Download Physics 7910: HW # 03.

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Matter wave wikipedia, lookup

Quantum entanglement wikipedia, lookup

Werner Heisenberg wikipedia, lookup

Bra–ket notation wikipedia, lookup

Renormalization wikipedia, lookup

Tight binding wikipedia, lookup

Density matrix wikipedia, lookup

Nitrogen-vacancy center wikipedia, lookup

Hidden variable theory wikipedia, lookup

Wave–particle duality wikipedia, lookup

Coherent states wikipedia, lookup

Lattice Boltzmann methods wikipedia, lookup

Bell's theorem wikipedia, lookup

Scalar field theory wikipedia, lookup

Wave function wikipedia, lookup

Quantum group wikipedia, lookup

EPR paradox wikipedia, lookup

Hydrogen atom wikipedia, lookup

Path integral formulation wikipedia, lookup

Particle in a box wikipedia, lookup

Copenhagen interpretation wikipedia, lookup

Renormalization group wikipedia, lookup

Spin (physics) wikipedia, lookup

Max Born wikipedia, lookup

Molecular Hamiltonian wikipedia, lookup

Quantum state wikipedia, lookup

Ising model wikipedia, lookup

Theoretical and experimental justification for the Schrödinger equation wikipedia, lookup

T-symmetry wikipedia, lookup

Relativistic quantum mechanics wikipedia, lookup

Ferromagnetism wikipedia, lookup

Canonical quantization wikipedia, lookup

Symmetry in quantum mechanics wikipedia, lookup

Transcript
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.