Strange metal state near a heavy-fermion quantum critical point

1.9 MB·

by  Yung-Yeh Chang

Strange metal state near a heavy-fermion quantum critical point
Yung-Yeh, Chang1, Silke Paschen2, and Chung-Hou Chung1,3
1Department of Electrophysics, National Chiao-Tung University,
1001 University Rd., Hsinchu, 300 Taiwan, R.O.C.
2Institute of Solid State Physics, TU Vienna, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria
3Physics Division, National Center for Theoretical Sciences, HsinChu, 300 Taiwan, R.O.C.∗
(Dated: January 18, 2017)
Recent experiments on quantum criticality in the Ge-substituted heavy-electron material
YbRh2Si2 under magnetic field have revealed a possible non-Fermi liquid (NFL) strange metal
7
(SM) state over a finite range of fields at low temperatures, which still remains a puzzle. In the
1
SM region, the zero-field antiferromagnetism is suppressed. Above a critical field, it gives way
0
to a heavy Fermi liquid with Kondo correlation. The T (temperature)-linear resistivity and the
2
T-logarithmic followed by a power-law singularity in the specific heat coefficient at low T, salient

n NFL behaviours in the SM region, are un-explained. We offer a mechanism to address these open
a issuestheoretically basedonthecompetition betweenaquasi-2dfluctuatingshort-rangedresonant-
J valence-bonds (RVB) spin-liquid and the Kondo correlation near criticality. Via a field-theoretical

7 renormalization group analysis on an effective field theory beyond a large-N approach to an anti-
1 ferromagnetic Kondo-Heisenbergmodel, we identify thecritical point, and explain remarkably well
both thecrossovers and the SMbehaviour.

]
l
e
- Introduction. Magnetic field tuned quantum phase the Kondo breakdown at the QCP?
r
t transitions (QPTs)[1] in heavy-fermion metals of both A short-rangedresonanting-valence-bond(RVB) spin-
s
pure and Ge-substituted YbRh Si (YRS) compounds liquid (SL) picture has recently been proposed to de-
. 2 2
at [2–6] are of great interest both theoretically and exper- scribe the metallic spin-liquid phase of heavy fermion
m imentally. Near a quantum critical point (QCP), these metalwithfrustratedantiferromagneticRKKYcoupling
- systems show exotic non-Fermi liquid (NFL) electronic [20]. A generic phase diagram was proposed in Ref. 20
d propertiesatfinitetemperatures,includingaT-linearre- in terms of magnetic frustration and Kondo correlation.
n
sistivity [3, 5, 7] and power law-in-T at low T followed There,asmallKondocouplingatlowfieldsmayco-exist
o
by a T-logarithmic specific heat coefficient at higher T with a small Fermi surface to form a (Kondo stablized)
c
[ [3], which still remain as outstanding open issues [8, 11– SL metal [21]. The disorder due to Ge substitution here

14, 16–19]. makes this proposal more attractive to account for the
1
v For the QPT in pure YRS, the “Kondo breakdown” SM state. It is of great interest to further explore the
9 scenario [8] offers a general understanding: competition mechanismofthis behaviour. Inthis work,we proposea
9 between the Kondo and antiferromagnetic RKKY cou- mechanism for the Kondo breakdown and the quantum
5 plings leads to a QCP, separating the antiferromagnetic criticality in Ge-substituted YRS at gc via a fermionic
4
metallic state from the paramagnetic Landau Fermi- large-N approach based on the symplectic group Sp(N)
0
. liquid state (LFL) with enhanced Kondo correlation. symmetry ona quasi-2dKondolattice. The competition
1
There, a magnetic phase transition and the emergence between the Gaussian fluctuating RVB spin-liquid and
0
(or the breakdown) of the Kondo effect occur simulta- the Kondo correlation (see Fig. 1) explains remarkably
7
1 neously and in the ground state the system undergoes a well both the transition and the NFL behaviours. The
: jump from a small to a large Fermi surface [9, 10]. The SM state indicated in experiments can be understood as
v
i understanding of the NFL properties near the QCP still quantumcriticalregionnearcriticalKondobreakdown.
