Quantum Criticality and Novel Phases: Summary and Outlook

0.29 MB·English

by  A. J. Schofield

physicastatussolidi, 24January2010
Quantum Criticality and Novel
Phases: Summary and Outlook
A.J.Schofield
SchoolofPhysicsandAstronomy,
UniversityofBirmingham,
Edgbaston,Birmingham,B152TT,
UnitedKingdom.
0
1 ReceivedXXXX,revisedXXXX,acceptedXXXX
0 PublishedonlineXXXX
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PACS 71.10.-w,71.10.Hf,75.10.-b,75.40-s
n
a ∗Correspondingauthor:[email protected],Phone:[phone]
J

4
2

] This conference summary and outlook provides a per- Indiscussingnovelorder,themagneticanaloguesofsu-
l
e sonal overview of the topics and themes of the August perconductivityareconsideredascandidatestatesforthe
- 2009Dresdenmeetingonquantumcriticalityandnovel hiddenorderthat sometimesdevelopsin the vicinity of
r
t phases. The dichotomy between the local moment and quantum critical points in metallic systems. These ana-
s
the itinerant views of magnetism is revisited and re- loguescanbethoughtofas“pairing”intheparticle-hole
.
t freshed in new materials, new probes and new theoret- channelandaretabulated.Thisanalogyisusedtooutline
a
m icalideas.New universalityandapparentzerotempera- aframeworktostudytherelationbetweenferromagnetic
ture phases of matter move us beyond the old ideas of fluctuationsandthepropensityofametaltonematictype
-
d quantum criticality. This is accompanied by alternative phases—which at weak coupling correspondto Pomer-
n pairinginteractionsandasyetunidentifiedphasesdevel- anchukinstabilities. This question can be related to the
o opinginthevicinityofquantumcriticalpoints. fundamentalrelationsofFermiliquidtheory.
c
[

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v
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1 Introduction Thevibrantgrowthofresearchactiv- (Stoner)magnetism.Thesmallorderedmomentandmetal-
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ity in the area of quantum criticality is amply illustrated liccharacteristicswhichpointtothelatterpicturestandat
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by the presentations at the Dresden meeting on Quantum oddswiththeirhightemperatureCurie-Weisssusceptibili-
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. CriticalityandNovelPhasesinAugust2009andthisasso- tiesandlargefluctuatingmomentmorecharacteristicofa
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ciatedproceedings.Itisasubjectareawhichiswell-served localmomentsystem.Theresolutionofthisparadoxisthe
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byreviews,bothtechnical[1]andnon-specialist[2,3],as inclusion of critical spin-fluctuations which renormalize
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wellastextbooks[4]andthisbriefconferencesummaryis the magnetic equation of state [5]. Numerousresearchers
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: certainlynotintendedtocompetewiththesenorevencover have played significant roles in understanding this—the
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all the activity of the Dresden meeting. Rather it gives a “conventional theory” of itinerant quantum criticality—
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X subjective pathway through some of the main themes of culminatingintherenormalizationgrouptreatmentofthe
themeetingandsomepersonalthoughtsonkeyquestions Hertz-Millisaction[6,7]
r
a outstandinginthefield.
toni2shQinugalnetguamcycwrithiiccahl,itiyn Qthueanctounmtecxrtitoicfaliittiynehraasnatnsyass-- S =Z Xωn dDq(cid:20)(x−xc)+q2+ q|ωz−n2||ψ|2+u|ψ|4···(cid:21).
tems, can be traced back to the 1960s with the puzzling (1)
behaviour of d-metals at or on the border of ferromag- As we have seen in this meeting, there are many cases
netism.Materialslikepalladium,Ni3Al,Ni3Ga,YNi3and where this theory seems to give both a qualitative (for
ZrZn2 combinedfacetsoftwo distinctviewsofmagnetic example, the work presented here by Shinsaku Kambe
behaviour:localmomentmagnetismanditinerantmoment et al. [8] on NMR of USn3) and sometimes a quantita-
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2 A.J.Schofield:QCNP09:Summaryandoutlook
critical behaviour for a range of parameters as presented
byThomasVojta[16].Itseemsunlikelythatthisexhausts
the list giventhe experimentsin MnSi at pressuresabove
thequantumphasetransitionwhichsuggestacriticalstate
ofmatter[17].
