Crossover from non-Fermi liquid to Fermi liquid behavior close to a quantum critical point: A brief review

ARTICLE IN PRESS Physica B 404 (2009) 2949–2954

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Crossover from non-Fermi liquid to Fermi liquid behavior close to a quantum critical point: A brief review P. Schlottmann  Department of Physics, Florida State University, Tallahassee, FL 32306, USA

a r t i c l e in fo

abstract

PACS: 71.27.+a 71.28.+d 72.15.Qm 75.20.Hr

The nesting of the Fermi surfaces of an electron and a hole pocket separated by a nesting vector Q and the interaction between electrons gives rise to itinerant antiferromagnetism. The order can gradually be  suppressed by mismatching the nesting and a quantum critical point is obtained as the Neel temperature tends to zero. We review our results on the specific heat, the quasi-particle linewidth, the electrical resistivity, the amplitudes of de Haas–van Alphen oscillations and the dynamical spin susceptibility. & 2009 Elsevier B.V. All rights reserved.

Keywords: Quantum critical point Non-Fermi liquid Nested Fermi surfaces Heavy fermions

1. Introduction Landau’s Fermi liquid (FL) theory has been successful in describing the low energy properties of most normal metals. Numerous U, Ce and Yb based heavy fermion systems [1–3] display deviations from FL behavior, which manifest themselves as, e.g., a logðTÞ-dependence in the specific heat over T, C=T, a singular behavior at low T of the magnetic susceptibility, w, and a power-law dependence of the resistivity, r, with an exponent close to one. These deviations from FL are known as non-Fermi liquid (NFL) behavior. The breakdown of the FL can be tuned by alloying (chemical pressure), hydrostatic pressure or the magnetic field. In most cases the systems are close to the onset of antiferromagnetism (AF) and the NFL behavior is attributed to a quantum critical point (QCP) [4–12]. Recently we studied the pre-critical region of a heavy electron band with two parabolic pockets, one electron-like and the other hole-like, separated by a wave vector Q using (i) a field-theoretical multiplicative renormalization group (RG) approach [9] and (ii) the Wilsonian RG that eliminate the fast degrees of freedom close to an ultraviolet cutoff and rewrite the Hamiltonian in terms of renormalized slow variables [12]. The interaction is the remaining repulsion between heavy quasi-particles after the heavy particles have been formed in the sense of a Fermi liquid and is assumed to

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be weak. The interaction between the electrons induces itinerant AF or charge density waves (CDW) due to the nesting of the Fermi surfaces of the two pockets. For perfect nesting (electron–hole symmetry) an arbitrarily small interaction is sufficient for a ground state with long-range order. The degree of nesting is controlled by the mismatch parameter, d ¼ 12jkF1  kF2 jvF [kF1 (kF2 ) is the Fermi momentum of the electron (hole) pocket]. In this way the ordering temperature can be tuned to zero, leading to a QCP. In this paper we review our main results. In the paramagnetic phase the effective mass, m (specific heat over T, C=T) and the magnetic susceptibility increase logarithmically as T is lowered and diverge at the critical point signaling the breakdown of the FL [9,12]. There is a crossover from the lnðTÞ dependence of C=T to constant g as T is lowered if the QCP is not perfectly tuned, in agreement with experiments on numerous systems. The quasiparticle linewidth shows a crossover from NFL (T) to FL (T 2 ) behavior with increasing nesting mismatch and decreasing temperature [13]. The electrical resistivity [14], the dynamical susceptibility [15] and the amplitudes of the de Haas–van Alphen oscillations [16] have also been studied. The response function to superconductivity diverges as TN is approached [17], but the dominating correlations are AF. NFL behavior, AF order and superconductivity in the neighborhood of a QCP have been observed in CePd2 Si2 and CeIn3 under pressure [18]. We have also investigated the renormalization of the electron–phonon coupling, the softening of the phonon with wave vector Q and the consequences of this softening on the thermal expansion [19].

