Attractive Hubbard model with disorder and the generalized Anderson theorem
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec Attractive Hubbard model with disorder and the generalized Anderson theorem
E.Z. Kuchinskii , N.A. Kuleeva , M.V. Sadovskii , Institute for Electrophysics, Russian Academy of Sciences, Ural Branch, Ekaterinburg 620016, Russia Institute for Metal Physics, Russian Academy of Sciences, Ural Branch, Ekaterinburg 620290, Russia
Using the generalized DMFT+Σ approach we have studied disorder influence on single–particleproperties of the normal phase and superconducting transition temperature in attractive Hubbardmodel. The wide range of attractive potentials U was studied — from the weak coupling region,where both the instability of the normal phase and superconductivity are well described by BCSmodel, towards the strong coupling region, where superconducting transition is due to Bose – Ein-stein condensation (BEC) of compact Cooper pairs, formed at temperatures much higher thanthe temperature of superconducting transition. We have studied two typical models of conduc-tion band with semi – elliptic and flat densities of states, appropriate for three-dimensional andtwo-dimensional systems respectively. For semi – elliptic density of states disorder influence onall single-particle properties (e.g. density of states) is universal for arbitrary strength of electroniccorrelations and disorder and is due only to the general disorder widening of conduction band. Inthe case of flat density of states universality is absent in general case, but still the disorder influ-ence is due mainly to band widening and universal behavior is restored for large enough disorder.Using the combination of DMFT+Σ and Nozieres – Schmitt-Rink approximations we have studieddisorder influence upon superconducting transition temperature T c for the range of characteristicvalues of U and disorder, including the BCS-BEC crossover region and the limit of strong coupling.Disorder can either suppress T c (in the weak coupling region) or significantly increase T c (in strongcoupling region). However in all cases the generalized Anderson theorem is valid and all changesof superconducting critical temperature are essentially due only to the general disorder widening ofthe conduction band. PACS numbers: 71.10.Fd, 74.20.-z, 74.20.Mn
I. INTRODUCTION
The problem of strong coupling superconductivity wasstudied for a long time, starting with pioneering papersby Eagles and Leggett [1,2]. Significant progress herewas achieved by Nozieres and Schmitt-Rink [3], who sug-gested an effective method to study the transition tem-perature crossover from weak coupling BCS-like behav-ior towards Bose – Einstein condensation (BEC) sce-nario in the strong coupling region. Recent progressin experimental studies of quantum gases in magneticand optical dipole traps, as well as in optical lattices,with controllable parameters, such as density and interac-tion strength (cf. reviews [4,5]), has increased the inter-est to superconductivity (superfluidity of Fermions) withstrong pairing interaction, including the region of BCS– BEC crossover. One of the simplest models allowingthe study of BCS – BEC crossover is the Hubbard modelwith attractive on site interaction. The most successiveapproach to the solution of Hubbard model, both in thecase of repulsive interaction and for the studies of BCS– BEC crossover in case of attraction, is the dynamicalmean field theory (DMFT) [6–8]. Attractive Hubbardmodel was studied within DMFT in a number of recentpapers [9–13]. However, up to now there are only fewstudies of disorder influence on the properties of normaland superconducting phases in this model, especially inthe region of BCS – BEC crossover. Qualitatively disor-der effects in this region were analyzed in Ref. [14], whereit was argued, that Anderson theorem remains valid inBCS – BEC crossover region in the case of s -wave pairing. Diagrammatic approach to (weak) disorder effects uponsuperconducting transition temperature and propertiesof the normal phase in crossover region was developedrecently in Ref. [15].In recent years we have developed the generalizedDMFT+Σ approach to Hubbard model [16–19], whichis very convenient for the studies of different external(with respect to those taken into account