X remainsanoutstandingopenissuethoughcertainaspects The Large-N Mean-Field Hamiltonian. Our start-
r have been addressed [8, 11–19]. ing point is the fermionic Sp(N) large-N mean-
a
Recent experiments on Ge-substituted YRS, however, field Hamiltonian of the Kondo lattice model [14]:
reveals intriguing distinct features. First, the magnetic HSMpF(N) = H0 + Hλ + HK + HJ, where H0 =
phase transition at gc′ occurs at a lower field than the hi,ji;σ tijc†iσcjσ +h.c. − iσ µc†iσciσ , Hλ =
Kondo destruction at g , leading to a decoupling of the h i
AFM and the LFL phacses (see Fig. 1(a)) [3]. Interest- Pi,σλ fi†σfiσ −2S , HJ =P hi,jiJijSiimp · Sijmp =
h i
ingly, similar NFL behaviour persists over a finite range P Φ Jαβf f +h.c.P+ N|Φij|2, H =
inmagneticfields atthe lowesttemperatures,suggesting hi,ji;α,β ij iα jβ hi,ji JH K
a possible exotic novel stable spin-disordered “strange- PJK iSiim(cid:2)p·sc = i, σh(cid:16)c†iσfiσ(cid:3)(cid:17)χPi+h.c.i+ i N|χJiK|2,
metal” (SM) ground state [5]. Open questions to be ad- whePre the antiferPromagnetic RKKY interactPion H is
J
dressedinclude: Dosethis SMbehaviourscomefromthe described by the fermionic Sp(N) spin-singlet with
SMgroundstateorfroma singleQCP?Whatisthe role [Sp(1)≃SU(2) for N =2], Jαβ = J = −Jβα be-
αβ
playedbythemagneticfield? Whatmechanismisbehind ingthegeneralizationoftheSU(2)antisymmetrictensor
2
(b)
FL* g LFL g
c
FIG. 1: (a) Schematic phase diagram of Ge-substituted YRS [3, 5]. The horizontal axis refers to either the applied magnetic
field B or g =g0−JΦ/Jχ with JΦ =JH, Jχ =JK. The solid curve T∗ labels the small-to-large Fermi surface crossover (see
(b)),whileTLFL (TFL∗)referstothecrossoverscalefromquantumcriticalregiontoLFL(FL∗),respectively. gc andgc′ denote
thequantumcriticalpointsforAF-FL∗ andFL∗-LFLtransitions,respectively. WhitedashedarrowsrefertotheRGflows. (b)
Schematic plot describing the competition between the RVB spin-liquid FL∗ phase and the Kondo LFL phase near gc at zero
temperature. The black heavy lines between two adjacent f-electrons refer to the RVB spin-singlets, while the helical lines
embedded in yellow stand for theKondo hybridization.
the fully screened Kondo effect [26, 27]. The mean-field
KondohybridizationandRVBspin-singletaredefinedas
χ ≡ hJK f† c i and Φ ≡ hJH J fα†fβ†i,
i N σ iσ σ ij N α,β αβ i j
respectivelPy [28]. Besides the RVB phPase, this approach
enables us to describe both the Kondo LFL and the
superconducting phases [29] via Bose-condensing χ-field
and the fermionic Sp(N) singlets, respectively [30]. Ex-
cluding superconductivity here (as it is likely suppressed
by magnetic field), HMF on a 2d lattice is shown to
Sp(N)
have a fractionalized Fermi-liquid (FL∗) phase [14] with
χ = 0, Φ 6= 0 for J ≪ J , and a Kondo-RVB spin-
i ij K H
liquid co-existing heavy-Fermi-liquid (LFL) phase with
χ 6= Φ 6= 0 for J ∼ O(J ) and a large Fermi sur-
i ij K H
face due to Kondo hybridization [14, 31]. The mean-
FIG. 2: The schematic RG flow diagram shows the competi-
field result captures qualitatively the competition be-
tion between Jχ and JΦ at fixeduχ =uΦ =0. Themagnetic tweenRVBsingletsandtheKondocorrelationsnearcrit-
field in experiments is proposed to follow the green dashed
icality (see Fig. 1(b)). However, to account for the NFL
line.
andcrossovers,weshallanalyzethedynamicsandfluctu-
ations beyond mean-field level via a perturbative renor-
ǫαβ = ǫαβ = −ǫβα = iσ2 [22–25]. H0 describes hop- malization group (RG) approach.
ping of conductionc-electrons,while HK denotes Kondo The Effective Field Theory Beyond Sp(N) Mean-Field.
interaction. We assume a uniform Jij = JH RKKY We consider here the Gaussian (amplitude) fluctuations
coupling on a lattice with i, j being nearest-neighbour around the mean-field order parameters χ and Φ with
i ij
sites, σ,α,β ∈ {−N2,··· ,N2} with N → ∞. Hλ de- dynamics in the FL∗ phase close to the QCP [32, 33].