Despite the successes of the “conventional theory”
of the itinerant quantum critical metal, a growing set of
counter examples has emerged. They seem to lie out-
side the theory at a fundamental level as typified by two
systems. CeCu6−xAux, thoughnominallyabovethecon-
ventional theory’s upper critical dimension, shows E/T
scaling and anomalous exponents, as well as evidence
of critical properties throughout the Brillioun zone [18].
YbRh2Si2 showsevidenceofmultipleenergyscales [19]
converging to zero at its magnetic-field-driven quantum-
Figure 1 Schematic phase diagram of a quantum critical criticalpoint.New “badactors” havebeenpresentedhere
metal with a novel phase (labelled “?”) centred over the including Ce3Pd20Si6 by Silke Paschen [20]. One com-
quantumcriticalpoint(qcp).Thepower-lawquantumcrit-
mon denominator amongst them is that these materials
icalmetalisapproachedalongpath(a).Path(b)allowsthe are heavy fermion systems where the quantum criticality
instabilitytothenovelphasetobestudiedasaFermiliquid emergesfromaFermiliquidstatewhichisalreadyexotic.
theoryinstability.
Likethed-metalswhereourstorybegan,thesef-metal
materials combine local moments and itinerancy, but un-
like them, they do this at microscopic level with distinct
tiveagreementwithexperiment(asshownbyGilbertLon- local moments which interact weakly with separate con-
zarich[9,10]). ductionbands.Yetdespitethismicroscopicseparationap-
In the latter case we saw the ferromagnetic metal parent at high energies, these degrees of freedom com-
demonstrating the same physics as the Reizer singular- bine at low energy. Together they form a metallic state
itywhicharisesfromtheweakcurrent-currentinteractions comprised of quasiparticles which reveal their local mo-
in all metals [11]. These unscreened long-range interac- mentconstituentsentropicallyintheirheavymassandnu-
tions would ultimately drive any metal into a non-Fermi merically in a Fermi volume that includes the magnetic
liquidstate—albeitatmicro-Kelvintemperatures.Theidea moments as if they were itinerant. The suggestion that a
that one might employ this mechanism and amplify it to newformofquantumcriticalityassociatedwithbreakupof
makesuchastatevisibletoexperimentonceinspiredideas thesecompositequasiparticlesandrevealedinthecollapse
like the U(1) gauge theory descriptionsof the resonating ofthelarge-volumeFermisurface[21]wasemphasizedby
valencebond(RVB)state.Thesewereconjecturedforthe QimiaoSi[22]andrecurredinmultiplepresentationshere.
