Critical quasiparticle theory; scaling near a quantum critical...

April 22, 2013

Elihu Abrahams

Critical quasiparticle theory; scaling near a quantum critical point.

1

Saturday, April 27, 2013

Introduction:Fermi liquid, quasiparticles, quantum criticality

Physics situation: strong coupling critical fluctuations associated with a quantum critical point

Extending the quasiparticle description into the non-Fermi liquid regime.

Scaling properties near the quantum critical point; comparisons to experiment and predictions.

Contents

2


Some topics in this talk:

Fermi and non-Fermi liquid: quasiparticles, Quantum criticality and scaling.

Self-consistent calculation of exponents

YbRh2Si2(YRS)

CeCu6-xAux

3=

= bosonic fluctuation


Quasiparticle weight Z

The true ground state

The overlap of the quasiparticlewith the bare particle excitation

The quasiparticle excitationof momentum p

Excitation of a “bare” particleof momentum p

Because of the interaction,quasiparticles can decayinto multiple quasiparticle-quasihole excitations

...+

=


What is required: Z is not 0 and quasiparticles do not decaybefore they establish their excitation energy: Z 6= 0 � < Ep

What makes a Fermi liquid?

quasiparticle peak, weight Z,effective mass: m*/m = 1/Z > 1

incoherent excitations, weight 1- Z

Ep

Ap(!)The spectral distribution

Ep

�(⇥ � Ep)

A

!

Non-interacting electronsspectral peak, weight 1

Landau Fermi liquid is derived from the interacting Hamiltonian by many-body perturbation theory.


•The quasiparticle excitations are in one-to-one correspondence with the excitations in the non-interacting system. The temperature dependence of thermodynamic and transport properties are similar, with renormalized parameters.

•The quasiparticle energy depends on the configuration of the surrounding quasiparticles. This is described by residual interactions called Landau parameters, which are related to the effective mass m* and to the quasiparticle weight Z.

Some properties of the Fermi liquid

C / m⇤T, ⇢ / T 2, � / m⇤

1 + F(T 0)

Ek = k2/2m⇤ � EF ⇡ v⇤F (k � kF ), v⇤F = kF /m⇤, Z = m/m⇤


G(k,!) = G0(k,!) + G0(k,!) ⌃(k,!) G(k,!)

=1

✏k � ! � ⌃(k,!)

Quasiparticle pole at ! = !p = Ek + i�

so G(k,!) = Z

! � Ek � i�, 1/Z = 1� @⌃/@!|p

NEED Z > 0,� < Ek

Quasiparticle self energy and weight Z

7


�( eEp)eEp

⇠ 1

ln |!c/ eEp|⌧ 1, Z( eEp) ⇠

1

ln |!c/ eEp|!

(⇢(T )� ⇢(0) / T

C/T / ln(T0/T )

What makes a non-Fermi liquid?

For example - Interaction of conduction electron (Fermi liquid) quasiparticles with 2D Gaussian antiferromagnetic spin fluctuations (like “marginal Fermi liquid” -PRL 1989):

Traditional definition of non-Fermi liquid: at the Fermi surface, EF :

But it could be that for non-zero excitation energy and/or non-zero temperature. Then the spectral distribution still exhibits a coherent quasiparticle peak.

Z(! = 0, T = 0) = 0

Z(!, T ) 6= 0


1. Interaction with 2D AFM fluctuations + disorder

2. Interaction with 3D FM fluctuations

3. Interaction with scale invariant fluctuations (marginal Fermi liquid)

4. Interaction with 3D transverse gauge field

5. ν = 1/2 quantum Hall state with Coulomb interaction

⌃ =X

g2G�

�(q, ⌫) / N0

M + (q�Q)2 + i�|⌫|/Q

�(q, ⌫) / N0

M + q2 + i�|⌫|/q

�(q, ⌫) / ⌫

T

⇥1� ln

|⌫|T

+⇡i

2sgn(⌫)

⇤

�(q, ⌫) / N0

q2 + �|⌫/q|

⌃(!, T ) / ! ln!0

T� i⇡T

⇢� ⇢0 / T

� =Cel

T/ ln(T0/T ) / m⇤ / 1/Z

Possible sources of non-Fermi liquid behavior

9


Ap(!) / ImGqp(p,!) = ImZ

! � eEp � i�

Re⌃ = eEp � ✏p, Im⌃ = ��/Z

Z = 1� @Re⌃/@!

