Megascopic Quantum Phenomena
335
(4) Lee-Low-Pines polarons and their self-energy [106]:
Neglecting two-electron terms in Eq. (11.39), one obtains a simple analytical
expression for the fermionic one-particle excitation energies
ε P =
r ∈V
⎛
⎝
A =P
|u
r
P A |
2
ε
0
P −ε
0
A −ω r
+
I =P
|u
r
P I |
2
ε
0
P −ε
0
I +ω r
⎞
⎠
=
r ∈V
⎛
⎝
R =P
u
r
P R
2
1
ε
0
P −ε
0
R −ω r
− 2
I =P
u
r
P I
2
ω r
(ε
0
A −ε
0
I ) 2 −(ω r ) 2
⎞
⎠
(11.47)
and in the solid state notation (r → q; P → k, σ ; R → k − q, σ ; I → k − q, σ with
the occupation factor f k−q )
ε k =
q =0
u
q
2
1
ε
0
k −ε
0
k−q −ω q
− 2
q =0
u
q
2 f k−q
ω q
(ε
0
k −ε
0
k−q ) 2 −(ω q ) 2
(11.48)
The electron energies ε
0
k with the corrections (11.48) represent the
quasiparticles/polarons that were originally derived via the Lee-Low-Pines transformation [106]. The first part of (11.48) refers to individual polarons, whereas
the second part represents correlations originating from the effective field of other
polarons.
The previous examples were based on a simplistic model of the electronvibrational/electron-phonon Hamiltonian, taking only Galilean broken symmetries
from the Goldstone theorem into account. Investigating the consequences of including all Goldstone bosons, descending from all three types of symmetry breakings
in condensed matter, our attention will be focused on three most interesting and
important examples: the Born-Huang ansatz, and ground state energy and excitation
spectra of the B-O degenerate systems [107, 109].
(5) The Born-Huang ansatz [7]:
In the adiabatic limit, which means that all non-adiabatic coefficients ˜
c will be equal
to zero, the change of the ground state energy (11.34) yields the adiabatic correction
E 0(ad) =
AIr
˜
ω r
c
r
AI
2 = 2
AI
r ∈V
1
2
ω r +
r ∈R
ρ r +
r ∈T
τ r
c
r
AI
2 (11.49)
that is exactly identical with the Born-Handy ansatz (11.28). We see that only the
field COM covariant theory can pass the test of the Born-Huang ansatz. Noting that
the roton terms in (11.49) has nothing to do with the energies of the rotational degrees
of molecular freedom as well as the translon terms with the de Broglie wave of the
translational freedom of the whole system, the contribution of roton and translon
335
(4) Lee-Low-Pines polarons and their self-energy [106]:
Neglecting two-electron terms in Eq. (11.39), one obtains a simple analytical
expression for the fermionic one-particle excitation energies
ε P =
r ∈V
⎛
⎝
A =P
|u
r
P A |
2
ε
0
P −ε
0
A −ω r
+
I =P
|u
r
P I |
2
ε
0
P −ε
0
I +ω r
⎞
⎠
=
r ∈V
⎛
⎝
R =P
u
r
P R
2
1
ε
0
P −ε
0
R −ω r
− 2
I =P
u
r
P I
2
ω r
(ε
0
A −ε
0
I ) 2 −(ω r ) 2
⎞
⎠
(11.47)
and in the solid state notation (r → q; P → k, σ ; R → k − q, σ ; I → k − q, σ with
the occupation factor f k−q )
ε k =
q =0
u
q
2
1
ε
0
k −ε
0
k−q −ω q
− 2
q =0
u
q
2 f k−q
ω q
(ε
0
k −ε
0
k−q ) 2 −(ω q ) 2
(11.48)
The electron energies ε
0
k with the corrections (11.48) represent the
quasiparticles/polarons that were originally derived via the Lee-Low-Pines transformation [106]. The first part of (11.48) refers to individual polarons, whereas
the second part represents correlations originating from the effective field of other
polarons.
The previous examples were based on a simplistic model of the electronvibrational/electron-phonon Hamiltonian, taking only Galilean broken symmetries
from the Goldstone theorem into account. Investigating the consequences of including all Goldstone bosons, descending from all three types of symmetry breakings
in condensed matter, our attention will be focused on three most interesting and
important examples: the Born-Huang ansatz, and ground state energy and excitation
spectra of the B-O degenerate systems [107, 109].
(5) The Born-Huang ansatz [7]:
In the adiabatic limit, which means that all non-adiabatic coefficients ˜
c will be equal
to zero, the change of the ground state energy (11.34) yields the adiabatic correction
E 0(ad) =
AIr
˜
ω r
c
r
AI
2 = 2
AI
r ∈V
1
2
ω r +
r ∈R
ρ r +
r ∈T
τ r
c
r
AI
2 (11.49)
that is exactly identical with the Born-Handy ansatz (11.28). We see that only the
field COM covariant theory can pass the test of the Born-Huang ansatz. Noting that
the roton terms in (11.49) has nothing to do with the energies of the rotational degrees
of molecular freedom as well as the translon terms with the de Broglie wave of the
translational freedom of the whole system, the contribution of roton and translon