ЯДЕРНАЯ ФИЗИКА, 2011, том 74, № 9, с. 1266-1295
ЯДРА
ADAPTATION OF THE LANDAU-MIGDAL QUASIPARTICLE PATTERN TO STRONGLY CORRELATED FERMI SYSTEMS
©2011 V. A. Khodel1),2), J. W. Clark2), M. V. Zverev1),3)*
Received December 27, 2010
A quasiparticle pattern advanced in Landau's first article on Fermi-liquid theory is adapted to elucidate the properties of a class of strongly correlated Fermi systems characterized by a Lifshitz phase diagram featuring a quantum critical point (QCP) where the density of states diverges. The necessary condition for stability of the Landau Fermi-Liquid state is shown to break down in such systems, triggering a cascade of topological phase transitions that lead, without symmetry violation, to states with multi-connected Fermi surfaces. The end point of this evolution is found to be an exceptional state whose spectrum of single-particle excitations exhibits a completely flat portion at zero temperature. Analysis of the evolution of the temperature dependence of the single-particle spectrum yields results that provide a natural explanation of classical behavior of this class of Fermi systems in the QCP region.
1. PREAMBLE
Once upon a time at a traditional meeting of nestlings of the Migdal school, A. B. began his speech with words that stunned the audience.
Until recent times I was proud that I have never published an incorrect article. But is it actually a true cause for pride? Can an experienced mountaineer take pride in not having broken any ribs, or a professional motorcyclist, that his legs are still intact? The correct answer is no! It just means that these people may have run to the best of their abilities, but certainly not more.
This memorial article, dedicated to a great physicist and a great man, is devoted to a problem first discussed around 20 years ago [1—3]. The cited works considered the possibility of a breakdown of the conventional Landau—Migdal quasiparticle pattern [4— 7] of phenomena observed in Fermi liquids (FL), associated specifically with rearrangement of the T = = 0 Landau quasiparticle momentum distribution ™f(p) = - P).
Over the last decade, experimental studies of non-Fermi-liquid (NFL) behavior of strongly correlated systems have extended the frontiers of low-temperature condensed matter physics [8—15]. During the same period, a number of theorists have engaged in efforts to extend the frontiers of FL
^Russian Research Centre Kurchatov Institute, Moscow.
2)McDonnell Center for the Space Sciences and Department
of Physics, Washington University, USA.
3)Moscow Institute of Physics and Technology, Russia.
E-mail: zverev@mbslab.kiae.ru
theory with the aim of explaining this anomalous behavior [16-28].
There is a famous Migdal Correspondence Criterion for judging new theories, which boils down to this:
First and foremost, the proposed theory must be able to match results of previous, well-tested theoretical descriptions. It is only of secondary importance that it matches the relevant experimental data. Why so? Because as a rule, the experiments of burning interest were performed yesterday, and therefore the results obtained may be flawed, whereas theoretical physics stands as body of knowledge created and honed by a multitude over three hundred years.
The extended quasiparticle picture reviewed and discussed here meets the Migdal Criterion. It reduces naturally to the standard FL picture when dealing with conventional Fermi liquids. At the same time, convincing experimental evidence supporting this extension of the Landau-Migdal vision, while present, remains scarce — possibly because of the short time the relevant experimental programs have been in operation [9, 14, 29, 30]. Even so, one cause for the lack of unambiguous correlations between available experimental data and corresponding theoretical results may be conceptual (or technical) errors made by the theorists in developing the new theory, errors that are yet to be exposed. In this case, we should not be too upset, since according to Migdal's provocative challenge to his assembled group, such errors tell us that the struggling theorists, being professionals, did, after all, run beyond the breaking point of their abilities.
2. ROUTES TO BREAKDOWN
OF STANDARD FERMI-LIQUID THEORY
Any theory has own limits of applicability, and FL theory is no exception. Conventionally these limits are imputed to violation of Pomeranchuk stability conditions [31]. Such violation is associated with second-order phase transitions that are accompanied with jumps of the specific heat C(T) and cusps of the spin susceptibility x(T). Upon approach to the transition point, spontaneous creation and enhancement of fluctuations suppresses the value of the z factor that determines the quasiparticle weight in the single-particle state at the Fermi surface. At the transition point, z vanishes, signaling a breakdown of the FL quasiparticle picture [32, 33].
