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Correlated electrons and generalized statistics

Several important generalizations of Fermi-Dirac distribution are compared to numerical and experimental results for correlated electron systems. It is found that the quantum distributions based on incomplete information hypothesis can be useful for describing this kind of systems. We show that the additive incomplete fermion distribution gives very good description of weakly correlated electrons and that the nonadditive one is suitable to very strong correlated cases.


I. INTRODUCTION
In this work, we attempt to relate generalized statistical theories to correlated electron or quasiparticle systems which, since 1980's, have been subjected to intensive theoretical and experimental studies due to their important roles in optical [1], mechanical [2], electric and magnetic [3][4][5][6][7][8] properties of metal compounds with partially filled d-or f-electrons. In general, with complex correlations, the conventional Boltzmann-Gibbs statistics (BGS) for free fermions is no longer valid. Most of the theoretical study up to now was numerical simulations based on approximate models including essential physics of the systems of interest. Although these theoretical works are very important and fruitful, the need of general understanding and theoretical formulation of the distribution laws begins to be felt. Due to the complexity of the correlations, we can conjecture that generalization of Fermi-Dirac (FD) theory may be useful. In this sense, some efforts have been made to understand localized electron behavior on the basis of fractal geometry (see [9] and references there-in), a generalization of normal space-time.
In what follows, I present a work in this direction based, not on the fractal or chaotic assumption, but on generalized statistical theories proposed for complex systems showing correlated phenomena. The theoretical results will be compared to numerical and experimental ones for correlated electrons. It is expected that this attempt can help to better understand different generalized theories and identify valid theories for this special circumstance.

II. SOME IMPORTANT GENERALIZATIONS OF BGS AND CORRELATED ELECTRONS
Indeed, in the last decade, BGS theory has experienced a turbulent period with the rapid development of some anomalous theories to treat complex systems. These theories are anomalous in the sense that they may perturb our old conceptions concerning, e.g., information, energy, quantum states, additivity etc. The main characters of these theories can be resumed as follows : 1) Consideration of nonadditivity in entropy, energy, quantum occupation number etc. depending on empirical parameters, e.g. 0 < q < ∞ in Tsallis nonextensive statistical mechanics (NSM) [10] and in its incomplete statistics (IS) version [11][12][13][14][15] as well as in quantum group theory (QGT) [16,17], κ ≥ 0 in κ-statistics (KS) [18].
The philosophy of these generalizations, explicit or not, is to introduce empirical parameters to "absorb" the effects of complex correlations. So interacting systems can be mathematically treated as noninteracting or conventional ones. For example, the total energy of an interacting "free particle" can be written as p 2 /2m where p is the momentum and m the mass of the particle. The extra interaction energy is in this way "absorbed" in the empirical parameter when they are different from the values corresponding to conventional cases.
About the generalization of FD statistics, Figure 1 shows the different fermion distributions at T =100 K of the generalized statistical mechanics mentioned above. We see that the discontinuity in occupation number n at Fermi energy e f depends almost only on temperature for all these distributions. The sharp n-drop hardly changes for whatever value of the empirical parameters. In addition, we note that the approximate distribution (AD) n = 1 [1+(q−1)β(e−e f )] 1/(q−1) +1 (0 < q ≤ 1 in order that the inverse temperature β can go to ∞) of NSM obtained with factorization approximation [22,23] by neglecting correlation energy between the particles is only slightly different from FD one. So its Fermi energy is almost identical to that of FD. The situation of KS is similar. At low temperature, the κ-distribution of standard fermion (η = 1) n = 1 [ √ 1+κ 2 β 2 (e−e f ) 2 +κβ(e−e f )] 1/κ +1 [18] is hardly different from FD for κ values which give finite internal energy (e.g. 0 ≤ κ < 1/2 for two dimensional fermion gas). For FES fermion distribution (1/n − α) α (1/n − α + 1) 1−α = e (e−e f )/kT [19] (or equivalently n = 1 e qβ(e−e f ) −1 − (1+α)/α e qβ(e−e f )(1+α)/α −1 [24]), the n-discontinuity depends only on temperature just as for FD. The influence of α smaller than unity (for FD) is to decrease e f (α times) and to increase the maximal occupation number (1/α). As for the nonextensive fermion distribution of IS n = 1 [1+(q−1)β(e−e f )] q/(q−1) +1 [14,15], though n drop at e f is always very sharp, e f increase considerably as q decreases from unity. And consequently, there is an important wide n decrease with decreasing q. It was shown [15] that, for q → 0, e f can increase up to two times the e f of FD with n = 1/2 (n = 0) for all energy below (above) e f . Another common character of NSM and IS distributions is that, due to the energy cutoff with q < 1, there are few electrons above e f at low temperature (large β).
Now let see the distributions of correlated electrons. From some results of experiments and numerical simulations based on low dimension Kondo lattice models (KLM) [5][6][7][8], we notice two important effects of correlations. First, the wide decrease of n and the sharp n drop at e f much larger than that without correlation have indeed been observed for very strongly correlated electrons (with the coupling parameter J ≥ 4) [6,8] (see Figure 3 and 4). So it is possible to describe correlated electron systems in strong coupling regime by IS fermion distribution with q very smaller than unity (close to zero) [15]. On the other hand, another effect of correlation, in the weak coupling regime, is the flattening of n drop at e f [5][6][7][8]. That is that the correlation, even at low temperature, drives electrons above e f so that the n discontinuity becomes less and less sharp as the correlation increases. Curiously, this flattening of n discontinuity at e f , though confirmed by experimental results [5], is completely absent in the fermion distributions given by all the statistics, extensive (FES) or nonextensive (NSM, IS and KS), mentioned above.
Fermi surface plays an essential role in the physical properties of metals and metal compounds. So it is of great interest for generalized statistical theories to take into account the essential physics relevant to Fermi energy. In what follows, we will show that it is possible to cover the observed weak correlation effects upon e f within an additive generalization of BGS based on the IS principles. The reader can find a complete review of IS theory in references [11][12][13][14][15].

