cos!0 =MB

2 " M 2 !K

! = 0

« spin-flop »

ferro Uferro = ! M2 ! 2MB

très haut champ

Uflop = ! M2 (2cos2"0 !1)! 2MBcos"0 +K(1! cos

2"0 )

Uflop = ! M2 cos2" ! 2MBcos" +K sin2"UNéél = ! ! M

2

Uflop !UNéél = (2 ! M2 !K )cos2"0 ! 2MBcos"0 +K

(1)

(2) θ0

θ0

Uflop !UNéél = !(MB)2

2 ! M 2 !K+K

champ critique de spin-flop Bflop =K(2 ! M 2 !K )

M 2 " 2K !Uflop !UNéél = 0

K

Bflop

UNéél

Uflop

transition spin-flop

Transition « spin flop »

Bflip = ! MUferro !UNéél = 2 ! M2 ! 2MB

?

Transition « spin flop »

magnetization measurements, to band-structure calculationsof the magnetic anisotropy energy in Gd5Ge4, finding goodagreement.

II. EXPERIMENTAL DETAILS

Single crystals of Gd5Ge4 were obtained from the AmesLaboratory Materials Preparation Center,15 which weregrown using the Bridgman technique. Appropriate quantitiesof gadolinium !99.996% metals basis" and germanium!99.999%" were cleaned and arc melted several times under

an argon atmosphere. The buttons were then remelted to en-sure compositional homogeneity throughout the ingot andthe alloy drop cast into a copper mold. The as-cast ingot waselectron beam welded in a tungsten Bridgman style cruciblefor crystal growth. The ingot was heated in a tungsten meshresistance furnace under a pressure of 8.8!10−5 Pa up to1925 °C then withdrawn from the heat zone at a rate of4 mm /hr. The as-grown crystal was oriented by backreflec-tion Laue technique. Samples were extracted from the ingot,and prepared with a polished surface perpendicular to the baxis with a size of approximately 2!2!3 mm3. The mag-netization was measured using a Quantum Design SQUIDmagnetometer.

The XRMS experiment was performed on the 4ID-Dbeamline at the Advanced Photon Source at an incident beamenergy corresponding to the maximum in the resonant dipolescattering cross section at the Gd L2 absorption edge.12 Thescattering geometry is shown in Fig. 4. A photon polarizedperpendicular to the plane of scattering is said to exhibit "polarization, while a photon polarized in the plane has #polarization. The incident beam was linearly polarized in thehorizontal scattering plane !# polarized" with a cross sectionof 0.22 mm !horizontal"!0.1 mm !vertical". The samplewas mounted on the cold finger of a helium flow variabletemperature insert !VTI" with the b axis parallel to the scat-tering vector Q, and the c axis perpendicular to the horizon-tal scattering plane. A vertical magnetic field was applied!perpendicular to the scattering plane" using a superconduct-ing 4-Tesla split-coil magnet. Pyrolytic graphite !0 0 6" func-tioned as both a polarization analyzer and to suppress thecharge background in the measurement of the magnetic scat-tering signal.

The resonant scattering of interest, at the Gd L2 absorp-tion edge, is due to electric dipole transitions between thecore 2p states and the 5d conduction bands. The 5d bandsare spin polarized through the exchange interaction with themagnetic 4f electrons. The #-# scattering geometry is real-ized when the scattering plane for the sample is horizontalbut that for the analyzer is vertical. In this geometry, themagnetic signal is sensitive to the component of the ordered

-15 -10 -5 0 5 10 15

-0 .6

-0 .4

-0 .2

0 .0

0 .2

0 .4

0 .6

spin-f lop

Hsf

= 8 .8 kOe

spin-flop

Gd5Ge

4

T = 10 KH||c

M(µ

B/G

d)

H (kOe)

FIG. 2. Field dependence of the magnetization of a zero-fieldcooled Gd5Ge4 single crystal measured at T=10 K with the mag-netic field parallel to the c axis.

0 20 40 60 80 100 120 140

0

2

4

6

8

10

12

14

Gd5Ge

4

H || c

Hs

f(k

Oe

)

T (K)

ZFAFM phase

SF phase

PM

FIG. 3. Temperature dependence of the spin-flop field Hsf de-rived from the field dependence of the magnetization measured atdifferent temperatures. !From Ref. 13" PM, SF, and ZFAFM repre-sent the paramagnetic phase, the spin-flop phase, and the zero-fieldantiferromagnetic phase, respectively. The dashed lines representthe two field-dependence measurements and two temperature-dependence measurements using XRMS in the present experiment.

