Pis'ma v ZhETF, vol. 101, iss. 6, pp. 448-454
© 2015 March 25
Occurrence of flat bands in strongly correlated Fermi systems and high-Tc superconductivity of electron-doped compounds
V. A. Khodela'b, J. W. Clarkb'c, K. G. PopoVi'e, y R. Shaginyanf'S1')
a National Research Centre "Kurchatov Institute 123182 Moscow, Russia b McDonnell Center for the Space Sciences and Department of Physics, Washington University, MO 63130 St. Louis,, USA c Centro de Ciéncias Matemáticas, Universidade de Madeira, 9000-390 Funchal, Madeira, Portugal dKomi Science Center UD of the RAS, 167982 Syktyvkar, Russia eDepartrnent of Physics, St.Petersburg State University, 199034 St. Petersburg, Russia
f Konstantinov Petersburg Nuclear Physics Institute of the RAS, National Research Centre "Kurchatov Institute",
188300 Gatchina, Russia
3 Clark Atlanta University, GA 30314 Atlanta, USA Submitted 9 February 2015
We consider a class of strongly correlated Fermi systems that exhibit an interaction-induced flat band pinned to the Fermi surface, and generalize the Landau strategy to accommodate a flat band and apply the more comprehensive theory to electron systems of solids. The non-Fermi-liquid behavior that emerges is compared with relevant experimental data on heavy-fermion metals and electron-doped high-Tc compounds. We elucidate how heavy-fermion metals have extremely low superconducting transition temperature Tc, its maximum reached in the heavy-fermion metal CeCoIns does not exceed 2.3 K, and explain the enhancement of Tc observed in high-Tc superconductors. We show that the coefficient Ai of the T-linear resistivity scales with Tc, in agreement with the experimental behavior uncovered in the electron-doped materials. We have also constructed schematic temperature-doping phase diagram of the copper oxide superconductor I^-zCe^CuCU and explained the doping dependence of its resistivity.
DOI: 10.7868/S0370274X15060119
An especially challenging task confronting present-day condensed matter theory is explication of the elusive origin of the non-Fermi-liquid (NFL) behavior observed in strongly correlated Fermi systems beyond a critical point where the low-temperature density of states N(T —> 0) diverges without breaking any symmetry inherent of the ground state. In homogeneous matter, the case to be addressed in this article, such a divergence is associated with the onset of a topological transition (TT) signaled by the emergence of an additional root p = pb of the condition [1, 2]
e(p,At;T = 0)=0 (1)
for vanishing of the single-particle energy measured from the chemical potential /x, otherwise satisfied only by p = pf, the Fermi momentum. Thus, if At is the critical coupling constant for onset of the posited TT, the curve e(p, At) must touch the axis p at the bifurcation
^e-mail: vrshag@thd.pnpi.spb.ru
momentum p = pb. Accordingly, at p —> pb the group velocity v(p) = de(p, Xt)/dp vanishes as \Je(p) to yield
N(T^0) <xT~1/2. (2)
In an original scenario advanced by I. M. Lifshitz more than fifty years ago [1] and recently applied to describe electron systems of heavy-fermion metals [3, 4], some TTs are assumed to occur in systems of noninteract-ing electrons moving in the external field of the crystal lattice. In such cl CclSG, the quasiparticle occupation numbers adhere to the discrete Fermi-liquid (FL) values n(p) = 0,1 beyond the Lifshitz topological transition (LTT) point, while the Fermi surface becomes multi-connected. Topological transitions may occur in systems of interacting fermions as well [5]. Within the FL approach, the feasibility of the Lifshitz ("bubble" or "pocket") scenario for the rearrangement of the Landau state was considered [6, 7] within the framework of the Landau equation [8]
de(p,T) pf,, ,,dn(p',T) j ,
-ej- = M+Jf( (3)
448 Письма в ЖЭТФ том 101 вып. 5-6 2015
for the single-particle spectrum e(p). Here M is the mass of a free fermion, dv = 2d3p/(2Tr)3 is the three-dimensional volume element,
n{p,T) = (i + ee^/T)-1 (4)
is quasiparticle momentum distribution, and /(p, p') is a phenomenological interaction function. Eq. (3) was derived by Landau [8] from the equality
J p n{p)d,v = M J ^ln(p)dv (5)
between the momentum of the system, moving with velocity Sv, its mass flow stemming from Galilean invariance. We note that Eq. (5) is also valid when the model of heavy-fermion liquid is applicable, as it is in our present case [9, 10]. Equation (3) is then obtained upon retaining only the leading terms in Sn(p) on both sides of Eq. (5) while invoking the FL relation
Se(p,n) = YJf(P,P')5n(p')- (6)
p'
As will be seen, Eq. (5) allows one to introduce and explore a different, interaction-induced type of rearrangement of the Landau state, often called fermion condensation and described more vividly as a swelling of the Fermi surface [11, 13, 12]. This phenomenon has its genesis in a proliferation of the number of roots of Eq. (1) to form a continuum. (For recent articles on this topic, see Refs. [9,10,14-20].)
