Pis'ma v ZhETF, vol.94, iss. 1, pp. 73-80
© 2011 July 10
Topological crossovers near a quantum critical point
V. A. Khodel+-1), J. W. Clark*, M. V. Zverev+t
+ Russian Research Centre Kurchatov Institute, 123182 Moscow,Russia * McDonnell Center for the Space Science and Department of Physics, Washington University, St.Louis, MO 63130, USA t Moscow Institute of Physics and Technology, 123098 Moscow, Russia Submitted 19 May 2011
We study the temperature evolution of the single-particle spectrum e(p) and quasiparticle momentum distribution n(p) of homogeneous strongly correlated Fermi systems beyond a point where the necessary condition for stability of the Landau state is violated, and the Fermi surface becomes multi-connected by virtue of a topological crossover. Attention is focused on the different non-Fermi-liquid temperature regimes experienced by a phase exhibiting a single additional hole pocket compared with the conventional Landau state. A critical experiment is proposed to elucidate the origin of NFL behavior in dense films of liquid 3He.
The study of non-Fermi-liquid (NFL) behavior of strongly correlated Fermi systems in the regime of a quantum critical point (QCP) is currently one of the most active and challenging areas of condensed matter physics [1, 2]. As a rule, such behavior is attributed to second-order phase transitions, and the QCP is identified with the end point of a corresponding line of transition temperatures, denoted by Tpf(H) in the prototype in which an external magnetic field H is the control parameter. In this case, NFL behavior is triggered by critical antiferro- or ferromagnetic fluctuations, which lead to violation of respective Pomeranchuk stability conditions (PSC). Ensuing NFL phenomena are presumably explained either within the Hertz-Millis theory [3, 4] or, in heavy-fermion metals, within a Kondo breakdown model [1, 2, 5, 6].
However, the widely promulgated fluctuation scenario is inconsistent with experimental data on a number of strongly correlated Fermi systems exhibiting NFL behavior:
(i) In dense 3He films where the emergent NFL behavior has been documented, experiment [7-10] has not identified any related second-order phase transition.
(ii) In several heavy-fermion metals [11, 12], concurrent divergence of the Sommerfeld ratio "f(T) = C(T)/T and the magnetic susceptibility x(T) is observed at a point that is separated by an intervening NFL-phase from termination points of any second-order phase transitions.
(iii) In many instances of well-pronounced NFL behavior, the order parameters required to specify associated second-order phase transitions are still elusive, casting further doubt on the fluctuation scenarios.
e-mail: vak0wuphys.wustl.edu
(iv) In external magnetic fields, thermodynamic properties demonstrate scaling behavior governed specifically by the ratio HfH/T where is the magnetic moment of constituent fermions.
These NFL-phenomena can be understood when one recognizes that standard FL-theory possesses its own quantum critical point, in the vicinity of which it fails. At this point, the necessary stability condition (NSC) for the T = 0 Landau state is violated [13-16], as opposed to violation of some PSC at a conventional QCP.
The NSC-states that an arbitrary admissible variation 5n(p) from the FL quasiparticle momentum distribution np(p) = 9(pf — p), while conserving particle number, must produce a positive change of the ground-state energy E0, i.e.,
5E0 = J e(p;nF(p))Sn(p)dv > 0. (1)
Here, e(p; np) denotes the spectrum of single-particle excitations measured from the chemical potential ¡j,(T = 0) and evaluated for the initial Landau state specified by the quasiparticle occupancy np(p). The reduction in energy due to breakdown of the NSC, which involves contributions linear in 5n, is clearly larger than that due to violation of any PSC, which involves bilinear combinations of 5n. We must conclude that any associated fluctuation scenario is irrelevant to the different type of QCP associated with violation of the NSC, which we shall call a Fermi-liquid QCP.
Violation of the NSC (1) is unambiguously linked to a change of the number of roots of equation
e(p,nF) = 0. (2)
In standard Fermi liquids, this equation has a single root at the Fermi momentum pp, and in that case the signs of
e(p) and 5n(p) coincide, ensuring satisfaction of the NSC (1). However, consideration of the full Lifshitz phase diagram anticipates the emergence of additional roots of Eq. (2). For example, such roots appear at a critical density p« where the function e(p,p«) attains either a maximum, with a bifurcation point pi, < pf, or a minimum, with pj > pf, so that e(p pb,p<>) oc (p — pj)2, (see the upper two panels of Fig. 1). Thus, vanishing
Fig. 1. Three scenarios of emergent bifurcation in Eq. (2): Pb < Pf (top left panel), pb > Pf (top right panel), Pb = Pf (bottom panel)
of e(pb,p<,) is always accompanied by vanishing of the group velocity v(pb,p<>) = (de(p,po)/dp)Pb. Beyond the critical density p«, the NSC-fails to hold, since e(p, rip; p) and 5n(p) have opposite signs close to pj.