A different domain where predictions of FL theory prove to be fallacious has been discovered and explored during the last decade. This is the regime of the so-called quantum critical point (QCP) revealed in experimental studies of two-dimensional (2D) liquid 3He and heavy-fermion metals. In this domain, the specific heat C(T) and spin susceptibility x(T), both proportional to the density of states N(T) (and hence to the effective mass M*), are found to diverge as the temperature T goes to zero [8—15].
The presence of a QCP is the hallmark that distinguishes strongly correlated Fermi systems from those with weak or moderate correlations. In this respect, three-dimensional (3D) liquid 3He, for which M* remains finite at any density, belongs to the class of systems with moderate correlations. Its 2D counterpart, on the other hand, is assigned to the class of strongly correlated Fermi systems based on evidence for the existence of a QCP provided by experimental studies of dense 3He films [8—12].
It is well to emphasize that the divergence of the specific heat C(T) that is observed in the QCP region has little in common with the jumps in C(T) inherent in second-order phase transitions. Notwithstanding this fact, theoretical explanations of the failure of FL theory in the QCP region have commonly linked its apparent breakdown with attendant second-order phase transitions [ 13, 32—35]. However, as they occur in the QCP region, such transitions are found to be quite atypical. In most cases, even the structure of the order parameters remains unknown, and the properties of states beyond the QCP defy explanation within the standard scaling theory of second-order phase transitions. One is led to conclude that the viability of such transitions as triggers of the observed NFL behavior is problematic.
In this article, we pursue another explanation of NFL behavior in the QCP region, attributing the breakdown of standard FL theory to violation
of the necessary stability condition for the Landau state [16]. The salient feature of the proposed scenario is rearrangement of single-particle degrees of freedom driven by topological phase transitions (TPT's). This mechanism stands in direct contrast to the violation of sufficient Pomeranchuk stability conditions [31], which entail rearrangement of collective degrees of freedom in second-order phase transitions.
Since the topological scenario does not implicate the violation of any Pomeranchuk stability condition, the catastrophic suppression of the quasiparticle weight posited in the collective scenario for the QCP does not take place, and failure of standard FL theory in the QCP region must have another explanation. Indeed, the fundamental FL formula
e(p) = Vf(p - Pf) = Pf(p - Pf)/M* (1)
for the single-particle spectrum e(p) measured from the Fermi surface, becomes powerless when the effective mass M * diverges. If such a divergence is present, additional terms of the Taylor expansion of e(p) must be added to the right side of Eq. (1) (see [22]), giving rise to a change of sign of the Fermi velocity. This in its turn signals a violation of the necessary stability condition for the conventional Landau state (as seen in Section 5.2), resulting in a topological rearrangement of the Fermi surface that necessitates modification of the standard FL formalism.
3. REPRISE OF THE STANDARD FL QUASIPARTICLE PICTURE
To set the stage for discussion of the required modifications, we recall that the heart of the Landau— Migdal quasiparticle picture is the postulate that there exists a one-to-one correspondence between the totality of real, decaying single-particle excitations of the actual Fermi liquid and a system of immortal interacting quasiparticles. There are two facets of this correspondence. First, the number of quasiparticles is equal to the given number N of particles, a condition expressed as
N
Tr / n{p)dv = — =p, (2)
where n(p) is the quasiparticle momentum distribution, p is the density, Tr implies summation over spin and isospin variables, and du = dp/(2n)D is a volume element in a momentum space of dimension D. Second, all thermodynamic quantities, notably the ground state energy E and entropy S, are treated as functionals of the quasiparticle momentum distribution n(p). In particular, the entropy S of the real
system is given by the ideal-Fermi-gas combinatorial expression
S f
— = -Tr / [n(p)lnn(p) + + (1 - n(p)) ln(1 - n(p))]dv.
(3)
In homogeneous matter, a standard variational procedure based on Eq. (3) and involving restrictions that impose conservation of the particle number and energy, leads to the result [4]
n(p) =
1 + e
<p)/t
-1
(4)
While this relation resembles the corresponding Fermi—Dirac formula for the ideal Fermi gas, the quasiparticle energy e(p) measured from the chemical potential does not coincide with the bare single-particle energy e'p = p2/(2M) — ц. Instead, it is given by the variational derivative
e(p) = 6Q/6n(p) (5)
of the thermodynamic potential Q = E — ¡j,N.
It is significant that the
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