Critical quasiparticle theory: Scaling, thermodynamic and transport properties

0.69 MB·English

by  Elihu Abrahams

Critical quasiparticle theory: Scaling, thermodynamic and transport properties
Elihu Abrahams1 and Peter W¨olfle2
1Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, CA 90095
2Institute for Theory of Condensed Matter and Center for Functional Nanostructures,
Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
(Dated: January 4, 2012)
We use the recently developed [2] critical quasiparticle theory to derive the scaling behavior
associated with a quantum critical point (QCP) in a correlated metal. This is applied to the
magnetic-field induced QCP observed in YbRh2Si2 (YRS), for which we also derive the critical
behavior of the specific heat, resistivity, Gru¨neisen coefficient, and the thermopower. The theory
2
accounts verywell for theexperimental results.
1
0
2
Introduction the f electronsare alsoresponsible for the antiferromag-
n
netism in a region of the phase diagram of the mate-
a
rial. Near the AFM critical point, critical spin fluctu-
J Recent advances in low-temperature experimental
ations are enhanced and interact with the heavy quasi-
3 techniques have stimulated much interest in quantum
critical phenomena, which comprise phase transitions at particles. This produces further mass enhancement and
] occurs in two stages. Above a certain temperature T
l zero temperature (“quantum critical point”) and asso- x
e butbelowT ,thecriticalfluctuationsareGaussian(i.e
ciated effects due to quantum fluctuations at very low KL
- non-interacting); those of two-dimensional AFM charac-
r temperatures. Reference[1]givesanintroductoryreview
t ter (or three dimensional ferromagnetic character) pro-
s of the subject.
. duce a logarithmic mass enhancement. A review of this
t These developments have generated a variety of dif-
a physics can be found in reference [3]. Below T , how-
m ficult theoretical questions; among them is the issue of x
ever,the critical fluctuations begin to interact with each
how to treat the regime of strongly-interacting quan-
- other and consequently the effects on the quasiparticles
d tum fluctuations. In a recent paper [2], we developed
change, e.g. the effective mass, or Z-factor, acquires a
n anextension ofthequasiparticleconceptofFermiliquid
o singularpower-lawfrequencydependence. Thisisthere-
theory to the non Fermi-liquid regime near a quantum
c giondescribedbythecriticalquasiparticletheory. Figure
[ critical point (QCP). In essence, the theory goes beyond
1 illustrates these regions in the phase diagram for the

the Gaussian regime of critical fluctuations by introduc-
1 heavy-fermion metal YRS. In YRS, an AFM quantum
inginteractionsamongthequantumfluctuationsintothe
v critical point is accessed by tuning a magnetic field H,
correlation function of the fluctuations. Central to the
3 but the theory is appropriatewhatever the nature of the
7 analysis is the concept of critical quasiparticles,which is
ordered phase and whatever the tuning parameter.
5 basedonthe recognitionthat the single-particlespectral
0 function candisplaya quasiparticlepeak atnon-zeroex-
. Figure 1 shows the phase diagram for YbRh2Si2 near
1 citation energy or temperature. This is expressed as a
the quantum critical point. Five different regions may
0 non-zeroquasiparticleweightZ(ω) for |ω| nottoo small,
be seen: (1) The antiferromagnetically ordered region
2
although, as in a non Fermi liquid, at the Fermi surface
1 attemperatures below the N´eeltemperature TN(H); (2)
Z(ω =0)=0.
: The Landau Fermi-liquid regime at T . T (H); (3)
v FL
We realized the critical quasiparticle theory for the A high temperature local-moment regime at T ≫ T ,
i KL
X case of an antiferromagnetic quantum critical point and where T is the characteristic temperature of the lat-
KL
r appliedittoseveralquantities,principallyresistivityand ticeKondoeffect,belowwhichcoherentheavyquasiparti-
a
specific heat, for successful comparison to experimen- cles formby hybridizationof f-electrons andconduction
talresultsonthe heavy-fermionmetalYbRh2Si2 (YRS), electrons; (4) A regime of quantum critical quasi-two-
therebyshowingthatthe theory,whichdescribesaphys- dimensional antiferromagnetic Gaussian fluctuations at
ically transparent scenario, is capable of accounting for T . T . T , characterized by moderate non-Fermi
x KL
experimental results on a quantum critical metal. liquid behavior; (5) The true critical regime, “QC”, for
The implementation of the theory for an antiferro- T .T governedbythree-dimensionalantiferromagnetic
x
magnetic (AFM) quantum critical point for a heavy- fluctuations interacting strongly with the heavy quasi-
fermion compound is based on the recognition that be- particles.