FIG. 4. The experimental arrangement consisting of the sample,analyzer, and detector. k and k! are the incident and scattered x-raywave vectors, respectively. The magnetic field H was applied alongthe vertical direction. The switch between #-" geometry !the de-tector arm in the horizontal plane" and #-# geometry !the detectorarm along the vertical direction" was accomplished by a motor-driven analyzer angle $an.

TAN et al. PHYSICAL REVIEW B 77, 064425 !2008"

064425-2

cos!0 =MB

" M 2 !K

MB =M cos!0 =M 2B

" M 2 !K

linéaire en B dans l’état spin-flop

Plan du cours

  Magnétisme sans interaction   Magnétisme atomique   Moments magnétiques localisés   Environnement

  Magnétisme localisé en interaction   Interactions d’échange   Modèle de champ moyen du ferromagnétisme   Anisotropie: hystérésis et métamagnétisme

  Au delà du champ moyen   Hamiltonien d’Heisenberg: du classique au quantique   Ondes de spin ferromagnétiques et antiferromagnétiques   Magnétisme frustré et liquides de spin

  Magnétisme itinérant   Paramagnétisme d’un gaz d’électrons libres   Instabilité magnétique de Stoner   Effets Hall quantiques

Hamiltonien d’Heisenberg quantique F

H = !J2

S!i S!i+!

i,!"

Hi! = !J2S!i S!i+!énergie du fondamental de paires

Hi! = !J4(S!i + S!i+! )2 ! Si

2 ! Si+!2"

#$%=

J2s(s+1)! J

4(S!i + S!i+! )2

minimum si S!i + S!i+! = 2s Ei!

0 =J2s(s+1)! J

42s(2s+1) = ! Js

2

2

E 0 = !J2

s2 = ! NzJs2

2i!"

J>0 et 1er voisins uniquement

J

S!i / /S!i+!

énergie du fondamental réseau: spins alignés

N sites et z premiers voisins

Hamiltonien d’Heisenberg quantique F

fondamental: état propre ?

J H = !J2

Siz

i,!" Si+!

z + SiySi+!

y + SixSi+!

x S+ = Sx + iS y

S! = Sx ! iS y

S± s,ms ! s,ms ±1

H = !J2

SizSi+!

z !J4i,!

" Si+Si+!

! + Si!Si+!

+

i,!"

FM = !!!.....!

Sx =S+ + S2

!

Sy =S+ ! S2i

!

FM Stotalz FM = Ns

H FM = !JNzs2

2FM !

J4

Si+Si+!

! FM + Si!Si+!

+

i,!" FM

Si+ FM = Si+!

+ FM = 0S+ s,ms = s = 0

0

état maximalement aligné selon z

FM = !!!.....! est bien l’état fondamental quantique ferromagnétique

0

Hamiltonien d’Heisenberg quantique AF: paires

H = !J2

S!i S!i+!

i,!"

Hi! =J2S!i S!i+!énergie du fondamental de paires

Hi! =J4(S!i + S!i+! )2 ! Si

2 ! Si+!2"

#$%= !

J2s(s+1)+

J4

2 !Si +!Si+!( )

2

minimum si Ei!0 = !

J2s(s+1)

J<0 et 1er voisins uniquement

J

S!i = !S!i+!

état de Néél

paires de Néél S!i,S!i+! selon z et anti // Ei!

Néél = !J s2

2> Ei!

0

paires de Néél pas le fondamental: fixer la direction des spins a un coût

H ...!"... = #J s2

2...!"... +

J4Si+Si+!

# ...!"... + Si#Si+!

+ ...!"...même pas un état propre!

! ..."#...0

Ei!Néél

Ei!0 =

SS +1 !

" #" 1

limite classique

Néél = ...!"...

Hamiltonien d’Heisenberg quantique AF: réseau

H = !J2

S!i S!i+!

i,!" J<0 et 1er voisins uniquement

E 0 = NzEi!0 = !

Nz J2

s(s+1)

passage au réseau = somme de paires

si toutes les paires sont satisfaites:

problème: pas toujours possible de satisfaire toutes les paires EAF ! E 0

J

état de Néél

énergie de l’état de Néél

H =J2

SizSi+!

z

i,!! ENéél =

!Nz J2

s2 fondamental classique sur réseau

EAF ! ENéél

E0 est une borne inférieure

ENéél est une borne supérieure

Hamiltonien d’Heisenberg quantique AF: réseau

Emin =J2

Eimin

i! = "

Nz J2

s(s+ 1z)

H = !J2

S!i S!i+!

i,!" =

J2

Hii"

Hi = S!i. S!i+!

