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Lecture VII 19

Lecture VII: Quantum Magnetism and the Ferromagnetic Chain

Following on from our investigation of the phonon and interacting electron system, we nowturn to another example involving bosonic degrees of freedom the problem of quantummagnetism.

Spin S Quantum Heisenberg Magnet spin analogue of discrete harmonic chain

H = JN

m=1

Sm Sm+1

periodic boundary conditions Sn+N = Sn

N 1 2

Sign of exchange coupling J depends on material parameters: Coulomb interactiontends to favour ferromagnetism J > 0 (cf. Hunds rule) while superexchange processesfavour antiferromagnetismJ < 0.

Aim: To uncover the ground states and nature of low-energy (collective) excitations

Classical ground states?

Ferromagnet: all spins aligned along a given (arbitrary) directioni.e. manifold of continuous degeneracy (cf. crystal)

Antiferromagnet: (where possible) all neighbouring spins antiparallel Neel state

Quantum ground states:

H = Jm

SzmS

zm+1+

12

(S+mSm+1 + S

mS

+m+1)

SxmSxm+1 + S

ymS

ym+1

where S = Sx iSy denotes spin raising/lowering operator

Ferromagnet: as classical, e.g. |g.s. = Nm=1|Szm = SNo spin dynamics in |g.s., i.e. no zero-point energy! (cf. phonons)Manifold of degeneracy explored by acting total spin lowering operator m Sm

Antiferromagnet: spin exchange interaction (viz. S+mSm+1) zero point fluctuationswhich, depending on dimensionality, may or may not destroy ordered ground state

Lecture Notes October 2005

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Lecture VII 20

Elementary excitations?

Formation of magnetically ordered state breaks continuous spin rotation symmetry low-energy collective excitations (spin waves or magnons) cf. phonons in a crystal

Example of general principle known as Goldstones theorem

However, as with lattice vibrations, general theory is nonlinearfortunately, low-energy excitations described by free theory

To see this, for large spin S, it is helpful to switch to a spin representation in whichdeviations from |g.s. are parameterised as bosons:

|Sz

= S |n = 0|Sz = S 1 |n = 1|Sz = S 2 |n = 2...

...|Sz = S |n = 2S

i.e. a maximum of n bosons per lattice site (softcore constraint)

For ferromagnetic ground state with spins oriented along z-axis,the ground state coincides with the vacuum state |g.s. |, i.e. am| = 0

Mapping useful when elementary spin wave excitation involves n 2S Mapping of operators: Sz, S = Sx iSy?

with = 1, operators obey quantum spin algebra

[S, S] = iS [S+, S] = 2Sz, [Sz, S] = S

cf. bosons: [a, a] = 1, n = aa

According to mapping, Sz = S aa;therefore, to leading order in S 1 (spin-wave approximation),S

(2S)1/2

a

,S

+

(2S)1/2

aIn fact, exact equivalence provided by Holstein-Primakoff transformation

S = a

2S aa1/2 , S+ = (S), Sz = S aa Applied to ferromagnetic Heisenberg spin S chain, spin-wave approximation:

H = JN

m=1

SzmS

zm+1 +

1

2(S+mS

m+1 + S

mS

+m+1)

=

Jm

S2 S(amam

am+1am+1) + S(ama

m+1 + a

mam+1) + O(S

0)= J

m

S2 2Samam + S

amam+1 + h.c.

+ O(S0)

Lecture Notes October 2005

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Lecture VII 21

with p.b.c. Sm+N = Sm and am+N = am

To leading order in S, Hamiltionian is bilinear in Bose operators;diagonalised by discrete Fourier transform (exercise)

ak =1N

Nn=1

eiknan, an =1N

B.Z.k

eiknak, [ak, ak] = kk

noting

n ei(kk)n = Nkk

H =

JNS2 +

B.Z.

k kakak + O(S

0), k = 2JS(1

cos k) = 4JSsin2(k/2)

cf. free-particle spectrum

Terms of higher order in S spin-wave interactions

Lecture Notes October 2005