CASINO manual - Theory of Condensed Matter

one band is partly filled, while in non-metallic systems all the bands are fully occupied or empty.
Exceptions arise in some systems in which the band is partially filled but the Coulomb interaction
between the electrons is sufficiently strong that the electrons are localized. An alternative view to band
theory would help to distinguish the character of the system. It is argued by Kohn [89] that the system
is insulating as a result of wave function localization in the configuration space. The development of
the Berry-phase theory of polarization [90, 91, 92, 93, 94] further advanced Kohn’s idea and provided
tools to measure the localization of the electrons and the polarization. This Berry-phase approach
solves the problem that the polarization is ill defined in an extended system by computing the quantity
directly from the many-body wave function instead of the electron positions. Souza and Martin [93]
provided the expressions for the localization tensor and the polarization in terms of a many-body
Wannier wave function. This wave function is linked to Kohn’s wave function: for an insulating
system it is localized in configuration space in the thermodynamic limit. Souza and Martin [93]
also showed that the localization tensor is related to a frequency integral of the conductivity. The
conductivity formula implies that for a system with non-vanishing conductivity the localization tensor
is infinite, otherwise it has a finite value. In [95], the off-diagonal elements of the localization tensor
are used to calculate the DC conductivity in the transverse direction for a quantum hall fluid.
The localization tensor and polarization are written in terms of a many body operator
〉
Z (α)
N
〈Ψ|e = iGα·X(Rm) |Ψ , (409)
where G α is a simulation cell reciprocal lattice vector and X is the sum of electron positions ∑ n
i=1 r i
of a configuration R m .
The Berry-phase polarization is given by
P α = N V 〈r α〉 c
, (410)
where 〈r α 〉 c
is the expectation value of the electron distribution given by
〈r α 〉 c
= 1
NG α
IlogZ (α)
N , (411)
the equation (410) measures the polarization current of the system in response to an adiabatic change
of the Hamiltonian by approximating the Coulomb interaction as a first-order perturbation.
The localization tensor can be interpreted as a measure of the quadratic spread of a charge distribution.
This gives an indication of how well the electrons are localized in the simulation cell according to the
wave function. This is written as
where N is the number of electrons in the simulation cell.
The off-diagonal elements of the localization tensor are defined by
〈
r
2
α
〉c = −1
NG 2 log |Z (α)
N |2 , (412)
α
〈r α r β 〉 c
= −1 log |Z(α) N
||Z(β) N |
NG α G β |Z (αβ)
N | , (413)
where Z (αβ)
N
is defined as 〈 Ψ|e −i(Gα−G β)·X(R m) |Ψ 〉 .
In QMC, the localization tensor and polarization are calculated from Z N with the periodic boundary
conditions imposed. This is done by summing the electron positions of each configuration R m to
calculate e −iGα·X(Rm) . This is then averaged over configurations generated by VMC or DMC to give
mean
¯z N = 1 M∑
Z N (R m ) , (414)
M
m=1
to ensure (414) tends to (409) and the statistical error is small, a large number of steps must be
taken to sample the configuration. The localization tensor diverges when Z N is zero. Numerically,
Z N would never be zero, but could become small so that the localization tensor becomes very large
when approaching metal-insulator transition from the insulating side [96]. As the difference between
¯z and Z N (R m ) becomes large when approaching the divergence, it is often useful to examine the error
bar in Z N to determine when the localization tensor diverges.
196