NORMAL PHASE AND SUPERCONDUCTING INSTABILITY IN THE ATTRACTIVE HUBBARD MODEL: A DMFT(NRG) STUDY
N. A. Kuleeva"*, E. Z. Kuchinskii"**, M. V. Sadovskiia>b***
" Institute for Electrophysi.es, Russian Academy of Sciences, Ural Branch 620016, Ekaterinburg, Russia
b Institute for Metal Physics, Russian Academy of Sciences, Ural Branch 620990, Ekaterinburg, Russia
Received January 10, 2014
We study the normal (nonsuperconducting) phase of the attractive Hubbard model within the dynamical mean field theory (DMFT) using the numerical renormalization group (NRG) as an impurity solver. A wide range of attractive potentials U is considered, from the weak-coupling limit, where superconducting instability is well described by the BCS approximation, to the strong-coupling region, where the superconducting transition is described by Bose condensation of compact Cooper pairs, which are formed at temperatures much exceeding the superconducting transition temperature. We calculate the density of states, the spectral density, and the optical conductivity in the normal phase for this wide range of U, including the disorder effects. We also present the results on superconducting instability of the normal state dependence on the attraction strength U and the degree of disorder. The disorder influence on the critical temperature Tc is rather weak, suggesting in fact the validity of Anderson's theorem, with the account of the general widening of the conduction band due to disorder.
DOI: 10.7868/S0044451014080094
1. INTRODUCTION
The study of superconductivity in the strong-coupling region attracts theorists for a rather long time fl], and the most important advance here was made by Nozieres and Schmitt Rink [2], who proposed an effective approach to describe crossover from the weak-coupling BCS limit to the picture of Bose Einstein condensation (BEC) of preformed Cooper pairs in the strong-coupling limit. The recent progress in experimental studies of ultracold gases in magnetic and optical traps, as well as in optical lattices, allowed a controlled change of parameters, such as the density and interaction strength (see reviews [3, 4]), increasing the theoretical interests in studies of superfluidity (superconductivity) in the case of a very strong pairing interaction, as well as in the BCS BEC crossover region. Probably, the simplest model allowing theoretical studies of the BCS BEC crossover is the attractive Hub-
E-mail: strigina'fliep.uran.ru E-mail: kuchinsk'fliep.uran.ru E-mail: sadovski'fliep.uran.ru
bard model. It is widely used also for the studies of the superconductor insulator transition (see a review in [5]). The most effective modern approach to the solution of the Hubbard model, both for strongly correlated electronic systems (SCES) with repulsive interaction and for the studies of the BCS BEC crossover in the case of attraction, is the dynamical mean field theory (DMFT), giving an exact solution in the limit of infinite dimensions [6 8]. The attractive Hubbard model was studied within the DMFT in a number of recent papers [9 12]. However, only few results were obtained for the normal (nonsuperconducting) phase of this model, for example, there were practically no studies of two-particle properties, such as the optical conductivity.
To describe the electronic properties of SCES, we obviously need to take different additional interactions, which are inevitably present in such systems into account (electron phonon interaction, scattering by fluctuations of different order parameters, disorder scattering, etc.). Recently, we have proposed the generalized DMFT^E approach [13 16], which is very convenient and effective for the studies of such additional interactions (external with respect to the Hub-
bard model itself, e. g., psoudogap fluctuations [13 16], disorder [IT, 18], and electron plionon interaction [19]). This approach was also successfully extended to the analysis of optical conductivity [17, 20]. In this paper, we apply the DMFT^E approach to the studies of the normal-state properties of the attractive Hubbard model, including the effects of disorder.
2. THE BASICS OF THE DMFT+S APPROACH
In the general case, we consider the nonmagnetic Hubbard model with site disorder. The Hamiltonian of this model can be written as
H = -*T. "I
{if}*
Y^ + U nnnu. (!)
where t > 0 is the transfer integral between nearest sites of the lattice, U is the onsite interaction (U < 0 in the case of attraction), ni(T = uj^«^ is onsite electron number operator, uicr ) is the annihilation (creation) operator for the electron with spin a on site i, and local energy levels are assumed to be independent random variables at different lattice sites. To simplify the diagram technique in what follows, we assume the Gaussian distribution of these energy levels:
V(€i) =
1
■ exp
2A5
(2)
The parameter A represents the measure of disorder, and this Gaussian random field (with the "white noise" correlation on different lattice sites) generates "impurity" scattering and leads to the standard diagram technique for the calculation of ensemble-averaged Green's functions [21].
