Re-entrant hidden order at a metamagnetic quantum critical end point

8 Pages·0.44 MB·English

by  N. Harrison

Re-entrant hidden order at a metamagnetic quantum critical end point
N. Harrison1, M. Jaime1 and J. A. Mydosh2,3
1National High Magnetic Field Laboratory, LANL, MS-E536, Los Alamos, New Mexico 87545
2Kamerlingh Onnes Laboratory, Leiden University, NL-2300 RA Leiden, The Netherlands
3Max-Planck Institut for Chemical Physics of Solids,
N¨othnitzer Str. 40, D-01187 Dresden, Germany
3 Magnetization measurementsofURu2Si2 inpulsedmagneticfieldsof44Trevealthatthehidden
0 orderphaseisdestroyedbeforeappearingintheformofare-entrantphasebetween≈36and39T.
0
Evidence for conventional itinerant electron metamagnetism at higher temperatures suggests that
2
there-entrant phaseis created in the vicinity of a quantumcritical end point.

n
a PACSnumbers: 71.45.Lr,71.20.Ps,71.18.+y
J

6
1 Recent studies of itinerant electron magnetism in URu2Si2 can be explained by a scenario in which the
strongly correlated d- and f-electron metals have shown magnetic field first destroys the HO phase before creat-

] that metamagnetism gives rise a new class of field- ing a new field-induced re-entrant phase [4] in the vicin-
l
e induced quantum phase transition [1, 2]. Sr3Ru2O7, ityofthemetamagnetictransition(seeFig.1foraphase
r- CeRu2Si2andUPt3[3]areallconsideredexamplesofsys- diagram). IEM is accompanied by a pronounced asym-
t tems that could posses a quantum critical end point, in metry between the occupancy of itinerant spin-up and
s
. which a notional line of first order phase transitions ter- spin-down f-electron states [15], brought on by the sud-
t
a minates at zero rather than finite temperature [1]. Here den population of the spin-up component as it sinks be-
m we propose that URu2Si2 may be the first example of a low the Fermi energy εF at a magnetic field BM. Mag-
- system in which thermodynamic instabilities associated netization measurements reveal that the magnetic field-
d
with the end point give rise to an ordered phase at high induced phase is accompanied by the opening of a gap
n
magneticfieldsandlowtemperatures[4]. Thisbehaviour in the spin-up f-electron band at BM. We argue that
o
is reminiscentofthe creationofsuperconductivity inthe sucha gapcouldbe compatible with a spin-singlet order
c
[ vicinityofanantiferromagneticquantumcriticalpointin parameter that breaks translational symmetry, of which

