Table Of ContentNormal phase and superconducting instability in attractive Hubbard model: the
DMFT(NRG) study
N.A. Kuleeva1∗, E.Z. Kuchinskii1 †, M.V. Sadovskii1,2 ‡
1Institute for Electrophysics, Russian Academy of Sciences, Ural Branch, Ekaterinburg 620016, Russia
2Institute for Metal Physics, Russian Academy of Sciences, Ural Branch, Ekaterinburg 620990, Russia
Westudythenormal(non-superconducting)phaseofattractiveHubbardmodelwithindynamical
meanfieldtheory(DMFT)usingnumericalrenormalization group(NRG)asimpuritysolver. Wide
rangeofattractivepotentialsU isconsidered,fromtheweak-couplinglimit,wheresuperconducting
instability is well described by BCS approximation, up to the strong-coupling region, where super-
conductingtransitionisdescribedbyBose-condensationofcompactCooperpairs,whichareformed
at temperatures much exceeding superconducting transition temperature. We calculate density of
states, spectral density and optical conductivity in the normal phase for this wide range of U, in-
4
1 cludingthedisordereffects. Alsowepresenttheresultsonsuperconductinginstabilityofthenormal
0 statedependenceontheattractionstrengthU andthedegreeofdisorder. Disorderinfluenceonthe
2 criticaltemperatureTc isratherweak,suggestinginfactthevalidityofAndersontheorem,withthe
account of thegeneral widening of theconduction band dueto disorder.
b
e
F
I. INTRODUCTION ations of different order-parameters, disorder scattering
5 etc), which are inevitably present in such systems. Re-
1 cently we have proposed the generalized DMFT+Σ ap-
The studies of superconductivity in the strong cou-
plingregionattractstheoristsforratherlongtime[1]and proach[13–16],whichisveryconvenientandeffectivefor
]
n most important advance here was made by Nozieres and the studies of such additional (external with respect to
o Hubbard model itself) interactions (e.g. pseudogap fluc-
Schmitt-Rink [2], who proposed an effective approach
c tuations [13–16], disorder [17, 18] and electron-phonon
to describe crossover from weak coupling BCS limit to
-
r the picture of Bose-Einstein condensation (BEC) of pre- interaction[19]). Thisapproachwasalsosuccessfullyex-
p tendedtothe analysisofopticalconductivity[17,20]. In
formed Cooper pairs in the strong coupling limit. The
u
thispaperweapplytheDMFT+Σapproachtothestud-
recentprogressofexperimentalstudiesofultracoldgases
s
. in magnetic and optical traps, as well as in optical lat- ies of the normal state properties of attractive Hubbard
t
a tices, allowed the controlled change of parameters, such model, including the effects of disorder.
m
as density and interaction strength (see reviews [3, 4]),
- increasing theoretical interests for studies of superfluid-
d II. THE BASICS OF DMFT+Σ APPROACH
ity (superconductivity) for the case of very strong pair-
n
ing interaction, as well as in BCS-BEC crossover region.