scribes the local impurity fiσ electrons with λ being The effective action Seff for a fixed N =1 reads:
the Lagrange multiplier to impose the local constraint
h f† f i = Nκ where a constant κ ≡ 2S ensures
σ iσ iσ
P
3
S = S +S +S +S +S +S +S ,
eff 0 χ Φ K J G 4
iω
S0 = Z dkσX=↑↓ c†kσ (−iω+ǫc(k))ckσ +fk†σ (cid:18)− Γ +λ(cid:19)fkσ, Sχ =Z dkσX=↑↓hχkfk†σckσ +h.c.i+Xi Z dτ|χi|2/JK,
SΦ = Z dkXαβ hΦkǫαβfkαf−βk+h.c.i+hXi,jiZ dτ|Φij|2/JH, SK =JχσX=↑↓Z dkdk′h(c†kσfk′σ)χˆ˜†k+k′ +h.c.i,
SJ = JΦα,Xβ=↑↓Z dkdk′hǫαβΦˆ˜kfkβ′fkα+k′ +h.c.i, SG =Z dkhχˆ˜†k (−iω+ǫχ(k)+mχ)χˆ˜k+Φˆ˜†k (−iω+ǫΦ(k)+mΦ)Φˆ˜ki
S = uχ dk dk dk χˆ˜† χˆ˜† χˆ˜ χˆ˜ + uΦ dk dk dk Φˆ˜† Φˆ˜† Φˆ˜ Φˆ˜ , (1)
4 2 Z 1 2 3 k1 k2 k3 −k1−k2−k3 2 Z 1 2 3 k1 k2 k3 −k1−k2−k3
with k = (ω,k) and dk = ddkdω and τ the imaginary renormalizationand the field (or the Green’s functions)-
time. The actions S (S ) and S (S ) represents renormalization[11, 12] (see Ref. 33) :
χ K Φ J
the Kondo hybridization and RKKY interaction at
dj d−z 1
((bSey)onredp)retsheentsmethane-fiaecltdionlevoefl,threespqeucatdivrealtyi,c wGhaiulessiSaGn dlχ =−(cid:18) 2 (cid:19)jχ+ 2jχ3 +2jΦ2jχ
4
χ(qku(aχˆ˜rtki)c)arfleutchteuaFtoinugriefire-tldrasn,srfeosrpmeecdtivmeelya.n-fiΦekld(vΦˆ˜akr)iabalneds ddjΦl =−d2jΦ+4jΦ3 ; ddulΦ =−(d−z)uΦ−3u2Φ
du
((Φaˆmpl≡itud1eΣflufcαtu†fatβi†n)ganfideldχs (aχˆbov≡e m1eΣanf-fi†ecld)), Φreij- dlχ =−(d−z)uχ−4jχ2uχ−3u2χ
dspiseipjcetrivseeldyN.kǫicαn(βekt)iic, ǫχejn(ke)rgaiensd oǫΦfi(kth)eiarietitnhNeeraqσnutaiσderliaeσctitcraolnlys ddl (cid:18)Γ1(cid:19)=−z(cid:18)Γ1(cid:19)+4jΦ2 ; dmdlχ =zmχ+jχ2
ck, local singlet Φˆ˜k and the Kondo hybridization χˆ˜k, dmΦ =zmΦ, (2)
respectively. Thequadraticformsofǫ (k) andǫ (k)are dl
χ Φ
derived via integrating out c-electrons away from Fermi wheredl=−dlnΛwithΛbeing therunningenergycut-
surface [33, 34]. We find χˆ˜-field is not Landau damped offwithinmomentum-shellRGisusedwhiletheconstant
since the imaginary part of its self-energy vanishes: λ¯istheeffectivechemicalpotentialofthelocalf-electron
ImΣχ(ω) ∝ dǫδ(ω + ǫ − λ¯) = 0 as λ¯ ≫ ǫ + ω [33], [33]. Here, jχ,Φ(Jχ,Φ) refers to the renormalized (bare)
leading to aRjump in the Fermi volume at the critical coupling [33]. At two stable phases,the mass termm
χ,Φ
point, consistent with experiments [9, 10]. flows to a massive fixed point, m∗ → ∞. Near the
χ,Φ
QCP, their bare values vanish linearly with distance to
RG Analysis: Within our RG scheme k, kF and ω criticality: mχ,Φ ∝ |g −gc|. Since the effective dimen-
are rescaled as: k′ = elk; k′ = elk ; ω′ = ezlω sion d+z > 4, greater than the upper critical dimen-
where the dynamical exponentF z is seFt to 2. This sion, the Gaussian fixed point u∗ = v∗ = 0 is stable
scheme, distinct from the conventional one [35, 36], [41], and violation of hyperscaling is expected [40]. Two
allows the Fermi momentum k to flow the same way non-trivialintermediatecriticalfixedpoints arefoundat
as the momentum variable k: [kF] = [kF] = 1, effectively (jχ2∗, jΦ2∗)=(η,0)(P)and(0, d/8)(Q)(seeFig. 2). The
capturing the mixture of electron population in small fixed point at Q controls the transition between the two
and large Fermi surface (or the continuous evolution of FL∗ fixed points, while P separates FL∗ phase at jχ =0
fromtheKondoco-existingwithspin-liquid(LFL)phase
the Hall coefficient) at finite temperatures [36–38]. At
tree-level, Jχ, JΦ, uχ and uΦ are irrelevant couplings at jχ →∞.