in DMFT) in-teractions (such as pseudogap fluctuations [16–19], disor-der [20,21], electron – phonon interaction [22]) etc. Thisapproach is also well suited to analyze two–particle prop-erties, such as optical (dynamic) conductivity [20,23]. InRef. [13] we have used this approximation to calculatesingle – particle properties of the normal phase and opti-cal conductivity in attractive Hubbard model. In a recentpaper [24] DMFT+Σ approach was used by us to studydisorder influence upon superconducting transition tem-perature, which was calculated in Nozieres – Schmitt-Rink approximation. In this paper for the case of semi –elliptic density of states of the “bare” conduction band,which is adequate for three – dimensional systems, wehave numerically demonstrated the validity of the gener-alized Anderson theorem, so that all the changes of crit-ical temperature are controlled only by general wideningof the conduction band by disorder.In this paper we present the analytic proof of such uni-versal influence of disorder (in DMFT+Σ approximation)upon single – particle characteristics and temperature ofsuperconducting transition for the case of semi – ellipticdensity of states, and also investigate disorder effects inthe case of the “bare” band with flat density of states,qualitatively appropriate for two – dimensional systems.It will be shown, that for the flat band model the uni-versal dependence of single – particle properties and su-perconducting transition temperature is also realized forthe case of strong enough disorder. II. DISORDERED HUBBARD MODEL WITHINDMFT+ Σ APPROACH
We consider the disordered nonmagnetic Hubbardmodel with attractive interaction with Hamiltonian: H = − t X h ij i σ a † iσ a jσ + X iσ ǫ i n iσ − U X i n i ↑ n i ↓ , (1)where t > U represents Hubbard – like on siteattraction, a iσ ( a † iσ ) is annihilation (creation) operator ofan electron with spin σ , n iσ = a † iσ a iσ particle number op-erator on lattice site i , while local on site energies ǫ i areassumed to be random variables (independent on differ-ent lattice sites). For the standard “impurity” diagramtechnique to be valid we take the Gaussian distributionof energy levels ǫ i : P ( ǫ i ) = 1 √ π ∆ exp (cid:18) − ǫ i (cid:19) (2)Parameter ∆ is the measure of disorder strength, whilethe Gaussian random field of random on site energy lev-els (independent on different sites – “white noise” corre-lation) induces “impurity” scattering, which is analyzedusing the standard formalism of averaged Green’s func-tions [25].The generalized DMFT+Σ approach [16–19] extendsthe standard dynamical mean field theory (DMFT) [6–8] taking into account an additional “external” self-energy part Σ p ( ε ) (in general case momentum depen-dent), which is due to some additional interaction outsideDMFT, and gives an effective method to calculate bothsingle – particle and two – particle properties [20,23].The success of this generalized approach is based uponthe choice of the single – particle Green’s function in thefollowing form: G ( ε, p ) = 1 ε + µ − ε ( p ) − Σ( ε ) − Σ p ( ε ) , (3)where ε ( p ) is the “bare” electronic dispersion, while thecomplete self – energy is assumed to an additive sum ofthe local self – energy of DMFT and some “external” self– energy Σ p ( ε ), due to neglect of the interference of Hub-bard and “external” interactions. This allows the conser-vation of the standard form of self – consistent equationsof the standard DMFT [6–8]. At the same time, at eachstep of DMFT iterations we consistently recalculate an“external” self – energy Σ p ( ε ) using the appropriate ap-proximate scheme, corresponding to the form of an addi-tional interaction, while the local Green’s function is also “dressed” by Σ p ( ε ) at each step of the standard DMFTprocedure.For “external” self – energy entering DMFT+Σ cyclefor the problem of random scattering by disorder we usethe simplest self – consistent Born approximation, ne-glecting diagrams with crossing “impurity” lines, whichgives: Σ p ( ε ) → ˜Σ( ε ) = ∆ X p G ( ε, p ) , (4)where G ( ε, p ) is the single – electron Green’s function (3)and ∆ is the amplitude of site disorder.To solve the effective single Anderson impurity prob-lem of DMFT we used the numerical renormalizationgroup approach (NRG) [26].In the following we consider two models of “bare” con-duction