low a “lattice Kondo temperature” T , hybridization
KL
betweenconduction(s,p,d)electronsandlocalmagnetic The basic input for the theory is a phenomenological
moments (f orbitals) produces a heavy-electron liquid form for the critical antiferromagnetic (AFM) spin fluc-
with an associated mass enhancement due to the orig- tuation correlator at low temperature below T , which
x
inally localized character of the f electrons. However, reflects that it is generated from quasiparticles having a
2
T ≫T to the heavy-fermion phase at T . T . The
KL KL
T (K)
experimentally determined values in YRS are T ≈ 25
KL
K and T ≈0.3 K.
x
The justification for the form of Eq. (1) is discussed
10.00 lattice Kondo effect
in detail in reference [2]. We neglect the momentum de-
5.00 pendence of the Z-factor by assuming that the critical
2 d
behaviorat the hot spots is spreadoverthe whole Fermi
Gaussian
surfacebyimpurityscatteringorotherinteractioneffects.
1.00
The Z-factor is determined by the quasiparticle self en-
0.50
Out[112]= ergy,whichinturnisdeterminedbytheinteractionwith
3 d
the spin fluctuations. This leads [2] to a self-consistency
non-Gaussian
0.10 relation in the form of a differential equation for Z(ω):
QC
0.05
d
Z−1 =1+λ (Z−3ω3/2). (2)
dω
0.01 AFM
FL
This equation has two different physically meaningful
H (T)
0.04 0.05 0.06 0.07 0.08 0.09 solutions. The first is a weak coupling solution, valid
provided the second term on the r.h.s. of Eq. (2) is
≪ 1 in the energy range considered. This corresponds
to the conventional spin-density wave scenario as dis-
cussedintheworksofHertz[5],Millis[6]andMoriya[7].
FIG.1. Phasediagram forYbRh2Si2in theneighborhood of
thecriticalmagneticfieldHc ≃0.06T:Thedashedlinesrep- There exists, however, a second solution in the strong-
resent crossovers. As the temperature is lowered below TKL, couplingdomain;itis accessibleprovidedthe initialZ−1
thelatticeKondoeffectandheavyquasiparticlesdevelopand at the scale when the three-dimensional antiferromag-
weakly-interacting(“Gaussian”) 2dAFMfluctuationsassoci- neticfluctuationregimeisentered(inthecaseofYRSat
ated with the quantum critical point cause non-Fermi liquid T ≈0.3K)is sufficiently large such that the secondterm
behavior. The Tx line represents the crossover to strongly- on the r.h.s. of Eq. (2) dominates. We conjecture that
interacting 3d fluctuations, which are dominant within the
in YRS, two-dimensional antiferromagnetic (assisted by
cone of quantum criticality (QC), bounded on the right by
the crossover at TFL(H) into the heavy Fermi-liquid state three-dimensionalferromagnetic)spinfluctuationsabove
and on the left by the curved dashed line within the anti- T ≈0.3K can provide the necessary growth of the effec-
ferromagnetic ordered state that sets in below TN(H). The tive mass everywhere on the Fermi surface, even in the
T∗(H) crossover (dot-dashed line) is discussed in the text, absence of impurity scattering. (In the case of quasi-2d
below Eq.(3). antiferromagneticfluctuationsina3dmetalthishasbeen
noticed first in reference [8]). Then the strong-coupling
solution is
non-zero frequency-dependent Z-factor:
Z−1 =3−1/2(k ξ )−3/2(ω/v Q)1/4. (3)
∗ F 0 F
(N /Z)(ω/v Q)
Imχ(q,ω)= 0 F , (1)
[r (H)+Z(q−Q)2ξ2]2+(ω/v∗Q)2 At non-zero temperature, Z is approximately given by
0 0 F Z(ω,T)∝[ω2+(πT)2]1/8.
