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 !