band. The first one is the band with semi –elliptic density of states (per unit cell and single spinprojection): N ( ε ) = 2 πD p D − ε (5)where D defined the band half-width. This model is ap-propriate for three – dimensional system. The second oneis the model with the flat density of states, appropriatefor two – dimensional case: N ( ε ) = (cid:26) D | ε | ≤ D | ε | > D . (6)In principle, for two – dimensional systems we shouldtake into account the presence of the weak (logarithmic)Van Hove singularity in the density of states. However,this singularity is effectively suppressed even by rathersmall disorder, so that the simple model of Eq. 6 is quitesufficient for our aims.All calculations in this work has been done for thecase of quarter – filled band (the number of electrons perlattice site n=0.5).The temperature of superconducting transition in at-tractive model was analyzed in a number of papers[9,10,12], both from the condition of instability of thenormal phase [9] (divergence of Cooper susceptibility)and from the condition of superconducting order param-eter going to zero [10,12]. In a recent paper [13] we havedetermined the critical temperature from the condition ofinstability of the normal phase, reflected in the instabil-ity of DMFT iteration procedure. The results obtained inthis way in fact coincided with those of Refs. [9,10,12].Also in Ref. [13] to calculate T c we have used the ap-proach due to Nozieres and Schmitt-Rink [3], which al-lows the correct (though approximate) description of T c in BCS – BEC crossover region. In a later work [24] wehave used the combination of Nozieres and Schmitt-Rinkand DMFT+Σ approximations for the detailed numeri-cal studies of disorder dependence of T c and the numberof local pairs in the model with semi – elliptic density ofstates. III. DISORDER INFLUENCE ON SINGLE –PARTICLE PROPERTIES FOR THE CASE OFSEMI–ELLIPTIC DENSITY OF STATES
In this section we shall analytically demonstrate, thatin DMFT+Σ approximation disorder influence upon sin-gle – particle properties of disordered Hubbard model(both attractive or repulsive) with semi – elliptic “bare”conduction band is completely described by effects of gen-eral band widening by disorder scattering.In the system of self – consistent equations DMFT+Σequations [17,19,20] both information on the “bare” bandand disorder scattering enter only on the stage of calcu-lations of the local Green’s function: G ii = X p G ( ε, p ) , (7)where the full Green’s function G ( ε, p ) is determined byEq. (3), while the self – energy due to disorder, in self– consistent Born approximation, is defined by Eq. (4).Thus, the local Green’s function takes the form: G ii = Z D − D dε ′ N ( ε ′ ) ε + µ − ε ′ − Σ( ε ) − ∆ G ii == Z D − D dε ′ N ( ε ′ ) E t − ε ′ , (8)where we have introduced the notation E t = ε + µ − Σ( ε ) − ∆ G ii . In the case of semi – elliptic density of states (5)this integral is easily calculated in analytic form, so thatthe local Green’s function is written as: G ii = 2 E t − p E t − D D . (9)It is easily seen that Eq. (9) represents one of the rootsof quadratic equation: G − ii = E t − D G ii , (10)corresponding to the correct limit of G ii → E − t for in-finitely narrow ( D →
0) band. Then G − ii = ε + µ − Σ( ε ) − ∆ G ii − D G ii == ε + µ − Σ( ε ) − D eff G ii , (11)where we have introduced D eff – an effective half-widthof the band (in the absence of electronic correlations, i.e.for U = 0) widened by disorder scattering: D eff = D r D . (12)Eq. (10) was obtained from (8), thus comparing (11) and(10), we obtain: G ii = Z D eff − D eff dε ′ ˜ N ( ε ′ ) ε + µ − ε ′ − Σ( ε ) , (13) Here ˜ N ( ε ) = 2 πD eff q D eff − ε (14)represents the density of states in the absence of inter-action U “dressed” by disorder. This density of statesremains semi – elliptic in the presence of disorder, sothat all effects of disorder scattering on single – parti-cle properties of disordered Hubbard model in DMFT+Σapproximation are reduced only to disorder widening ofconduction band, i.e. to the replacement D → D eff . IV. DISORDER INFLUENCE ONSUPERCONDUCTING TRANSITIONTEMPERATURE