o
c Probablythe simplestmodelallowingtheoreticalstudies In general case we shall consider non-magnetic Hub-
[ of BCS-BEC crossoveris the attractive Hubbard model. bard model with site disorder. The Hamiltonian of this
Itiswidelyusedalsoforthestudiesofsuperconductor— model can be written as:
2
insulator transition (see review in [5]). The most effec-
5v tive modernapproachto the solutionofHubbardmodel, H =−t a†iσajσ + ǫiniσ +U ni↑ni↓, (1)
9 both for strongly correlated electronic systems (SCES) hXijiσ Xiσ Xi
2 with repulsive interaction and for the studies of BCS-
2 BEC crossover in the case of attraction is the dynami- where t>0 is the transfer integralbetween nearestsites
. of the lattice, U is the onsite interaction (for the case of
1 cal mean field theory (DMFT), giving an exact solution
0 in the limit of infinite dimensions [6–8]. The attractive attractionU <0),niσ =a†iσaiσ isonsiteelectronnumber
4 Hubbard model was studies within DMFT in a number operator,a (a† )isannihilation(creation)operatorfor
iσ iσ
1 of recent papers [9–12]. However only few results were electron with spin σ on site i, local energy levels ǫ are
i
:
v obtained for the normal (non-superconducting) phase of assumedtobeindependentrandomvariablesatdifferent
i this model, e.g. there were practically no studies of two- latticesites. Tosimplifydiagramtechniqueinthefollow-
X
particle properties, such as optical conductivity. ing we assume the Gaussian distribution of these energy
r
a TodescribeelectronicpropertiesofSCESweobviously levels:
need to take into account different additional interac-
1 ǫ2
tions (electron-phonon interaction, scattering by fluctu- (ǫ )= exp i (2)
P i √2π∆ −2∆2
(cid:18) (cid:19)
Parameter∆representsherethemeasureofdisorderand
this Gaussian random field (with “white noise” correla-
∗E-mail: [email protected]
†E-mail: [email protected] tion on different lattice sites) generates “impurity” scat-
‡E-mail: [email protected] tering and lead to the standard diagram technique for
2
calculation of the ensemble averaged Green’s functions for ε =ε ω, and
± ± 2
[21].
Generalized DMFT+Σ approach [13–16] extends the Φ0RR(RA)(ω,q) Φ0RR(RA)(ω,0)
φ0RR(RA)(ω)= lim ε − ε ,
standard DMFT [6–8] introducing an additional “exter- ε q→0 q2
nal” self-energy Σp(ε) (in general case momentum de- (6)
pendent), which is due to some interaction mechanism where the two-particle Green’s function Φ0RR(RA)(ω,q)
ε
outsidetheDMFT.Itgivesaneffectiveproceduretocal-
containallvertexcorrectionsfrom“external”interaction,
culate both single- and two-particle properties [17, 20].
butdonotincludevertexcorrectionsfromHubbardinter-
Thesuccessofthisapproachisconnectedwiththechoice
action. This considerably simplifies calculations of opti-
of the single-particle Green’s function in the following
cal conductivity within DMFT+Σ approximation, as we
form:
have only to solve the single-particle problem determin-
1 ing the local self-energy Σ(ε±) via the DMFT+Σ pro-
G(ε,p)= , (3) cedure. Non-trivial contribution from non-local correla-
ε+µ ε(p) Σ(ε) Σ (ε)
− − − p tionsentersonlyviaΦε0RR(RA)(ω,q),whichcanbecalcu-
lated in appropriate approximation, taking into account
where ε(p) is the “bare” electronic dispersion, while the
only “external” interaction. To obtain the loop contri-
total self-energy neglects the interference between the
butions Φ0RR(RA)(ω,q), determined by disorder scatter-
Hubbard and “external” interaction and is given by the ε
ing, we can either use the “ladder” approximation for
additive sum of the local self-energy Σ(ε) of DMFT and
the case of weak disorder, or following Ref. [17], we can
“external” self-energy Σ (ε). This conserves the stan-
p
use the generalization of the self-consistent theory of lo-
dardstructureofDMFTequations[6–8]. However,there
calization [23, 24], which allows us to treat the case of
are two important differences with standard DMFT. At
strong enough disorder. In this approach conductivity
each iteration of DMFT cycle we recalculate the “exter-
is determined mainly by the generalized diffusion coeffi-
nal” self-energy Σ (ε) using some approximate scheme
p
cientobtainedfromthegeneralizationofself-consistency
for the description of “external” interaction and the lo-
equation [23, 24] of this theory, which is to be solved in
cal Green’s function is “dressed” by Σ (ε) at each step
p
combination with DMFT+Σ procedure.
of the standard DMFT procedure.