Z 6= 0, � < eEp

Critical quasiparticle scheme - examples

Quasiparticles beyond Landau Fermi liquidExamples:

10

Power law non-Fermi liquid (NFL):

⌃NFL / �i(�i!)↵

Z(!) ⇠ !1�↵

�/| eE| = cot(⇡↵/2) < 1, (1 > ↵ > 1/2)

Z ! Z(!, T ) 6= 0,

when(!, T ) ! 0, Z ! 0,

but �/|E| < 1

Marginal Fermi liquid (MFL):⌃MFL / ![ln(!c/!)� i(⇡/2)]

1/Z(!) ⇠ ln(!c/!)

�/| eE| / ln(!c/| eE|)�1 < 1

eEp


Quantum phase transition•Thermodynamic phase transition: macroscopic order is destroyed by thermal fluctuations as the temperature (the control parameter) is raised.

•Quantum phase transition: zero-temperature transition between distinct ground states induced by variation of a non-thermal control parameter like pressure, magnetic field, chemical composition.

•Quantum critical behavior influences measured quantities even at non-zero temperature.

•Phase diagrams of many substances exhibit quantum critical points.

T

xc

dρ/dT >0

x

dρ/dT < 0

M I

T

?

T*

x

Metal-insulator transition

HIgh-Tc superconductor

possible quantum critical points


Classical and quantum phase transition

Typical quantum antiferromagnet phase diagram.High-Tc superconductors

Heavy-fermion metals

Iron-based pnictide superconductors

Cs doped into buckyballs - Cs3C60

TN

T

x (e.g. p, H, ...)

AF

xc

TN marks the second-order thermal phase transition from a paramagnetic phase to an antiferromagnetic phase.

But at very low T, varying x through xc, quantum fluctuations destroy antiferromagnetic long-range order at the quantum critical point.

Near TN, spatial fluctuations of antiferromagnetic regions determine the critical behavior as described in the classical theory of second order phase transitions.

When , quantum fluctuations enter in the critical behavior.

Quantum critical behavior may influence measurable quantities over a wide region of the phase diagram.

� ⇥ |x� xc|�� ⇥fluc ⇥ |x� xc|z� z = dynamical exponent


Typical properties near a quantum critical point

32

1

5

T

xxc

4

1. Ordered phase2. Classical critical region3. Thermally disordered region4. Quantum critical region5. Quantum disordered region

5. well-defined quasiparticles (Fermi liquid in a metal).

3. thermal fluctuations destroy the ordered state.

4. critical fluctuations characteristic of xc are thermally excited and produce power-law temperature behavior of physical properties.

T < |x� xc|�z :

T < |x� xc|�z :

T > |x� xc|�z :

13

Critical Cone


Z /Z

D[�(~r)]e��F [�], � = 1/kBT

F [�] =

Zd~r

⇢a(x� xc)�(~r)(1 + cr2)�(~r) + b�4(~r)

�

�(~r) ! �(~r, ⌧) F [�(~r)] !Z ~�

0d⌧S[�(~r, ⌧)], S = F + dynamics

Z /Z

D[�(~r, ⌧)]e�R ~�0 d⌧S[�(~r, ⌧)]

Order parameter fluctuations

Order parameter Φ distinguishes the ordered phase from the disordered phase:Magnet: Φ = staggered or uniform magnetization.Superconductor: Φ ~ superconducting energy gap.

Landau formulation for second-order phase transitions:

Generalization for quantum phase transitions:

If Gaussian fluctuation regime.In the “non-Gaussian regime,” fluctuations interact via the Φ4 term.

deff = d+ z > dc = 4 !


S =X

k

TX

!n

�k2 + �|!n|+ r}|�(k,!n)|2 +O(|�|4) + . . .

