ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 9, с. 1122-1137
ЯДРА
FERMION CONDENSATE GENERATES A NEW STATE OF MATTER
BY MAKING FLAT BANDS
© 2014 V. R. Shaginyan1),2)*, K. G. Popov3), V. A. Khodel4),5)
Received July 24, 2013
This short review paper is devoted to 90th anniversary of S.T. Belyaev birthday. Belyaev's ideas associated with the condensate state in Bose interacting systems have stimulated intensive studies of the possible manifestation of such a condensation in Fermi systems. In many Fermi systems and compounds at zero temperature a phase transition happens that leads to a quite specific state called fermion condensation. As a signal of such a fermion condensation quantum phase transition (FCQPT) serves unlimited increase of the effective mass of quasiparticles that determines the excitation spectrum and creates flat bands. We show that the class of Fermi liquids with the fermion condensate forms a new state of matter. We discuss the phase diagrams and the physical properties of systems located near that phase transition. A common and essential feature of such systems is quasiparticles different from those suggested by L.D. Landau by crucial dependence of their effective mass on temperature, external magnetic field, pressure, etc. It is demonstrated that a huge amount of experimental data collected on different compounds suggest that they, starting from some temperature and down, form the new state of matter, and are governed by the fermion condensation. Our discussion shows that the theory of fermion condensation develops completely good description of the NFL behavior of strongly correlated Fermi systems. Moreover, the fermion condensation can be considered as the universal reason for the NFL behavior observed in various HF metals, liquids, compounds with quantum spin liquids, and quasicrystals. We show that these systems exhibit universal scaling behavior of their thermodynamic properties. Therefore, the quantum critical physics of different strongly correlated compounds is universal, and emerges regardless of the underlying microscopic details of the compounds. This uniform behavior, governed by the universal quantum critical physics, allows us to view it as the main characteristic of the new state of matter.
DOI: 10.7868/S0044002714090086
1. INTRODUCTION
S.T. Belyaev's contribution to theoretical physics is very impressive. In 1958 S.T. Belyaev has published his classical works on the theory of nonideal Bose-gas. In these works, S.T. Belyaev demonstrates that peculiarity of Bose systems comes from a microscopic number of particles in the condensed state with the momentum p = 0 [1, 2]. To deal with such a system, he suggests splitting the system into two parts or subsystems, the condensate with p = 0 and the rest with p > 0. It turns out that it is the condensate that creates the "flavor" of Bose liquid, generating its vivid properties. One may try to figure out if physics like that of Bose systems could be represented in Fermi systems. Belyaev's daring ideas of the two macroscopic parts in Bose systems are adopted by a theory
'■'Petersburg Nuclear Physics Institute, Gatchina, Russia.
2) Clark Atlanta University, Atlanta, USA.
3)Komi Science Center, Ural Division, RAS, Syktyvkar.
4)Russian Research Centre Kurchatov Institute, Moscow.
5)McDonnell Center for the Space Sciences and Department of Physics, Washington University, USA.
E-mail: vrshag@thd.pnpi.spb.ru
of fermion condensate that permits to construct the new class of strongly correlated Fermi liquids with the fermion condensate (FC) [3—7], which quasiparticle system is also composed of two parts. One of them is represented by FC located at the chemical potential ¡, and giving rise to the spiky density of states (DOS) like that with p = 0 of Bose system. Figure 1 a shows DOS of a Fermi liquid with FC located at the momentum pi < p < pf and energy e = ¡.In contrast to the condensate of a Bose system occupying the p = = 0 state, quasiparticles of FC with the energy e = i must be spread out over the interval pi < p < pf.
The quasiparticle distribution function n(p) of Fermi system with FC is determined by the ordinary equation for a minimum of the Landau functional E [3—5]. In contrast to common functionals of the number density x [8, 9], the Landau functional of the ground state energy E becomes the exact functional of the occupation numbers n. In case of homogeneous system a common functional becomes a function of x = Y1P n(p), while E remains a functional, E =
О Q
Fermion condensate
8 = Ц
Pi < P < Pf
E
ад
<D
Ö
pq
Г = 0
Fermion condensate
Laudau Fermi liquid
«o( P)
Pi Pf Pf
Fig. 1. (a) Schematic plot of the density of states (DOS) of quasiparticles versus energy E at the momentum pi < p < pf of a Fermi liquid with FC. (b) Schematic plot of two-component Fermi liquid at T = 0 with FC. The system is separated into two parts shown by the arrows: The first part is a Landau—Fermi liquid with the quasiparticle distribution function no (p < pi) = 1, and no (p > pf) = 0; the second one is FC with 0 < no (pi < p < pf) < 1 and the single-particle spectrum e(pi < p < pf) = M. The Fermi momentum pF satisfies the condition pi < pF < pf.