Inthefollowingweshallconsiderthethree-dimensional
For “external” self-energy due to disorder scattering
system with “bare” semi-elliptic density of states (per
entering DMFT+Σ cycle below we use the simplest ap-
elementary cell and one spin projection), which is given
proximationneglecting the diagrams with “intersecting”
by:
interactionlines,i.e. theself-consistentBornapproxima-
tion, For the Gaussian distribution of site energies it is
2
momentum independent and is given by: N (ε)= D2 ε2 (7)
0 πD2 −
p
Σ (ε) Σ˜ =∆2 G(ε,p), (4) with the bandwidth W = 2D. All calculations below
p
→
p are done for quarter-filled band (n=0.5). The value of
X
conductivityonallfigureswillbegiveninuniversalunits
where G(ε,p) is the single-particle Green’s function (3), of σ = e2 (where a is the lattice spacing).
0 ha
while ∆ is the strength of site energy disorder.
To solve the single Anderson impurity problem of
DMFT we have employed the reliable algorithm of III. MAIN RESULTS
the numerical renormalization group [22], i.e. the
DMFT(NRG) approach..
In Fig.1 we show densities of states obtained for
WithinDMFT+Σapproachwecanalsoinvestigatethe T/2D = 0.05 and quarter filling of the band (n = 0.5)
two-particle properties. In particular, the real part of for different values of attractive (U < 0) Fig.1(a) and
dynamical (optical)conductivity in DMFT+Σ we have repulsive (U > 0) Fig.1(b) interaction. It is well known
the following general expression [17, 20]: that at half-filling (n =1) density of states of attractive
and repulsive Hubbard models just coincide (due to ex-
e2ω ∞ act mapping of these models onto each other). This is
Reσ(ω)= dε[f(ε ) f(ε )]
2π − − + × not so when we deviate from half-filling. From Fig.1 we
Z−∞
cansee thatthe density ofstates close to the Fermi level
ΣR(ε ) ΣA(ε ) 2
Re φ0RA(ω) 1 + − − dropswiththegrowthofU,bothforattraction(Fig.1(a))
× ( ε (cid:20) − ω (cid:21) − and repulsion (Fig.1(b)), but significant growth of U
| |
ΣR(ε ) ΣR(ε ) 2 in repulsive case leads only to vanishing quasiparticle
φ0RR(ω) 1 + − − , (5) peak and density of states at the Fermi level becomes
− ε (cid:20) − ω (cid:21) ) practically independent of U, while in attractive case
the growth of U leads to superconducting pseudogap
| |
where e is electronic charge, f(ε ) — Fermidistribution opening at the Fermi level (curve 3 in Fig.1(a)) and for
±
3
0,10
1 |U|/2D=0.2
2 0.6
3 1.0
4 1.4
T/2D=0.05
5 1.8
21 -0U.6/2D=0.2 1 1 21 U0./62D=0.2 Re0,05 2 3
1,0 34 11..04 2 1,0 2 34 11..04 1
5 1.8 5 1.8 4
5
S S
O O
D0,5 D0,5 3 4 5 0,000 1 2 3
a 5 4 3 b /2D
FIG. 2: Optical conductivity for different values of Hubbard
0,0 0,0
-2 -1 0 1 2 -1 0 1 2 3 attraction. Temperature T/2D=0.05.
/2D /2D
FIG. 1: Densities of states for different values of Hubbard
attraction (a) and repulsion (b). Temperature T/2D=0.05. 1 |U|/2D=0.2 4 5
2 0.6
2
3 1.0
4 1.4
5 1.8
U /2D > 1.2 we observe the full gap opening at the )
F|er|mi level (curves 4, 5 in Fig.1(a)). This gap is not re- A(
1
lated to the appearance of superconducting state, but is 3
due to the appearanceofpreformedCooperpairs,as the
2
temperature for which the results shown in Fig.1 were 1
obtained is larger than superconducting transition tem- 0
-2 -1 0 1 2
perature (cf. Fig.7 below). Thus we observe the impor-
/2D
tant difference from repulsive case, where the deviation
from half-filling leads to metallic state for arbitrary val- FIG. 3: Spectral density at the Fermi surface for different
ues of U, while insulating gap at large U opens not at valuesof Hubbardattraction. Temperature T/2D=0.05.
the Fermi level.
