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Patrick Bruno, ESRF Theory Group
Summary Introduction: classical magnets vs. quantum magnets
Majorana’s stellar representation of quantum spin states
spin wavefunction as a (fictitious) thermodynamics partition function
Berry’s geometric phase in Majorana representation
Towards a geometric theory of quantum many-spin systems
Geometric theory of quantum spin systems
Introduction: Classical magnets vs. quantum magnets
“Traitté de l´aiman” Dalencé (1691)
object with the property of being able to point into a certain direction (= dipole moment) dynamics of a magnet = changes of its orientation
Petrus Peregrinus de Maricourt (13th cent.), William Gilbert (16th cent.)
classical concept for a magnet:
dynamics described by classical equations (Landau-Lifshitz-Gilbert)
classical magnets
m=m u
damping term
1
eff
eff
dmdt
dEm d
γ= × +
= −
u B u
Bu
m = constant
= equation of motion of a classical gyroscope = dynamics of a point on a sphere (phase space of dimension 2)
u
“0”
“0” “1”
( ) ( )( )
8 2 812 6Fe O OH tacn Br
tacn 1,4,7-triazacyclononane
=
8Fe
( )( )
3+
3+
6 ions Fe 5 / 2 10
2 ions Fe 5 / 2
SS
S
= ↑ ⇒ == ↓
( )2 2 2ˆz x yKS D S S= − ⋅ − + −H S
molecular magnets
genuine quantum effects: tunneling, quantum interferences, entanglement ...
Quantum spin systems with exotic ordering spin nematics (magnets without dipole moments)
Example:
21 1 2 2 1 2
,i j
H J J
S S S S
spin 1S
1,1 0 (dipole moment)
1, 1
1,0 0, 0 (quadrupole moment)zzQ
m
m
1 20 J J
→ spontaneous quadrupolar ordering
cf. “hidden” order in heavy-fermions systems (hexadecapole ordering, or even higher multipolar ordering, has been proposed)
cf. ultracold spin 1 gases
Majorana’s stellar representation of quantum spin states
AGEWMETRHTOS MHDEIS EISITW (PLATWN)
“Let no one ignorant of geometry enter here” (inscription above the entrance of Plato’s Academy in Athens)
Plato (427-347 B.C.)
x
y
z
S
N
Ω ( )ψ ψ=Ω Ω
( ) ( ) 2Qψ ψ=Ω Ω
quantum state: ψ
( ) ˆH H=Ω Ω Ω
traditional description of spin systems
( )basis set eigenstates of :
, 1, 2, ,zJ
JM M J J J J=
= − − −
( )M JMψ ψ=
matrix element of the Hamiltonian:ˆMMH JM H JM′ ′=
• convenient for numerical calculations • physically not very insightful (except for particular cases) • needs to single out some arbitrary axis
spin coherent states
2 1 d4Jπ+
= ∫ Ω Ω Ω1
( ) rotated to the axisJJ=Ω Ω
• “geometrically” more satisfactory description (no need to single out some axis) • the quantum state is entirely characterized the so-called Husimi function • the Hamiltionian is entirely described by its diagonal matrix elements
( ) ( ) 2Qψ ψ=Ω Ω
( ) ˆH H=Ω Ω Ω
select some axis z
stereographic projection
x
y
z
S
N
n
( ) ( ) 2Qψ ψ=n n ( )1Log
Qψ
n
cubic anisotropy (ground state)
20S =
( )( )4 4 4
223
1x y z
KH S S SS S
−= + +
+
Ettore Majorana (1906 – 1938 (?))
Majorana’s “stellar” representation
the wave function or the Husimi distribution of a system with spin J has exactly 2J zeros (= stars)
the quantum state is entirely characterized by the location of its 2J stars (= constellation)
( )ψ ψ=Ω Ω( ) ( ) 2Qψ ψ=Ω Ω
ψ
x
y
z
S
N
Ω
for a coherent state , the Majorana stars are all degenerate and located at the antipode of Ω coherent are quasi-classical states (smallest transverse spread)
Ω
entangled (non-classical) states are characterized by Majorana stars that are not all degenerate ↔ a generic spin J state is obtained a as a unique fully symmetrized product of 2J spin ½ states
x
y
z
S
N
spin wavefunction as a (fictitious) thermodynamics partition function
spin ½ coherent state: iˆ ˆcos sin e2 2
ϕθ θ = + −
Ω z z
spin J coherent state: ( )2
2 factors
J
J
⊗≡ ⊗ ⊗ ⊗ ≡Ω Ω Ω Ω Ω
( ) ( )( )
2
1 perms 2
12 !
