научная статья по теме THE LOW-TEMPERATURE COLLAPSE OF THE FERMI SURFACE AND PHASE TRANSITIONS IN CORRELATED FERMI SYSTEMS Физика

Текст научной статьи на тему «THE LOW-TEMPERATURE COLLAPSE OF THE FERMI SURFACE AND PHASE TRANSITIONS IN CORRELATED FERMI SYSTEMS»

Pis'ma v ZhETF, vol.94, iss.8, pp.697-703

© 2011 October 25

The low-temperature collapse of the Fermi surface and phase transitions in correlated Fermi systems

V. A. Khodel^

National Research Centre "Kurchatov Institute", 123182 Moscow, Russia

McDonnell Center for the Space Science and Department of Physics, Washington University, MO 63130 St.Louis, USA

Submitted 13 September 2011

A topological crossover, associated with the collapse of the Fermi surface in strongly correlated Fermi systems, is examined. It is demonstrated that in these systems, the temperature domain where standard Fermi liquid results hold dramatically narrows, because the Landau regime is replaced by a classical one. The impact of the collapse of the Fermi surface on pairing correlations is analyzed. In the domain of the Lifshitz phase diagram where the Fermi surface collapses, splitting of the BCS superconducting phase transition into two different ones of the same symmetry is shown to occur.

Introduction. For the past few years, the investigation of topological transitions in correlated Fermi systems that dates back to a pioneer work by I. M. Lifshitz, published in 1960 [1], has become one of hot topics in condensed matter physics [2-15]. These transitions are responsible for non-Fermi-liquid (NFL) behavior that manifests itself in singularities of thermodynamic characteristics of Fermi systems. E.g. in FL-theory, the spin susceptibility x(T) remains unchanged at T —> 0. Contrariwise, at the transition point, x(T —y 0) diverges as with the critical index a > 0, somewhat depending on the shape of the single-particle spectrum e(p) [16, 17, 11]. In case the function e(p) is measured from the chemical potential ¡j,, the topological rearrangement of the Landau state is associated with the change in the number of its zeroes [1, 18]. In conventional nonsuper-fluid homogeneous Fermi systems, whose Fermi surfaces are singly connected, equation

e(p) = 0 (1)

has the single real root, the Fermi momentum pp. In the original article, Lifshitz analyzed topological transitions, occurring in noninteracting electron gas of metals at high pressures. Scenarios for such transitions, entailed by interactions between quasiparticles, one of which is addressed in this article, have emerged thirty years later [19-23], (for a recent review, see Ref. [24]). NFL behavior of Fermi systems near one of the critical points of Eq. (1) associated with the divergence of the effective mass M* (this point is called the quantum critical point (QCP)), has been studied extensively in recent years [25, 26].

e-mail: vak0wuphys.wustl.edu

Usually Eq. (1) is analyzed at zero temperature. However, the spectrum e(p) depends on T, being a functional of the quasiparticle momentum distribution n(p,T) that has the standard Fermi-like form

n(p,T)= (l + e^^y1 , (2)

normalized in 3D by ordinary condition

Jn(p)dv = p =il (3)

with the volume element in 3D momentum space dv = = p2dp/ir2.

The key point of this article is that at T > 0 where topological transitions are replaced by crossovers, there is a different route of the topological rearrangement, associated with the collapse of the Fermi surface. The collapse occurs in case Eq. (1) has no real roots at all. A classical Maxwell reconstruction of the T = 0 ideal-Fermi-gas momentum distribution is the simplest example of such a topological rearrangement. In this case where e(p) = p2 ¡2M — n(T), the trajectory of the single root of Eq. (1), denoted further p1/2 due to relation n(Pi/2) = 1/2, stemming from Eq. (2) at e(p) = 0, is easily traced. At T —y 0, one has Pi/2(0) = pp. With the temperature rise, Pi/2(T) moves toward the origin, attaining it at temperature where p(T) changes sign. At higher T, real roots of Eq. (1) no longer exist, and the function e(p) becomes positive defined, the gap in the spectrum e(p) increasing with the further increase of T that entails rapid spread of the quasiparticle momentum distribution n(p, T).

In systems with weak and/or moderate correlations, such an option is of no interest, since temperature Tm,

at which the Maxwell-like crossover takes place, is comparable with the Fermi energy ep = pp/2M. However, with strengthening correlations, the effective mass M* increases, and correspondingly, the FL-spectrum e(p) = Pf(p — Pf)/M* becomes flatter and flatter. As a result, temperature Tm goes down that leads, in its turn, to the dramatic shrinkage of the temperature region where the standard FL behaviors: C(T) oc T, for the specific heat, and x(T) = const, for the spin susceptibility, hold. Importantly, at T ~ Tm -C Cp, the Landau-Migdal quasiparticle pattern remains applicable to evaluation of thermodynamic properties. This allows one to apply the quasiparticle formalism to the investigation of relevant problems, like splitting of the BCS superconducting phase transition into two ones, (see below), notwithstanding discrepancies between experimental data and standard FL-predictions.