This picture of density of states evolution with the
growth of U is supported by the behavior of dynamic whereξk representskineticenergyofelectrons. Itisseen,
| |
(optical) conductivity shown in Fig.2. We see that with that this distribution changes from more or less defined
the growth of U Drude peak at zero frequency (curves Fermi step-function at small U (curves 1, 2 in Fig.4)
| | | |
1, 2 in Fig.2) is replaced by pseudogap dip (curve 3 in to effective constant at large values of U (curves 4, 5
| |
Fig.2) and wide maximum of conductivity at finite fre- in Fig.4), due to formation of Cooper pairs with binding
quency, connected with scattering across the pseudogap. energy of the order of U .
| |
The further growth of U leads to the appearance of
| |
the full gap in optical conductivity due to formation of
Cooper pairs (curves 4, 5 in Fig.2).
Similar evolution with growth of U is also observed
1,0
in spectral density. In Fig.3 we sh|ow| spectral density 1 1 U/2D=0.2
A(ε,p) = 1ImGR(ε,p) at the Fermi surface (p = p ) 2 2 0.6
−π F 3 1.0
for different values of attractive interaction U. With the 4 1.4
5 1.8
growth of U a narrow peak in spectral density at the
Fermi level| (c|urves 1, 2 in Fig.3) is smeared and with n()k0,5 43
the further growth of U the pseudogap dip appears at
5
| |
the Fermi level (curve 3 in Fig.3). At still larger U
| |
this dip is transformed into the real gap (curves 4, 5
in Fig.3). This behavior of spectral density correlates 0,0
-0,2 0,0 0,2 0,4 0,6
well with qualitative change (with the growth of U ) of
distribution function n(ξk) (Fig.4), defined as: | | ( k- F)/2D
FIG.4: Distribution function fordifferent valuesofHubbard
∞
attraction. Temperature T/2D = 0.05. ξF – kinetic energy
n(ξ )= dεA(ε,ξ )f(ε), (8)
k k of electrons at theFermi surface.
Z−∞
4
sity of states (curve 3 in Fig.6(a),(b)), which leads to
drop of static conductivity. Finally, with the further
growth of disorder Anderson localization effects become
important. At T =0 Anderson transition takes place at
∆/2D = 0.37 [17]. However, here we consider the case
of high enough temperature T/2D=0.05, so that static
conductivity (see curves 4, 5 in Fig.6(b)) remains finite,
though at finite frequencies we clearly observe localiza-
tion behavior with σ(ω) ω2. At larger value of attrac-
∼
tive interaction U /2D=1, the evolution of the density
| |
of states and optical conductivity is more or less similar
(Fig. 6(c,d) ). However, in the absence of disorder we
observe here superconducting pseudogap in the density
of states and disorder growthsuppressesit, leading both
to the growth of the density of states at the Fermi level
and appropriate growth of static conductivity. Finally,
at still larger attraction U /2D = 1.6 (Fig.6(e),(f)) in
| |
FIG.5: SpectraldensitymapsfordifferentvaluesofHubbard the absence of disorder there is the real Cooper gap in
attraction. Color represent the intensity of spectral density. the density of states. This gap is also clearly observed
Temperature T/2D=0.05. in optical conductivity. With the growth of disorder
Cooper gap both in the density of states and conduc-
tivity becomes narrower (curves 1-3). Further growth
The formation of superconducting pseudogap and of disorder leads to complete suppression of Cooper gap
Cooper pairing gap with the growth of U is also well and restoration of metallic state with finite density of
| |
demonstrated by the maps of spectral density, shown in states at the Fermi level and finite static conductivity.