III. ADDITIVE INCOMPLETE STATISTICS
The concept of IS is inspired by incomplete probability distribution (i.e. Trρ = Q = 1) [25] as well as the theoretical difficulties one encounters with Trρ q = 1 in NSM [10,13]. The basic assumption of IS is that our knowledge about physical systems is in general incomplete due to unknown interactions or their effects which can not be studied explicitly. In this case, there are always missing informations about the physical systems. So probability distribution is in general inexact and incomplete and can not sum to one. One should write Tr(ρ/Q) = TrF (ρ) = 1 where F is certain function to be determined. In the case of complete or approximately complete distribution (such as BGS), F is identity function. In my previous paper, in order to keep Tsallis framework of NSM, I proposed F (ρ) = ρ q so that where q is Tsallis entropy index [11][12][13]. Since ρ < 1, we have to set q ∈ [0, ∞]. When q = 1, ρ becomes complete, which implies a complete or quasi-complete knowledge about the system of interest. The nonextensive IS formalism [11,13] on the basis of Tsallis entropy yields Tsallis q-exponential distributions and the fermion distribution [14,15] discussed above. An successful application of Eq.(1) to the study of some power laws on the basis of Rényi entropy [25] can be found in reference [26]. One can even find a plausible justification of Eq.(1) in a work of Tsallis on the probability distribution on some simple fractal supports like Koch or Cantor sets [27]. Indeed, the phase space of the systems including complex interactions is in general chaotic or fractal structure. In these structures, exact probability distribution is impossible due to the fact that, first, the structures are not differentiable and integrable, second, as mentioned above, some state points in the structures are not accessible. So the treatment of the incomplete distributions, not in the fractal space of d f dimension, but in a differentiable and integrable d-dimension space, is inevitable. This consideration yields in a natural way the incomplete normalization Eq.(1) [15]. Now we suppose that the system of interest has additive information and physical quantities (e.g. H = i H i where H is the hamiltonian of the compound system and H i that of the i th subsystem). This case may happen if there are only weak and short range interactions, so that the Hartley formula ln(1/ρ) for information measure should apply. The entropy of the system can be definedà la Shannon as follows [12] : S = kTrρ q ln(1/ρ). (2) S obviously becomes Gibbs-Shannon entropy (S GS ) when q = 1, which identifies k to Boltzmann constant. It is straightforward to see that S verifies all the properties of S GS . For microcanonical ensemble (ρ q = 1 w ), we have S = k q lnw which decreases with increasing q value. In general, ∆S = S − S GS < 0 (or > 0) if q > 1 (or q < 0). For canonical ensemble, we postulate where H is the Hamiltonian and U the internal energy of the system. The maximum entropy of S with Eq. (1) and (3) as constraints leads to : with Z = (Tre −qβH ) 1/q . The Lagrange parameter β can be determined by ∂Sq ∂Uq = kβ = 1 T where T is the absolute temperature. It is easy to see that all thermodynamic relations in this theory are identical to those in BGS.
We indicate in passing that this generalization of BGS is obviously the consequence of the "power-normalization" Eq.(1). Other useful normalizations, if any, of incomplete distributions may lead to different generalization of BGS. For example, if we use Hartley formula as information measure and define expectation value for additive entropy and energy with F (p i ) satisfying ∂ ln F (x) ∂x = 1/x α−ln x−(− ln Zx) 1/γ where γ is an empirical parameter, maximum entropy will lead to the famous stretched exponential distribution [28] p i = 1 Z e −(βEi) γ where E i is positive energy of the system at state i, α and β the Lagrange multipliers related respectively to normalization and energy constraint x as in BGS.