high-Tc cuprates to account for, among other things, the The definitive proof of a volume change in the Fermi
linear T normal state resistivity [12,13]. Yet, as the fer- surface at a quantum critical point is arguably still lack-
romagnetic metal illustrates, quantum criticality can also ing. Quantum oscillation experiments have been the pre-
generate long-ranged, unscreened interactions and asso- eminent technique for the pinning down the fermiology
ciated power-law resistivities. This has led many to see of correlated electron systems. In that vein, the work
the cuprates as quantum critical systems. The precise na- of Shishido et al. [23] in CeRhIn5 is very promising
ture andlocationof the putativequantumcriticalpointin and so too is the research presented here by Stephen Ju-
the cuprates is hotly debated [14]. Fresh insight by Subir lian[24]beginningtheelucidationoftheFermisurfaceof
Sachdev presented at this meeting [15] addresses this by YbRh2Si2withthesametechnique.Inthisconferencewe
showingthatsuperconductivityandspin-densitywaveor- wereshowntwoothertechniqueswhichhavebeenusedto
derinterferetopushtheassociatedquantumcriticalpoint greateffectincorrelatedoxidematerialsbutarenowbeing
intotheunderdopedregion. appliedtotheheavyfermionsystems.Photoemissionspec-
Nevertheless,thereremainsthemoregeneralquestion troscopyofferstheprospectofimagingtheelectronstruc-
of whether power-law temperature dependencies of scat- ture of quasiparticles and we saw, in its angle-integrated
tering down to the lowest temperatures, often now pre- form, the emergenceof the Kondo resonance [25]. When
sented in the form of colour coded scattering maps, are angle-resolved, one might hope to see the Fermi surface
always indicative of an associated quantum critical point itself[26].OfcourseobservingtheputativeFermisurface
(Fig.1).Obviouscounterexamples,suchastheLuttinger volumechangeat a quantumcriticalpointis a significant
liquidstatein1D,werediscussedatthemeetingbyThierry challenge for ARPES. The second technique which has
Giamarchi. So too were the Griffiths singularities in dis- been employed to great effect in the correlated oxides is
ordered quantum magnets which again can give singular low temperature STM and at this conference we saw the
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resultsofitsapplicationtoheavyfermionsystemsbySea-
musDavis and his group.We were shownthe STM view
of how local moments and conductionelectronscombine
to form composites and many were struck by how much
more insight this new experimental development might
soongiveus.
One should not overlook the role that transport mea-
surements, particularly magneto-transport, can make in (a) (b)
characterizingthenatureofthequantumcriticalpoint[27].
Figure 2 (a) A schematic of the lowest order spin-
The appearance of a linear-in-magnetic-field transverse
fluctuationinteractionsin the particle-particlechannelre-
magnetoresistanceshouldcharacterizeaconventionalden-
sponsible for pairing-making and pair-breaking. (b) Ro-
sity wave transition [28]. This has now been observed
tating the direction of the propagator indicates that these
experimentally in Ca3Ru2O7 by Kikugawa et al. [29].
sameinteractionscould“pair”intheparticle-holechannel.
The complexity of density wave and other transitions in
Asingularityherewoulddenoteaninstabilitytoaformof
this material was presented to us by Malte Grosche [30].
magnetism(orchargeorder).
Other work presented in this conference by Friedemann
andco-workersarguesthatevidenceforE/T scalingcan
be seen in the Hall effect of YhRh2Si2, thereby linking
Nevertheless the prospect of being able to address single
for the first time the anomalous properties of this mate-
atomic sites means that when we reach the appropriate
rial with the other canonical anomalous quantum critical
regimes we can expect true quantum simulation of many
system,CeCu6−xAux.
ofthemodelswhosephysicscontinuestoeludetheoretical
Thetheoreticalideastoaccountfortherichbehaviour
description. Yet, as Achim Rosch pointed out in a com-
of metallic quantum critical matter are far from settled
plementary theory talk, not only can experiments on the
and experiments may even suggest that they are still too
Mottregionbewellmatchedtodynamicmean-fieldtheory
conservative. This is not just the apparent phase of non-
predictions [38], but exploiting the metastability intrinsic
Fermi liquid matter in MnSi, but the observation that the
tocold-atomicgasesallowsanovelformofpairinginthe
two collapsing energy scales seen in YbRh2Si2 do not repulsive Hubbardmodel[39]. This naturally links to the
need to hit zero at the same point [31] but could open
secondkeythemeofthemeeting.
a spin-liquidwindowbetweenthe classically-orderedand
quantum-disorderedphases.
3 Novel Phases One of the incentives for studying
The inspiration for new theories will not just come
quantumcriticalmatteristhatatornearthequantumcriti-
from heavy fermionexperiments.In this meeting we saw
calpointweoftenseetheappearanceofneworderedstates
two other classes of quantumsystem, each of which pro-
(shownschematicallyinFig.1).Thenaturalexplanationis
vide a new window into correlated quantum behaviour.
that the fluctuations induced by the quantum melting of
Quantum magnetism offers systems which are often con-
an orderedphase can drive other instabilities. The classic
ceptuallysimplerthantheiritinerantcounterparts(without
exampleistheappearanceofsuperconductivitynearanti-
the damping effects of other low lying degrees of free-
ferromagneticquantumcriticalpoints[40,41].