Fig.5 for different values of U. Colors represent the in- This closure of Cooper gap is related to the widening of
tensity of spectral density. We observe that the growth effective bandwidth Weff due to disorder, which leads
of U leads to transformation of initially well defined to the diminishing ratio U /W , which controls the
eff
| | | |
dispersion of Fig.5(a) to dispersions with pseudogap re- formation of Cooper gap. Situation here is similar to
gion, shownin Fig.5(b,c), which transforms into the real the closure of Mott gap by disorder in repulsive Hub-
Coopergap,showninFig.5(d,e) withthe further growth bard model [17]. However, at larger disorder (curve 5 in
of U . Fig.6(f))weclearlyobservelocalizationbehavior,sothat
| |
the growth of disorder at T = 0 will first lead to metal-
lic state (the closure of Cooper gap), while the further
growth of disorder will induce Anderson metal-insulator
A. Disorder effects
transition. Similar picture is observed for large positive
U at half-filling (n = 1) [17], where the growth of disor-
InFig.6we showevolutionofthe densityofstatesand
derleadstoMottinsulator-correlatedmetal-Anderson
optical conductivity with changing disorder. At weak
insulator transition.
enough attraction (U /2D = 0.8, Fig.6(a),(b)), we see
| |
that the growth of disorder smears density of states,
leading to some widening of the band. This smearing
masks peculiarities of the density of states due to cor- B. Superconducting transition temperature
relation effects. In particular, quasiparticle peak and
“wings”duetoupperandlowerHubbardbandsobserved
Superconducting transition temperature T in attrac-
c
inthedensityofstatesinFig.6(a)intheabsenceofdisor-
tive Hubbard model was studied in a number of papers
der completely vanish at strong enough disorder. There
[9, 10, 12], both from the criterion of instability of nor-
are no singularities in the density of states due to An-
mal phase (divergence of Cooper susceptibility) [9] and
derson metal-insulator transition, which takes place at
from the condition of vanishing superconducting order
∆/2D = 0.37 [17], as density of states does not feel
parameter[10, 12]. In Fig. 7 black squares,white circles
Anderson localization. Evolution of optical conductiv-
and white squares show the results of Refs. [9],[10],[12]
ity with the growth of disorder ∆, shown in Fig.6(b),
correspondingly, for the case of quarter-filling n = 0.5
corresponds in general to evolution of density of states. (1).
The growth of disorder, while it remains weak enough,
(curves 1, 2 in Fig.6(b)), leads to some growth of static
conductivity,whichisconnectedwithsuppressionofcor-
relation effects at the Fermi level, noted above (curves 1 In Ref. [10] it was claimed that n = 0.75 was considered, but
1, 2 in Fig.6(a). The further growth of disorder leads
results are obtained practically coincide with those of Ref. [9]
to significant widening of the band and the drop of den- obtainedforn=0.5
5
1,0 0,2
1 /2D=0 1 1 /2D=0 0,05
0,8 32 00..1215 2 2 32 00..1215
4 0.37 3 4 0.37 0,04
DOS0,6 5 0.5 45 Re0,1 1 3 5 0.5 D 0,03 BCS
0,4 |TU/2|/D2D=0=.00.58 4 5 |TU/2|/D2D=0=.00.58 T/2c
0,2 0,02
a b
0,0 0,0 0,01
-2 -1 0 1 2 0 1 2
/2D /2D
0,00
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0
1,0 1 U/2D
12 0./121D=0 2 0,10 3 12 0./121D=0
3 0.25 3 0.25 FIG. 7: Dependence of superconducting critical temperature
4 0.37 4 0.37
S 5 0.5 3 4 5 0.5 on attractive interaction strength. Black squares, white cir-
O0,5 4 e cles and white squares show the results of Refs. [9],[10],[12]
D 5 R0,05 5 respectively for quarter-filled band with n = 0.5. Stars rep-
|U|/2D=1.0 2 |U|/2D=1.0 resent theresultsobtained from thecriterion ofinstability of
T/2D=0.05 T/2D=0.05
thenormalphase. FilledcirclesshowTc obtainedNozieres—
c 1 d
Schmitt-Rink approximation. Continuous black curve repre-
0,0 0,00
-2 -1 0 1 2 0 1 2 3 sents theresult of BCS theory.