As indicated in the lower panel of Fig. 1, the condition (1) is also violated at a critical density p^ where the effective mass M* (p) diverges. In this case, standard manipulations based on the Landau relation connecting the single-particle spectrum and the quasiparticle momentum distribution (see Eq. (9) below), yield
vF(p)
M
M*(p)
= 1
^i(p),
(3)
where vg = pF/M and F°(p) = /i(pf,Pf; p)pfM/it2 is the dimensionless first harmonic of the Landau interaction function, normalized with the density of states N0 = pFM/-K2 of the ideal Fermi gas. Evidently, F^(p) is a smooth function of the density p, and F°(p) = 3 at p = poo. Then beyond the critical point, one has F1(p) > 3, and the Fermi velocity i^f(p) becomes negative. This behavior conflicts with the fluctuation scenario for the QCP, in which such a sign change is impossible.
To summarize, we infer that at any point where the NSC is violated, the density of states, given by
N(T) =
T
J n(€)[ 1
n(e)]|de,
diverges at T —t 0 due to vanishing of the group velocity de(p)/dp. One has [17, 15]
N(T 0, Poo) cx T-2/3, N(T 0, p.) cx T-1'2. (5)
The difference in critical indexes is associated with the fact that dp/de oc e-2/3 at the critical density whereas dp/de oc |^| 1 /2 at the critical density p«.
Significantly, the Sommerfeld-Wilson ratio iZsw = x{T)/"f{T) cannot diverge at these points. Indeed, the density of states N(T) cancels out in the ratio Rsw, while the Stoner factor entering x(T) maintains a finite value, since, as we have seen, the PSC and NSC cannot fail at the same point. This conclusion is in agreement with experimental data [7, 18, 19] on dense films of liquid 3He, the two-dimensional electron gas of MOSFETs, and the majority of heavy-fermion metals.
Since no symmetry is violated at a Fermi liquid QCP, and hence no hidden order parameters are involved, the transition ensuing from the violation of the NSC (1) is topological in character [20, 21]. Beyond the bifurcation point, Eq. (2) usually has two additional roots pi and P2 situated near each other (however, cf. Refs. [22-24]). It is for variations 5n(p) involving momenta pi < p < P2, at which 5n(p) and e(p) have opposite signs, that the NSC (1) breaks down.
The analysis of topological rearrangements triggered by the interaction between quasiparticles began twenty years ago [22] with important subsequent developments reported in Refs. [17,23-32]. In this article, we address the Fermi-liquid QCP in homogeneous matter and focus on the case where the new roots pi and P2 emerge near the Fermi momentum pp. The physics of this phenomenon is captured if we keep the three first terms,
I \ ( , Vl , V2 2A
e(x) = ppx I vf + -z-x + —x I
\ ¿i u '
/x t?2 2
v(x) = vp + v\x + —x ,
(6)
in the Taylor expansions of the spectrum e(x) and its group velocity v(x), where x = (p — Pf)/pf- To some extent, this approach is reminiscent of that employed by Landau in his theory of second-order phase transitions. In an ideal Fermi gas, v$ = v\ = Up = (2Me^)1/2. The case vi = 0, V2 >0 was considered in Ref. [17]. Here we assume that v\ > 0, V2 > 0, and V1/V2 -C 1, the situation addressed in the numerical calculations of Ref. [15].
To find the bifurcation momentum pj = pf(1 + %b) one must solve the set of equations e(p) = 0 and v(p) = 0, i.e.
(4)
, Vl , V2 2 n VF + YX>> + YXb =
v2 2
vf + vi Xb + —xb = u.
(7)
V
F
This system has the solution Xf, = —Svi/2v2 provided the critical condition
8v2vf(p)
3v?
= 1
(8)
is met. Thus in the case v\ ^ 0, the critical Fermi velocity i>f is still positive, and therefore the Landau state becomes unstable before the system reaches the point at which the effective mass diverges - as was first discovered and discussed in Refs. [25].
The prerequisite Xb ■C 1 for applicability of the expansion (6) is satisfied provided vi/v2 -C 1, implying that the critical Fermi velocity is small: «f = = 3vf /8t?2 -C t>p. Given this situation, upon accounting for the dependence of «f on the temperature T and control parameters such as the external magnetic field H that do not change the form of Eq. (8), one can establish a critical line T = To(H) separating phases with different topological structure.
Evaluation of relevant T- and If-dependent corrections to the Fermi velocity «f is based on the Landau equation [33, 34] for the single-particle spectrum e(p), which in 3D has the form
de(p) p 1 f dn(p-
■ dv 1
(9)
dp m 3 J dpl
with dv = p2dp/n2. This relation provides a nonlinear integral equation for self-consistent determination of e(p,T,H) and the momentum distribution
-l
n(p,T,H)= [l
pe(p,T,H)/T
(10)
with the Landau interaction function /(p, pi) (hence its first harmonic /i) treated as phenomenological input.
Our goal is to evaluate the T- and If-dependence of the key quantity vp(p,T,H). In the simplest case H = 0, the overwhelming T-dependen
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