Temperature of superconducting transition T c is nota single – particle characteristic of the system. Cooperinstability, determining T c is related to divergence of two– particle loop in Cooper channel. In the weak couplinglimit, when superconductivity is due to the appearanceof Cooper pairs at T c , disorder only slightly influencessuperconductivity with s -wave pairing [27,28]. The socalled Anderson theorem is valid and changes of T c areconnected only with the relatively small changes of thedensity of states by disorder. Th standard derivationof Anderson theorem [27,28] uses the formalism of exacteigenstates of an electron in the random field of impu-rities. Here we present another derivation of Andersontheorem, using the exact Ward identity, which allows usto derive the equation for T c , which will be used to cal-culate T c in Nozieres – Schmitt-Rink approximation indisordered system.In general, Nozieres – Schmitt-Rink approach [3] as-sumes, that corrections due to strong pairing attractionsignificantly change the chemical potential of the sys-tem, while possible correction due to this interaction toCooper instability condition can be neglected, so that wecan always use here the weak coupling (ladder) approxi-mation. In such approximation the condition of Cooperinstability in disordered Hubbard model takes the form:1 = U χ ( q = 0 , ω m = 0) (15)where χ ( q = 0 , ω m = 0) = T X n X pp ′ Φ pp ′ ( ε n ) (16)represents the two – particle loop (susceptibility) inCooper channel “dressed” only by disorder scattering,and Φ pp ′ ( ε n ) is the averaged two – particle Green’sfunction in Cooper channel ( ω m = 2 πmT and ε n = πT (2 n + 1) are the usual Boson and Fermion Matsub-ara frequencies).To obtain P pp ′ Φ pp ′ ( ε n ) we use the exact Ward iden-tity, derived by us in Ref. [23]: G ( ε n , p ) − G ( − ε n , − p ) == − X p ′ Φ pp ′ ( ε n )( G − ( ε n , p ′ ) − G − ( − ε n , − p ′ )) , (17)Here G ( ε n , p ) is the impurity averaged (but not con-taining Hubbard interaction corrections!) single – par-ticle Green’s function. Using the obvious symmetry ε ( p ) = ε ( − p ) and G ( ε n , − p ) = G ( ε n , p ), we obtain fromthe Ward identity (17): X pp ′ Φ pp ′ ( ε n ) = − P p G ( ε n , p ) − P p G ( − ε n , p )2 iε n , (18)so that for Cooper susceptibility (16) we have: χ ( q = 0 , ω m = 0) == − T X n P p G ( ε n , p ) − P p G ( − ε n , p )2 iε n == − T X n P p G ( ε n , p ) iε n . (19)Performing now the standard summation over Matsubarafrequencies [25], we obtain: χ ( q = 0 , ω m = 0) == − πi Z ∞−∞ dε P p G R ( ε, p ) − P p G A ( ε, p ) ε th ε T == Z ∞−∞ dε ˜ N ( ε )2 ε th ε T , (20)where ˜ N ( ε ) is the density of states ( U = 0) “dressed” bydisorder scattering. In Eq. (20) the energy ε is reckonedfrom the chemical potential and if we reckon it from thecenter of conduction band we have to replace ε → ε − µ ,so that the condition of Cooper instability (15) leads tothe following equation for T c :1 = U Z ∞−∞ dε ˜ N ( ε ) th ε − µ T c ε − µ , (21)where ˜ N ( ε ) is again the density of states (calculatedfor U = 0) “dressed” by disorder scattering. At the sametime, the chemical potential of the system at different val-ues of U and ∆ should be determined from DMFT+Σ cal-culations, i.e. from the standard equation for the numberof electrons (band-filling), determined by Green’s func-tion given by Eq. (3), which allows us to find T c forthe wide range of model parameters, including the BCS-BEC crossover and strong coupling regions, as well asfor different levels of disorder. This reflects the physicalmeaning of Nozieres – Schmitt-Rink approximation — inthe weak coupling region transition temperature is con-trolled by the equation for Cooper instability (21), whilein the limit of strong coupling it is determined as thethe temperature of BEC, controlled by chemical poten-tial. Thus, the joint solution of Eq. (21) and equation for the chemical potential guarantees the correct interpo-lation for T c through the region of BCS-BEC crossover.This approach gives the results for the critical tempera-ture, which are quantitatively close to exact results, ob-tained by direct numerical DMFT calculations [13], butdemands much less numerical efforts.It should be stressed, that we have used the exact Wardidentity, which allows the use of Eq. (21) also in the re-gion of strong disorder, when the effects of Anderson lo-calization may become relevant. Eq. (21) demonstrates,that the critical temperature depends on disorder onlythrough the disorder dependence of the density