Download The Quantum Heisenberg Ferromagnet

Topic 3
Spinors, Fermion Fields, Dirac Fields
Lecture 16
The Quantum Heisenberg Ferromagnet
Soon after Schrödinger discovered the wave equation of quantum mechanics, Heisenberg and Dirac
developed the first successful quantum theory of ferromagnetism W. Heisenberg, Mehrkörperproblem
und Resonanz in der Quantenmechanik, Z. f. Phys. 38, 411–426 (1926), P.A.M. Dirac, On the
Theory of Quantum Mechanics, Proc. R. Soc. Lond. A 112, 661–677 (1926) They found that the
laws of quantum mechanics imply the existence of an effective interaction between electron spins on
neighboring atoms with overlapping orbital wave functions. This exchange interaction is caused by
the combined effect of the Coulomb repulsion and the Pauli exclusion principle. It plays a key role in
the microscopic theory of ferromagnetism and other cooperative phenomena involving electron spin.
The Heisenberg model is a set of quantum mechanical spins Ŝm located at fixed lattice points with
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Spinors, Fermion Fields, Dirac Fields
Lecture 16
exchange interactions between nearest-neighbors
Ĥ = −J
X
Ŝm · Ŝn ,
hmni
It was introduced by W. Heisenberg, Zur Theorie des Ferromagnetismus, Z. f. Phys. 49, 619636 (1928), who studied the properties of the ground state and low-lying excitations based on the
commutation relations
h
i
j
k
Ŝm , Ŝn = −iδmn jk ` Ŝn` , Ŝ2 = S(S + 1) .
The figure from Altland-Simons shows a spin-wave or Magnon excitation of a one-dimensional chain
of spins.
If the coupling J > 0, neighboring spins will prefer to align. The ground state of a chain of N spins
will have all spins aligned,
for example in the z direction of spin space. This is an eigenstate of total
P
z
z
z component Ŝ = m Ŝm with eigenvalue N S and energy E0 = −J(N S)2 .
The ground state is highly degenerate because the Hamiltonian is invariant under global rotation
in spin-space of all spins simultaneously. This symmetry results in zero-mass excitations called spin
waves, similar to the density (sound) waves in the continuum limit of the oscillator chain.
To find the low-lying excitations of the system, define spin ladder operators
±
Ŝm
PHY 510
≡
x
Ŝm
±
y
iŜm
,
h
i
z
±
Ŝm , Ŝn = iδmn Ŝn± ,
2
h
+
Ŝm
, Ŝn−
i
z
= 2δmn Ŝm
.
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Spinors, Fermion Fields, Dirac Fields
Lecture 16
This algebra can be simplified by introducing bosonic ladder operators with commutation relations
am , a†n = δmn ,
and making a Holstein-Primakoff transformation
−
Ŝm
= a†m 2S − a†m am
1/2
,
+
Ŝm
= 2S − a†m am
1/2
am ,
z
Ŝm
= S − a†m am ,
introduced by T. Holstein and H. Primakoff, Field Dependence of the Intrinsic Domain Magnetization
of a Ferromagnet, Phys. Rev. 58, 1098–1113 (1940). The bosonic ladder operators are used to
construct the Fock space for magnon excitations.
Low Energy Excitations
The Holstein-Primakoff transformation is exact but non-linear in the number operator a† a. In the
low-energy and long-wavelength limit, there are a small number of spin wave excitations, and their
wavefunctions are spread out over the chain. The expectation value ha† ai will be a small fraction of
the spin S, and the non-linear square root operator can be approximated
2S −
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1/2
†
am am
'
√
1
2S + √ a†m am + · · ·
2 2S
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Spinors, Fermion Fields, Dirac Fields
Lecture 16
The Hamiltonian operator can be expanded in powers of 1/S
X
1
z z
+ −
− +
Ĥ = −J
Ŝm
Ŝm+1 +
Ŝm
Ŝm+1 + Ŝm
Ŝm+1
2
m
i
Xh
†
2
†
†
= −JN S − JS
(am am+1 + am+1 am − 2am am + O(S 0 )
m
2
= −JN S − JS
X
a†m+1
−
a†m
†
am+1 − am + O(S 0 )
m
.
The quadratic operator sum can be diagonalized by Fourier transforming to wavenumber space and
imposing periodic boundary conditions for convenience
N
1 X
am eikm ,
ak = √
N m=1
k = 0, ±1, . . . , ±K ,
where K = πN/2 is the Brioullin zone (B.Z.) boundary. The spin-wave Hamiltonian is
2
Ĥ = −JN S +
B.Z.
X
~ωk a†k ak
0
+ O(S ) ,
h
i
†
ak , ak0 = δkk0 ,
k
with dispersion relation
k
~ωk = 2JS(1 − cos k) = 4JS sin2
.
2
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Lecture 16
The Bethe Ansatz
H. Bethe, Zur Theorie der Metalle, Zeitschrift für Physik 1931, Volume 71, Issue 3-4, pp 205–226
(1931) presented a method for obtaining the exact eigenvalues and eigenvectors of the one-dimensional
S = 1/2 Heisenberg model.
The webpage, G. Müller et al., Introduction to the Bethe Ansatz, has a series of tutorial articles on
Bethe’s method.
The Bethe ansatz is an exact method for the calculation of eigenvalues and eigenvectors of a class of
quantum many-body model systems. It is useful for two reasons. (1) The eigenstates are characterized
by a set of quantum numbers which can be used to distinguish them according to specific physical
properties. (2) In many cases the eigenvalues and the physical properties derived from them can be
evaluated in the thermodynamic limit.