Simple model of the collapse of the Fermi surface at low T. Within the quasiparticle picture, all the thermodynamic properties of correlated Fermi systems are evaluated in terms of the single-particle spectrum e(p). This spectrum can be calculated on the base of FL-equation, connecting e(p) with the FL quasiparticle momentum distribution (2) in terms of the first harmonic /1 of the interaction function /, taken as a phenomenological input. In the 3D case, this equation has the form:

dejp) dp

P_

M

J h(p,pi)

dn(pi dpi

-dvi

(4)

The interaction function /(p,pi) is known to coincide with a specific limit of the scattering amplitude T(pi,p2;q,w) where q p - Pi ^ 0,w 0;q/w 0. Generally F contains two different components. The first that prevails in the vicinity of second-order phase transitions changes rapidly near the Fermi surface. The second varies smoothly until momenta reach values much larger than pp. Neglecting irregular components, the scattering amplitude F can be written in the standard form F oc l/(ffl-1 — req2 ¡2) where a is the scattering length, and re, the effective range. This form has to be supplemented by an exchange term to yield

4-tt

T(p,pi,q) =

a/2

^areq2/2 l^are(p^pi+q)2/2.

(5)

To facilitate the analysis we expand this expression into the Taylor series. Retaining only two first terms in this expansion, inherent in an effective mass approximation, where

27ra r are i

(6)

/(P,Pi) = r(p,pi,q = 0) = -j¡f[1~ ^-(P - Pi)

the group velocity de(p)/dp is evaluated from Eq. (4) in the closed form:

de(p) dp

P

= 17 Í1 - 2^2rep)

M

(7)

The spectrum e(p) is then calculated straightforwardly. In doing so an immaterial constant, associated with the first term in Eq. (6) that contains the scattering length a, is absorbed into the chemical potential p to yield

e(p) =

2M*

M , „ 2

M, = 1 - 2Tra rep.

(8)

Upon accounting for a contribution a from long-wavelength spin fluctuations, relevant e.g. in the case of 3D liquid 3He discussed below, these formulas change. Nevertheless, as it was shown in microscopic calculations of the spectrum e(p), performed in Ref. [27], the effective mass approximation remains adequate, merely the effective mass is modified to

M M

—— = 1 — 2ita rep

(9)

with cr ~ 0.5. With these results, the temperature evolution of the single root pi/2 (T) turns out to be identical to that in ideal Fermi gas.

Temperature Tm, at which the chemical potential changes sign, is straightforwardly evaluated from the normalization condition (3) and formula (8) to yield

Tm

Pf

2M*(p) '

(10)

As correlations are strengthening, the ratio M* /M increases and consequently, temperature Tm goes down.

Unfortunately, in the vicinity of the QCP where the effective mass M* diverges, the effective mass approximation begins to fail, because other terms of the Taylor expansion come into play. For illustration, let us retain only the next term oc (pi — P2)4- In this case, the group velocity de/dp acquires the form

MP) P . P3

with v > 0 and a, being dimensionless quantities, whose values are supposed to be small. As long as a keeps positive sign, the bifurcation momentum pj equals 0. At the critical point where a vanishes, one has de(p pp)/dp = pp/M* = vpp/M, so that M/M* = v -C 1. Repeating the same manipulations that give rise to Eq. (10), one then finds Tm oc vp^/M, and therefore Tm/cp -C 1.

P

a

When a changes sign, the single-particle spectrum, found with the aid of simple integration of Eq. (11), can be conveniently rewritten in the form:

e(x) = vm (x

■xbf-n(T),

(12)

where x = p2/p| and Xf, = \o\/v. Again the effective mass M* turns out to be enhanced: M*/M ~ v~x.

The single root pf of T = 0 equation (1) persists as long as Xf, < 1/2, otherwise, this equation acquires the second root p<(0) = pp(2xi, — 1), and the Landau state is rearranged. The single-particle states with p< < p < pf, where pf is a new Fermi momentum, determined by the normalization condition (3), remain filled, while states with p < p< and p > pf turn out to be empty. With the rise of temperature, both the roots of Eq. (1) move to meet each other at the bifurcation momentum pj = pp-i/xt- Critical temperature Tm is straightforwardly evaluated from Eq. (3). In the relevant case Xf, ~ 1, one finds

TM ~ i'p|/M « 4,

(13)

implying again that the elevation of temperature results in the rapid shrinkage of the FL-domain.

Discussion. All the analytically solved models, addressed above, have a common feature: strengthening interactions results in the shrinkage of the temperature domain where standard FL results hold. This conclusion remains valid even if flattening of the single-particle spectrum e(p) takes place only for occupied states, the situation, inherent in many strongly correlated systems where the spectrum e(p) has a more complicated structure, (see e.g. Ref. [4]). The shrinkage becomes especially pronounced beyond the QCP [22, 23, 4], where as mentioned above, the Fermi surface becomes multi-connected. For illustr

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