J
i P iiP JJ =∈
Ψ ≡ −
∑ ⊗n n
(unnormalized) spin 2J state parametrized by its 2J Majorana stars : ( )1, ,2i i J=n
( ) 2
22
1
2 1norm of state : d4
12 1 d4 2
J
Ji
i
JZ
J
π
π =
+Ψ Ψ ≡ Ψ Ψ = Ψ Ψ
− ⋅+ =
∫
∏∫
Ω Ω Ω
Ω nΩ
some notation conventions
( )
( ) ( )
1/21 21 2
1 2
1 2 1 2
ˆ, ,1scalar product of 2 spin 1/2 states: exp i2 2
ˆ ˆ , , oriented area of spherical triangle , ,
Σ + ⋅ = Σ =
z Ω ΩΩ ΩΩ Ω
z Ω Ω z Ω Ω
( )2D-pl 1 2 ln CU q q D∝ − +
( )2D-sph 1 2 chln CU q q D∝ − +
chchordal distance: 2sin2
D θ =
planar 2D Coulomb interaction
spherical 2D Coulomb interaction
D
uniformly charged infinite wires
Dch
uniformly charged semi-infinite wires
( )21 d 14
Pπ Ψ =∫ Ω Ω
( ) ( )( )
( )( )
2,22
1sin
12
2 2 1 i
J J
J
i
J JP
Z Zθ
=Ψ
+ +≡ Ψ
Ψ
=Ψ ∏ Ω nΩ Ω
( )( ) ( )( )2 1
expJ
JZ
Vβ Ψ+
= −Ψ
Ω
( ) ( )( )22 1 d exp4
partition function of a fictitious gas of independent particles interacting with 2 fixed test charges
JJZ V
J
βπ Ψ+
Ψ = −
=
∫ Ω Ω
( ) ( )β Ψ=
= ≡∑
2
1with inverse "temperature" 1 and ,
J
ii
V VΩ Ω n
( )θ
≡ −
,spherical "Coulomb repulsion" : , ln 2 sin2
iiV Ω nΩ n
( ) ( )( )1fictitious free energy: ln JF Zβ
Ψ ≡ − Ψ
2J fixed test charges
x
y
z
S
N
Ω
ni
probability of finding our system in the coherent state :Ω
Majorana stars
N S
probability density = fictitious gas density
(random state of spin J = 25)
→ expression of the (real !) energy in terms of the Majorana stars: e.g.:
( )11 1/ 22
JJ n= − =
( )1 2
12
1 12 1
2
Jn nJ σ+
= − =−
( )23 13 12
1 2 3
12 13 23
1 1 11 3 3 3 3 / 22 1
3
Jn n n
J
σ σ σ
σ σ σ
− + − + − = − =
+ +−
2 1sin
2 2ij i j
ij
θσ
− ⋅ ≡ =
n n
from known expressions of ZJ one can derive explicit expressions for the expectation value of the dipole moment for arbitrary value of J :
etc...
+ expressions of higher multipoles (quadrupole, octupole, etc...)
( ) HH
Ψ ΨΨ ≡
Ψ Ψ
( ) ( ) 2 2 2z x y iH K J D J J HB J nΨ = − ⋅ + + − + =
(systematic diagram technique)
quantum metric
( )2
iji j
F∂ Ψ=∂ ∂
gn n
allows to measure the quantum (so-called Fubini-Study) distance between quantum states
random Hamiltonian
40J =
(classically chaotic system)
Leboeuf et al. (1990); Hannay (1996)
→ statistical (Coulomb) repulsion of Majorana stars = consequence of the bosonic character of the Majorana stars
stereographic representation
( )the metric becomes degenerate, i.e., the
volume measure det goes to zero,
whenever two or more Majorana stars coincide
g
the fictitious free-energy determines the quantum metric:
Berry’s geometric phase in Majorana representation
Sir Michael V. Berry
Berry phase in a nutshell
R0
ˆ ( )RHamiltonian:
set of external parameters
ˆ ( ) ( ) ( ) ( )n n nEΨ = ΨR R R R( )nΨ Rbasis states:
( )( )0 0 0( ) ( ) exp ( )Cn n n n ni γδ ′Ψ → Ψ = + Ψ R R Radiabatic transformation:
( ) ( ) ( )n nC
n i dCγ = Ψ ∇ Ψ ⋅∫ RR R R
non-integrable (Berry) phase:
dynamical phasenδ =
E
R
what is the value of this phase ???