dom) and has a long history of inspiring new theoretical
Spin-fluctuation mediated superconductivity and su-
ideasfromRVB[32,14]todeconfinedcriticality[33].Gi-
perfluidity has, like quantum criticality itself, been much
amarchi stressed the utility of quantum spin systems as
studied theoretically and experimentally from the 1960s
simulators of Bose condensates[34] as well as providing
onwards.Condensingthiswealthofknowledgeintoasin-
detailedtestsofoneofthefewnon-Fermiliquidphaseswe
glesetofprinciplesischallengingyet,indiscussionatthe
thinkweunderstand—theLuttingerliquid.Infacttwodis-
meeting, Gil Lonzarich articulated what might be termed
tinctrecentexperimentssuggestLuttingerliquidproperties
“Lonzarich’s Rules” [47,48,49] in homage to the older
may extend over a wider range than naively thought [35,
“Matthias’ Rules”[50] which inspired a previous genera-
36].
tion’s search for phonon mediated superconductors. Like
The other more recent quantum playground comes in “Matthias’ Rules” these have not explicitly appeared in
the form of trapped ultra-cold atomic gases held in opti- printbutareotherwiseunliketheminalmosteveryrespect.
callattices.Thisrapidlydevelopingfieldwassurveyedfor So in looking for higher superconducting transition tem-
usbyImmanuelBlochandAchimRosch[37].Thepresent peraturesoneshould:
stateoftheartexperimentsinFermigasesgodowntotem-
peraturesthatarestillabout15%oftheFermitemperature –Lookontheborderofmagnetism(i.e.nearaquantum
and, in the Mott insulating region, are a factor of two or criticalpoint),
so higher than the expected Nee´l temperature. Thus we –Preferantiferromagnetismtoferromagnetism,
are not yet at a stage where we can simulate directly the –Ifferromagnetic,favouruniaxialanisotropy,
quantum critical systems being studied in the solid state. –Prefer2Delectronicstructuresover3D,
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4 A.J.Schofield:QCNP09:Summaryandoutlook
Table1UnconventionalsuperconductorswheretheCooperpairshaveinternalangularmomentumand/orcentreofmass
momentumoraspacialtexturehaveexoticmagneticanaloguesintheparticle-holechannel.Theanalogyisexploredhere
whereV isthepairinginteractionandg denotesasymmetryfactorwhichcandependonspinandorientationaroundthe
Fermisurface.
Superconductor(particle-particlechannel) Magnet(particle-holechannel)
Conventionals-wave Stonerferromagnet
∆∗ =XVhcˆ†k′σcˆ†−k′σi Mσ = X gσ,σ′hcˆ†k′σ′cˆk′σ′i
k′,σ k′,σ′
Unconventional(d-waveetc.) Pomeranchuk(d-wavenematic[42]etc.)
∆∗(k)=XVkk′hcˆ†k′σcˆ†−k′σi Mσ(k)= X gk,k′;σ,σ′hcˆ†k′σ′cˆk′σ′i
k′σ k′,σ′
Inhomogeneous(s-waveFFLO[43,44]) Inhomogeneous(spiral,density-wave)
∆∗(q)=XVhcˆ†k′+q/2,σcˆ†−k′+q/2,σi Mσ(q)= X gσ,σ′hcˆ†k′+q/2,σ′cˆk′+q/2,σ′i
k′σ k′,σ′
Inhomogeneous(d-waveFFLOetc.) Inhomogeneous(d-density-wave[45]etc.)