of states˜ N ( ε ), which is the main statement of Anderson theo-rem. In the framework of Nozieres – Schmitt-Rink ap-proach Eq. (21) is conserved also in the region of strongcoupling, when the critical temperature is determined byBEC condition for compact Cooper pairs. In this case thechemical potential µ , entering Eq. (21), may significantlydepend on disorder. However, in DMFT+Σ approxima-tion this dependence of chemical potential (as well as anyother single – particle characteristic) in the model withsemi – elliptic density of states is only due to disorderwidening of conduction band. Thus, both in BCS – BECcrossover and strong coupling regions the generalized An-derson theorem actually remains valid. Correspondingly,in the model of semi – elliptic band Eq. (21) leads to uni-versal dependence of T c on disorder, due to the changeof D → D eff . Such universality is fully confirmed bynumerical calculations of T c in this model, performed inRef. [24] (cf. also the results presented below). V. MAIN RESULTS
Let us now discuss the main results of our numericalcalculations, explicitly demonstrating the universal be-havior of single – particle properties and superconductingtransition temperature with disorder. We shall see, thatall disorder effects are effectively controlled, in fact, onlyby the growth of half-width of conduction band, whichfor the case of semi – elliptic density of states are givenby Eq. (12). In case of the band with flat density ofstates, the growth of disorder changes the shape of thedensity of states, making it semi – elliptic in the limit ofstrong enough disorder, while the effective half-width ofthe band is given by (cf. Appendix A): D eff D = r D + 12 ∆ D ln q ∆ D + 1 q ∆ D − . (22)As an example of the most important single – particleproperty we take the density of states. In Fig. 1 weshow the evolution of the density of states with disorderin the model of semi – elliptic band [13]. We can see,that the growth of disorder smears the density of statesand widens the band. This smearing somehow masks thepeculiarities of the density of states due to correlationeffects. In particular, both quasiparticle peak and lowerand upper Hubbard bands, observed in Fig. 1 in the ab-sence of disorder are completely destroyed in the limit ofstrong enough disorder. However, we can easily convinceourselves, that this evolution is only due to the generalwidening of the band due to disorder (cf. (12), (22)), asall the data for the density of states belong to the sameuniversal curve replotted in appropriate new variables,with all energies (and temperature) normalized by theeffective bandwidth by replacing D → D eff , as shownin Fig. 2(a), in complete accordance with the generalresults, obtained above. In the case of conduction bandwith flat density of states, there is no complete univer-sality, as is seen from Fig. 2(b) for low enough values ofdisorder. However, for large enough disorders the dashedcurve shown in Fig. 2(b) practically coincides with uni-versal curve for the density of states¡ shown in Fig. 2(a).This reflects the simple fact, that at large disorders theflat density of states effectively transforms into semi –elliptic (cf. Appendix A).Going now to the analysis of superconducting transi-tion temperature, in Fig. 3 we present the dependence of T c (normalized by the critical temperature in the absenceof disorder T c = T c (∆ = 0)) on disorder for differentvalues of pairing interaction U for both models of ini-tial “bare” density of states (semi – elliptic — Fig.3(a)and flat — Fig. 3(b)). Qualitatively the evolution of T c with disorder is the same for both models. We cansee, that in the weak coupling limit ( U/ D ≪
1) dis-order slightly suppresses T c (curves 1). At intermedi-ate couplings ( U/ D ∼
1) weak disorder increases T c ,while the further growth of disorder suppresses the criti-cal temperature (curves 3). In the strong coupling region( U/ D ≫