The emergence of such a phase transition in homogeneous matter can be elucidated within the original Landau approach to FL theory, in which the ground-state energy E is postulated to be a functional of the quasiparticle momentum distribution n(p). At T = 0, the onset of fermion condensation in homogeneous matter is attributed to the occurrence of a nontrivial solution n*(p) of the variational condition [11]
SE(n, A)
——--/x = 0, pG[pi,pu J, (7)
on(p)
in a finite momentum interval surrounding p^, the chemical potential /x being determined from the Landau postulate that the quasiparticle and particle numbers coincide. It is Eq. (7) that describes a flat band pinned to the Fermi surface, here also referred to as the fermion condensate (FC). Outside the FC domain [pi,pu] the usual FL occupancies still apply: n*(p) = 1 at p < pi and n*(p) = 0 at p > Pu- However, the occupation numbers inside the FC, evaluated through Eq. (7), change continuously between 1 and 0 with increasing p. The volume occupied in momentum space by quasiparticles
with nonzero probability is no longer just p|/37r2 but is instead given by the relation [15]
Pi Pu 3
J dv +Jn,Xp)dv = j^. (8)
0 Pi
We now turn to the demonstration that Landau equation (3) is to be modified in dealing with Fermi systems harboring a FC. It is revealed by examination of analytic properties of solutions of this equation in systems where the interaction function /(p,p') has no singularities in momentum space. For this case, it has been established [7] that the solutions of Eq. (3) are in fact analytic functions of momentum p in the full momentum space. However, this property is lost if the system hosts a FC. Indeed, the left side of Eq. (7) is nothing but the quasiparticle energy measured from the Fermi surface, which vanishes identically in the FC domain. Evidently, the FC domain cannot occupy the full momentum space - a fact confirmed in analytically soluble models of fermion condensation. Thereby we arrive at the strong conclusion that the single-particle spectrum e(p, n*) of the problem with FC present must be a non-analytic function of momentum p, for if an any analytic function vanishes identically in some domain, it must vanish everywhere. This implies that in the case where the interaction function / has no singularities in momentum space, the FC solutions e(p, n*) cannot meet Eq. (3), and therefore in systems with a FC, Landau equation (3) must be modified to allow for such an eventuality. This modification can be made on the basis of Eq. (5) along the same lines as Eq. (3) was obtained in Landau theory. The profound difference is that the system harboring a FC is, in fact, a two-component system, as made quite evident in Eq. (8). Correspondingly, variations of the quasiparticle momentum distribution n(p) lead to the closed equation
P'P' G [PhPu] (9)
for determining n(p) = n*(p) inside the FC domain derived with accounting for the fact that outside the FC region, dn*(p)/dp = 0. In arriving at this relation, we have ensured consistency with the central feature of the FC-inhabited state that the group velocity vanishes identically in the interval \pi,pu], whose range is found with the aid of the requirement n*(p) < 1. The quasiparticle spectrum of the normal component obeys the Landau-type equation
and in effect p' € [pi,pu], derived in the same way as Eq. (3) from Eq. (5). Equation (9) can also be derived by differentiation of the basic equation (7) with respect to momentum p, then adopting the relation (6) between interaction-induced variations of relevant quantities, rewritten as 5e(p) = Seo + fSn(p), where Seo is the variation of the spectrum of noninteracting quasiparti-cles. In homogeneous matter, where deo/dp is simply p/M, we are led to Eq. (9). The special convenience of this route lies in the opportunity to generalize Eq. (9) for analysis of the FC phenomenon in electron systems of solids. In doing so, one needs to replace p/M by the corresponding derivative deo/dp, evaluated, say, within the tight-binding model, or within a more advanced microscopic description of the electron spectrum eo(p). A definite mathematical signature identifies those interacting many-fermion systems whose single-particle spectrum exhibits a flat portion. This is the topological charge (TC) of the corresponding ground state. For a system containing a FC, the TC must take a half-odd-integral value, whereas the TC of any unorthodox state featuring one or more Lifshitz pockets is always integral [12].
It is instructive to compare changes that occur in the fundamental equation (3) beyond the point of fermion condensation with those that occur at a second-order phase transition. In the latter case, some symmetry inherent in the Landau ground state is broken, and a corresponding order parameter comes into play (notably, the gap A(p) in superfluid FL's), dramatically rearranging Eq. (3) and both the key FL quantities, i.e., the momentum distribution n^(p) = 0(p
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