∆∗(k,q)=XVkk′hcˆ†k′+q/2,σcˆ†−k′+q/2,σi Mσ(k,q)= X gk,k′;σ,σ′hcˆ†k′+q/2,σ′cˆk′−q/2,σ′i
k′σ k′,σ′
Abrikosovfluxlattice Skyrmionspin-texture[46]
–Lookformaterialswithalargespin-fluctuationscale, densationenergy)isan outstandingquestion.Itisa ques-
–Prefersinglebandmaterialsornestedmultibandmate- tionthatisrelevantnotjustfortheheavyfermionsystems
rials, butalsotheironbasedsuperconductorswhereissuessuch
–Avoidcompetingantiferromagneticandferromagnetic aslocalmomentphysicsversusitinerancyareactivelybe-
fluctuations. inginvestigatedasweheardfromNan-LinWang[62]and
The question of how the order parameters of magnetism fromMaw-KuenWu[63].
and superconductivitycan compete and even coexist was Superconductivityisnottheonlynovelphaseseennear
posed by Georg Knebel [51]. In essence these “rules” at- quantumcriticalpoints.Theenigmatic17Ktransition[64]
tempttocapturethatcompetitionbetweentheCooperpair- inURu2Si2continuestobeacauseoffascinationandnew
makingandCooperpair-breakingeffectsofthe spinfluc- ideasandexperimentsweredescribedinnumerousposters
tuations [see Fig. 2(a)] and how the resulting gap struc- presentedhere.AsstressedbySeamusDavisindescribing
ture interplays with underlying electron structure to opti- STMworkonthisproblem,thisisnot“hiddenorder”—its
mize the condensation energy. Each of these rules come presenceinthephasediagramisunmistakable.Ratheritis
with caveatsand counterexamplesyet new superconduc- moreappropriately“darkorder”inthatitistransparentto
torswhichseemto conformto theseguidelineswere pre- our usualprobes,yet new theoryand experimentseem to
sentedheresuchasCe2PdIn8 [52].PascoalPagliusoalso beonthevergeofilluminatingit[65,66,67,68].
stressed the role of low dimensionality and hybridization Theothermysteriousphasewhichwasdescribedinthe
as key ingredients in the quest for new superconducting meetingwasthatseeninthevicinityofthemetamagnetic
materials. Moreover Satoru Nakatsuji has discovered su- quantumcriticalend-pointinSr3Ru2O7 [69,70].Thelat-
perconductivity in the hole system β−YbAlB4 [53,54] estresultspresentedbyAndyMackenziedemonstratethat
whichpointsallthemoretoauniversalmechanismforsu- this is a true phase surrounded by thermodynamic phase
perconductivityagainfromanovelmetallicstate. transitions with an intriguing entropy landscape [71,72].
However,theconferencealsosawarangeofsupercon- Thesuggestionthatitisanelectronicnematicphasefinds
ductorswhichweresuggestiveofotherformsofpairingin- some experimental support [70] but is certainly not the
teraction.Ifavalencetransitioncouldbemadecriticalthen onlysuggestion.Equallyimportantisthequestionofwhat
ShinjiWatanabeargueditcouldcausepairing[55],asper- isdrivingitsformation:isitformedbythequantumcriti-
hapsisseeninCeCu2(Si1−xGex)2 [56].Thecorrespond- calfluctuationsofthemetamagneticquantumcriticalend-
ing magnetic analogue would be a pairing from metam- point, or is the suppression of metamagnetism simply al-
agnetic fluctuations—perhapsthe case in UGe2, [57,58]. lowing a sub-leading order instability to become promi-
TusonParkraisedtheintriguingquestionastowhetherthe nent?
local character of some of the anomalous heavy fermion Howthenarewetoaddressthesetwokeyissues?One
quantumcriticalpointscouldalsodrivesuperconductivity promisingavenueto investigateboth the identity of these
in, for example,CeRhIn5 [59,60]. Indeed,given the rich “darkorder”phasesandtheir originis to exploitpossible
magnetic phase diagram of materials like CeCoIn5 [61], parallels with superconductivity[73]. Taking our inspira-
determining the dominant driving mechanism for super- tionfromthatbodyofworkonspin-mediatedCooperpair-
conductivity(i.e.theonethatcontributesmosttothecon- ingwecould“rotate”onearmofthepairpropagatorround
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pssheaderwillbeprovidedbythepublisher 5
~q+~k
Γ = Γ˜ + Γ˜ Γ
~q
Figure 3 ByapproachingthequantumcriticalpointfromwithintheFermiliquidstate(alongpath(b)inFig.1)wecan
considerinstabilitiesintheparticle-holechannelusingthestandardmicroscopicrelationsinFermiliquidtheory[75].In
particular,thequasiparticlepoleinthefullyrenormalizedsingle-particleGreen’sfunctionmeansthatthevertexfunction
Γ can be expressed as in integral equation involving the non-singular part Γ˜ as k → 0. This can then be used to re-
late the scattering amplitude to the Landau parameters and hence the relationship between divergentscattering and the
Pomeranchukinstability.
