Spin and Charge Correlations in Quantum Dots: An Exact Solution
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Pis’ma v ZhETF
Spin and Charge Correlations in Quantum Dots: An Exact Solution
I.S. Burmistrov + , ‡ , Yuval Gefen (cid:3) , and M.N. Kiselev △ + L.D. Landau Institute for Theoretical Physics RAS, Kosygina street 2, 119334 Moscow, Russia ‡ Department of Theoretical Physics, Moscow Institute of Physics and Technology, 141700 Moscow, Russia (cid:3)
Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel △ International Center for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy
Submitted October 30, 2018
The inclusion of charging and spin-exchange interactions within the Universal Hamiltonian description ofquantum dots is challenging as it leads to a non-Abelian action. Here we present an exact analytical solutionof the probem, in particular, in the vicinity of the Stoner instabilty point. We calculate several observables,including the tunneling density of states (TDOS) and the spin susceptibility. Near the instability point theTDOS exhibits a non-monotonous behavior as function of the tunneling energy, even at temperatures higherthan the exchange energy. Our approach is generalizable to a broad set of observables, including the a.c.susceptibility and the absorption spectrum for anisotropic spin interaction. Our results could be tested innearly ferromagnetic materials.PACS: 73.23.Hk, 75.75.-c, 73.63.Kv
The physics of quantum dots (QDs) is a focal pointof research in nanoelectronics. The introduction of the“Universal Hamiltonian” [1, 2] made it possible to sim-plify in a controlled way the intricate electron-electroninteractions within a QD. This provided one with aconvenient framework to calculate physical observables.Within this scheme interactions are represented as thesum of three spatially independent terms: charging,spin-exchange, and Cooper channel. Notably, even theinclusion of the first two terms turned out to be non-trivial: the resulting action is non-Abelian [3, 4]. At-tempts to account for those interactions in transportinvolved a rate equation analysis [5, 6] and a perturba-tion expansion [4]. Alhassid and Rupp [5] have analyzedsome aspects of the problem (see below) exactly. It isknown that in the presence of significant spin-exchangeinteraction such systems can become Stoner unstable.More precisely, one distinguishes 3 regimes of behavioras function of increasing the strength of the exchange in-teraction: paramagnetic (no zero field magnetization),mesoscopic Stoner regime (finite magnetization whosevalue increases stepwise with the exchange) and ther-modynamic ferromagnetic phase (magnetization is pro-portional to the volume) [2]. Both the mesoscopic andthermodynamic phases manifest (Stoner) instabilitiestowards ferromagnetic ordering. The presence of en-hanced quantum and statistical fluctuations underlyingsuch instabilities calls for a full-fledged quantum me-chanical treatment of the problem. e-mail: [email protected], [email protected],[email protected] Here we present an exact analytic algorithm to tacklethis challenging problem. We employ our approach toa few physical variables within the mesoscopic Stonerregime, but it can be used to tackle the broad range ofproblems involving spin and charge on a QD, and beextended to the thermodynamic ferromagnetic regimetoo. As examples we calculate the following quanti-ties: the partition function, the magnetic susceptibility,the distribution function of the total spin, the tunnel-ing density of states (TDOS), and the sequential tun-neling conductance. Our approach allows us to obtainanalytic results as one approaches the Stoner instabil-ity. Below we list possible applications of our methodto