IV. ADDITIVE IS FERMION DISTRIBUTION
For grand canonical ensemble, the same machinery as above leads to with Z = [Tre −qβ(H−µN ) ] 1/q . Supposing U = j n j e j and N = j n j in accordance with the additivity assumption, the average occupation number n of one-particle state of energy e can be straightforwardly calculated [12] : n = 1 e q(e−e f )/kT + 1 (6) which recovers FD distribution for q = 1. The Fermi energy can be calculated in the standard way. For 2-D systems, e.g., we have e f = e f0 + k q T ln(1 − e −e f /kqT ). In Figure 2 is plotted the T -dependence of e f for different q values for 1-D fermion systems. We see that e f considerably changes with decreasing q. From Eq. (6), we see that q-dependence of e f at given T will be similar to the T -dependence. The effect of q on other physical quantities (specific heat, electrical conductivity, susceptibility, effective mass etc.) is discussed in [12]. The n distribution around the Fermi momentum k f can be estimated from Eq.(6) using e =h 2 k 2 /2m. We get : which is similar to the analog given by a Monte-Carlo calculation based on a tight-binding Kondo lattice (KL) [7]. A comparison with [7] leads to Z = qhk f 2 , where Z is the quasiparticle weight of photoemission spectrum. Note that Z is k f -dependent here. Eq.(7) tells us that decreasing q yields a flattening of n drop at e f .
In Figure 3 and 4, we compare the momentum distribution given by Eq.(6) for T =50 K to some numerical results of the momentum distribution of 1-D correlated electrons [6,8]. We see that, for about J < 1 and q > 0.003, KLM calculations can be well reproduced by Eq. (6). But for smaller q or stronger coupling with J > 1, a long tail in the KLM distributions begins to develop at high energy and can not be reproduced by Eq. (6). At the same time, a new Fermi surface at k f +π/2 starts to appear and a sharp n drop (energy cutoff) takes place at the new Fermi momentum. All these strong correlation effects turn out to be completely absent within the present IS Fermion distribution. It is clear that this discrepancy marks the limit of this formalism and perhaps implies that nonextensivity is no more negligible in very strong-coupling case. It is worth noticing that the strong correlation effects, absent in additive IS fermion distribution, are just what we noted in nonextensive IS fermion distribution [15] plotted in Figure 1. This suggests that a combination of these two IS generalizations of BGS, representing respectively two different aspect of correlation, may be an interesting approach leading to more generally valid statistical mechanics suitable for correlated particle systems.

V. CONCLUSION
We have compared several important generalizations of FD distribution to numerical and experimental results for correlated electron systems. It is found that the generalized fermion distributions based on incomplete information can be useful for describing these systems. It is shown that the extensive IS fermion distribution gives a good description of correlated electrons in weak coupling regime. On the other hand, it fails to describe strongly correlated conduction electrons and localized f -electrons which are possible to be described by the nonextensive IS fermion distribution. The latter, though powerless for the description of weakly correlated electrons, shows similar behavior as the heavy electrons in strong coupling regime : strong increase of e f and n-cutoff accompanied by a general decrease in n at all energy up to e f . In a current work, we are trying to combine these two partially valid generalizations. Further results will be presented in a future paper. . The density of electrons is chosen to give k f 0 = 0.25π in the first Brillouin zone. We note that IS distribution reproduces well the numerical results for about J < 1. When coupling is stronger, a long tail in the KLM distributions begins to develop at high energy. At the same time, a new Fermi surface at k = k f 0 + π/2 = 0.75π starts to appear and a sharp n drop takes place at the new Fermi momentum. At J = 4, KLM distribution (x-marks) is very different from that of IS (e.g. q = 0.0011). The solid line fitting better the J = 4 KLM distribution is given by the IS version of fractional exclusion distribution (1/n − α) α (1/n − α + 1) 1−α = e qβ(e−e f ) [19,20] or n =