⇢(T )� ⇢(0) / T

C/T / ln(T0/T )

��1(k,!n)

Order parameter fluctuations in metals

Magnetic quantum phase transition in Fermi liquid:“Hertz (1976) - Millis (1993) - Moriya (1995) theory”

1. Fermions coupled to order-parameter degrees of freedom Φ (spin fluctuations).2. Integrate out the fermions and keep the low-energy Φ fluctuations.3. Expand in powers and gradients of Φ

The interaction with magnetic fluctuations causes “heavy-fermion” behavior of quasiparticles; m* grows and e.g.

Landau damping tuning parameter ➝ 0 at the QCP


The critical theory

16

The spin fluctuations are a collective mode of the quasiparticles.

What if the quasiparticles develop critical behavior, described by a scale-dependent Z(ω) ~1/m*, due to coupling to order parameter fluctuations near a QCP?

The Hertz-Millis-Moriya theory above may be extended by feeding back the critical behavior of the quasiparticles into the spin fluctuation spectrum.

The coupling of fermionic (quasiparticle) and bosonic (spin fluctuation) degrees of freedom requires a self-consistent treatment of their critical behaviors


�(q,!) =N0�

r(T, x)� i!/vFQ+ ⇠

20(q�Q)

2

+

! heavy quasiparticles with m

⇤= 1/Z

! new �

qp(=

XGG�) depending on Z

! new⌃{Z}! 1/Z = 1� @⌃/@! = (self consistent equation forZ)

new�

qp(q,!) =N0/Z

r(T, x)� i!/ZvFQ+ Z⇠

20(q�Q)2

Implementation of self-consistency

17

= �(Q) = ZF (Q)/N0 ⇡ �1/N0·

This gives correct behavior of small (q-Q)correction to and large (q-Q) behavior of

�(q, 0)

�coh = Z2�qp

The Z factors in are due to feedback of renormalized quasiparticles.

�qp(q⌫)


Im�(q,!) / !/Z2

[r + Zq2]2 + (!/Z)2, Z / !1/4 ! z = 4, ⌫ = 1/3

! ⇠ qz, ⇠ / r�⌫

1/Z(!) = 1� @⌃

@!= 1 +

const

(vFQ)

1/2

!1/2

Z3(!)

! Z / !↵

Critical behavior

18

new ��1 = {r + Zq2 + i(�/Z)|!|/Q}/ Z

N0

! new ⌃ ! new Z(!)

New critical behavior and quasiparticles beyond the Fermi liquid!


Im⌃(!) = �2

Zdq d⌫ ImGqp(k� q,! � ⌫) Im�qp(q, ⌫) ⇥thermal(!, ⌫)

�qp(�Q)

�E ⇠ hJS · S; JS · Si ⇠ hS · SihS · Si ⇠ �qp�qp

Momentum dependence?

19

The theory is based on momentum-independence of the self energy. So m* is uniform . But portions of Fermi

surface (FS) connected by Q (“hot lines”) acquire large contributions to the self-energy, which then becomes highly anisotropic in k-space. Unless:

1. Impurity scattering distributes hot lines over the FS or2. Exchange energy fluctuations (q=0) dominate in contributing to critical behavior of m* all over the FS.

Q

�qp(Q)


9>>=

>>;! Z ! �qp ! �E ! ⌃ ! Z

2. ⌃ = G · �qp(Q) ·G · �qp(�Q) ·G

Momentum dependence

20

1. ⌃(k) = �2X

q

Gk� q) · �qp(q) = �

= �E ⇠ � ·G · �

⌃(k) = �2E

X

q

G(k� q) · �E(q)

�E(q) =X

p

�qp(q� p) · �qp(p)

q� p ⇠ Q, p ⇠ �Q ! q ⇡ 0


�E(q) =X

p

�qp(q� p) · �qp(p)

Im�E(q,!) /(!/Z2)d/2+1

(Zq2 + r)2 + (!/Z)2

⌃ =X

G · �E !

Z�1 = 1� @Re⌃

@!⇡ 1 +

!d�3/2

Z1+2d

1/m⇤ / Z(E) / E↵, ↵ = (d� 3/2)/2d =

(1/4 (d = 3)

1/8 (d = 2)

Energy fluctuation spectrum ➝ m*

AND: Z( E, or T) > 0, with Γ< Ε ⟶ quasiparticles beyond Landau Fermi liquid and

new critical behavior


��1 ⇠ [r + Z(!)q2]2 + [!/Z(!)]2, Z(!) ⇠ !↵

Scaling results- antiferromagnetic QCP

22

32

15

T

xxc

4

1. AFM ordered phase)2. Classical critical region3. Thermally disordered region4. Inside cone (critical region) 5. Outside outside (FL?)