for d > z = 2, while J , u and u become marginal To more precisely locate the QCP at finite values of
χ χ Φ
j , j ,the RGequationsforj ,j areobtainednearthe
for d = z = 2 ([Jχ] = (z −d)/2,[uχ] = [uΦ] = z −d), Φ χ χ φ
which allows a controlled perturbative RG analysis on fixedpointsP (withJχbeingfixedatJχ∗)andQ(withJΦ
the effective action on a quasi-2d lattice: d=2+η with being fixed at j∗) (see Ref. 33): d˜jΦ = −d + η ˜j +
Φ dl 2 2 Φ
η =0+ [39]. 4˜j3, d˜jχ = −(η +d/4−2(j∗)2)˜j + 1˜j(cid:0)3. The c(cid:1)ritical
Φ dl 2 Φ χ 2 χ
point C in Fig. 2, which controls the FL∗-LFL QPT,
r
The RG β-functions in the weak-coupling limit, J = is located at the intersect of the above two RG flows:
χ
J → 0, are readily obtained via diagrammatic per- ((j∗)2, (j∗)2) = (η,d/8). Note that C is an interacting
Φ χ Φ r
turbative approaches, which include coupling constants QCP due to the presence of Kondo interaction. As a
4
result,theω/T scalingindynamicalobservablesisfound tained via the conductivity [46] σ(T) = ρ−1(T) =
there via the Kondo breakdown scenario [10, 33]. −2e2 dk v2τ(k) ∂f with v being the electron group
Critical Properties and Crossovers. The correlation velo3ciRty,(2aπn)3dkτ−1(ω)∂ǫtkhe scattkering rate of the electron,
length ξ diverges near C : ξ ∼ |g−g |−ν with an expo-
r c given by the imaginary part of the conduction electron
nfleonwtoνf.jW−ejfi∗ndviathβa(t˜jν)is[3s3o].leTlyhdisetyeiremldisnνed=b1y/tzh,eleRadG- T−matrix ImT(ω) : τ−1(ω) = −cim2p kImTkk(ω+).
Φ Φ Φ Remarkably,wefindthattheT−matrixcPontributedfrom
ingtothelinearSM-LFL(SM-FL∗)crossoverscaleT
LFL thequasi-2dbosonicfluctuationsintheKondohybridiza-
(TFL∗) below which jΦ < jχ∗ (jΦ > jχ∗): TLFL,TFL∗ ∝ tion χˆ˜−field leads to the observed T-linear resistivity at
|g − gc|, in perfect agreement with the experiment on lowtemperatures: ρ(T)=a−1(1+cT/a),whereaandc
Ge-substituted YRS [3, 5]. This suggests that the main
areconstantpre-factors[33]. Wefindtheratioinresistiv-
effect of magnetic field B in experiment (represented by
ity at low temperatures: ∆ρ/ρ(T = 0)≡ (ρ(T)−ρ(T =
the coupling g = g −J /J ) near the QCP is to sup-
0 Φ χ 0))/ρ(T = 0), in reasonable agreement with that mea-
press J while J is near its critical value J ∼ J∗ (see
Φ χ χ χ sured in experiment [5].
