The appearance of the fermion condensation, which can be compared to the Bose-Einstein condensation, in different Fermi liquids is considered; its properties are discussed; and a large amount of experimental evidence in favor of the existence of the fermion condensate (FC) is presented. We show that the appearance of FC is a signature of the fermion condensation quantum phase transition (FCQPT), which separates the regions of normal and strongly correlated liquids. Beyond the FCQPT point, the quasiparticle system is divided into two subsystems, one containing normal quasiparticles and the other, FC, localized at the Fermi level. In the superconducting state, the quasiparticle dispersion in systems with FC can be represented by two straight lines, characterized by effective masses M FC * and M L * and intersecting near the binding energy E 0, which is of the order of the superconducting gap. The same quasiparticle picture and the energy scale E 0 persist in the normal state. We demonstrate that fermion systems with FC have features of a “quantum protectorate” and show that strongly correlated systems with FC, which exhibit large deviations from the Landau Fermi liquid behavior, can be driven into the Landau Fermi liquid by applying a small magnetic field B at low temperatures. Thus, the essence of strongly correlated electron liquids can be controlled by weak magnetic fields. A reentrance into the strongly correlated regime is observed if the magnetic field B decreases to zero, while the effective mass M* diverges as $$M^ * \propto {1 \mathord{\left/ {\vphantom {1 {\sqrt B }}} \right. \kern-\nulldelimiterspace} {\sqrt B }}$$ . The regime is restored at some temperature T∗∝B . The behavior of Fermi systems that approach FCQPT from the disordered phase is considered. This behavior can be viewed as a highly correlated one, because the effective mass is large and strongly depends on the density. We expect that FCQPT takes place in trapped Fermi gases and in low-density neutron matter, leading to stabilization of the matter by lowering its ground-state energy. When the system recedes from FCQPT, the effective mass becomes density independent and the system is suited perfectly to be conventional Landau Fermi liquid.
We discuss shape (Pomeranchuk) instabilities of the Fermi surface of a two-dimensional Fermi system using bosonization. We consider in detail the quantum critical behavior of the transition of a two-dimensional Fermi fluid to a nematic state which breaks spontaneously the rotational invariance of the Fermi liquid. We show that higher dimensional bosonization reproduces the quantum critical behavior expected from the Hertz-Millis analysis, and verify that this theory has dynamic critical exponent z=3. Going beyond this framework, we study the behavior of the fermion degrees of freedom directly, and show that at quantum criticality as well as in the quantum nematic phase (except along a set of measure zero of symmetry-dictated directions) the quasiparticles of the normal Fermi liquid are generally wiped out. Instead, they exhibit short-ranged spatial correlations that decay faster than any power law, with the law $\ensuremath{\mid}x{\ensuremath{\mid}}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\mathrm{const}\phantom{\rule{0.3em}{0ex}}\ensuremath{\mid}x{\ensuremath{\mid}}^{1∕3})$ and we verify explicitly the vanishing of the fermion residue utilizing this expression. In contrast, the fermion autocorrelation function has the behavior $\ensuremath{\mid}t{\ensuremath{\mid}}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\mathrm{const}\phantom{\rule{0.3em}{0ex}}\ensuremath{\mid}t{\ensuremath{\mid}}^{\ensuremath{-}2∕3})$. In this regime we also find that, at low frequency, the single-particle fermion density of states behaves as ${N}^{*}(\ensuremath{\omega})={N}^{*}(0)+B{\ensuremath{\omega}}^{2∕3}\mathrm{ln}\ensuremath{\omega}+\ensuremath{\cdots}$, where N∗(0) is larger than the free Fermi value, N(0), and B is a constant. These results confirm the non-Fermi liquid nature of both the quantum critical theory and of the nematic phase.
We argue that the gauge-fermion interaction in multiflavor quantum electrodynamics in (2+1) dimensions is responsible for non-Fermi-liquid behavior in the infrared, in the sense of leading to the existence of a nontrivial (quasi)fixed point that lies between the trivial fixed point (at infinite momenta) and the region where dynamical symmetry breaking and mass generation occurs. This quasifixed-point structure implies slowly varying, rather than fixed, couplings in the intermediate regime of momenta, a situation which resembles that of (four-dimensional) "walking technicolor" models of particle physics. The inclusion of wave-function renormalization yields marginal O(N1) corrections to the "bulk" non-Fermi-liquid behavior caused by the gauge inter-action in the limit of infinite flavor number. Such corrections lead to the appearance of modified critical exponents. In particular, at low temperatures there appear to be logarithmic scaling violations of the linear resistivity of the system of order O(N1). The connection with the anomalous normal-state properties of certain condensed-matter systems relevant for high-temperature superconductivity is briefly discussed. The relevance of the large (flavor) N expansion to the Fermi-liquid problem is emphasized. As a partial result of our analysis, we point out the absence of charge-density-wave instabilities from the effective low-energy theory, as a consequence of gauge invariance.