M.V. Berry, Proc. Roy. Soc. London A 392, 45 (1984)
Berry phase as a fictitious flux in parameter space (3D)
( ) ( ) ( )
( )
( ) ( ) ( )
geometric (Berry) phase:
Berry connec
with tio ( )n
n
n
n
i n n d
d
i n n
γ = ∇ ⋅
= ⋅
= ∇
∫
∫
R
R
R R R
A R R
A R R R
( ) ( )
( ) ( )
( )
Stokes' theorem: ( = )
with
Berr
y curvatu
e
rn nn
n
m n
dS
i n n
i n m m n
γ
≠
= ⋅
= ∇ ×
= ∇ × ∇
= ∇ × ∇
∫∫
∑
R
R R
R R
B R n B
B R A R
( )( ) ( )( )( )
2
ˆ ˆn
m n m n
n m m ni
E E≠
∇ × ∇=
−∑
R RB R
R R
( ) is large if is close to a degeneracy (flux of a Dirac monopole located at )
nB R RR
R∗
∗
Berry phase: , 1, ,
C MM J J Jγ = − Ω
= − −
z
y
solid angle Ω
Dirac monopole of unit magnetic charge
path C
Berry sphere
n
Canonical example of Berry phase: spin in a magnetic field
( )ˆHamiltonian: B= − ⋅ = − ⋅B J Jn n
external parameter = unit vector n
= Aharonov-Bohm phase of an electric charge 2M in the magnetic field of a Dirac monopole of unit magnetic charge
quantization of the Dirac monopole ↔ topology
i2
A α αα
Ψ ∂ Ψ − ∂ Ψ Ψ ≡
Ψ Ψ
2i2
f α β β α α β β ααβ
∂ Ψ ∂ Ψ − ∂ Ψ ∂ Ψ ∂ Ψ Ψ Ψ ∂ Ψ − ∂ Ψ Ψ Ψ ∂ Ψ = −
Ψ Ψ Ψ Ψ
f A Aαβ α β β α≡ ∂ − ∂
here label the 4 coordinates of the 2 Majorana stars iJ Jα n
Berry connection (= “vector potential”):
Berry curvature (= “flux” density):
2
1
dJ
B i ii=
Φ = ⋅∑∫ A n
geometrical (Berry-Aharonov-Anandan) phase for a round trip in state space:
x
y
z
ni Ai
plays the role of a vector potential for i iA n
geometrical phase and Berry curvature (Majorana rep.)
( ) ( ) ( )ˆ1 1dˆ4 2 1
ii i i
i i
FP
π Ψ
∂ Ψ×= = − × − ⋅ ∂
∫z nA Ω Ω A Ω n
z n n
( ) ( )1 d4i iPπ Ψ= ∫A Ω Ω A Ω
( ) ˆ1ˆ2 1 1
i ii
i i
× ×= − − ⋅ − ⋅
z n Ω nA Ωz n Ω n
with
x
y
z
ni Ai
Ω
= vector potential for a Dirac string carrying unit flux quantum, entering along z and exiting along Ω
fictitious force acting test charge ni
( )the flux density is given by PΨ Ω
magnetic monopole
( )2
1
dd
Jj
ijj i
Ht=
∂ Ψ⋅ =
∂∑n
fn
0 11 0
x
p
HxHp
∂ − = ∂
quantum dynamics
symplectic structure of the projective Hilbert space
energetics (interactions)
= generalization for a spin J of the Bloch equation of motion of a spin ½ = quantum version of the Landau-Lifshitz (classical) equation for spin dynamics
quantum spin J ↔ ensemble of 2J classical gyroscopes coupled through their symplectic structure, as well as dynamically (Hamiltonian)
Towards a Geometric Theory of Quantum Many-Spin Systems
1partition function at temperature : expnn
Z D S Ω Ω
Standard method to treat many-spin systems: path integral over coherent states
with the boundary condition 0n n Ω Ω
unit vector of the coherent state for a spin at site n nJΩ R
path integral over all the closed paths for the nΩ
solid angle described by between 0 and n n Ω Ω
0
with the action i nn
S J d H
Ω Ω Ω
Berry phase
imaginary time
describes successfully: - spin waves in ferromagnets and antiferromagnets - Mermin-Wagner theorem (no ordering for T>0, D ≤ 2) - Haldane gap for AF chain of integer spin - ...
inconvenient for describing spin sytems with exotic (quadupolar) ordering such as spin nematics
Path integral method with Majorana representation
(Majorana constellation at site )n i n nN n R R
1partition function at temperature :
expnn
Z D S
N N
with the boundary condition 0n n N N x
y
z
ni Ai
0 0
1with the ac dettion i lnB n nn n
S d d H
N N NgN
Berry phase
[ ]2
1
dJ
B n i ii
N A n=
Φ = ⋅∑∫
appropriate to describe spin sytems with exotic ordering because the quantum state of each each spin is treated exactly
quantum metric dynamic action
x
y
z
x
y
z
spin 1 nematic
spin 3/2 nematic
Concluding remarks and outlook Majorana’s stellar representation allows a novel, fully geometrical,
interpretation of the concept of geometrical phase for a spin system
the geometrical representation is the most natural setting to describe the dynamics of systems such as molecular magnets
→ quantum spin systems (chains, planes, etc...) with J > ½ and unconventional magnetic order (quadrupole, octupole, etc...)
→ study the interaction of X-rays and neutrons with spin systems having quadrupolar ordering (suggest methods for experimental investigation of spin nematics)
if you want to learn more about this: P.B., Phys. Rev. Lett. 108, 240402 (2012) see also: Physics Viewpoint “A Quantum Constellation”, Physics 5, 65 (2012)
Thank you for your attention !