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15

2. Two-pocket model The model consists of two pockets, one electron-like and the other one hole-like, separated by a wavevector Q. The kinetic energy of the carriers is given by [9,12] i Xh y y e1 ðkÞc1k ð1Þ H0 ¼ s c1ks þ e2 ðkÞc2ks c2ks ;

kk0 qss0

þU

X

y y c1kþq s c2k0 qs0 c1ks0 c2k0 s :

ð2Þ

kk0 qss0

Here V and U represent the interaction strength for small (jqj5jQ j) and large (of the order of Q ) momentum transfer between the pockets, respectively. The limit of the Hubbard model is obtained by choosing V ¼ U. The leading order corrections to the vertex are the bubble diagrams of the zero-sound type (antiparallel propagator lines), which are logarithmic in the external energy o. Assuming that o is small compared to the cutoff energy D, and that the density of states for electrons and holes is constant, rF , we have V~ ¼

V ; 1  rF V x

2U~  V~ ¼

ð2U  VÞ ; 1 þ rF ð2U  VÞx

ð3Þ

where x ¼ ln½D=ðjoj þ 2T þ dÞ [12]. A divergent vertex indicates strong coupling and signals an instability [9,12]. Within the logarithmic approximation the linear response to a staggered magnetic field, wS ðQ ; oÞ, and to a CDW, wc ðQ ; oÞ, are given by [9]

wS ðQ ; oÞ ¼ 2xrF V~ =V; wc ðQ ; oÞ ¼ 2xrF ð2U~  V~ Þ=ð2U  VÞ:

5

δ = 0.08 δ = 0.10 δ = 0.25

0 −10

temperature TN ¼ 12 Dexp½ðrF VÞ1   12d, and if 2UoV a CDW can be formed at Tc ¼ 12 Dexpf½rF ðV  2UÞ1 g  12d. The condition for a QCP is TN ¼ 0 or Tc ¼ 0, and if TN o0 and Tc o0 long-range order has not developed. Thus, for sufficiently large Fermi surface mismatch the renormalization does not lead to an instability [12]. The QCP is an unstable fixed point and can only be reached by perfectly tuning the system [9]. In the disordered phase the g-coefficient of the specific heat is given by the effective thermal mass [9,12] ð5Þ

where g0 refers to the non-interacting system. Here we kept only the leading logarithmic contributions, and x is to be taken with o ¼ 0. The T-dependence of C=T as a function of lnðTÞ is shown in Fig. 1. Here d0 ¼ 0:07 corresponds approximately to the critical mismatch. For the tuned QCP, C=T increases logarithmically as T is lowered and diverges at the critical point signaling the breakdown of the Fermi liquid [9,12]. If d4d0 there is a crossover

δ = 0.12 −8

−6

−4

ln (T/D) Fig. 1. Enhancement of the thermal mass as a function of lnðTÞ for V rF ¼ U rF ¼ 0:2, D ¼ 10, and several mismatch parameters d. d0  0:07 is approximately the critical mismatch. Note the crossover from NFL to FL for d4d0 as T is lowered.

from the logarithmic dependence (NFL) to a constant C=T (FL) as T is lowered [13].

3. Quasi-particle linewidth In an FL the damping of the quasi-particles is proportional to T 2 , while the nesting condition changes this behavior to a quasilinear dependence in T. The linewidth G is calculated following a procedure outlined by Virosztek and Ruvalds [20] in the context of high-Tc superconductivity. In the disordered phase G is given by the imaginary part of the electron self-energy, which can be expressed as a convolution of a staggered susceptibility wS00 ðo=2TÞ with a fermion Green’s function [13], Z h  1 o i GNFL ðo; TÞ ¼ T dx cothðxÞ  tanh x  2T 2 ð6Þ  w 00 ðxÞ½3jV~ j2 þ j2U~  V~ j2 r ; S

ð4Þ

Hence, if V40 a spin density wave is possible with a Ne el

xr2 m ðTÞ g ¼ 1 þ F ½3V V~ þ ð2U  VÞð2U~  V~ Þ; ¼ g0 4 m

(C/T)/(C0/T)

ks

where k is measured from the center of each pocket, and assumed to be small compared to the nesting vector Q. Here e1 ðkÞ ¼ vF ðk  kF1 Þ and e2 ðkÞ ¼ vF ðkF2  kÞ, and for simplicity we assume that the Fermi velocity is the same for both pockets. A strong interaction between electrons gives rise to heavy fermion bands. In the spirit of the FL theory, there are weak remaining interactions between the heavy quasi-particles after the heavy particles are formed. The heavy electron bands are described by Eq. (1) and the weak interactions between quasiparticles are given by [9,12] X y y c1kþqs c1ks c2k H12 ¼ V 0 qs0 c2k0 s0

