Download Physics 7910: HW # 03.

Physics 7910: HW # 03.
(Dated: February 15, 2013)
Homework is due Thursday, February 28.
1. Hamiltonian of the classical Heisenberg chain is given by
X
~n · S
~n+1 + J2 S
~n · S
~n+2 }.
H=
{J1 S
n
The nearest-neighbor exchange is ferromagnetic, J1 < 0, while the next-nearest neighbor one is antiferromagnetic,
~n · S
~n = S 2 for all n.
J2 > 0. Treat spins as classical vectors of magnitude S, S
Find the ordering momentum and the energy of the ground state configuration as a function of the dimensionless
ratio w = −J2 /J1 in the full possible range 0 ≤ w ≤ ∞.
[The problem is motivated by recently discovered frustrated ferromagnets LiCuVO4 and LiCu2 O2 .]
~ and the ground state energy E0 of the classical Heisenberg antiferromagnet on
2. Find the ordering momentum Q
triangular lattice.
X
~r · S
~r0 .
H=J
S
(r,r 0 )
Here J > 0 and the sum counts every nearest-neighbor bond (r, r0 ) of the triangular lattice once.
~r as a function of two-dimensional position
Describe also the magnetic order that obtains, by explicit result for S
vector r.
3. Holstein-Primakoff representation of the spin operator S is given by
r
r
√
√
a+ a
a+ a
−
+
+
S = 2S a
, S = 2S 1 −
a, S z = S − a+ a.
1−
2S
2S
Show that if the operators a, a+ satisfy boson commutation relations,
[a, a+ ] = 1,
the spin operators defined above satisfy the spin commutation relations [S a , S b ] = iabc S c .
4. Consider quantum antiferromagnet on a simple cubic lattice and, following discussion in the class, find the
temperature dependence of its internal energy, specific heat and magnetization. Compare your results with the case
of the quantum ferromagnet discussed in the class and explain the origin of the differences between the two cases.