[see Fig. 2(b)]. In effect we are now asking whether crit- phaseshoulddevelopbeforethequantumcriticalpoint.It
icalfluctuationsmightcause“pairing”singularitiesinthe wouldbeaninstabilityintheFermiliquidstateandhence
particle-holechannel[74].Thesewouldcorrespondtoex- addressableundertheassumptionsofFermiliquidtheory.
otictypesofmagnetismorchargeorder. Tracingtheeffectofthefluctuationsintheparticle-hole
The idea that there is an analogy between supercon- channelmeansstudyingvertexfunction.Inthecaseofthe
ductivityandmagnetismstemsfromtheveryearliestdays Fermiliquidtheself-consistentequationsfortheparticle-
of BCS theory [76]. It is also one which has re-emerged holevertexfunction(seeFig.3),arewellknown[75].They
in the intervening years. Christian Pfleiderer provided us describe the connection between the on-shell (scattering)
with an excellent example of this in describing his work partofthevertexfunctionandtheoff-shellLandauparam-
onthenovelmagnetictextureseeninMnSi[77].Itseems eters
i
likelythatthisisa skyrmionlatticewhichcanbe thought i Fl
ofasthemagneticanalogueofthemixedstateofasuper- Bl = 1+ Fli . (2)
2l+1
conductor [46]. The mixed state is a rather conventional
superconducting state whose magnetic analogue is more Here l is the angular momentum channel and i = s or
exotic. What if we now consider more exotic supercon- a for spin symmetric or antisymmetric components. This
ductors and ask what magnetic states they correspond tellsusthatthescatteringamplitudeBli inagivenangular
to? Unconventional superconductors which pair at finite momentumchanneldivergesatthepointwhentheLandau
angularmomentumcorrespondto Pomeranchukdistorted parameterFli inthesameangularmomentumchannelap-
nematicfluidswhichbreakrotational,butnottranslational, proachesthePomeranchukinstabilitycondition
symmetries [78]. The Fulde-Ferrel-Larkin-Ovchinikov i
F
(FFLO)state[43,44]statemapstoaspirallingspin-density l →−1. (3)
wave[79].Otherstateslikead-densitywave[45]statecan 2l+1
alsobeviewedasoneelementofarathergenericmapping Taken at face value this suggests that say, an instabil-
(seeTable1). ity in the Stoner channel (l = 0, i = a), is uncoupled
Yetourcruderotationfromtheparticle-particletothe from Pomeranchukdistortions in higher angular momen-
particle-holechannelsuggeststherecouldbemoretothis tumchannels(l 6= 0)soferromagneticfluctuationsdonot
analogythansimplyoneofenumeratingpossibleorderpa- promote such a distortion. This contrasts with supercon-
rameters.Itmightbeusedtoindicatethepropertiesofsuch ductivity where magnetic (particle-hole) fluctuations in-
phases.Forexample,wehaverecentlyexploitedthismap- duce particle-particlepairing in other angular momentum
ping to show that the sensitivity to disorder that charac- channels. Yet this is not quite the end of the story since
terizesunconventionalsuperconductorsisparallelledbya the antisymmetry of the wavefunction couples the differ-
Pomeranchuk(nematic)phase[80]. ent angular momentum together in the Landau relation
Thisanalogymayalsobe ableto giveinsightintothe 0= (Bs+Ba).Thustoproperlyanswerthisquestion
l l l
mechanism behind the formation of such novel magnetic amicProscopicexampleneedstobestudiedinmoredetail.