δ = 0.07 10

F

  r 1 G o  2ðd  d0 Þ wS00 ðo=2TÞ  F Im c þ NFL þ i 2 2 2pT 4pT   rF 1 GNFL o þ 2ðd  d0 Þ Im c þi ; þ þ 2 2 2pT 4pT

ð7Þ

where Im c is the imaginary part of the digamma function, o is the external frequency, and d0 is the nesting mismatch corresponding to the QCP. The frequency in the vertices is 2Tjxj þ joj=2 and we use the analytic continuation of the vertex functions, i.e. ip=2 is added to x. The frequency of GNFL in Im c is 2Tjxj. The selfconsistent solution of Eqs. (6) and (7) yields the quasi-particle NFL linewidth as a function of o and T [13]. There is also a FL contribution to the quasi-particle linewidth given by [13]

GFL ðo; TÞ ¼

p 8

½o2 þ ðpTÞ2 ½3V 2 þ ð2U  V 2 Þr3F ;

ð8Þ

which is added to FNFL assuming that Matthiessen’s rule is valid. The vertices in GFL are not dressed, since this contribution does not arise from the nesting condition. The o and T dependence of the self-consistent GNFL can be understood from some limiting cases [13]. First, consider the perfectly tuned QCP, i.e. d ¼ d0 , set o ¼ 0 and neglect GNFL in the

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digamma function, as well as the vertex renormalizations. The integral on the right-hand side of Eq. (6) is then independent of T and hence GNFL pT, and not T 2 as for a FL. Similarly, as T-0, neglecting GNFL in the digamma function and the vertex renormalizations, we obtain, for d ¼ d0, that the right-hand side of Eq. (6) is proportional to joj, which again differs from the FL behavior (po2 ). The vertex renormalizations yield additional logarithmic corrections, so that to logarithmic order we have approximately

GNFL p½3jV~ j2 þ j2U~  V~ j2 r2F maxðjoj; TÞ:

ð9Þ

In the presence of an instability the vertex corrections strongly enhance GNFL . The self-consistent solution, as a consequence of 0.020

Γ

0.015

2951

the logarithmic corrections, yields a GNFL that has a slightly sublinear T- and joj-dependence [13]. Second, for dad0, neglecting again the self-consistency and the vertex corrections, GNFL is exponentially activated at low T and gradually crosses over to a linear T-dependence with increasing T. Hence, at low T the FL contribution (proportional to T 2 ) dominates, but at higher T there is NFL behavior (Fig. 2). Third, at T ¼ 0, GNFL vanishes identically for jojo2ðd  d0 Þ and is proportional to joj  2ðd  d0 Þ at larger frequencies. Hence, again the FL contribution (proportional to o2 ) dominates at low energies [13]. The self-consistent solution of G ¼ GNFL þ GFL for zero frequency is displayed in Fig. 2. The crossover from FL to NFL is indicated by arrows, where the two contributions are equal. The crossover region is not a precisely defined quantity, especially at intermediate temperatures. In the specific heat, the crossover from the lnðTÞ dependence of C=T to constant g agrees with that of the linewidth. Both, TN ¼ ðd  d0 Þ=2 (separating the AF and paramagnetic phases) and the crossover temperature (shaded area) are shown in Fig. 3.

4. Resistivity

0.010 δ = 0.07

0.09 0.11

0.005

0.13

0.15

0.000 0.00

0.01

0.02

0.03

T Fig. 2. Quasi-particle linewidth G ¼ GNFL þ GFL as a function of T for V rF ¼ U rF ¼ 0:2, D ¼ 10 and several values of d. d0 ¼ 0:07 corresponds approximately to the tuned QCP. The arrows indicate the crossover from NFL to FL behavior with decreasing T.