where N is the bare density of states at the Fermi sur- In the following, we describe the phase diagram ob-
0
∗ ∗
face, v = (m /m )v is the renormalized quasiparticle tained within our theory as consisting of the ordered
F b F
Fermi velocity, ξ ≃ k−1 is the microscopic AFM cor- phaseboundedbytheN´eeltemperatureT (H),thecrit-
0 F N
relation length and r (H) = 1+F(Q,H) ∝ H/H −1. icalregime(“criticalcone”)atT >T >T (H)andthe
0 c x FL
Here F(Q,H) is a dimensionless generalized Landau pa- Landau Fermi liquid regime at T < T (H) (see figure
FL
rameter,which → −1 at the critical point. For conve- 1). In addition to these phase boundaries and crossover
nience, we denote the underlying tuning parameter by lines, Hall effect and other measurements in YRS [9–11]
∗
H, which for YRS is the magnetic field. The interaction showanomaliesatatemperatureT (H)inthecriticalre-
∗
between quasiparticles and spin fluctuations I is related gion. TheoriginoftheT (H)lineisnotclearatpresent.
totoF(Q,H)byI =ZF(Q,H)/N . The“bare”quanti- Recent experiments on YRS under pressure [12] or YRS
0
tiesm , N , v herearetobeunderstoodasthosequan- doped with Co or Ir [13] have shown that while the crit-
b 0 F
tities already renormalized at T by hybridization and ical field for the magnetic transition is found to shift,
KL
∗ ∗
the lattice Kondo effect, as discussed above. The addi- the T line stays unchanged. This suggests that the T
tional (frequency-dependent) mass enhancement caused featureisonlyweaklyornotatalltiedtotheAFMtran-
by interaction with critical spin fluctuations is denoted sition. The present theory ascribes the critical behavior
∗
as m /m = 1/Z. We do not consider questions rele- to interaction of renormalized quasiparticles with renor-
b
vant to the crossover[4] from the localmoment phase at malized antiferromagnetic spin fluctuations, rather than
3
∗
withfluctuationsassociatedwithT (H),ashasbeensug- Note that at the critical wave vector (at q = 0), ω/T
gested elsewhere [11, 14, 15]. scaling holds.
In this paper, we elaborate the critical quasiparticle The scaling form for the free energy density may be
theory further by deriving the scaling behavior of quan- deduced by dimensional analysis [3]:
tities near the QCP and we add the critical behavior
of the Gru¨neisen coefficient, thermopower, thermal ex- f(H,T)=ξ−(d+z)Φf(r0ξ1/ν,Tξz)
pansion, magnetization, and susceptibility to the exper- d+z r0
imental consequences of the theory. Again, we compare →T z Ψ(cid:18)Tz1ν (cid:19) (6a)
these results of the theory to experimental information
T
on YbRh2Si2, which has a magnetically tuned QCP. or r0ν(d+z)Ψ(cid:18)rzν(cid:19). (6b)
The deduction of the critical behavior of several ther- 0
e
modynamic quantities from the scaling form of the free
In our case, d=3, ν =1/3, z =4.
energy has been discussed by Zhu, et al [16]. The corre-
Wemaygeneralizetheseresultstoarbitrarydimension
sponding exponents for these quantities are determined
d. First,wenote(fromthederivationofthequasiparticle
by the correlation length exponent ν, the dynamical ex-
weightfactorZ presentedinreference1)thatd=4isthe
ponent z and the dimensionality d.