! ⇠ qz, z = 4d/3 =

(4 (d = 3)

8/3 (d = 2)

Dynamical exponent z:

At x=xc, r = 0 (region 4) ➝

Correlation length exponent ν: (region 5) ➝

r + Z(!)q2 ⇠ r + q

z↵+2 ! ⇠ ⇠ r

�1/(z↵+2) ⇠ (x� xc)�⌫,

⌫ = 3/(3 + 2d) =

(1/3 (d = 3)

3/7 (d = 2)


Thermodynamics and transport

23

For transport: Quasiparticle scattering rate has a scaling form

� = ⇠�z��(r⇠1/⌫ , T ⇠z)

and ⇢ / (m⇤/m)� /(T 3/4 (d = 3)

T 7/8 (d = 2)inside

From the critical Σ, determine entropy and free energy:

f(r, T ) = ⇠�(1+2d)�f (r⇠1/⌫ , T ⇠z)

e.g. C/T = @2f/@T 2 /(T 2d+1�2z)/z, inside

r⌫(2d+1�2z), outside

z and ⌫ : T > |x� xc|z⌫The boundary of the critical cone is determined by the exponents

Next: comparison to experiments onheavy-fermion materials


REVIEW ARTICLE FOCUS

FLAF

0.010.01

0.1

1

10

0.1 1 10 50H (T)

T (K

)

T *

H0

T0

YbRh2Si2

Figure 6 The temperature-versus-magnetic-field phase diagram for YbRh2Si2. Themagnetic field (H ) is applied perpendicular to the crystallographic c-axis. Dashedand solid grey lines show the initial Kondo crossover scale T0 = 25 K (at H= 0) andthe corresponding H0, which suppresses the heavy quasiparticle behaviour127. Thefine black line indicates the boundary of the AF phase87. The broad red line specifiesthe position of crossover in the isothermal Hall resistivity60, which agrees with thatobserved in the isothermal magnetostriction, magnetization and longitudinalresistivity44 and marks an additional low-energy scale. The regime at H > Hc, wherethe electrical resistivity follows the Landau FL behaviour 1⇢ / T 2, is shown inblue87. A double-log representation has been used for clarity.

quantum-mechanical excitations that are in addition to criticalfluctuations of the order parameter. We can therefore expect theoccurrence of more than one energy scale, all of which vanish asthe QCP is reached. The Hall-eVect measurement in YbRh2Si2,described earlier, had in fact proven the existence of a T⇤(H) line(Fig. 6) (ref. 60).

The Hall-eVect crossover is accompanied by changes in theslope of the isothermal magnetization and magnetostriction44. Asshown in Fig. 5b, the latter shows a kink-like structure whoseposition in field matches the position of the Hall-eVect crossover.This proves that the low-energy scale T⇤(H) is intrinsic to theequilibrium excitation spectrum. The energy scale is separate fromboth the Neel temperature and TFL, the boundary of the FL regime(Fig. 6). The distinction between T⇤ and TFL is reinforced bythe diVerences in how the physical properties behave44: variousisothermal properties show a temperature-smeared jump across theT⇤ line, but are always smooth across the TFL line. In the T ! 0limit, the lines defined by the three energy scales merge at the samepoint, the field-induced44,60 QCP.

The comparison between the thermodynamic results with theHall-eVect data suggests that T⇤(H) is the finite-T manifestationof the localization of the f electrons at the QCP. This inturn suggests that the measured T⇤(H) scale characterizes theFermi-surface collapse at zero temperature44,60. It correspondsto the E⇤

loc scale of the local quantum-critical picture12, shownin Fig. 2a. Multiple vanishing energy scales also arise in the‘deconfined’ quantum-criticality scenario for insulating quantummagnets31; in its extension to itinerant HF systems14,65, though,the additional energy scale vanishes at a point away from themagnetic QCP.