other physical observables and extensions beyond theUniversal Hamiltonian. The physics discussed here canbe best tested in quantum dots with materials whichare close to the thermodynamic Stoner instability, e.g.,Co impurities in Pd or Pt host, Fe dissolved in varioustransition metal alloys, Ni impurities in Pd host, and Coin Fe grains, as well as new nearly ferromagnetic rareearth materials [7, 8].The main reason why, in this context of a QD, thetreatment of the exchange term is non-trivial, is thenon-Abelian nature of the action. One needs to tackletime ordered integrals of the form A ( p ) γ = T exp (cid:18) i Z t p dt ′ θ p s γ (cid:19) . (1)Here θ p is a dynamical, quantum field operating on thespin s γ (whose x component is proportional to the Paulimatrix σ x etc.); p and γ are indices to be elaboratedbelow; T is a time ordering operation. Wei and Nor-1 I.S. Burmistrov, Yuval Gefen, and M.N. Kiselev man [9], addressing the problem of a quantum spin sub-ject to a prescribed classical time-dependent magneticfield, have elegantly shown that by preforming a non-linear transformation from θ xp , θ yp , θ zp to a set of othervariables (cf. Eq. (15)), Eq. (1) can be written as aproduct of 3 Abelian terms (cf. Eq. (16)). Even so, thatproblem could not be solved. The problem of a quan-tum field appears to be even more intricate. To solveit we employ here a generalized Wei-Norman-Kolokolov(WNK) method [10].We consider a quantum dot of linear size L in theso-called metallic regime, whose dimensionless conduc-tance g Th = E Th /δ ≫
1. Here E Th is the Thoulessenergy and δ is the (spinless) mean single particle levelspacing. We account for the following terms of the Uni-versal Hamiltonian H = H + H C + H S , H = X α,σ ǫ α a † α,σ a α,σ . (2)Here, ǫ α denotes the spin ( σ ) degenerate single particlelevels. The charging interaction H C = E c (ˆ n − N ) ac-counts for the Coulomb blockade, with ˆ n ≡ P α ˆ n α = P α,σ a † α,σ a α,σ being the particle number operator; N represents the positive background charge. The term H S = − J S represents spin interactions within the dot( S = P α s α = P α a † α,σ σ σσ ′ a α,σ ′ ), with the compo-nents of σ comprising of the Pauli matrices.The imaginary time action for this system reads: S tot = Z β L dτ = Z β hX α ¯Ψ α ( ∂ τ + µ )Ψ α − H i dτ. Here µ is the chemical potential, β = 1 /T , T the tem-perature, and we have introduced the Grassmann vari-ables ¯Ψ α = ( ¯ ψ α ↑ , ¯ ψ α ↓ ) T , Ψ α = ( ψ α ↑ , ψ α ↓ ) to representelectrons on the dot.Employing a Hubbard-Stratonovich transformationleads to a bosonized form L = X α ¯Ψ α (cid:20) ∂ τ − ǫ α + µ + iφ + σ · Φ (cid:21) Ψ α + Φ J + φ E c − iN φ where φ and Φ are scalar and vector bosonic fields re-spectively. The SU (2) non-Abelian character of the ac-tion poses a serious difficulty. It prevents one from per-forming a gauge transformation [11] which works effi-ciently in the Abelian U (1) (charging only) case [11–13].Employing the Wei-Norman-Kolokolov trick we are ableto overcome this difficulty. Results. —
Below we present our main results. TheTDOS is given by the following exact expression ν ( ε ) = 1 + e − βε Z X n ↑ , ↓ ∈ Z e − βE c ( n − N ) + βµn + βJm ( m +1) × X α δ h ε − ǫ α + µ − E c (2 n − N + 1) − J (cid:0) m + 14 (cid:1)i × ( m h Z n ↑ ( ǫ α ) Z n ↓ − Z n ↑ +1 Z n ↓ − ( ǫ α ) i +(2 m + 1) h Z n ↑ Z n ↓ ( ǫ α ) − Z n ↑ ( ǫ α ) Z n ↓ i) . (4)Here n ↑ ( n ↓ ) represents the number of spin-up (spin-down) electrons, the total number of electrons n = n ↑ + n ↓ , m = ( n ↑ − n ↓ ) /
2. Note that for m > m < S = m ( S = − m −