zerofield[5]. Weshowthatthepresenceofmultiplemag- theorbitalantiferromagnetic(OAF)phase(recentlypro-
2
v netictransitionsinURu2Si2 atlowtemperatures[6,7,8] posed by Chandra et al. [9] to explain the origin of the
can be ascribed to re-entrant phenomena arising from HO) is one such example.
4
4 the interplay between itinerant electron metamagnetism
Figure 2a shows examples of the magnetization M
2 (IEM)andthehiddenorder(HO)parameterrecentlyat-
1 tributed to orbital antiferromagnetism [9]. of URu2Si2 measured in pulsed magnetic fields of up
to 44 T at several different temperatures. The data
0
3 URu2Si2 belongs to a class of stronglycorrelatedmet- areobtainedusingawire-woundsample-extractionmag-
0 als in which f-electrons, rather than being localised and netometer in which the sample is inserted or removed
/ giving rise to magnetism, develop a distinctly itinerant from the detection coils in-situ. While the experimen-
t
a character[10]. Coulombinteractionscause the quasipar- tal curves in Fig. 2 appear similar to those measured
m
ticle effective masses to be heavily renormalised, mak- by other groups [6, 7, 8], the phase diagram obtained
- ing the energetic rewards for forming ordered ground- in Fig. 1 upon extracting the positions of the maxima
d
n states substantially higher than in normal metals [11, in the differential susceptibility χ = µ0∂M/∂B|T at dif-
12]. Indeed, in addition to forming the HO phase at ferent temperatures is markedly different. In a recent
o
c To 17.5 K [9], URu2Si2 becomes superconducting at study, Jaime et al [4] noted that the magnetocaloric ef-
≈
: Tc 1.2 K [10]. The presence of an itinerant f-electron fect can cause severe variations in sample temperature
v ≈
i Fermi surface [13, 14] also furnishes URu2Si2 with the in pulsed magnetic field experiments if the sample can-
X essential preconditions for IEM [15], by which the f- not exchangeheat with the bath as the magnetic field B
r electronsreverttoalocalisedbehaviourupontheiralign- changes. Thiseffectisparticularlyseriousifthesampleis
a
ment in strong magnetic fields. IEM is considered to ac- too large, a poor thermal diffusivity isolates the sample,
countfortheincreaseinthemagnetizationby 1µBper or if the field is swept too rapidly. Adequate isothermal
≈
U atom at magnetic fields between 35 T and 40 T, equilibrium in pulsed magnetic fields could, however, be
≈ ≈
although the existence of multiple magnetic transitions achieved by using a long-pulse magnet (with a field de-
has remained controversial[6, 7, 8]. Recent observations cay constant of 0.25 s) combined with a sample thick-
≈
that local moment antiferromagnetism occurs within a ness of 150 µm [4]. It is by making such provisions in
≃
minority phase that is destroyed by fields in excess of the present study that we obtain a phase diagram that
15 T [9, 16] call for a re-examination of the bulk high agrees more closely with specific heat measurements in
magnetic field phenomena in URu2Si2. static magnetic fields [4].
In this paper, we show that multiple transitions in TheexistenceofIEMofasimilartypetothatobserved
2
in Sr3Ru2O7 [17], UPt3 [18, 19] and CeRu2Si2 [20, 21] is itinerantf-electronband is shifted by the Zeeman inter-
evidencedattemperaturesabove 6KinFig.2bbythe actiontoenergiesjustbelowεF atBM (seeFig.3),caus-
≈
presence of a single broad maximum in χ . The dashed ingM toundergoadramaticincreasebyasmuchas1µB
lineinFig.1indicatesthatthelocationofthisfeatureat per f-electron atom [15]. As a result, f-electrons that
BM 37.9Tdoesnotchangesignificantlywithtempera- were mostly itinerant below BM become mostly aligned
≈
ture. The rapid increase in the χ at BM with decreasing and localised at fields above BM. The field BM corre-
temperature, shown in Fig. 2c (filled squares), implies sponds to a situation where the Fermi energy εF inter-
that the jump in M sharpens with decreasing tempera- sectsthemiddleofthespin-upf-electronbandcausingit
ture. Suchbehaviourisconsistentwiththeexistenceofa tobehalfoccupied. Thisleadstoanapproximatelytem-
firstordercriticalendpointatafieldBMattemperatures perature independent M at BM (see Fig. 2a) but with χ
wellbelow6Kthatisbroadenedbythermalfluctuations increasing dramatically with decreasing temparture (see
athighertemperatures[2]. Ratherthandivergingindefi- Fig. 2c). The continuation of the temperature indepen-
nitely,however,themaximuminχvanishesbelow 6K dence of M at BM below 6 K, accompanied by an
≈ ≈
on entering the field-induced ordered phase recently in- abrupt reduction in χ, indicates that the field-induced
dentifiedinspecificheatmeasurements[4]. Thefactthat orderedphase stabilisesa situationwhereapproximately
BM occurs within the field-induced ordered phase im- half of the 5f-electrons become localised while the other
plies that fluctuations associatedwiththe metamagnetic halfremainitinerant. Thistype ofbehaviourimplies the
critical end-point [1] could play a role in its formation. existence of a charge gap in the spin-up itinerant 5f-
Strong fluctuations in the vicinity of quantum critical electron band at εF, at BM.