It is instructive to contrast the case of YbRh2Si2 with whathappens in the V-doped Cr. There, the changes in the HallcoeYcient occur only at the Neel transition62,66. By extension,there is no indication for a separate T⇤ scale60. These observations

are consistent with the spin-density-wave QCP description ofV-doped Cr.

The hexagonal compound YbAgGe is another Yb-based HFsystem that undergoes field-induced magnetic quantum phasetransitions67. It shares some similarities with YbRh2Si2. Inparticular, Hall-eVect measurements in this material have alsoshown an anomaly that defines a crossover line in the temperature-versus-field phase diagram68. Unlike the case for YbRh2Si2, however,the features associated with the ‘Hall line’ have not yet alloweda linkage with a Fermi-surface jump in the zero-temperaturelimit. There are also some other important diVerences betweenthe two systems. Compared with YbRh2Si2, TN in YbAgGe atambient pressure is much larger (about 1 K) and, correspondingly,the critical field is much greater (about 5 T). The temperatureversus field phase diagram is considerably more complex, withadditional first-order transitions inside the AF region and ametamagnetic signature in M(H) when crossing the Hall line69.Thermodynamic and transport measurements in YbAgGe haveraised the possibility that an NFL phase occurs over an extendedrange of magnetic fields67. Given the hexagonal crystal structure,the underlying spin system may very well be frustrated; it ishence natural to raise the question about the potential role thatgeometrical frustration plays in influencing the phase diagramof YbAgGe.

SUPERCONDUCTIVITY

In the past few years growing evidence has been collected formore than one non-phononic pairing mechanism operating indiVerent HF superconductors. These include Cooper pairingpossibly mediated by nearly critical valence fluctuations70 in bothCeCu2Ge2 (ref. 71) and CeCu2Si2 (ref. 72) under high pressure(Fig. 7), by AF magnons in UPd2Al3 (ref. 73) and ferromagneticones in pressurized UGe2 (ref. 74) as well as URhGe (ref. 75) andUCoGe (ref. 76).

CeCu2Si2 (at ambient and low pressure) is the primecandidate for antiparamagnon-mediated superconductivity near a3D spin-density-wave QCP72. This is illustrated in Fig. 7, wherea low-pressure dome of superconductivity occurs near the AFQCP for 10-at.%-Ge-doped CeCu2Si2. From neutron-diVractionmeasurements on an AF (‘A-type’) single crystal77, the latter wasfound to be of the spin-density-wave variety. In addition, the1⇢ / T 1.5 and �(T) = �0 � ↵T 0.5 dependencies found earlier inthe low-temperature normal state in ‘S-type’ samples located onthe strong-coupling side of the QCP56 suggested that the correlatedcritical fluctuations are 3D. Ongoing inelastic-neutron-scatteringexperiments carried out on ‘S-type’ CeCu2Si2 single crystals78

are devoted to searching for quantum-critical spin-density-wavefluctuations, which were proposed to act as superconducting gluein this case79,80. Future neutron-diVraction experiments on otherNFL superconductors like CePd2Si2 (ref. 79) are urgently needed togain more insight into the relationship between quantum criticalityand superconductivity.

The available results, however, highlight a surprisingly richdiversity in physical scenarios. For instance, superconductivity mayoccur from the proximity to a field-induced QCP, as in CeCoIn5

(ref. 81) and perhaps also in UBe13 (ref. 82).Microscopic theories of HF superconductivity were mostly

constructed before the period of active studies on quantumcriticality. Given that a large number of HF superconductors arenow recognized as being close to a QCP, it is important that theinterplay between quantum criticality and superconductivity besystematically studied in the future. More generally, it is likely thatHF superconductors will have an important role to play in the largerunderstanding of strongly correlated superconductors83.

192 nature physics VOL 4 MARCH 2008 www.nature.com/naturephysics

YbRh2Si2 - a “heavy-fermion” metalThe phase diagram - P. Gegenwart, F. Steglich, Q. Si, Nature Physics (2008)

conduction bandhybridizes withnarrow f band

with strong local repulsion

⎫⎬⎭

Anderson lattice model

Hc= 0.06 TTN = 70 mK

TN

A magnetic-field tunedquantum critical point

Hc

AF : antiferromagnetism

Gaussian regime : Resistivity Specific heat

FL : “Heavy” Fermi liquid, m*/m ~ 40 due to theKondo lattice effect.

