Notes on Spin Operators and the Heisenberg Model Physics 880.06: Winter, 003-4 David G. Stroud In these notes I give a brief discussion of spin-1/ operators and their use in the Heisenberg model. 1. Spin Operators for Spin-1/ Particles. We denote the spin angular momentum operator for a single spin by S = (S x, S y, S z ). These operators satisfy the commutation relations [S x, S y ] = i hs z (1) and cyclic permutations of these relations. For spin-1/ particles, these can be written S i = h σ i, () where i = x, y, z, and the σ is are the Pauli spin matrices. These spin matrices have the convenient representations, σ x = 0 1 1 0, 1
0 i σ y =, i 0 σ z = 1 0 0 1. We can also introduce the raising and lowering operators S ± = S x ± is y. (3) In terms of these, S x = 1 (S + + S ) (4) S y = 1 i (S + S ) (5) S = 1 (S +S + S S + ) + S z. (6) S z has two eigenstates, which may be denoted and, with eigenvalues ± h/. Thus and S z = h (7) S z = h. (8) If the spin operators are represented by Pauli matrices, then and are column vectors: = 1 0
and = 0 1. It is also easily shown, using the Pauli spin matrices, that S + and S satisfy S + = (9) S = (10) S + = 0 (11) S = 0. (1) Using all these results, one can show that the states and are also eigenstates of the operator S :. Spin Operators for Two Spin-1/ Particles. S = 3 h (13) 4 S = 3 h. (14) 4 If one has two identical spin-1/ particles, one may introduce the spin operators S 1i and S i (i = x, y, z) for spins 1 and. The i th component of total spin is The spin operators satisfy the commutation relations S i = S 1i + S i. (15) [S αi, S βj ] = i hs αk δ αβ, (16) 3
where i, j, k are cyclic permutations of x, y, z and α and β label the spins. One can also introduce the square of the total spin, S = S x + S y + S z. (17) It can easily be shown that S = 1 (S 1+S + S 1 S + ) + S 1z S z. (18) If there are two spins, then there are four possible spin states. A convenient basis, in obvious notation, is the four states,,, and. These can be arranged into a set of three triplet states, which we denote for short 10 and 1 ± 1, and a singlet state, which we denote 00. The triplet states are given by 11 = (19) 10 = 1 ( + ) (0) 1 1 =. (1) The singlet state is 00 = 1 ( + ). () The first number is the total spin quantum number and the second is the S z quantum number. A state with total spin quantum number S has a square of total spin angular momentum h S(S + 1), and one with quantum number 4
S z has z component of total spin angular momentum hs z. Thus, the state 10 satisfies S 10 = h 10 (3) S z 10 = 0. (4) Note that the S = 1 states are symmetric under spin exchange, while the singlet S = 0 state is antisymmetric. But the total wave function (space + spin) for fermions must be antisymmetric, and all spin-1/ particles are fermions. Therefore, if we write the total wave function as a product of a spin wave function and a space wave function, then a spin-1 state must go with an antisymmetric space wave function, and a spin-0 with a symmetric wave function. To deal with the Heisenberg model (discussed below) it is useful to calculate expectation values of the operator S 1 S. The operator S 1 S can be written as S 1 S = 1 ( ) S S1 S. (5) Using this representation, it is easily shown that 1i S 1 S 1i = (1/) h ( 3/4 3/4) = (1/4) h (6) where i is the S z quantum number, and 00 S 1 S 00 = (3/4) h. (7) 5
3. Heisenberg Model. The Heisenberg model is defined by the Hamiltonian H = ij J ij S i S j i S i B. (8) Here J ij is the exchange interaction, and the first sum runs over distinct pairs of spins. The external field is denoted B. In what follows, we consider the nearest-neighbor Heisenberg model, where J ij = J if i and j are nearest neighbor sites, and J ij = 0 otherwise. We shall also consider the spin-1/ Heisenberg model, so that the quantities S i are operators for a spin-1/ particle. The model is called ferromagnetic if J > 0, because parallel spins are favored energetically in this case; it is called antiferromagnetic if J < 0. We generally consider the spin sites i to form a Bravais lattice, though of course one could consider a Heisenberg model on a lattice with a basis, or even on a non-periodic collection of sites. Finally, one need not consider only a spin-1/ Heisenberg model; one could also consider higher spins, or even a classical Heisenberg model, in which the spins are classical vectors. What is the origin of the interaction energy J? As implied in class, it is a purely quantum-mechanical interaction arising from the fact that electrons are fermions, and is actually a certain integral involving single-particle wave functions of two electrons, known as the exchange integral. The true magnetic dipole-dipole interaction is much smaller than the usual size of J in ferromagnetic materials, and can usually be neglected in comparison to the 6
exchange energy. Let us consider the spin-1/ nearest-neighbor ferromagnetic Heisenberg model with zero external field. The Hamiltonian can be expressed in terms of raising and lowering operators for spin, as H = J ij [ 1 (S i+s j + S i S j+ ) + S iz S jz ]. (9) The ground state of H is the state with all spins pointed down, and is denoted.... It is readily verified that this state is an eigenstate of H with eigenvalue E g = (Nz/)( h /4). (Note that we are retaining the factor of h in the definition of the spin operator here.) If we denote this state by 0 for short, then we have H 0 = E g 0. (30) 4. Excited states of the Heisenberg model; spin waves. One might think that the state with all spins but one pointed down, and one spin (say the l th ) pointed up, would be an excited eigenstate of H, but one would be wrong, because this state is not an eigenstate of H at all. To see this, first introduce a notation for this state: call it l. Then, operating on l with H and using the properties of the raising and lowering operators, one finds that H l = (E g + J h J h ) l l + δ, (31) δ 7
where δ denotes a nearest neighbor lattice vector, and the sum runs over all nearest neighbors. For example, in a simple cubic lattice with lattice constant a, the six possible values of δ are δ = ±aˆx, ±aŷ, ±aẑ. (3) The form of eq. (31) suggests that perhaps a suitably chosen linear combination of the states l might be an eigenstate of H. Indeed, this is the case. In fact, the suitable linear combination is just k = 1 exp(ik R l ) l. (33) N To prove this, we just operate on k with H, to get l H k = (E 0 + ɛ k ) k, (34) where ɛ(k) = J h [z δ cos(k δ)]. (35) This equation is derived by using the properties of the raising and lowering operators together with the definition of the state k. We also make use of the fact that for every Bravais lattice vector δ there is another Bravais lattice vector δ. Note that the above spin waves have a wave-vector/frequency relationship ω k at small values of k. 5. Physical Interpretation of Spin Waves. 8
The above definition of a spin wave may seem a bit strange: a linear combination of states, each with one spin up and all the rest down. You can think of this as a result of quantum mechanics. If you have a spin-1/ particle, and you measure its spin in the z direction, you must measure it as being either up or down. In a spin wave state, any given spin has only a 1/N probability of pointing down. Thus, we have to represent this state using a linear combination as mentioned above. The idea of spin wave also exists classically, and this may be easier to understand conceptually. Think of the ground state of the Heisenberg model as a collection of pendula, all at rest in their down position. In a state with a spin wave of wave vector k, any given spin is precessing with small amplitude around its equilbrium down position, and each pendulum has a definite phase with respect to the others. Finally, the wave is traveling with a frequency ω. 9