phases, at least in certain cases. One such case, perhaps In fact since this meetingMaslov and Chubukovhave in-
relevantforSr3Ru2O7, is whetherferromagneticfluctua- dependentlymadeprogressinthisregard[82].Whetheror
tions(likethoseofmetamagnetism)favouraPomeranchuk notthereisaunifiedframeworkaccountingfornovelphase
typeinstability?Oneimaginesapproachingtheferromag- formationatquantumcriticalpoints,thereisclearlymuch
neticquantumcriticalpointfromtheparamagneticsideat remainingtobediscovered.
T =0[i.e.alongpath(b)inFig.1].(Weassumesufficient
uniaxial anisotropy to prevent the transition being driven 4 Conclusions Quantum criticality and associated
firstorderbysofttransversemodes[81].)Anyputativenew novelphases representsome of the most significant chal-
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6 A.J.Schofield:QCNP09:Summaryandoutlook
lengestoourunderstandingofcondensedmattersystems. [17]N.Doiron-leyraud,I.R.Walker,L.Taillefer,M.J.Steiner,
Theveracityof this was capturedby the paneldiscussion S.R.Julian,andG.G.Lonzarich,Nature425(6958),595–
whichwaschairedbyPiersColemanandsummarizedelse- 599(2003).
whereintheseProceedings[83].Capturingalltheactivity, [18]A.Schro¨der,G.Aeppli,R.Coldea,M.Adams,O.Stockert,
discussion,argumentanddebateisanimpossibletaskand H.v.Lo¨hneysen,E.Bucher,R.Ramazashvili,andP.Cole-
this summary is woefully incomplete. However, drawing man,Nature407(6802),351–355(2000).
thethreadstogether,wehavebeeninspiredbynewtypesof [19]P. Gegenwart, T. Westerkamp, C. Krellner, Y. Tokiwa,
S.Paschen,C.Geibel,F.Steglich,E.Abrahams,andQ.Si,
materialsandevennewcontrollableexperimentalsystems
Science315(5814),969–971(2007).
in our quest to explore these intriguing phenomena. The
[20]S.Paschen,LuandLasubstitutedCe Pd Si ,Thesecon-
continuingsurprisesateverytwistandturninthisstoryin- 3 20 6
ferenceproceedings.
dicatethatnatureisclearlymoreimaginativethanweare!
[21]P.Coleman,C.Pe´pin,Q.Si,andR.Ramazashvili,J.Phys.:
Thismeetinghasalsoseen theapplicationofnewexperi-
Condens.Matter13,R723–R738(2001).
mentaltoolsandtechniquestoinvestigatethesesurprises.
[22]Q. Si, Quantum criticality and global phase diagram of
Thetheoreticalquestionsthatarebeingraisedandthenew
magnetic heavy fermions, These conference proceedings
ideasrequiredtoanswerthemareinspiringthis,andanew,
(arXiv:0912.0040).
generationofscientiststotakeupthesechallenges. [23]H.Shishido,R.Settai,H.Harima,andY.O¯nuki,J.Phys.
Soc.Jpn.74(4),1103–1106(2005).
Acknowledgements Iacknowledgemanyusefulconversa- [24]A.B. Sutton, P.M.C. Rourke, V. Taufour, A. McCollam,
tionswithfellow attendeesof themeetingbut Iamparticularly G.Lapertot,G.Knebel,J.Flouquet,andS.R.Julian,Ob-
indebted to conversations with Gil Lonzarich. I also gratefully
servationoftheJ-sheetoftheFermisurfaceofYbRh Si ,
2 2
acknowledgethecontinuingsupportoftheRoyalSocietyandthe
Theseconferenceproceedings(arXiv:0910.1082).
EPSRC,aswellastheorganizersofQCNP09.
[25]M.Klein,A.Nuber,F.Reinert,J.Kroha,O.Stockert,and
H.v.Lo¨hneysen,Phys.Rev.Lett.101(26),266404(2008).
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