Yb3+ = 4f13, 1 f hole

/ T

/ T lnT

2D AFM and/or 3D FM fluctuations

Non-Fermi liquid

3D AFM


TK ⇠ 25 K

Tcrit ⇠ 0.3 K

Fluctuation regimes in YRS

2D AFM independent fluctuations: Gaussian regime

Small Z, large m*/m ~ 40

C/T ⇠ lnT, �� 0 ⇠ T

(2D/3D crossover: Garst, Fritz, Rosch, M. Vojta, PRB ’08)

C/T ⇠ T�↵, ⇢� ⇢0 ⇠ T 1�↵

3D AFM fluctuations: Strong coupling, new power laws

25


YbRh2Si2 -thermodynamics and transport

Tcr ⇠ 0.3 K

Trovarelli et al.,PRL 85, 626 (2000)

Tcr ⇠ 0.3 K

�(T )� �(0) / T

Tcrit < T < T0

Tcrit < T < T0

�(T )� �(0) / T

C/T / ln(T0/T )} Interaction of conduction electron

(Fermi liquid) quasiparticles with antiferromagnetic spin fluctuations (2D).

Philipp Gegenw

art, Univ. St. A

ndrews (K

ITP QPT C

onf 1-18-05) Magnetic properties of the H

eavy Fermion state in Y

bRh2Si2

Page 2

ρ (µΩ

Custers et al.,Nature 424, 524 (2003)

26


Comparison to YbRh2Si2

27

Specific heatResistivity

Critical region, 0.03 K < T < 0.3 K

Gegenwart et al, Nature Phys. 2008 Oeschler et al, Physica B. 2008

Also thermopower, susceptibility, resistivity, Grüneisen ratio, see Abrahams & Wölfle, PNAS (2012)


r = x� xc / H �Hc

Mcrit = �✓

@f

@H

◆

T=0

= � 1

Hc⌫(d+ z)r⌫(d+z)�1 e�f (0) / r4/3

! �crit / (H �Hc)1/3

C/T / |H �Hc|�1/3

Some predictions

Outside the critical region

⇢(T,H)� ⇢(0) / (H �Hc)�5/3T 2

Resistivity outside the critical cone (Fermi liquid region 5)

Magnetization outside the critical cone

Comparison to YbRh2Si2 -predictions

Predictions!

Specific heat outside the critical cone

V < 20.3 K, and a surprisingly large effective moment m eff < 1.4mB per Yb3!, indicating the emergence of coupled, unquenchedspins at the QCP. The electronic specific heat coefficient, C el(T)/T,exhibits a pronounced upturn below 0.3 K.

We now discuss the field dependence of the electronic specificheat in YbRh2(Si0.95Ge0.05)2 in more detail. In these measurements,magnetic fields were applied perpendicular to the crystallographicc axis, within the easy magnetic plane (Fig. 2b). At fields above 0.1 T,C el/T is almost temperature-independent, as expected in a LFL12. A

weak maximum is observed in C el(T)/Tat a characteristic tempera-ture T0(B) which grows linearly with the field (inset of Fig. 3a),indicating that entropy is transferred from the low-temperatureupturn to higher temperatures by the application of a fieldB . B c " 0.027 T. As the field is lowered, the temperature windowover which C el(T, B)/T " g0(B) is constant shrinks towards zero

Figure 2 Low-temperature electronic specific heat of YbRh2(Si12xGex )2 single crystals as

Cel /T versus T in semi-logarithmic plots at zero field and at low values of the applied

magnetic field B. Insets show low-T, B " 0, a.c.-susceptibility as x 21 versus T (a) andmagnetization as M versus B (b). Cel is obtained by subtracting the nuclear quadrupolarcontribution, CQ " aQ /T