The generalized DMFT^E approach [13 16] extends the standard DMFT [6 8] by introducing an additional "external" self-energy Sp(e) (in the general case, momentum dependent), which is due to some interaction mechanism outside the DMFT. It gives an effective procedure to calculate both single- and two-particle properties [17, 20]. The success of this approach is connected with the choice of the single-particle Green's function in the form
G(e,p) =
1
(3)
where e(p) is the "bare" electronic dispersion, while the total self-energy neglects the interference between the Hubbard and "external" interaction and is given by the additive sum of the DMFT local self-energy E(e) and the "external" self-energy Sp(e). This preserves
the standard structure of DMFT equations [6 8]. But there are two important differences from the standard DMFT. At each iteration of the DMFT cycle, we recalculate the "external" self-energy Sp(e) using some approximate scheme for the description of "external" interaction, and the local Green's function is "dressed" by Sp(e) at each step of the standard DMFT procedure.
For the "external" self-energy due to disorder scattering entering the DMFT^E cycle below, we use the simplest approximation neglecting the diagrams with "intersecting" interaction lines, i.e., the self-consistent Born approximation. For the Gaussian distribution of site energies, it is independent of the momentum and is given by
(4)
where G(e,p) is the single-particle Green's function (3) and A is the strength of site energy disorder.
To solve the single Anderson impurity problem of DMFT, we have used the reliable algorithm of the numerical renormalization group [22], i.e., the DMFT(NRG) approach.
Within the DMFT^E approach, we can also investigate the two-particle properties. In particular, the real part of the dynamical (optical) conductivity has the following general expression in DMFT^E [17,20]:
Re<r(u;) = ^ I de [/(e_) - ,/(e+)] x
x Re j.;/,,nU) - olU<"^)
U>
1
u)
(5)
where e is electronic charge, f(e±) is the Fermi distribution for e± = e ± ui/2, and
o0l<l<:l<uU) =
= lim
q^Q
r
(6)
where the two-particle Green's functions contain all vertex corrections from the "external" interaction, but do not include vertex corrections from the Hubbard interaction. This considerably simplifies calculations of optical conductivity within the DMFT—I; approximation, because we only have to solve the single-particle problem determining
7 >K9T<£>, libiii. 2 (8)
305
DOS
DOS
s/'2D s/'2D
Fig. 1. Densities of states for different values of (a) Hubbard attraction and (b) repulsion. Temperature T¡2D = 0.05
the local self-energy via the DMFT—I; pro-
cedure. The nontrivial contribution from nonlocal correlations enters only via RA^ , q), which can
be calculated in an appropriate approximation, taking only the "external" interaction into account. To obtain the loop contributions RA^ , q), determined
by disorder scattering, we can either use the "ladder" approximation in the case of weak disorder, or, following Ref. [17], use the generalization of the self-consistent theory of localization [23, 24], which allows treating the case of sufficiently strong disorder. In this approach, the conductivity is determined mainly by the generalized diffusion coefficient obtained from the generalization of the self-consistency equation [23, 24] of this theory, which is to be solved in combination with the DMFT—I; procedure.
In what follows, we consider the three-dimensional system with a "bare" semi-elliptic density of states (per elementary cell and one spin projection), which is given
by
A^fl^V^ (7)
with the bandwidth W = 2D. All calculations below are done for a quarter-filled band (n = 0.5). The value of conductivity on all figures is given in universal units <r0 = e2//iu (where u is the lattice spacing).
3. MAIN RESULTS
In Fig. 1, we show the densities of states obtained for T/2D = 0.05 and quarter filling of the band (n = 0.5) for different values of the attractive (U < 0, Fig. la) and repulsive (U > 0, Fig. 16) interaction. It is well known that at half-filling (n =1), the density of states of the attractive and repulsive Hubbard models coincide (due to the exact mapping of these models onto each other). This is not so when we deviate from half-filling. From Fig. 1, we can see that the density of states close to the Fermi level decreases as U increases, for both attraction (Fig. la) and repulsion (Fig. 16), but a significant increase in |Z7| in the repulsive case leads only to the vanishing of the quasiparticle peak, and the density of states at the Fermi level becomes practically independent of U, while in the attractive case, the increase in |Z7| leads to the superconducting pseu-dogap opening at the Fermi level (curve 3 in Fig. la); for \U\/2D > 1.2, we observe the
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