/2D /2D
0,06
1 /2D=0 1 /2D=0 such strong values of attractive interaction the critical
2 0.11 2 0.11
1,0 3 0.25 1 5 3 0.25 temperature is determined by the condition ofBose con-
45 00..357 2 0,04 45 00..357 densation of preformed Cooper pairs and transfer am-
S 3 plitude of these pairs appears only in the second order
O e
D0,5 |TU/2|/D2D=0=.10.365 4 R0,02 4 1 |TU/2|/D2D=0=.10.365 of perturbation theory and is proportional to t2/|U| [2].
2 Stars in Fig.7 show the critical temperature, obtained
5 3
e f from the criterion of normal phase instability. For large
0,0 0,00 enough U lowering temperature leads to instability of
-2 -1 0 1 2 3 0 1 2 3
DMFT(NRG)iterationprocedure—athighenoughtem-
/2D /2D
peratures DMFT(NRG) procedure converges to a single
solution,whilefortemperaturesbelowsomecriticaltem-
FIG. 6: Evolution of the density of states (left panels) and
peratureweobservetwodifferentstablesolutionsforodd
optical conductivity (right panels) with disorder for different
oreveniterations. We suggest,thatthis instability ofit-
values of U (|U|/2D =0.8 - a,b; |U|/2D =1 - c,d; |U|/2D =
erationprocedure correspondsto the physicalinstability
1.6 - e,f).
of the normal phase. Unfortunately, for U /2D < 1,
| |
theobservedinstabilityisratherweak(thedifferencebe-
Actually, the overall picture of T dependence on U is tweenthe odd andeven iterationsis too small), thus the
c
well approximated by filled circles curve shown in Fig. accuracy of our calculations is insufficient to determine
7 and obtained from Nozieres — Schmitt-Rink [2] ap- Tc in this way. Surprisingly enough, the results for Tc
proach, which gives the correct (approximate) descrip- obtained from the approximate approachof Ref. [2] and
tion of BCS-BEC crossover. Then for critical tempera- frominstabilityofDMFT(NRG)cyclearerathercloseto
ture T we have the usual BCS-like equation: each other. This is especially surprising for large values
c
of U/2D ratio, where pseodigap (or even the real gap)
U D thε−µ develops in the density of states.
1= | | dεN (ε) 2Tc , (9)
2 0 ε µ In Fig.8 we show the dependence of critical tempera-
Z−D − ture, obtained from the criterion of normal state insta-
whilethechemicalpotentialfordifferentvaluesofU isto bility,ondisorderstrength∆for U /2D=1.6. Atsmall
| |
be determined from DMFT calculations (for fixed band- ∆ we observe weak suppression of T by disorder, which
c
filling). From Fig. 7 we can see, that in the weak cou- is apparetnly due the general smearing of the density of
pling region of U /2D 1 the critical temperature in statedandbandwidthwideningbydisorderscatteringAt
| | ≪
this approach is close to the usual result of BCS the- large enough.disorder we observe the significant growth
ory (see appropriate curve in Fig.7). For U /2D 1 ofT withthegrowthof∆. Thisisrelatedtothe growth
c
| | ∼
the criticaltemperature T has the maximalvalue, while ofeffectivebandwithW duetodisorder,leadingtoef-
c eff
for U /2D 1 it drops as T 1/U [2], because for fectivedropoftheration U /W ,controllingthevalue
c eff
| | ≫ ∼ | | | |
6
disorderinthestrongcouplingregionleadstotheclosure
of the Cooper gap and restoration of the metallic state,
0,035
U/2D=1.6 whileintheintermediatecouplingregiondisordersmears
0,030 n=0.5
superconducting pseudogap and increases the density of
2D 0,025 0,045 states at the Fermi level. In both cases this is related to
T/c0,020 0,040 n0=.60.8 the general widening of the band (in the absence of U)
D 0.5
0,015 T/20c,035 by disorder.