The Ansatz was applied by Bethe to the Heisenberg model of spins Sn = (Snx , Sny , Snz ) with quantum
number S = 1/2 on a 1-dimensional lattice of N sites with periodic boundary conditions SN +1 = S1
and Hamiltonian
H = −J
N
X
n=1
Sn · Sn+1 = −J
N X
1
n=1
2
−
+
z
Sn+ Sn+1
+ Sn− Sn+1
+ Snz Sn+1
.
H acts on a Hilbert space of dimension 2N spanned by the orthogonal basis vectors |σ1 . . . σN i,
where σn =↑ represents an up spin and σn =↓ a down spin at site n.
The Bethe ansatz is a basis transformation of the Hilbert space that diagonalizes the Hamiltonian.
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Lecture 16
Block Diagonalization
The rotational
symmetry about the z-axis in spin space implies that the z-component of the total
P
N
z
z
spin STz ≡
n=1 Sn is conserved: [H, ST ] = 0. According to Table 1, the operation of H on
|σ1 . . . σN i yields a linear combination of basis vectors, each of which has the same number of down
spins. Sorting the basis vectors according to the quantum number STz = N/2 − r, where r is the
number of down spins, block diagonalizes H.
The block with r = 0 (all spins up) consists of a single vector |F i ≡ | ↑ . . . ↑i eigenstate of
H|F i = E0 |F i, with E0 = −JN/4.
The N basis vectors in the invariant subspace with r = 1 (one down spin) are labeled by the position
of the flipped spin:
|ni = Sn− |F i n = 1, . . . , N.
It is diagonalized using lattice translation invariance and a wavenumber basis
N
1 X ikn
e |ni ,
|ψi = √
N n=1
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k = 2πm/N, m = 0, . . . , N − 1 ,
6
E − E0 = J(1 − cos k).
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Spinors, Fermion Fields, Dirac Fields
Lecture 16
The Bethe ansatz deals with the invariant subspaces with 2 ≤ r ≤ N/2, where the translationally
invariant basis does not completely diagonalize the Hamiltonian matrix even if additional symmetries
such as the full rotational symmetry in spin space or the reflection symmetry on the lattice are used.
For r = 1, the ansatz (starting point) for any eigenvector H|ψi = E|ψi is a superposition
|ψi =
N
X
2[E − E0 ]a(n) = J[2a(n) − a(n − 1) − a(n + 1)] ,
a(n)|ni ,
a(n + N ) = a(n) .
n=1
This system of difference equations has N linearly independent solutions
a(n) = eikn ,
k=
2π
m,
N
m = 0, 1, . . . , N − 1 ,
which gives the same result as before.
Bethe ansatz for r = 2
The r = 2 subspace has dimension N (N − 1)/2 and an eigenvector can be expanded
|ψi =
X
a(n1 , n2 )|n1 , n2 i .
1≤n1 <n2 ≤N
Bethe’s ansatz for the coefficients is
1
1
a(n1 , n2 ) = ei(k1 n1 +k2 n2 + 2 θ12 ) + ei(k1 n2 +k2 n1 + 2 θ21 ) ,
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Lecture 16
where the phase angle θ12 = −θ21 ≡ θ depends on two wavevectors k1 , k2 that satisfy
2 cot
θ
k1
k2
= cot
− cot
.
2
2
2
Two additional relations between k1 , k2 , and θ follow from the requirement that the wave function
be translationally invariant, which implies that a(n1 , n2 ) = a(n2 , n1 + N ). This condition is satisfied
provided that
eik1 N = eiθ ,
eik2 N = e−iθ ,
N k1 = 2πλ1 + θ ,
N k2 = 2πλ2 − θ ,
where the integers λi ∈ {0, 1, . . . , N − 1} are called Bethe quantum numbers.
The eigenstates are found by solving this system of algebraic relations. Each solution k1 , k2 , θ determines a set of expansion coefficients a(n1 , n2 ), and thus determines an eigenvector with energy
E = E0 + J
X
(1 − cos kj ) ,
k = k1 + k2 =
j=1,2
2π
(λ1 + λ2 ) .
N
For r = 2, the solutions divide into three classes:
(1) A class C1 of states for which one of the Bethe quantum numbers is zero, λ1 = 0, λ2 =
0, 1, . . . , N − 1.
(2) A class C2 of states with nonzero λ1 , λ2 which differ by two or more: λ2 − λ1 ≥ 2. There are
N (N − 5)/2 + 3 such pairs.
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1 + 2 = N
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1 + 2 = N=2
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1
0
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1 + 2 = 3N=2
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(3) A class C3 of states has nonzero Bethe quantum numbers λ1 , λ2 which either are equal or differ
by unity. There exist 2N − 3 such pairs, but only N − 3 pairs yield solutions.
The figure shows allowed pairs of Bethe quantum numbers (λ1 , λ2 ) that characterize the N (N − 1)/2
eigenstates in the r = 2 subspace for N = 32. The states of class C1 , C2 , and C3 are colored red,
white, and blue, respectively.
It is not possible to write down exact analytic formulas for the energies and eigenfunctions. However,
the ansatz can be generalized systematically in algorithmic form. Exact solutions for any number of
spins can in principle be generated by a computer program, which is straightforward to code.
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