uppercriticaldimension,sothatthe spinfluctuationsdo
notdestroytheFermiliquidstateford>4. Atd<2,we
Scaling findthatthe criticalquasiparticlesareno longerwellde-
fined. Wethereforeconsiderhereonlydimensionswithin
the interval2≤d≤4. Usingthe exponentα to describe
Asdiscussedinreference[2],thecriticalregimeischar-
the frequency power law of the quasiparticle weight Z,
acterizedbyfluctuationsofdimensionalityd=3. Wecan
we find, from Eq. (4), that the dynamical critical expo-
determine ν and z from the critical quasiparticle theory
nentisgivenbyz =2/(1−2α)andthecorrelationlength
as follows: It can be shown that when the frequency de-
exponent is ν =1/(2+zα)
pendence of Z is a power less than 1/2, the frequency-
dependent Z-factorsappearing in Eq. (1) have the argu-
ment ω. Then from Eq. (1) and the result Z(ω) ∝ ωα,
Specific heat
where we found α=1/4, we can determine the dynami-
cal exponent z =4 since at the critical point
The critical part of the specific heat at the criti-
Imχ(q,ω)−1 ∼Z2(ω)q4+ω2/Z2(ω). (4) cal tuning parameter r = 0 is found from Eq. (6a):
C ∝Td/z =T3/4 as we found before [2]. We can obtain
Here and in the following, q denotes the deviation from c
the behavior of the specific heat near T = 0 as a func-
the ordering wave vector Q. The scaling exponent of
tion of r from Eq. (6b) as follows [16]: Near x=0, Ψ(x)
the spatial correlationlength ξ is found as ν =1/3 from
behaves as Ψ(x→0)=Ψ(0)+bxµ. Therefore
r (H) ∼ Zq2 ∼ (qz)1/4q2 ∼ q3 so that ξ ∼ r−1/3. At e
0 0
T =0 we thus find ξ ∝(H −Hc)−1/3.The characteristic Ce(r,T)∝r(d+ez)ν−µzνµ(µ−1)Tµ−1
correlation time is given by τ ∼ξz ∼(H −H )−4/3.
c c =µ(µ−1)Tµ−1r−1/3(4µ−7). (7)
When H = H (i.e. r = 0), the approach to the
c 0
T = 0 critical point is described by a temperature de-
This behavior is valid in the region of the T,H phase
pendence ofthe controlparameter,whichcanbe written
diagram defined by T/rzν < 1, which is T < T (H) ∝
r(H ,T)∼T1/zν. Theconditionr(H ,T)≈r(H,T =0) FL
c c |H −H |zν.
orT =T (H)∝|H−H |zν definestheboundaryofthe c
FL c
As weshowedinreference[2],the theoreticalbehavior
“critical cone” of quantum criticality. Then, within the
of γ = C /T at criticality found above agrees very well
quantum critical regime, the temperature dependence of c
the correlation length is found as ξ ∝ T−1/z. It may with the data. For the low T-dependence of γ on r =
|H−H |,wenotethatinYRSoneithersideoftheQCP,
be shown that the above critical temperature dependent c
Fermi liquid behavior obtains [17] so that µ = 2. Then,
correction to r is caused by the interaction of spin fluc-
from Eq. 7, γ = C/T varies as |H −H |−1/3. This is
tuations treated at the Hartree level [21], giving rise to c
precisely the behavior observed in experiment on YRS
r(H ,T)∼T/Z(T).
c
[17, 18].
From these results (ν = 1/3, z = 4), we may write
Imχ(q,ω), or equivalently the structure factor S(q,ω),
in the scaling form
Resistivity
x1/2
S(q,ω)=ξ4Φ(ωξ4,qξ), Φ(x,y)∝ .
(1+x1/4y2)2+x3/2 The scaling analysis may be applied to determine the
(5) temperature and magnetic field power-law dependences
4
oftheresistivity,which(assumingimpurityandumklapp we predict S ∝ T3/4. The comparison of our result to
scattering) is given by: the experiment reportedin reference [18] is shownin fig-
ure 2. Unfortunately, the data wastakenatH =0.06T,
ρ(T,H)−ρ(0)∝Γ(T,H)/Z(T,H), below the critical field of 0.064 T. At the measurement
field,theAFMN´eeltemperatureT isabout20mK[18],
N
where Γ is the quasiparticle relaxation rate, which we
so we do not expect good agreement at the very lowest
found [2] at d = 3 and H = H , to be proportional to
c temperatures, as seen in the figure.
ω3/2/Z2(ω). SinceΓhasdimensionsofenergy,itsscaling
form is Γ(T,H) = ξ−zΦ (Tξz) as in Eq. (6b). With
Γ
Z(ω)∝ωα ∝ξ−zα, we have
ρ(T,H)−ρ(0)∝ξ(α−1)zΦ (Tξz)
Γ
To insure that ρ is finite at H = H and T > 0, we
c
require ΦΓ(x)∝x1−α as x→∞, so that S=0.6−10.7T3/4
ρ(T,H )−ρ(0)∝T1−α. (8) K)
c /
V
µ
As wediscussedaboveforthe specific heat,awayfrom (
S
criticality, at low temperature outside either side of the
critical cone defined by T =T , Fermi liquid behavior,
FL
ρ(T)−ρ(0)∝T2 obtains, so that Φ (x)∝x2. Then
Γ
ρ(T,H)−ρ(0)∝ξ(α+1)zT2. (9) T (K)
Using the values α = 1/4,z = 4, which we found in the
FIG. 2. Thermopower: Comparison of theory and data of
self-consistent critical quasiparticle theory [2], we find
reference [18] near the critical magnetic field and below the
that Eqs. (6,7) become
critical temperature for quantumcritical scaling.