1) respectively. Thenormalization factor Z = X n ↑ , ↓ ∈ Z (2 m + 1) Z n ↑ Z n ↓ e − β [ E c ( n − N ) − µn − Jm ( m +1)] (5)coincides with the grand canonical partition functionfor the Hamiltonian (2) [5]. The quantity Z N ≡ R π dθ π e − iθN Q γ (cid:0) e iθ − βǫ γ (cid:1) is the canonical partitionfunction of N noninteracting spinless electrons, and Z N ( ǫ α ) ≡ R π dθ π e − iθN Q γ = α (cid:0) e iθ − βǫ γ (cid:1) determinesthe canonical partition function of a system of N non-interacting spinless electrons under the constraint thatlevel α is not occupied.Eqs. (4) and (5) allow us to study a host of physicalobservables for a given spectrum of single-particle levels { ǫ α } . At low temperatures, T . δ , these observablesare sensitive to details of the spectrum; their statisticalaverages would depend on the symmetry group of thespectral distribution [14].We now discuss a few quantities of interest. Thestatic spin susceptibility can be computed as χ =(1 / ∂ ln Z/∂J . At high temperatures, δ ≪ T ≪ µ/ ln( J ⋆ /T ), J ⋆ = Jδ/ ( δ − J ), the average static spinsusceptibility is given by χ = 12 1 δ − J + 112 T δ ( δ − J ) − T . (6)This expression, underlining the divergence at theStoner instability point, differs from that found by Kur- pin and Charge Correlations in Quantum Dots: An Exact Solution and by Schechter [15] . Near theStoner instability, δ − J ≪ δ , it is the first (second)term of Eq. (6) that dominates when T ≫ J ⋆ ( T ≪ J ⋆ ).For T ≫ J ⋆ the susceptibility behaves like a param-agnetic Fermi liquid (with an upward renormalized g-factor). As the system is driven towards the Stonerinstability limit one crosses over to the low temperatureregime, T ≪ J ⋆ , and a non-Fermi liquid (Curie) behav-ior, sets in, χ ∼ h S i /T , where the average spin scales as p h S i ∼ J ⋆ /δ . Note that the latter approximates thediscontinuous growth of the ground state spin of a spe-cific single electron spectrum (e.g. uniformly spaced),when J/δ is increased in the mesoscopic Stoner regimetowards 1. No dynamical spin response χ ( ω = 0) existsunless the dot is connected to reservoirs, or anisotropicspin interaction is considered.The average moments of the total spin can befound from the partition function Z as h [ S ] k i = T k Z − ∂ k Z/∂J k . It can be characterized by the dis-tribution function of S , P S ( x ) which can be foundfrom Eq. (5). Near the Stoner instability δ − J ≪ δ ,and for the same range of temperatures as in Eq. (6),the distribution becomes P S ( x ) = 2 s βδ πJ ⋆ e − βJ ⋆ / sinh( βδ √ x ) e − βδ x/J ⋆ . (8)The broad asymmetric non-Gaussian nature of the dis-tribution becomes manifest in the high temperaturelimit, and is not due to statistical fluctuations of thesingle particle levels but rather due to the effect of theexchange interaction. The average static spin susceptibility has been calculated inRef. [2] near the Stoner instability, δ − J ≪ δ . In our notations,the result of Ref. [2] at T ≫ J ⋆ becomes (see Eqs.(4.8), (4.13b),(4.15) of Ref. [2]) χ = c δ − J (cid:20) c √ J ⋆ √ T + c J ⋆ T + . . . (cid:21) where numerical coefficients c = 1 / c = √ π/
4, and c ≈ . c = 1 / c = √ π/
4, and c ≈ . c = 1 / c = 0 and c = 1 / T = 0, (see Eq.(4.19) of Ref. [2]) χ ∝ [ δ/ ( δ − J )] .As one can see from Eq. (6), our result for T ≪ J ⋆ smoothlyinterpolates into the result of Ref. [2] for T = 0. Our result for χ implies that the magnetic field tends tozero first (before, e.g., temperature). The result found bySchechter [15] is valid in the limit of vanishing temperature butfinite magnetic field (provided an additional coarse graining isperformed). Generalization of Eq. (6) to finite magnetic fieldresembles the result of Schechter at magnetic fields larger thantemperature [14]. - ¶ - E c J * Ν H ¶ L(cid:144) Ν Fig. 1. TDOS in the Coulomb valley. The solid(dashed) line corresponds to
J/δ = 0 . δ/T = 0 . J ⋆ /T = 3 .
95 (
J/δ = 0 . δ/T = 0 .