1) the growth of disorder leads to significantincrease of the critical temperature (curves 5). However,we can easily see, that such complicated dependence of T c on disorder is completely determined by disorder widen-ing of the “bare” ( U = 0) conduction band, demonstrat-ing the validity of the generalized Anderson theorem forall values of U . In Fig. 4 curve with octagons show thedependence of the critical temperature T c / D on cou-pling strength U/ D in the absence of disorder (∆ = 0)for both model of “bare” conduction bands (semi – el-liptic — Fig. 4(a) and flat — Fig. 4(b)). We can see,that in both models in the weak coupling region super-conducting transition temperature is well described byBCS model (in Fig. 4(a) the dashed curve represents theresult of BCS model, with T c defined by Eq. (21), withchemical potential independent of U and determined byquarter – filling of the “bare” band), while in the strongcoupling region the critical temperature is determinedby BEC condition for Cooper pairs and drops as t /U with the growth of U (inversely proportional to the effec-tive mass of the pair), passing through the maximum at U/ D eff ∼
1. The other symbols in Fig. 4(a) show theresults for T c obtained by combination of DMFT+Σ andNozieres – Schmitt-Rink approximations for the case ofsemi – elliptic “bare” band. We can see, that all data (ex- pressed in normalized units of U/ D eff and T c / D eff )ideally fit the universal curve, obtained in the absenceof disorder. For the case of flat “bare” band, results ofour calculations are shown in Fig. 4(b) and we do notobserve the complete universality — data points, corre-sponding to different degrees of disorder somehow deviatefrom the curve, obtained in the absence of disorder. How-ever, with the growth of disorder the form of the bandbecomes close to semi – elliptic and our data points movetowards the universal curve, obtained for semi – ellipticcase and shown by the dashed curve in Fig. 4(b), thusconfirming the validity of the generalized Anderson the-orem. VI. CONCLUSION
In this paper, in the framework of DMFT+Σ gener-alization of dynamical mean field theory, we have stud-ied disorder influence on single – particle properties (e.g.density of states) and temperature of superconductingtransition in attractive Hubbard model. Calculationswere made for a wide range of attractive interactions U ,from the weak coupling region of U/ D eff ≪
1, whereboth instability of the normal phase and superconduc-tivity is well described by BCS model, up to the strongcoupling limit of U/ D eff ≫
1, where superconductingtransition is determined by Bose – Einstein condensationof compact Cooper pairs, forming at temperatures muchhigher than the temperature of superconducting transi-tion. We have shown analytically, that in the case ofconduction band with semi – elliptic density of states,which is a good approximation for three – dimensionalcase, disorder influences all single – particle propertiesin a universal way — all changes of these properties aredue only to disorder widening of the band. In the modelof conduction band with flat density of states, which isappropriate for two – dimensional systems, there is nouniversality in the region of weak disorder. However,the main effects are again due to general widening ofthe band and complete universality is restored for highenough disorders, when the density of states effectivelybecomes semi – elliptic.To study the superconducting transition temperaturewe have used the combination of DMFT+Σ approachand Nozieres — Schmitt-Rink approximation. For bothmodels of conduction band density of states disorderingmay either suppress the critical temperature T c (in theregion of weak coupling) or significantly increase it (inthe strong coupling region). However, in all these caseswe have actually proven the validity of the generalizedAnderson theorem. so that all the changes of transitiontemperature are, in fact, controlled only by the effectsof general disorder widening of the conduction band. Incase of initial semi – elliptic band disorder influence on T c is completely universal, while in the case of initial flatband such universality is absent at weak disorder, but iscompletely restored for high enough disorder levels.Finally we should like to present some additionalcomments on the methods and approximations used.Both DMFT+Σ and Nozieres – Schmitt-Rink approachesrepresent cetrain approximate interpolation schemes,strictly valid only in corresponding limiting cases (e.g.small disorder or small (large) U ). However, bothschemes demonstrate their effectiveness also in the caseof intermediate values of U and intermediate (or evenstrong) disorder. Actually, the effectiveness of Nozieres –Schmitt-Rink (neglecting U corrections in Cooper chan-nel) approximation was verified by comparison with di-rect DMFT calculations [13]. The use of DMFT+Σ toanalyze the disorder effects in repulsive Hubbard modelwas shown to produce reasonable results for the phasediagram, as compared to exact numerical simulations ofdisorder in DMFT, including the region of large disorder(Anderson localized phase) [19–21]. However, the role ofapproximations made in DMFT+Σ, such as the neglectof the intrference of disorder scattering and correlationeffects, deserves further studies.This work is supported by RSF grant No. 14-12-00502. Appendix A
