Zero-Energy Majorana Modes in Condensed-
Matter Systems
For P268r.
November 2013

Localized zero-energy Majorana modes
•
Extra
“zero energy” degrees of freedom
that have been
hypothesized to occur at isolated point defects in some
special
correlated-electron systems.
•
Majorana modes
should have some
very peculiar
properties
, have generated much interest.
•
So far, only modest evidence of their realization in
experiments on actual physical systems.

Localized Majorana Modes
Defining Properties
Localized Majorana modes are associated with point
defects in a system that otherwise would have an energy
gap for electronic excitations.
In a
system with
N
point defects (Majorana sites) will
have a “degenerate” ground state.
Set of ground states
form
a Hilbert space with
dimension
2
N/2
.
Ground-state degeneracy is “robust” to local
perturbations.

Comparison:
Localized spins in an insulator
Magnetic impurities in an insulator
have localized low-energy degrees of
freedom.
Simplest case has S=1/2; low-energy states of an isolated
impurity form a two dimensional Hilbert space.
System with
N
impurities has low energy Hilbert space with dimension
2
N
.
Basis
states by specifying S
z
= 1/2 or -1/2 at each site.
If no applied
magnetic field
, and no interactions between spins,
states are
degenerate in energy.
Take into account exchange interactions,
degeneracy is split. Exchange interactions fall off exponentially with
distance for spins in an insulator, so splitting can be negligible if spins
are far apart.
But degeneracy is not robust,
even if spins are far apart.
Spins have
magnetic momen
t, observable quantity, couples to any local magnetic
field.
Magnetic moments give rise to dipole interaction between spins,
falls off as 1/r
3
,
not exponentially.

Majorana Modes
System with
N
point defects (Majorana sites):
Low energy Hilbert
space has dimension 2
N/2
.
Effectively:
one S=1/2 degree of freedom for every
two Majorana
sites.
If sites are far apart
, no local observable can distinguish between states
in the Hilbert space; energies are precisely degenerate.
For finite
separation r,
energy splittings fall off exponentially:
E
split
∝
e
- r /
ξ
.
(
ξ
≈
microscopic coherence length)
Splitting can be negligible if r is sufficiently large, and no local
perturbation can split this degeneracy.
In theoretical discussions, often
assume that separations are
sufficiently large that
E
split
can be set equal to zero.

Manipulation of Majorana states
Suppose there is a way to physically move defects around “adiabatically”
as
a function of time. (Adiabatically means that motion is
slow
on the frequency
scale defined by the lowest
finite-energy excitations
in the system, but
fast
on
the scale of the
exponentially small energy splittings
of states in the
“zero-
energy” Hilbert space.
Electron
s
ystem will stay in low-energy Hilbert space.

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- Fall '13
- BertrandI.Halperin
- Physics, Electron, Energy, Quantum Field Theory, Majorana, majorana states