points can cause metals to become highly susceptible to
The formation of a charge gap in the spin-up 5f-
orderas a means of loweringenergy [1, 2]. A wellknown
electron band is consistent with the existence of a spin-
exampleisprovidedbythecreationofsuperconductivity
singlet order parameter that breaks translational sym-
in the vicinity of an antiferromagnetic quantum critical
metry. Orderparametersof the charge-densitywave[22]
point [5].
andOAF[9]typebothpossessthisessentialproperty;the
The intense magnetic fields combined with formation latteralsobreakstimereversalsymmetry. Theybothin-
ofthe field-inducedphasebelow 6 KinURu2Si2 make volve singlet pairing of quasiparticles at a characteristic
≈
the process of identifying whether the critical end point translational wavevector Q, where Q is determined by
would otherwise terminate at T = 0 less certain than details of the Fermi surface topology [9, 22, 23]. In fact,
with Sr2Ru3O7 [1], UPt3 [18, 19] or CeRu2Si2 [20, 21]. regardlessofthe pairingsymmetry,any singletorderpa-
This normallyrequires evidence for non-Fermiliquid be- rameterthatinvolvesspatialvariationsinchargedensity,
havior. Fortunately, the transitionin the specific heat C orrelativechargedensities betweenone ormoreelectron
at 5 K [4] appears to be first order (being of 0.25 K channels, will lead to such a gap. Order parameters of
≈ ≈
in width, albeit without observable hysteresis),implying thistypearealsoamenabletothepossibilityofre-entrant
that the region above 6 K is free from thermal fluc- behaviour (see below). Evidence that the HO and field-
≈
tuations of the field-induced HO parameter. This region induced HO phases have a common origin may be pro-
cantherefore be investigatedfor non-Fermiliquid effects videdbythefieldandtemperaturedependenceofχ. The
associated with IEM [3]. Transport studies are thus far transitionintotheHOphaseisdevoidofanypronounced
incomplete, presently yeilding only a broadmaximum in features in the temperature and field dependence of χ at
the magnetoresistanceatBM [4]. Thestronglydivergent fields below 34 T, while those into the field induced
≈
behaviour of χ above 6 K in Fig. 2c together with the phase exhibit similar behaviour over a narrow interval
≈
approximately linearly decreasing variation in C/T with between 36.8 and 37.1 T (see Fig. 1). Furthermore, all
T [4] at 38 T could, nevertheless, be possible indi- transitions into (or out of) both phases evolve into ones
≈
cations of non-Fermi liquid behaviour. Similar types of that are first order in the limit T 0, although actual
→
behaviourinotherf-electronsystemshavebeenascribed magnetic hysteresis remains undetected [4]. First order
to the presence of a non-Fermi liquid [3]. transitions give rise to pronounced maxima in χ and/or
magnetocaloricheatingasthefieldisswept[4]. Somede-
ChangesinthevalueofM throughthetransitionspro-
greeofsimilaritybetweenthelowandhighmagneticfield
vides clues as to the nature of the ordered phase. For
B < 25 T, M is weakly dependent on temperature, ex- phasesisalsoapparentinthetemperaturedependenceof
hib∼iting a predominantly Pauli paramagnetic response the χ in Fig. 2c at 34.0 T and 37.9 T respectively.
typical of heavy Fermi liquids [11, 12]. This, together In order to understand how re-entrance of the HO pa-
withspecificheatmeasurementsabovetheorderingtem- rameter can occur, it is instructive first to consider the
perature [4] and de Haas-van Alphen measurements be- density of electronic states (DOS) within the ordered
low the ordering temperature [13, 14], unambiguously phase for B < 35 T, which has received most atten-
establishes the existence of a heavy Fermi liquid with an tion thus far [1, 9]. Figures 3a-e show a schematic of
effective Fermi energy of order 10 meV. In the itinerant the evolution of the total DOS with B with (black lines)
f-electronpicture,aheavyFermiliquidresultsfrommix- and without (red lines) ordering. At B = 0 (Fig. 3a
ing of the f-electrons with regular conduction electron red line), the spin-up and and spin-down Fermi surfaces
states [11]. When IEM occurs, the spin-up component are degenerate. The introduction of a periodic charge
3
potential V(r Q0) within the HO phase must therefore developsinthevicinityofBM,enablingtherealisationof
·
resultintheindependentformationofbandgapsforboth a re-entrant hidden order (RHO) phase with a modified
spins (black line). This process is only efficient at min- translationalvectorQ . Strong magnetic fluctuations at
↓
imising the energy of the systems if a significant part BM can be associated with the vanishingly small energy
of the DOS is gapped at εF. The introduction of B in thatseparatesspin-upelectronsinlocalisedanditinerant
Fig. 3b, however, causes the energies of the spin-up and states at εF [2]. This, combined with the weak disper-
spin-down bands to split, leading to spin-up and spin- sion of the spin-up f-electron band, makes the system
down Fermi surfaces of different sizes. The efficiency by especially vulnerable to forming an ordered phase. Or-