95, and J ⋆ /T = 10 . We next consider the average TDOS at δ ≪ T . Themost interesting regime seems to be that of interme-diate temperatures, T ≪ J ⋆ . Under the assumption µ ≫ T ln J ⋆ /T , Eq. (4) can be simplified, leading to ν ( ε ) ν = X n,σ = ± e − βE c ( n − N ) "(cid:18) J J ⋆ (cid:19) f F ( σε − σ Ω − σn ) − J J ⋆ F (cid:18) σε − σ Ω σk J ⋆ , βJ ⋆ (cid:19) n e − βE c ( n − N ) . (9)Here Ω σn ≡ E c ( n − N + σ/ ν is the averaged TDOSfor noninteracting electrons, and F ( x, y ) ≡
12 sgn (cid:16) cos πx (cid:17) e − y ( x − + yπ cos πx (10) × (cid:20) − Φ (cid:18) √ yπ (cid:12)(cid:12)(cid:12) cos πx (cid:12)(cid:12)(cid:12)(cid:19)(cid:21) + e y ( x −| x | ) × X m > ( − m e − y | x | m + ym ( m +1) θ ( | x | − m − .θ ( x ) is the Heaviside step function ( θ (0) ≡ z ) ≡ (2 / √ π ) R z exp( − t ) dt . As x isvaried for a fixed y , F ( x, y ) exhibits damped oscillationswith a period 4 (equivalent to an energy scale 4 J ⋆ ). Inthe limit y ≫ p h S i ∼ J ⋆ /δ indicatesthat they are due to precession of the spin of the in-jected electron about the effective magnetic moment inthe dot. This additional structure in the TDOS reflectsenhanced electron correlations due to the exchange in-teraction. At higher temperatures, T ≫ J ⋆ , there is nointeresting signature of spin-exchange on the TDOS. I.S. Burmistrov, Yuval Gefen, and M.N. Kiselev ¶ - E c J * - ¶ J * - ¶ E c Ν H ¶ L(cid:144) Ν Fig. 2. TDOS at the Coulomb peak. The parametersare the same as in Fig.1. The insets depict the non-monotonic behavior.
One can compute the sequential con-ductance through the QD employing G = G R dε ( − ∂f F ( ε ) /∂ε )( ν ( ε ) /ν ), where G is theconductance of the non-interacting QD. The maximalvalue of G will be enhanced by a factor 1 + J/ J ⋆ due to the exchange term. Much more interestingly,the non-linear conductance at the Coulomb peak willexhibit non-monotonic behavior, similar to Fig. 2 [14]. Derivation. -
Below we describe the main steps ofthe derivation. Further details will be given in [14].The TDOS, ν ( ε ) = − (1 /π ) Im P α,σ G Rασ ( ε ), is de-termined via the imaginary part of the retarded Green’sfunction, G Rασ ( t, t ′ ) = − iθ ( t − t ′ ) h{ a α,σ ( t ) , a † α,σ ( t ′ ) }i H of the Hamiltonian (2). The imaginary timeGreen function is given by G ασ ( τ , τ ) = −h T τ ψ α,σ ( τ ) ¯ ψ α,σ ( τ ) i S tot .The exact one-particle Green function for the Hamil-tonian (2) can be written as G ασ ( τ , τ ) = πT Z − πT dφ Z ( φ ) Z D ( τ , φ ) G ασ ( τ , φ ) , (11)where τ = τ − τ , φ is the static componentof φ , the grand canonical partition function Z = R πT − πT dφ D (0 , φ ) Z ( φ ), and the so-called Coulomb-boson propagator reads [11, 13] D ( τ, φ ) = e − E c | τ | X k ∈ Z e iφ ( βk + τ ) − βE c ( k − N ) − E c ( k − N ) τ . The one-particle Green function G ασ ( τ , τ , φ ) ap-pearing in Eq. (11) is defined as G ασ ( τ , τ , φ ) = −h T τ ψ α,σ ( τ ) ¯ ψ α,σ ( τ ) i S . Average is taken with respectto the action S = R β dτ hP α ¯Ψ α ∂ τ Ψ α − H i . Here H = H + H S with H in which ǫ α is replaced by˜ ǫ α = ǫ α − µ + iφ . Remarkably, the charge and spin degrees of freedom are almost disentangled in the action S . The latter involves only the spin-interaction part ofthe Hamiltonian (2). Traces of the charging-interactionare encoded in the variable φ , leading to a small imag-inary shift of the chemical potential. Subsequently, theone-particle Green function can be written as G ασ ( τ > τ ) = −Z − K ασ ( − iτ , − iτ + iβ ) K ασ ( t + , t − ) = Tr e − it + H a † α,σ e it − H a α,σ (13)and Z ( φ ) = Tr exp( − β H ). In order to evaluatethe trace we perform Hubbard-Stratonovich transfor-mations of the terms e ∓ it ± H S in the evolution operatorsand obtain K ασ ( t + , t − ) = Y p = ± Z D [ θ p ] e − ip J R tp dt ′ θ p (14) × Tr " e − it + H Y γ A (+) γ a † α,σ e it − H Y η A ( − ) η a α,σ . Here A ( p ) γ is defined in Eq. (1). We have definedthe bosonic fields θ p , p = ± . In order to employ theWNK trick we use a Hamiltonian evolution of our op-erators rather than a path untegral representation of G . Note that while H is time independent, the factors A ( p ) γ involve time ordering ( T ). This is due to the non-commutativity of