0.010

A proper definition of quasi-particles requires that their linewidth at low T is small compared to their energy. This is satisfied in a FL, where the width grows proportional to ðo=DÞ2 , with D being the bandwidth. However, for the tuned QCP, the linewidth of the quasi-particles grows as fast as their energy, and the quasi-particles are not well-defined. The resistivity is then not necessarily proportional to GNFL . An appropriate approach to obtain the resistivity is the Kubo equation. The dynamical conductivity can be expressed in terms of a memory function MðzÞ [21],

sðzÞ ¼ ðio2p =4pÞ=½z þ MðzÞ:

ð10Þ

Here o2p ¼ 4pe2 ðN1 =m1 þ N2 =m2 Þ is the plasma frequency for the two-pocket model, and N1 ðN2 Þ and m1 ðm2 Þ are the particle (hole) density and mass, respectively. After a lengthy calculation [14] we obtained the imaginary part of the memory function. The full set of equations determining M 00 ðoÞ will not be presented here. In Fig. 4

0.008

0.20

NFL

0.006 T

0.15

ρNFL

0.004

0.002

AF

0.10

0.08 δ = 0.07

FL

0.12 0.17

0.05

0.000 -0.02

0.25

0.10

0.00

0.02

0.04

0.006

δ - δ0 Fig. 3. Phase diagram as a function of the nesting mismatch d. The QCP corresponds to d0 . The system orders AF for dod0 and TN is indicated by the solid straight line separating the AF and NFL phases. The crossover region from FL to NFL behavior as obtained from the specific heat and the quasi-particle linewidth is shown as the shaded area [13].

0.00 0.00

0.02

0.04

0.06

0.08

0.10

T Fig. 4. Self-consistent resistivity in arbitrary units as a function of T for V rF ¼ U rF ¼ 0:2, D ¼ 10, and several values of d. d0 ¼ 0:07 corresponds approximately to the tuned QCP. rNFL is qualitatively similar to GNFL .

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we show the results for the resistivity rNFL ðTÞ ¼ 4pM 00 ð0Þ=o2p . Note that there is also a FL contribution to the resistivity, which strongly depends on impurity scattering [14]. The electrical resistivity is roughly proportional to GNFL þ GFL for zero frequency, i.e. rðTÞ is slightly sublinear in T for the tuned QCP. For Fermi surface mismatch larger than the critical one, the resistivity displays a crossover from NFL (T) to FL (T 2 ) behavior with decreasing temperature, in agreement with experiments [14]. This behavior is qualitatively similar to the quasi-particle line width. At low T the dynamical conductivity for the tuned QCP has strong deviations from the usual Lorentzian Drude behavior. This is the consequence of the NFL-dependence of M00 ðoÞ, which is roughly proportional to joj, rather than to a constant (impurity scattering) as for a FL. Hence, instead of falling off as o2 , sðoÞ decreases roughly as o1 [14]. CeRu0:48 Fe1:52 Ge2 is a system with QCP for which inelastic neutron scattering has revealed a linear T-dependence of the linewidth of the quasi-elastic peak [22]. At low T the quasi-elastic peak deviates from a Lorentzian (similarly to the conductivity in the present model) and when the Ru/Fe concentration is tuned away from quantum criticality, FL behavior and a Lorentzian peak are recovered.

0.5

0.4

Ar/Ar0

2952

r=1

0.3

0.2

r=2 r=3

0.1 r=5

r=4

0 0

0.0005

0.001

0.0015

0.002

T/D Fig. 5. dHvA amplitudes for the first five harmonics r as a function of T for fixed B for the tuned QCP, d0 ¼ 0:07, and the same parameters as before. The amplitudes are normalized to those of the non-interacting system. For D ¼ 1000 K the magnetic field is 40 T for m =m ¼ 20, m being the free electron mass.

6. Dynamical spin susceptibility 5. Amplitudes of de Haas–van Alphen oscillations Although the quasi-particles are not properly defined for the tuned QCP, there is a one-to-one correspondence between the excitations with those of a FL. Varying the Fermi surface mismatch parameter we can continuously interpolate between the states of the FL and the tuned QCP. Since all the states have free Fermi gas statistics, the extended Lifshitz–Kosevich equation can be applied [23]. There are two circular orbits corresponding to extremal crosssectional areas of the Fermi surface of radii kF1 (electrons) and kF2 (holes), respectively, and hence two fundamental frequencies of oscillation. The amplitude of the oscillations modified by the quasi-particle linewidth is [23] Ar ¼

X

" exp 

xn 40

2pr x 1þ _oc n

Z

D

D

do GðoÞ

p o2 þ x2n

!# :