!

! Hi =12

S!i + S

!i+!

!

!"

#$

%

&'

2

( Si2 + S

!i+!

!

!"

#$

%

&'

2)

*++

,

-..

)

*

++

,

-

.

.

Hi : spin i avec z voisins

(1) maximise: (zSi)2 (2) -zSi

Eimin =

12(z!1)s (z!1)s+1( )! s(s+1)! zs(zs+1)"# $% Ei

min = !zs(s+ 1z)

avec

orientations pas fixées a priori fluctuations quantiques

ENéél =!Nz J2

s2 Emin < ENéél

Emin ! ENéélz!"

s!"si

limite classique

prise en compte de z premiers voisins

= !zs2 ! s

Hamiltonien d’Heisenberg quantique AF: réseau

Emin = !Nz J2

s(s+ 1z) ENéél =

!Nz J2

s2E 0 = !Nz J2

s(s+1)

limite classique borne inférieure paires indépendantes

borne inférieure avec premiers voisins

énergie du fondamental

EAF

voie possible: partir de l’état de Néél puis inclure les corrections quantiques

inclure les fluctuations de l’orientation des spins: ondes de spin

Z=1: Dimères quantiques

état à liaisons de valence 2D

J0>>J1

Réseau Shastry-Sutherland 2D (SrCu2(BO3)2) 

Quantum spin liquids 505

Figure 6. Spin ladder. The ellipses represent singlets on the rungs.

Since the first observation of a spin gap in a spin-1 chain, several systems with similarproperties have been discovered. The example that has been most extensively studied in themid 90’s is that of the S = 1/2 spin ladders (figure 6) defined by the Hamiltonian:

H = J‖∑

i

∑

α=1,2

Siα · Si+1α + J⊥∑

i

Si1 · Si2. (19)

In that case, the presence of a gap is very intuitive in the limit J⊥ # J‖. In the limit J‖ = 0, theground-state wavefunction is just a product of singlets constructed on the rungs of the ladder:

|ψ〉 =∏

i

[i1, i2] (20)

where [i1, i2] is the singlet constructed on rung i:

[i1, i2] = | ↑i1↓i2〉 − | ↓i1↑i2〉√2

. (21)

When J‖ is small but not equal to zero, the ground-state wavefunction can be expanded inpowers of J‖, and it essentially retains the form of equation (20). To make an excitation, onehas to break a singlet, and this costs a finite energy equal to J⊥ minus a small correction dueto the kinetic energy to be gained thanks to J‖. It is however possible to show, using quantumfield theory arguments, that there is a gap for all values of J⊥ as long as J⊥ > 0. Severaloxides (SrCu2O3, CaV2O5,. . . ) are very good realizations of spin ladders [11]. Amongthe recent developments in that field, one can cite the study of an organo-metallic ladderCu2(C5H12N2)2Cl4 in a magnetic field [12]. The interesting effect is that a strong magneticfield can close the spin gap by pulling down one of the first triplet excitations, similarly to in theexample of two spins depicted in figure 5. In the same spirit, 2D systems with a spin gap havebeen discovered. The system CaV4O9 is particularly interesting because the building bricksare not dimers but four-site plaquettes [13]. The physics is nevertheless quite similar. Themost recent example is the compound SrCu2(BO3)2 [14]. It can be seen as a 2D arrangementof dimers, and the low-temperature physics is again quite similar. The properties under a highmagnetic field are quite remarkable, however. The magnetization curve does not rise smoothlybetween the field that closes the gap and the saturation field but exhibits plateaux at somerational values of the magnetization. Work is in progress to explain this effect.

4. Frustrated magnets and low-lying singlets

From the previous discussion, one might feel that the alternative ‘ordered or gapped’ exhauststhe physics of quantum magnets. This is far from the truth, however. It has been known for along time that, in addition to lowering the dimensionality, there is another way to increasequantum fluctuations, namely by introducing frustration, that is a competition betweenexchange integrals. The classical example is the AF Heisenberg model on the triangular lattice.There is frustration because it is impossible to satisfy fully and simultaneously the three bondsof a given triangle, and one must settle for a compromise. In the case of the triangular lattice,recent numerical results strongly suggest that the system still develops helical long-range orderby adopting a three sublattice configuration [15]. So in that case the quantum system retainsthe classical order in spite of strong quantum fluctuations.

Echelles de spin 2D (SrCu2O3)

E 0 = !Nz J2

s(s+1)

Il existe des systèmes pour lesquels on a un fondamental de paires singulet (z=1)

J0

J1