which V(r Q0) can gap both spins therefore becomes deringisespeciallyeasytorealiseiftheperiodicpotential
·
progressively worsened as B is increased, leading to the V(r Q ) becomes comparable to the bandwidth of the
↓
·
weakening of the gap and, eventually, to its destruction spin-up f-electrons, because it will succeed at gapping
inamanneranalogoustothatofreachingthePaulilimit the entire spin-up density of f-electron states regardless
ofasingletsuperconductor[24, 25,26]. Apreviousmag- of the value of Q and regardless of the absence of well
↓
netoresistancestudy hasprovidedexperimentalevidence definedspin-upmomentumquantumnumbers. Asignifi-
for weakening of the gap in fields of 25 T [27]. Ulti- cantamountofenergyisgainedbyopeningsuchagapat
≈
mately,theorderedphasemustbe destroyedatacritical εF, and this wouldthen accountfor the observednarrow
field Bc BM, whereupon the spin-up and spin-down gap in the spin-up f-electron band (see Fig. 3d). The
≤
Fermisurfacesbecomeextremelyasymmetric. Theeffect value of Q need only be optimised to match the topol-
↓
ofB ontranslationalsymmetry-breakingspinsinglet or- ogy of the spin-down Fermi surface, which continues to
derparametershasbeenextensivelymodeledusingmean be present at BM. Once B > BM, ordinary Fermi liq-
field theory [24, 25, 26]. One possibility is that the tran- uid behaviour is expected to be restored, but with the
sition evolves into one that is first order that terminates spin-up f-electrons being fully polarized as depicted in
at a critical field Bc = ∆0/√2σgµB. Figure 3c depicts Fig. 3e. The pronounced asymmetry between spins in
the density of electronic states at Bc where the spin-up theFermiliquidmakestheformationofanorderedphase
andspin-downf-electronsstateshavebecomeclearlyre- unlikely, enabling the emergence of a Schottky anomaly
solved and singlet gap formation is no longer favoured. in the specific heat [4].
Upon estimating the size of the order parameter using In summary, we presentM data which shows that the
the BCS relation 2∆0 = 3.52kBTo [24, 26] and inserting HOparameteris firstdestroyedby Zeemansplitting ina
free electron parameters for the spin σ = 1 and g-factor magneticfieldbutthenrestoredinare-entrantphase[4].
2
g = 2,we obtainBc 32 T.This is of comparableorder The T and B dependence of χ reveals that IEM plays a
≈
to the first order-like transition at 35 T [4] obtained role in its re-entrance,possibly indicating that HO is re-
≈
fromthecurrentmeasurements(seeFig.1)aswellasspe- storedinresponsetomagneticfluctuationsinthevicinity
cific heat[4]. Giventhat the product σgµB in f-electron of a metamagnetic quantum critical end point [1, 2]. If
systems candepartfromthe free electronvalue [11], this true, this could be the first observation of the creation
agreement may be merely circumstantial. However, a of an ordered phase in the vicinity of a magnetic field-
further prediction of mean field theory is that both the induced quantum critical end point. We propose the ex-
transitiontemperatureTo(B)asafunctionofB [24]and istenceofseparateHOandRHOphasescharacterisedby
the critical field Bc(T) as a function of T [26] intersect a common spin-singlet translational symmetry-breaking
the axesina perpendicularmanner,andbothcanbe ex- order parameter with slightly different translationalvec-
panded in series of even powers of B and T respectively. tors Q0 and Q↓.
A plot of T2 versus B2 should therefore yield a line that This work is supported by the National Science Foun-
o
intercepts both axes in an approximately linear fashion. dation,theDepartmentofEnergyandFloridaState. We
This is confirmed in Fig. 3f upon making such a plot thankChristianBatista,Kee-HoonKim,GregBoebinger
with actual URu2Si2 data. An interesting situation then and John Singleton for useful discussions.
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5
FIG. 3: a-e Schematics of the evolution of the total DOS
in URu2Si2 with B (as indicated) before (red lines) and
after (black lines) formation of the HO or RHO phases.
Prior to ordering, mixing between conduction electron states
and f-electron states gives rise to a large “Abrikosov-Suhl
resonance”-like feature [11]. f A plot of the transition tem-
perature squared To2 versus the magnetic field squared B2
taken from specific heat and transport data in Reference [4].
3.5E-2
B
10
URu Si B || [001]
M
2 2 itinerant localised
c
f-electrons f-electrons
2.0E-2
)
K
5
(
B
T
c
HO RHO
QCEP?
0
0
30 40
B (T)
Figure 1 of Harrison et al.
B (T)
0 10 20 30 40
1.5
a
URu Si B || [001] 0.46 K
2 2
)
U
1.0

/

B
7 K

m(
0.5 8 K
M
B B

c M
0.0
B
B b 7 K c
M
-2
d 1x10
8 K B ~34 T
/
M
9 K -2
1x10
d
0
-3
5x10
11 K
m
= c
B
M
c
0
0
35 40 0 10
B (T) T (K)
Figure 2 of Harrison et al
a b
B = 0 0 < B < B
c
f-electrons
) ) f-electrons
e( e(
n n broaden

HO gap
gap narrows
e e e e
F F
c d
B = B < B B = B
c M M
) (HO destroyed) )
e(
e( n
n

RHO gap
e e e e
F F
e
B > B 300 f
M
) (RHO destroyed)
e(
n ) 200 B
2
K M
(
B
2
o100 c
T

0
e e 0 500 1000 1500
F 2 2
B (T )
Figure 3 of Harrison et al