ð11Þ

Here r ¼ 1; 2; 3; . . . labels the harmonics, oc ¼ eB=m , m is the effective heavy fermion mass before renormalization, and xn ¼ pTð2n þ 1Þ for n ¼ 0; 1; 2; . . . are the fermionic Matsubara frequencies. To simplify we assumed that both pockets have the same linewidth, GðoÞ ¼ GNFL ðoÞ þ GFL ðoÞ. For G ¼ 0, i.e. no interactions, Ar reduces to A0r ¼ 1=½2sinhð2p2 kB Tr=_oc Þ. For interacting electrons Ar is always reduced with respect to the non-interacting system. Fig. 5 shows the oscillation amplitude for the first five harmonics as a function of T for the tuned QCP, d0 ¼ 0:07 normalized to the amplitude of the non-interacting system. The overall reduction of the amplitudes is largest close to the QCP [16]. A very low Dingle temperature is necessary to observe the fundamental frequency and the NFL effects affect the oscillations far away from the QCP. The dHvA-oscillations are periodic as a function of B1 , and are measured over a magnetic field interval. Hence, the amplitude of oscillation cannot be associated with a given B. The magnetic field frequently also acts as a tuning parameter for the QCP. The present discussion of the amplitudes is only meaningful if B, within the regime of measurement, does not affect the tuning of the QCP.

The dynamical susceptibility is a function of the energy and momentum transfers, o and Q þ q. We first evaluate the response function for q ¼ 0 in the absence of interactions to get qualitative insight, and incorporate then the interactions by summing the ladder diagrams (leading order logarithms) and inserting the quasi-particle linewidth (self-energy). For the non-interacting system the imaginary part of the dynamical susceptibility is given by      pr o þ 2d o  2d þ tanh ; ð12Þ w000 ðQ ; oÞ ¼ F tanh 4 4pT 4pT where for simplicity we used parabolic dispersions with m1 ¼ m2 ¼ m and vF1 ¼ vF2 ¼ vF . The nesting condition is e2 ðkÞ ¼ 2d  e1 ðkÞ and rF ¼ kF1 m=ð2p2 Þ is the density of states. Eq. (7) is antisymmetric in o, and at low T gapped for 2dror2d. The gap is the consequence of the momentum and energy conservation, which cannot be satisfied simultaneously unless joj42d. With increasing T the gap gradually closes. We now incorporate the quasi-particle linewidth into the calculation. Eq. (7) is the bubble diagram with antiparallel propagator lines. These propagators are now broadened into Lorentzians. The momentum integrations can be carried out, yielding another Lorentzian with linewidth given by the sum of the widths of the two original Lorentzians, i.e. GLW ¼ GNFL ðo0 þ o=2; TÞ þ GNFL ðo0  o=2; TÞ. Here o0 is the energy integration variable for the bubble. At low T the o0 -integration is limited to the interval ðo=2; o=2Þ. Hence, GLW predominantly depends on o and not on the integration variable o0 [13,20]. The o0 -integration is then straightforward and we obtain approximately   r 1 G o  2d w000 ðQ ; oÞ ¼ F Imc þ LW þ i 2 4pT 2 4pT   rF 1 GLW o þ 2d Imc þi ð13Þ þ þ 2 4pT 2 4pT with Im denoting imaginary part and GLW 2GNFL . Note that expression (13) is different from wS00 ðo=2TÞ in Eq. (7). We now choose opposite spins for the propagators, so that the correlation function involves a spin-flip (transversal susceptibility).

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80

Zp ¼

T<0.003

10−3 χ’’/ω

60

0.007

40

0.010

20

0.015

0 0.00

0.050 0.02

0.04

0.06

ω Fig. 6. Quasi-elastic peak in the imaginary part of the dynamical spin susceptibility divided by o as a function of frequency for d0 ¼ 0:07 (critical mismatch) and the same parameters as before.

