Feynman Path Integral Formulation

236 7 Analytical Lattice Expansion Methods
⎛
⎞
μ 1 1 1 1 2− μ
8π √ 1 μ 1 2− μ 1 1
aλ
1 1 μ 1 2− μ 1
9
⎜ 1 2− μ 1 μ 1 1
, (7.44)
⎟
⎝ 1 1 2− μ 1 μ 1 ⎠
2 − μ 1 1 1 1 μ
where μ = 2(5π 2 − 6 √ 3π + 54)/9π 2 ≈ 1.5919. (the λ/a dependence has disappeared
since the couplings a and λ only appear in the dimensionless combination
√
aλ). The eigenvalues of the above matrix (apart from the constants in front of it)
are 0 (with multiplicity 2), 2(μ −1) (with multiplicity 3) and 6 (with multiplicity 1).
The zero modes correspond to flat directions, for which deformations of the edge
lengths leave the lattice geometry unchanged. Their explicit form in the weak field
limit was given in Eq. (7.9).
1
Fig. 7.3 Octahedral tessellation
of the two-sphere, with
arbitrary edge length assignments.
2
l 15
l 25 l 45
5
l 12
l 14
l 23
3
l 13
4
l 34
l 56
l 46
l26
l 36
6
For the octahedron [see Fig. (7.3)] one obtains instead the following coefficients
of the small fluctuation matrix
ε12 2 → 2 √ aλ (216 − 12 √ 3π + 5π 2 )/27π
ε 12 ε 13 → 8 √ aλ (−27 − 3 √ 3π + 2π 2 )/27π
ε 12 ε 14 → 4 √ aλ (54 + π 2 )/9π
ε 12 ε 34 → 8 √ aλ (−54 + 3 √ 3π + π 2 )/27π
ε 12 ε 46 → 4 √ aλ (108 + 12 √ 3π + π 2 )/27π ,
(7.45)
again with the remaining coefficients being determined by symmetry. Up to a common
factor of 2 √ aλ/27π, the eigenvalues of the 12 × 12 small fluctuation matrix
are given by 36π 2 (with multiplicity 1), 972 (with multiplicity 2), and 8(3 √ 3 − π) 2
(with multiplicity 3), and zero (with multiplicity 6).