2 (with aQ " 5.68 £ 1026 J K mol21, calculated from

recent Mossbauer results17) (a) and, in addition, the nuclear Zeeman contributionChf " a(B ) /T 2 (b), from the raw data. Here, a(B ) has been deduced by plotting CT 2

versus T 3. The magnetization,M versus magnetic field B (black points in the inset to b), iscalculated via (Bhf 2 B )/a, with a the hyperfine coupling constant for Yb in this compound

and the hyperfine field Bhf " ##a#B$2aQ $=adip$1=2; adip represents the strength of the

nuclear magnetic dipolar interaction and amounts to 7.58 £ 1028 J Kmol21 T22

(ref. 19). With the assumption of a " 120 T/mB, the data points agree perfectly with the

measured magnetization curve at 40mK (red line in the inset to b). The B " 0 results

shown in a reveal an upturn in Cel(T ) /T for paramagnetic YbRh2(Si12xGex )2 (x " 0,

TN " 70mK; x " 0.05, TN " 20mK) below T " 0.3 K. In the same temperature range

the susceptibility x(T ) shows a Curie–Weiss law, x 21 / (T 2 V) (inset of a). For bothsamples very similar values are found for the Weiss temperature,V < 2 0.3 K, as well

as for the large effective moment, meff < 1.4mB per Yb3!. For YbRh2(Si0.95Ge0.05)2,

entropy is shifted from low to higher temperatures when a magnetic field is applied (b).The cross-over temperature between the field-induced LFL state (Cel(T )/T < const.) and

the NFL state at higher temperature is depicted by the position of the broad hump in

Cel(T )/T which shifts upwards linearly with the field, B # 0.8 T (B ’ c).

Figure 3 Field dependences of the Sommerfeld coefficient g0, of the electronic specificheat (a) and of the ratio of the A coefficient in the T 2 term of the electrical resistivity and

g20 (b) for YbRh2(Si0.95Si0.05)2. Note that g0 and A are proportional to the effective

quasiparticle mass and the effective quasiparticle–quasiparticle scattering cross-section,

respectively. The magnetic field was applied perpendicular to the c axis, and the applied

field values are corrected, on the abscissae, by the value of the critical field,

Bc " 0.027 T. The g0 values in a were obtained from two different samples: three

independent measurements on sample 1 are displayed by closed symbols (circles, up and

down triangles). The open diamonds show the results of measurements on sample 2. As

B ! Bc, g0 diverges / (B 2 Bc)20.33 (red line), that is much more strongly than

logarithmically (black dashed line). The symbols used in the semi-logarithmic plot K "A=g20 versus (B 2 Bc) of b correspond to values for the electronic specific-heat coefficientshown in a. Half-filled circles (or squares) display data for which the A coefficient of the

electrical resistivity was determined by extrapolating (or interpolating) A(B 2 Bc) with

respect to (B 2 Bc). The half-filled diamond represents a point for which the g0 value was

obtained by interpolation. The black dashed line indicates K SDW / %#B 2 Bc$ ln2#B 2

Bc$&21 for the 2D SDW scenario13. This is at strong variance from the (at

B 2 B c , 0.3 T) experimentally observed K / (B 2 B c)21/3, arising from the stronger

than logarithmic increase of g 0 upon cooling (red line in a). For (B 2 B c) . 0.3 T, K

becomes field-independent within the error bars at a constant value of 5.4mQ cmmol2 K2

J22 (green horizontal line in b). A similar high-field behaviour has been reported

previously6 on pure YbRh2Si2. We note that an almost identical value for K was found.

Insets show the scaling behaviour of the low-T electronic specific heat where, according

to equation (1), the ordinate is displayingF(B, T ) " (B 2 Bc)0.33Cel /T, a, as well as of the

temperature derivative of the electrical resistivity, dr/dT, b, as a function of T/(B 2 Bc).

letters to nature

NATURE |VOL 424 | 31 JULY 2003 | www.nature.com/nature 525© 2003 Nature Publishing Group

H-Hc

Custers et al, Nature(2003)

10.1

5


29

CeCu6-xAux - a “heavy-fermion” metalThe phase diagram

A. Schröder, et al, Nature (2000)

AF : antiferromagnetism

Heavy-fermion regime : Resistivity Specific heat

FL : “Heavy” Fermi liquid, m*/m ~ 140 due to the Kondo lattice effect.