Fig. 2andFig.1(a)). Carefulanalysisonthepre-factors
NFL: Specific Heat Coefficient. We further compute
gives [33] TFL∗ ∼ (Jχ)4 ∼ L−2η ≪ 1 with L being sys- the (normalized) scaling function of electronic specific
TLFL |χ|
tem size, giving rise to the difficulty to observe TFL∗ in heat coefficient in the SM region:
experiments (see Fig. 1(a)) [45]. We further identify T∗
as the crossover scale for the onset of the condensate χ γ T¯ ≡ e−ηl ∂E¯G(l) = T¯η2 TΛ¯ dx x2+η/2
gfie−ldgbeilnowfouwnhdic:hTj∗Φ∝<(gχ.−Ag )szu/bd-lwinietharzd/edp∼end1e−ncηe/izn, (cid:0) (cid:1) T(l)A(l) ∂T(l) (cid:12)(cid:12)l=l0 4 ZT1¯ sinh2(x/2)
c c (cid:12) (3)
and T∗ ∼ |χ| > 1, consistent with the experiments
[3,5].TLTFhLeexpJeχrimentallyobservedinverse-in-fielddiver- (where T¯ ≡ TLTFL, A(l0)∝|g−gc|−ζ with ζ ∼3η/2+η2
genceintheA-coefficientoftheT2termtotheresistivity near the QCP, and W(JΦ) is a non-universal constant),
ρ ∼ AT2 close to the FL∗-LFL QPT is also reproduced: contributed dominantly from the kinetic energy E¯G =
A ∝ ξ2 ∼ |g −gc|−2ν ∼ |g −gc|−1. The SM region in k eβǫǫΦΦ((kk))−1 of Φˆ˜ fields [33]. Here, l0 is a scale at which
Ref.5 onGe-substituted YRScanbe interpretedhereas Pm (l )∼1, and x=ǫ (l)/T(l) while T(l)=Tezl is the
Φ 0 Φ
theextendedquantumcriticalregimedowntoTFL∗ →0. scale-dependentdimensionlesstemperatureviathefinite-
temperature RG scheme [47]. As shown in Fig. 3, γ(T¯)
bears a striking similarity to that observed in Ref. 3: it
exhibits a power-law scaling behaviour at low tempera-
tures before it saturates at T = 0, i.e. γ(T¯) ∝ T¯−α,
followed by a logarithmic tail at higher temperatures
(a) (b)
γ(T¯) ∝ −lnT¯ with exponents α ∼ ζ ∼ 0.3(1) for an
estimated η ∼ 0.18, in excellent agreement with the ex-
perimental values α ∼ γ ∼ 1/3 [33]. The T-logarithmic
behaviourinγ(T¯)comesasaresultoftheGaussianfixed
point for d=2+η [48].
NFL: Local Spin Susceptibility. Finally, the observed
anomalousexponentinthedivergenttemperaturedepen-
dence of the zero-field local spin susceptibility χ ∼
loc
T−0.75 ∼ 1 for Ge-substituted YRS [49–51] with
T1−ηχ
η ∼ 0.25 is reasonably accounted for within our ap-
χ
proach [33]: χ (T) ∝ Jχ4(T) ∝ T2η−1, giving an esti-
loc T
mated η ∼0.36.
χ
FIG.3: Thelog-log (a)and log-linear (b)plotsfor theratio
Conclusions. We have theoretically addressed the
γ = CV/(TA(l0)) (normalized to 1 at T/D0 = 0 with D0 non-Fermi liquid and quantum critical properties of Ge-
being the bandwidth cut-off) at different |JΦ −JΦ∗| for η =
0.18 with the critical RKKY coupling J∗ = 0.5 and J∗ = substituted YbRh2Si2 by a field-theoretical renormaliza-
Φ χ
0.2. Theinset in (a) shows theexperimental data Φ(B,T)∝ tion group analysis on an effective field theory based
γ(T/TLFL) taken from Ref. 3 for Ge-substituted YRS, inset on the Sp(N) approach to the Kondo-Heisenberg lat-
in (b) shows TLFL v.s. |JΦ −JΦ∗|. Here, we fix the critical tice model. The quantumphase transitionand crossover
KondocouplingJχandevaluatethespecificheatwithvarious scales are well captured in terms of a competition be-
|JΦ−JΦ∗|=0.2, 0.25, 0.3, 0.35, 0.4, 0.45. tween a short-ranged fermionic resonant-valence-bond
spin liquid and the Kondo effect near a quantum critical
NFL: Electrical Resistivity. The finite tempera- point. The agreement of our predicted critical proper-
ture electrical resistivity ρ(T) near the QCP is ob- ties with experiments is remarkable. The strange metal
5
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104-2112- M-009 -004 -MY3, the MOE-ATU program,
Inthespin-liquidFL∗ phase,however,κ<1isexpected
the NCTS of Taiwan, R.O.C. (CHC), and the Austrian
duetolackofmagneticlong-rangedorder,andλtakesa
Science Fund project FWF P29296-N27(SP).
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the particle-hole channel: hij = hPσ=↑,↓fi†σfjσi, and
the Kondo hybridization in the particle-particle chan-
∗ Electronic address: [email protected]; Elec- nel: ξi = hPσc†iσfi†σi. However, ξi can be eliminated
tronic address: [email protected]; Electronic ad- by choosing a particular gauge, and hij is absent due to
dress: [email protected] a different (Sp(N)) symmetry exhibited in our model.
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