the spin-operators s γ .In order to overcome the intricacy of time-orderingwe use the following transformation of variables [16] inthe functional integral in Eq. (14) [10], θ zp = ρ p − κ pp κ − pp , θ xp − ipθ yp κ − pp ,θ xp + ipθ yp − ip ˙ κ pp + ρ p κ pp − ( κ pp ) κ − pp , (15)which recasts the time-ordered exponent as a productof simple Abelian ones: A ( p ) γ = e p ˆ s − pγ κ pp ( t p ) e i ˆ s zγ R tp dt ′ ρ p ( t ′ ) (16) × exp (cid:20) i ˆ s pγ Z t p dt ′ κ − pp ( t ′ ) e − ip R t ′ dτρ p ( τ ) dt ′ (cid:21) . Here we employ the initial condition κ pp (0) = 0 [9], and s ± γ = s xγ ± is yγ . We stress that Eqs. (15) and (16) arevalid for a general spin operator. In order to preserve thenumber of field variables (three) we impose the follow-ing constraints on the otherwise arbitrary new complexvariables: ρ p = − ρ ∗ p and κ + p = ( κ − p ) ∗ . The quantity K ασ ( t + , t − ) can be then evaluated as K ασ ( t + , t − ) = Y p = ± Z D [ ρ p , κ pp ] e − ip J R tp dt ( ρ p − ip ˙ κ pp κ − pp ) × e ip R tp dtρ p ( t ) C ασ ( t + , t − ) Y γ = α B γ ( t + , t − ) , (17) pin and Charge Correlations in Quantum Dots: An Exact Solution C ασ and B γ given in terms of single-particle traces: C ασ = tr h e − iε α ˆ n α t + A (+) α ( t + ) a † ασ e iε α ˆ n α t − A ( − ) α ( t − ) a ασ i , B γ = tr h e − iε γ ˆ n γ t + A (+) γ ( t + ) e + iε γ ˆ n γ t − A ( − ) γ ( t − ) i . (18)The expression for Z can be obtained from Eq. (17) bythe substitution of B α for C ασ . We can now evaluatethe single-particle traces in B γ and C ασ . The fields κ − p , κ + p appear in B γ . It turns out that the integration over κ pp first, and then ρ p , can be performed exactly, yielding K α ↑ (= K α ↓ ), K α ↑ = e − βJ − iε α t + J √ πβJ ∞ Z −∞ dh sinh( βh ) Y γ = α Y σ = ± h e β ( σh − ε γ ) i × X s = ± e iε α t s + isJts e − (2 βh + isJts )24 βJ (2 βh + isJt − s ) . (19)Next, we perform the integration over h in Eq. (19),substitute it into Eq. (13) and calculate the exchange-only Green function, G ασ . Then, integrating over φ inEq. (11) we obtain the full Green’s function G ασ . Em-ploying the general expression [18] ν ( ε ) = − π cosh βε X α ∞ Z −∞ dt e iεt G α ↑ (cid:18) it + β (cid:19) , (20)we, finally, find the TDOS (4). In a similar way weobtain the partition function Z (5).Within WNK method one may still have some free-dom in selecting regularization of the functional inte-grals. It is thus useful to check the validity of our re-sults against some benchmarks. Our non-trivial checksare: i) Eq.(5) for Z agrees with the exact derivationin Ref. [5]. ii) The TDOS (4) satisfies the sum rule: R dε ν ( ε ) f F ( ε ) = T ∂ ln Z/∂µ [17]. iii) For J = 0 ourresults for the TDOS coincide with those of Ref. [13].iv) Our results for Z and ν ( ε ) agree with a direct cal-culation for single and double level QDs.In summary, we have addressed here the interplay ofcharging and spin-exchange interactions of electrons ina metallic quantum dot. Even within the simple Univer-sal Hamiltonian framework, this problem leads to a non-Abelian action, and necessarily requires the evaluationof non-trivial time-ordered integrals. Our method is ap-plicable to the vicinity of the Stoner instability (well in-side the mesoscopic Stoner unstable regime), and couldbe extended to the ferromagnetic regime. Other ex-tensions include the study of anisotropic spin-exchange(where the non-vanishing a.c. susceptibility, absorptionand TDOS are of particular interest), cotunneling con-ductance, and an explicit inclusion of the leads. As a demonstration of the usefulness of our exactsolution we have calculated several quantities: the par-tition function, the magnetic susceptibility, the distri-bution function of the spin, the TDOS, and the linearand non-linear conductance at the Coulomb peak. Someof these quantities are amenable to experimental tests.Examples: the broad distribution of the spin would im-ply significant sample-to-sample fluctuations of the mea-sured susceptibility; the latter can be used to determinethe distance (1 − J/δ ) from the Stoner instability; therelative magnitude of the predicted non-monotonicitiesin the TDOS and the conductance may exceed 5 − J/δ = 0 .