b
a
= E(n(p)) [4, 5],
fffcM =*(,,)=,.; if: 0 < J?.(p) < 1. (1,
Equation (1) represents an ordinary one to search the minimum of functional E. In the case of Bose system the equation SE/Sn(p) = ¡i describes a common instance. In the case of Fermi systems such an equation, generally speaking, is not correct. Thus, it is the binding constraint 0 < n(p) < 1, taking place over some region p^ < p < pf, that makes Eq. (1) applicable for Fermi systems. Because of the binding constraint, Fermi quasiparticles in the region pi < < p < pf can behave as Bose one, occupying the same energy level e = ¡, and Eq. (1) yields the quasiparticle distribution function n0(p) that minimizes the ground-state energy E. A possible solution n0 (p) of Eq. (1) and the corresponding single-particle spectrum e(p) are depicted in Fig. 1b. As seen from Fig. 1b, n0(p) differs from the step function in the interval pi < p < pf, where 0 < n0 (p) < 1, and coincides with the step function outside this interval. Thus, the Fermi surface at p = pF transforms into the Fermi volume at pi < p < pf suggesting that the band is absolutely "flat" within this interval, giving rise to the spiky DOS. The existence of such flat bands formed by inter-particle interaction has been predicted for the first time in [3]. Quasiparticles with momenta within the interval (pi < p < pf) have the same single-particle energies equal to the chemical potential i and form FC, while the distribution n0(p) describes the new state of the Fermi liquid with FC, and the Fermi system is split up into two parts: a Landau—Fermi liquid (LFL) and the FC part, as it is shown in Fig. 1b [3—7].
In contrast to the Landau, marginal, or Luttinger— Fermi liquids, which exhibit the same topological structure of the Green's function, in systems with FC, where the Fermi surface spreads into the Fermi volume, the Green's function belongs to a different topological class. The topological class of the Fermi liquid is characterized by the invariant [6, 7]
N = tr (f p)diG~1(iuj1 p), (2)
C
where "tr" denotes the trace over the spin indices of the Green's function and the integral is taken along an arbitrary contour C encircling the singularity of the Green's function. The invariant N in (2) takes integer values even when the singularity is not of the pole type, cannot vary continuously, and is conserved in a transition from LFL to marginal liquids and under small perturbations of the Green's function. As shown by Volovik [6, 7], the situation is quite different for systems with FC, where the invariant N becomes a half-integer and the system with FC transforms into an entirely new class of Fermi liquids with its own topological structure.
In contrast to Bose liquid, which entropy S — 0 at temperature T — 0, a Fermi liquid with FC possesses finite entropy S0 at zero temperature [5, 10]. Indeed, as it is seen from Fig. 1b, at T = 0 the ground state of a system with a flat band is degenerate, and the occupation numbers n0 (p) of single-particle states belonging to the flat band are continuous functions of momentum p, in contrast to discrete standard LFL values 0 and 1. Such behavior of the occupation numbers leads to a T-independent entropy term S0 = = S(T — 0,n = n0) with the entropy given by
S(n) = ^[n(p) lnn(p) + (1 - n(p)) x (3)
P
x ln(1 — n(p))].
Since the state of a system with FC is highly degenerate, FC serves as a stimulator of phase transitions that could lift the degeneracy of the spectrum and make S0 vanish in accordance with the Nernst theorem. For instance, FC can excite the formation of spin density waves, antiferromagnetic state and ferromagnetic state, etc., thus strongly stimulating the competition between phase transitions eliminating the degeneracy. The presence of FC facilitates a transition to the superconducting state, because both phases have the same order parameter [3, 5]. Thus, in contrast to Bose systems with Bose condensate, entropy of which at lowering temperatures S ^ 0, the So peculiarity of Fermi systems with FC incites to the emerging of great diversity of states. Being generated by the same driving motive — S0 — these, as we shall see, exhibit a universal behavior, and form a new state of matter demonstrated by many compounds.
In this paper we briefly review the theory of FC that permits to describe a tremendously broad variety of experimental results in different systems. We assume that these systems are located near the fermion condensation quantum phase transition (FCQPT), leading to the emergence of FC [3, 5]. The rest of the paper is organized as follows. In Section 2, we examine a scaling behavior of the effective mass and heavy fermion (HF) metals based on the extended quasiparticle paradigm that is employed to renovate the Landau quasip
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