417 (2004). Y. Alhassid, Rev. Mod. Phys. , 895 (2000). Y. Alhassid and S. Malhotra, Phys. Rev. B , 245313(2002). H.E. Tureci and Y. Alhassid, Phys. Rev. B , 165333(2006). S. Rombouts, D. Van Neck and J. Dukelsky, Phys. Rev. C , 061303 (R) (2004). Y. Alhassid, L. Fang, and S. Schmidt, cond-mat/0702304. E. A. Yuzbashyan, A. A. Baytin, and B. L. Altshuler, Phys.Rev. B71