Ce3+ = 4f13, 1 f electron

/ T

/ T lnT

A quantum critical point tuned by alloying -

(or H, or p)

xc= 0.1

TN

QCP

Neutron scattering :d=2 spin fluctuations

Theory: z = 8/3, ν = 3/7


C/T /(T�1/8

inside

|p� pc|�1/7outside

M /(�T inside

|H �Hc|8/7 outside

30

Comparison to CeCuAu - predictions

H.v. Löhnheysen et al Physica B (1996)

Predictions

T (K)

M/H

(µB/T

)

M/H = 0.223� 0.07T + 0.02T 2

(at H = 0.1T)

data

0.2 0.4 0.6 0.8 1.00.17

0.18

0.19

0.20

0.21

Also dynamic susceptibility, resistivity, Grüneisen ratio, see Abrahams, Schmalian & Wölfle, ArXiV (2013)

C/T = �3 + 4T�1/8

T (K)

C/T

(J/m

olK

2)

data

Out[57]=

0.1 0.2 0.5 1.0 2.0

0.5

1.0

1.5

2.0

2.5


T�00 / E/T⇣�2

1 + (E/T ⇣)2,

Z ! ⇣ = [(E/T )2 + ⇡2]1/16

Dynamic susceptibility CeCu5.9Au.1

31

E/T

T�00(µ

B)

theory

data

1 2 5 10 20 50 100 200

0.01

0.02

0.05

0.10

0.20

0.50

1.00

Data from Almut Schröder, et al, PRL (1998) and private communication


A collaboration with Peter Wölfle and Jörg Schmalian (K. I. T., Karlsruhe)

“Strong coupling theory of heavy fermion criticality”, Elihu Abrahams, Jörg Schmalian, and Peter Wölfle. arXiv 1303.3962

“Critical quasiparticle theory applied to heavy fermion metals near an antiferromagnetic quantum phase transition,” Elihu Abrahams and Peter Wölfle, Proc. Natl. Aca. Sci. 109, 3238 (2012).

“Quantum Critical Behavior of Heavy Fermions: Quasiparticles in the Gaussian fluctuation regime” Ann. Phys. (Berlin) 523, 591 (2011).

“Quasiparticles beyond the Fermi liquid and heavy fermion criticality,” Peter Wölfle and Elihu Abrahams, Phys. Rev. B 84, 041101 (2011).

“Electron spin resonance in Kondo systems,” Elihu Abrahams and Peter Wölfle, Phys. Rev. B 78, 104423 (2008).

References


Introduction to Fermi liquid theory and quantum criticality.

Quasiparticles can be well-defined in certain classes of non-Fermi liquid states near quantum critical points.

That is, Z=0 at EF, but can be non-zero for T > 0 and/or non-zero excitation energy.

Marginal Fermi liquid behavior [m* ~ log(TK/T)] seen in many heavy fermion compounds, especially YbRh2Si2.

Close to quantum critical point, stronger deviations from Fermi liquid are observed.

Mutual interaction of antiferromagnetic fluctuations and quasiparticles leads to stronger mass enhancement.

Critical quasiparticle theory: Self-consistent determination of Z(E, T)

Comparison to YbRh2Si2 experiment below T = 0.3 K: resistivity, specific heat, etc.

Summary I


Near an AFM QCP: Spin fluctuations at large Q lead to energy fluctuations at small q ➝ momentum independent self energy and Z.

3D fluctuations: Comparison to YbRh2Si2 experiment below T = 0.3 K: resistivity, specific heat, etc.

2D fluctuations: Comparison to CeCu5.9Au.1 experiment: specific heat, magnetization, dynamic susceptibility, etc.

Some open questions/directions:

? Derivation of assumed magnetic fluctuation susceptibility in the critical region.

? Origin of constant term in YRS specific heat - low frequency oscillators (?).

? Apply to other compounds and systems near an antiferromagnetic quantum critical point.

? Apply to FM QCP.

? RG treatment of mutual fermionic and bosonic criticality.

? Superconductivity.

Summary II


A collaboration with Peter Wölfleand Jörg Schmalian

(K. I. T., Karlsruhe)

Acknowledgement

35


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