ρ(T,H )−ρ(0)∝T3/4 (10a)
c
ρ(T,H)−ρ(0)∝ξ5T2 ∝|H −H |−5/3T2. (10b)
c
Magnetization and susceptibility
TheavailabledataagreeverywellwithEq.(10a)asfound
in reference [2] by direct calculation. At the time of this The temperature dependence of the magnetization
writing, for the field dependence of the coefficient of T2 may be found from the derivative of the free energy in
inEq.(10b),thereisinsufficientdatacloseenoughtoHc Eq. (6a) with respect to field:
to allow a quantitative comparison.
∂f 1 ∂f
M =− =−
crit
(cid:18)∂H(cid:19) H (cid:18)∂r (cid:19)
T c 0 T
Thermopower 1 d+z ′ r0 − 1
=− T z Ψ T zν. (12)
Hc (cid:18)Tz1ν (cid:19)
At very low temperatures, when impurity scattering
′
Assuming M is regular at r = 0 so that Ψ(0) = const,
dominates,thethermopowerSmaybeexpressedinterms 0
we get
of the electrical conductivity σ (µ), depending on the
imp
chemical potential µ, by the Mott formula d+z−1/ν
M(T)−M(0)∝−T z =−T. (13)
π2T ∂lnσimp This result agrees fairly well with the experimental data
S = . (11)
3 e ∂µ atlowT (below400mK)asreportedinreference[19]. A
very good fit is obtained if the critical T-dependence of
Since the mean free path ℓ is not renormalized, Eq.(13)isaugmentedbythesub-leadingT2contribution
σ (µ) ∝ k2ℓ, where k is the Fermi wave vector, is
imp F F due to Fermi-liquid effects. This is shown on figure 3,
independent of the effective mass. The derivative with
where the theoreticalbehaviorat H =0.06T is givenby
respecttoµmaybeexpressedbytheonewithrespectto
the Fermi wave vector as ∂/∂µ = (1/v∗)(∂/∂k ) and M(T)−M(0)=−0.68T +0.63T2. (14)
F F
∗
∂σ /∂k is independent of m . Therefore, we find
imp F For the field dependence at small T, we use Eq. (6b):
∗
S ∝ Tm (T). In the Gaussian fluctuation regime, we
get S ∝ T ln(T), which has been observed [18]. In the M =− ∂f =− 1 ν(d+z)rν(d+z)−1Ψ(0),
critical regime proper, at the critical field H = 0.064 T crit (cid:18)∂H(cid:19) H 0
T=0 c
e
5
which for d=3, z =4, ν =1/3, r =H/H −1 is from Eq. (12) as
0 c
7 1 H 4/3 χ= ∂M = 1 ∂M ∝Td+zz−2/ν ∝T1/4, (17)
M(H)−M(Hc)=− −1 Ψ(0). (15) ∂H Hc ∂r0
3H (cid:18)H (cid:19)
c c ′′
where we assumed Ψ (0)=const. We can compare this
e
Therefore, the critical field dependence of the suscepti- resulttotheexperimentreportedbyGegenwartetal[22],
bility is whereχac wasmeasuredbelow0.2K atafieldof0.05T.
Since T is about 30 mK at this field, we do not include
N
χ = ∂M ∝−(H −H )1/3. (16) data below 40 mK in the comparison shown on figure 4.
crit c
∂H
The available data [20] indicate a cusp-like behavior of Gru¨neisen ratio
χ(H) when approaching H , but do not extend close
c
enough to allow for a quantitative comparison.