83) or YFe Zn ( J/δ = 0 .
94) [8].Previously, Alhassid et al. have calculated exactlythe partition function, matrix elements of a † ασ , a ασ [5],and many-body eigenstates which are also eigenstatesof the total spin operator [19]. That approach could beemployed for the calculation of other observables. Ourindependent approach is more manageable for the cal-culation of higher correlators, the inclusion of exchangeanisotropy, as well as to further generalizations, as in-dicated above.We acknowledge useful discussions with I. Aleinerand V.Gritsev. We thank Y. Alhassid for explainingto us his method and the results of his analysis. Weare grateful to I. Kolokolov for providing us with notesof his calculations and a detailed explanation. Thiswork was supported by RFBR Grant No. 09-02-92474-MHKC, the Council for grants of the Russian PresidentGrant No. MK-125.2009.2, the Dynasty Foundation,RAS Programs “Quantum Physics of Condensed Mat-ter”, “Fundamentals of nanotechnology and nanomate-rials”, CRDF, SPP 1285 “Spintronics”, Minerva Foun-dation, German-Israel GIF, Israel Science Foundation,and EU project GEOMDISS.
1. I. Aleiner, P. Brouwer, and L. Glazman, Phys. Rep. , 309 (2002); Y. Alhassid, Rev. Mod. Phys. ,895 (2000).2. I.L. Kurland, I.L. Aleiner, B.L. Altshuler, Phys. Rev.B , 14886 (2000).3.
50 years of Yang-Mills theory , ed. by G. ’t Hooft,World Scientific, Singapore (2005).4. M.N. Kiselev and Y. Gefen, Phys. Rev. Lett. ,066805 (2006).5. Y. Alhassid and T. Rupp, Phys. Rev. Lett. , 056801(2003).6. G. Usaj and H. Baranager, Phys. Rev. B , 121308(2003). I.S. Burmistrov, Yuval Gefen, and M.N. Kiselev
7. P. Gambardella et al., Science , 1130 (2003); G.Mpourmpakis, G.E. Froudakis, A.N. Andriotis, M.Menon, Phys. Rev. B , 104417 (2005).8. S. Jia, S. L. Budko, G. D. Samolyuk, P. C. Canfield,Nat. Phys. , 334 (2007).9. J. Wei and E. Norman, J. Math. Phys. , 575 (1963).10. I.V. Kolokolov, Ann. Phys. (N.Y.) , 165 (1990); M.Chertkov and I.V. Kolokolov, Phys. Rev. B , 3974(1994); Sov. Phys. JETP , 1063 (1994); for a re-view see I.V. Kolokolov, Int. J. Mod. Phys. B , 2189(1996).11. A. Kamenev and Y. Gefen, Phys. Rev. B , 5428(1996).12. K.B. Efetov and A. Tschersich, Phys. Rev. B ,174205 (2003).13. N. Sedlmayr, I.V. Yurkevich, I.V. Lerner, Europhys.Lett. , 109 (2006).14. I. Burmistrov, Y. Gefen, M. Kiselev, and L. Medve-dovsky, in preparation.15. M. Schechter, Phys. Rev. B , 024521 (2004).16. The Jacobian of the transformation is given as J = Q p = ± exp ip R t p dtρ p ( t ).17. The sum rule is the consequence of the follow-ing relations, h ˆ n i = − iG < ( t, t ), and G < ( ε ) =2 if F ( ε ) Im G R ( ε ).18. K.A. Matveev and A.V. Andreev, Phys. Rev. B ,045301 (2002).19. H.E. T¨ureci and Y. Alhassid, Phys. Rev. B74