10

10−2 χ’’/ω

8

6

4

ð17Þ

where c is a constant, p ¼ 71 and Re denotes real part. Here GNFL and GFL are functions of o and T. The additive constant c arises from the cutoff in the Cauchy transformation of w000 and is in principle arbitrary. The constant is determined by the quantum criticality condition, i.e. 1 ¼ V w00 ðQ ; o ¼ 0Þ for T ¼ 0, and d ¼ d0 . For the parameters used in this paper, we have c ¼ 0:038152. Two features should be pointed out in w00 ðQ ; oÞ=o for the critical Fermi surface mismatch d0 : (i) The quasi-elastic peak around o ¼ 0 and (ii) the shoulder at o ¼ 72d0 at low T. The shoulder is reminiscent of the non-interacting case, Eq. (12), and is gradually smeared with increasing temperature. The central peak, on the other hand, arises from the interaction through the denominator in Eq. (14) and is displayed in Fig. 6 as a function of o for several temperatures. As o-0 the imaginary component of the susceptibility tends to zero and also the real part, 1  V w0 , becomes small, thus giving rise to the peak. The maximum, defined as limo-0 w00 ðQ ; oÞ=o, is proportional to T 3 at low T. The form of the resonance is approximately Lorentzian with the halfwidth of the peak at half of its height being proportional to T. For non-critical nesting mismatch the behavior of w00 ðQ ; oÞ=o is different, since the denominator in Eq. (14) does no longer vanish at o ¼ 0. Hence, only for the critical mismatch d0 the peak is quasi-elastic, while for other d the peak is inelastic. As seen in Fig. 7 the height of the peak is strongly reduced with increasing non-critical mismatch.

7. Conclusions 0.09 δ = 0.07

x3x10−5 0.11

2

δ = 0.13

0 0.00

0.10

0.20

0.30

ω Fig. 7. Imaginary part of the dynamical spin susceptibility divided by o as a function of frequency for T ¼ 0:001 and several values of d. The parameters are the same as before. Note the strongly reduced scale for the central peak for d ¼ 0:07 (critical mismatch).

The most important terms contributing to w are the ladder diagrams in V, i.e. the random phase approximation (RPA) diagrams,

wðQ ; oÞ ¼ w0 ðQ ; oÞ=½1  V w0 ðQ ; oÞ:

ð14Þ

Note that to leading order the U-interaction does not contribute to the transversal susceptibility (in RPA). The dissipative part of the transversal susceptibility is given by

w00 ðQ ; oÞ ¼

o þ 2pd 1 GNFL þ GFL þi ; þ 2pT 4pT 2

2953

½1  V w

w000 ðQ ; oÞ : oÞ2 þ V 2 w000 ðQ ; oÞ2

0 0 ðQ ;

ð15Þ

The imaginary part for w0 ðQ ; oÞ to be inserted in Eq. (15) is Eq. (13) and the expression for w00 ðQ ; oÞ is r X w00 ðQ ; oÞ ¼ lnðD=2pTÞ þ c þ F Re cðZp Þ; ð16Þ 2 p

The nesting of a heavy electron Fermi surface can give rise to itinerant AF long-range order. The degree of nesting is controlled by a mismatch parameter. This way the ordering temperature can be tuned to zero, leading to a QCP. The QCP is an unstable fixed point and can only be reached by perfectly tuning the system. Otherwise, the RG flow will deviate to a phase with long-range order or the compound remains a heavy electron paramagnet. The lack of characteristic energy scale is probably responsible for the lack of universality in NFL compounds. We investigated the quasi-particle lifetime as the QCP is approached. Landau’s FL theory predicts that for normal metals the linewidth is proportional to o2 and T 2 . The QCP modifies this behavior to a linear joj and T dependence with logarithmic corrections. As a function of T and the Fermi surface mismatch there is a crossover from FL to NFL behavior. This crossover is also reflected in the specific heat over temperature and the electrical resistivity, leading to a phase diagram that is in qualitative agreement with experiments. For the tuned QCP the resistivity is approximately linear in T. The dynamical susceptibility (inelastic neutron scattering) is not in agreement with the experimental observations. The present model can also not account for the restructuring of the Fermi surface at d0 for T-0. A two-band model (Anderson lattice) is necessary to capture these features. We calculated the amplitudes of the dHvA oscillations using a modified Lifshitz–Kosevich expression. As expected the amplitudes are strongly reduced and the oscillations are difficult to observe. The dynamical spin susceptibility was self-consistently calculated using the linewidth of the quasi-particles and the random phase approximation. For the tuned QCP a Lorentzian quasi-elastic peak with a width linear in T is obtained. For non-critical mismatch of the Fermi surface the peak becomes inelastic, i.e. it is shifted to oa0. The susceptibility does not agree with the experimental findings. The results are valid in the disordered phase for weak and intermediate coupling. However, since the renormalization group

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does not allow a return to a weak-coupling fixed point once the system is strongly coupled, the present approach qualitatively describes the entire pre-critical regime. Acknowledgment The support by the U.S. Department of Energy under Grant no. DE-FG02-98ER45707 is acknowledged. References [1] [2] [3] [4] [5]

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