For the band with flat density of states (at U = 0 and∆ = 0) disorder leads both to widening of the band andto the change of the form of the density of states. Takingthe density of states in the form given by Eq. (6) wecalculate the local Green’s function as: G ii = 12 D Z D − D dε ′ ε − ε ′ − ∆ G ii =12 D ln (cid:18) ε − ∆ G ii + Dε − ∆ G ii − D (cid:19) , (A1)where energy ε is reckoned from the middle of the “bare”band. Let us introduce auxiliary notations, writing G ii = R − iI . At the band edges I →
0, so that expanding ther.h.s. of Eq. (A1) up to linear terms in I , we get: R − iI ≈ D ln (cid:18) ε − ∆ R + Dε − ∆ R − D (cid:19) − iI ∆ ( ε − ∆ R ) − D (A2)Equating the real parts in (A2) we obtain R = D ln (cid:16) ε − ∆ R + Dε − ∆ R − D (cid:17) . Similarly, equating the imaginaryparts at the band edges we get ε − ∆ R = ±√ D + ∆ , and substituting this expression into logarithm in the pre-vious expression, we find R and band edges positions at: ε = ± p D + ∆ + ∆ D ln √ D + ∆ + D √ D + ∆ − D !! (A3)Thus, the half-width of the band D eff widened by disor-der in this model is determined by Eq. (22) used above.We should note, that the Born approximation for dis-order scattering used by us, though formally valid onlyfor small disorder ∆ ≪ D , the effects of Anderson lo-calization at large disorders ∆ ∼ D do not qualitativelychange the density of states [27], so that Born approxima-tion gives qualitatively correct results also in the regionof large disorder. Actually, this approximation neglectsonly the appearance exponentially small “tails” in thedensity of states, outside the “mean field” band edges[27] and gives more or less correct results inside such aband.At large enough disorders almost any “bare” bandwidth bandwidth 2 D and arbitrary density of states N ( ε ) acquires semi – elliptic density of states. In thelimit of very large disorder ∆ ≫ D almost in the wholeband, widened by disorder, we have | ε − ∆ R | ≫ D andin the expression for the local Green’s function we can ne-glect ε ′ -dependence in the denominator of the integrand: R − iI = G ii = Z ∞−∞ dε ′ N ( ε ′ ) ε − ε ′ − ∆ G ii ≈ ε − ∆ R + i ∆ I (A4)Then we immediately get: ε − ∆ R = ε I = 12∆ p − ε (A5)so that the density of states “dressed” by disorder N ( ε ) = − π ImG ii = Iπ = 2 π (2∆) p (2∆) − ε (A6)becomes semi – elliptic (5) with half-width D eff = 2∆.Thus, at strong enough disorder any “bare” band be-comes semi – elliptic, restoring universal dependence ofsingle – particle properties on disorder discussed above.In this sense, the model of the “bare” band with semi– elliptic density of states is most appropriate for thestudies of the effects of strong disorder. D.M. Eagles. Phys. Rev. , 456 (1969) A. J. Leggett, in Modern Trends in the Theory of Con-densed Matter, edited by A. Pekalski and J. Przystawa(Springer, Berlin 1980). P. Nozieres and S. Schmitt-Rink, J. Low Temp. Phys. ,195 (1985) I. Bloch, J. Dalibard, and W. Zwerger. Rev. Mod. Phys.
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Figures -2 -1 0 1 20,00,20,40,60,81,0
1 /2D=02 0.053 0.114 0.195 0.256 0.377 0.5 D * N () /2D |U|/2D=0.8T/2D=0.05 FIG. 1: Dependence of the density of states on disorder in the model with semi – elliptic band. -2 -1 0 1 20,00,20,40,60,81,0-2 -1 0 1 20,00,20,40,60,81,0 b /2D=0 0.37 D e ff * N () /2D eff flat "bare" DOSU/2D eff =0.8T/2D eff =0.05 /2D=0 0.37 D e ff * N () /2D eff semi-elliptic "bare" DOSU/2D eff =0.8T/2D eff =0.05 a FIG. 2: Universal dependence of the density of states on disorder: (a) — the model of semi – elliptic “bare” density of states;(b) — the model of flat “bare” density of states. b
1 |U|/2D=0.62 0.83 1.04 1.45 1.6 T c / T c /2D flat DOS