In the case of a magnetic field tuned QCP,as in YRS,
the magnetic Gru¨neisen ratio [16] is studied:
(∂M/∂T) (∂S/∂H)
H T
Γ =− =− (18)
M
C C
H H
The critical part of the Gru¨neisen ratio has been dis-
M=0.39−0.68T+0.63T2 cussedinreference[16]andmeasuredinYRSbyTokiwa,
ol) (atH=0.06T) etal[19]. Asinreference[16],thebehaviorofΓM maybe
m determinedfromthe scalinganalysis. We find,forz =4,
2
T
J/ ν =1/3 and µ=2, that [∂S/∂H]r0=0 ∝T0. Then
(
M/H ΓM(r =0,T)∝T−3/4 (19a)
−1/3
Γ (r,T →0)= . (19b)
M
H −H
c
The factor −1/3 = ν(d−z) in Eq. (19b) is universal as
the scaling functions in the ratio defining Γ cancel.
T(K) M
These results as well as those for the thermodynamic
FIG. 3. Magnetization: Comparison of theory and data of Gru¨neisen parameter can also be obtained [21] by direct
reference [19] near the critical magnetic field. A T2 Fermi- calculationfromderivativesoftheself-energyexpressions
liquid contribution has been added to the critical behavior. given in reference [2].
Just as for the specific heat and the resistivity [2],
there is excellent agreement between the critical quasi-
particle theory and the experiment [19] for the magnetic
Gru¨neisenratio. Thefactorν(d−z)=−1/3inEq.(19b)
is measured as −0.30±0.01 and the comparison of our
result of Eq. (19a) of the T−3/4 dependence at H = H
) c
ol to the experiment is shown on figure 5.
m
/ Wemayevaluatetheconsistencyofourresultsbyusing
3
m
thetheoryfitsforM(T),Eq.(14)andforthespecificheat
6
− shownonfigure1ofreference[2](C =0.034+0.454T3/4)
0
1
( in Eq. (18). There are no further adjustable parameters
c χ=8.1−3.3T1/4−10.5T2
χa and the agreement is shown in the inset to figure 5.
The volume thermal expansion coefficient β(T) enters
the conventional thermodynamic Gru¨neisen ratio Γ =
T
T(K) β(r0,T)/C(r0,T). Thecriticalbehaviorhereistunedby
pressure P, so that r ∝ (P −P)/P . The critical part
0 c c
of β is determined by derivatives of the free energy as
FIG. 4. Magnetic susceptibility: Comparison of theory, Eq.
(17)anddataofreference[22]nearthecriticalmagneticfield 1 ∂2f
and below the critical temperature for quantumcritical scal- β (r ,T)= ∝T0,
c 0
ing. Similarly to Eq. (14) and figure 3, we have added a T2 Pv∂T∂r0
Fermi-liquid contribution. wherev isthespecificvolumeandwehaveusedEq.(6a).
To this we may add the conventional Fermi-liquid con-
ThecriticalT-dependenceofthesusceptibilityisfound tribution ∝T so that β =a+bT in the critical region.
6
behavioris different depending onthe energyscale. Fur-
thercandidatesystemslikeCeCoIn alsoappeartoshow
5
quasi-2dantiferromagneticGaussianfluctuations (for an
overview,seereference[3]). Inallofthesesystemsweex-
pectacrossoverto3dfluctuationsatlowertemperatures,
−1) in which case our theory should apply.
T
(
M
Γ WethankJ¨orgSchmalianforvaluablesuggestionsand
)
1
− Alexander Balatsky, Andrey Chubukov, Max Metlitski,
T
(M T(K) SubirSachdev,andQimiaoSiforusefuldiscussions. This
Γ work was supported in part by the DFG research unit
960”Quantumphasetransitions”. Partofthis workwas
Γ =−0.9+1.0T−3/4 carried out at the Aspen Center for Physics (NSF grant
M
1066293),LosAlamosNationalLaboratoryandasaCarl
Schurz Memorial Professor at the University of Wiscon-
T (K) sin,Madison(PW).Wethanktheseinstitutionsfortheir
hospitality and support.
FIG. 5. Magnetic Gru¨neisen ratio: Comparison of theory,
Eq. (19a) and data of reference [19] at the critical magnetic
field and below the critical temperature for quantum critical
scaling. The inset shows the comparison of the data and Eq.
(18) when thepreviously fit M(T) and C(T) are used.
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