We havedeterminedthe criticaltemperatureofsuper-
0,010 0,030
conductingtransitionT fromtheconditionofinstability
c
0,005 0,025 of the normalphase. Two methods to find suchinstabil-
0,00 0,05 0,10 0,15 0,20
0,000 /2D ity were used, demonstrating quantitatively similar re-
0,00 0,05 0,10 0,15 0,20 0,25
sults. In the weak coupling region T is well described
c
D by BCS theory, while in the strong coupling region it
is related to Bose-condensation of (preformed) Cooper
FIG. 8: Dependence of superconducting critical temperature pairs and drops as 1/U with the growthof U , passing
ondisorderfor|U|/2D =1.6. Attheinsert—Tc suppression throughthemaximum|at| U /2D 1. Wehav|ea|lsostud-
by weak disorder for different values of band-filling: n=0.5, | | ∼
iedtheeffectsofdisorderonT . Itwasshown,thatdisor-
n=0.6, n=0.8. c
derinfluence ofT isratherweak. Inthe strongcoupling
c
region, e.g for U/2D = 1.6 we observe both weak sup-
pression of critical temperature, as well as some growth
of criticaltemperature in this model. The growthof dis-
ofT withthegrowthof∆forstrongenoughdisorder. In
order leads to to the drop of U /W from the value of c
eff
| | fact, this behaviorsuggeststhe validity ofAndersonthe-
1.6 at ∆ = 0 to U /W 1 for ∆/2D 0.4, which
eff
| | ∼ ∼ orem (as was conjectured for BCS-BEC crossoverregion
leads to the appropriate growth of the critical tempera-
in Ref. [25]), with changes of T related to the widening
ture (cf. Fig.7). This behavior is similar to the growth c
of conduction band by disorder. These results are also
of the critical value of repulsion in Hubbard model for
consistent with recent lowest order perturbation theory
Mott metal-insulator transition with the growth of dis-
analysis of the effects of disorder throughout BCS-BEC
order (cf. Ref. [17, 18]). The dropof the ratio U /W
eff
| | crossoverregion [26].
with the growthof disorder does not allow us to guaran-
This work was partly supported by RFBR grant 14-
teethesufficientaccuracyofthevaluesofT inthecaseof
c
02-00065 and was performed within the Program of
U /2D 1fordisordervalueslargerthan∆/2D=0.11.
| | ∼ Fundamental Research of the Ural Branch of the Rus-
For such small values of disorder and for U /2D 1
| | ∼ sian Academy of Sciences “Quantum macrophysics and
the critical temperature is weakly suppressed by disor-
nonlinear dynamics”(projects No. 12-Π-2-1002, 12-T-2-
der,similarlyto the behaviorshowninFig.8for the case
1001).
of U /2D=1.6. At the insertin Fig.8we show the sup-
| |
pression of the critical temperature by weak disorder for
differentvaluesofband-filling: n=0.5,n=0.6,n=0.8.
IV. CONCLUSIONS
WithinthegeneralizedDMFT+Σgeneralizationofdy-
namical mean field theory we have studied the proper-
ties ofthe normal(non-superconducting)state ofattrac-
tive Hubbard model for the wide region of values of on-
site attractive interaction U. The results for the density
of states, spectral density, distribution function and dy-
namic (optical) conductivity demonstrate the formation
of superconducting pseudogap at the Fermi level for in-
termediate values of coupling strength U /2D 1 and
| | ∼
formation of the real Cooper gap in the strong coupling
region U /2D > 1. The appearance of Cooper gap is
| |
relatedtotheformationofcompactCooperpairsattem-
peratures, which are significantly higher. than the criti-
cal temperature of superconducting transition T , which
c
is determined as Bose-condensationtemperature of such
(preformed) pairs. Within our DMFT+Σ approach we
havealsostudiedtheinfluenceofdisorderonthe proper-
tiesofthenormalphase. Itwasshown,thatthegrowthof
7
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