r. jackiw (mit) a bit of physics history about dirac and his equation: dirac...
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Fractional and Majorana Fermions
R. Jackiw (MIT)
A bit of physics history about Dirac and his equation:
Dirac was looking for a relativistic equation for electrons and
eventually arrived at the following first-order matrix equation
i γµ ∂µΨ(x) +mΨ(x) = 0
(α · p+ βm)Ψ = i ∂tΨ
α = γ0 γ, β = γ0, p = 1i ∇
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To expose properties of the Dirac equation, we make the
usual decomposition
Ψ = e−iEtψ ⇒ [α · p+ βm]ψ = Eψ
and discover that there exist positive energy solutions E > 0,
which can describe electrons.
But because “square roots” come in both signs, there exist
also negative energy solutions, E < 0. These need interpre-
tation because they cannot describe electrons, which carry
positive energy. After some hesitation, Dirac concluded that
the negative energy solutions correspond to anti-electrons,
viz. positrons, which soon were discovered.
In a further conceptual leap, Dirac posited that in the ground
state all negative energy levels are filled, but nevertheless the
charge of the ground state is zero. This means that the
Dirac equation is really a many particle equation, where the
particles populate the energy levels.
3
The Dirac equation is a beautiful equation, with a rich hidden
physical structure. It goes beyond a single particle interpreta-
tion and it predicts new (anti-)particles: the positrons. These
characteristics make it a beautiful equation for physics.
Also it is a beautiful equation for mathematics because by
using matrices, it succeeds in taking a square root of a second
order differential equation.
The mathematical and physical beauty of the Dirac equation
suggests to mathematicians and physicists that deformations
of the equation may also yield beautiful and interesting re-
sults.
4
Which deformations should we consider?
One alteration that can be made is to reconsider the equa-
tion in dimensions different from the three spatial and one
time dimension. If we take fewer dimensions we gain the
mathematical advantage of simplicity and also have the pos-
sibility of describing physical systems that are confined to
lower dimensions, for example to a line or to a plane. Such
configurations can occur in condensed matter physics. There
one encounters situations where the low energy dynamics is
well described by a matrix equation, linear in the momenta.
Also there may be a constant mass term which separates
the positive energy solutions from the negative energy ones
by a “gap.” Depending on the nature of the material, the
equation may describe excitations on a line, on the plane in
addition to those in the three-dimensional bulk.
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Dirac Equations
(First-order matrix equations)
[α · p+ βm]ψ = Eψ
continuum solutions E > |m| and E < -|m|
“vacuum”:
Particle interpretation: E < 0 states filled (antiparticles)E > 0 states empty (particles)
Condensed matter: E < 0 states filled (valence band)interpretation: E > 0 states empty (conduction band)
m produces gap 2 |m|“vacuum” carries no net charge
One dimension, physics on a line, Dirac matrices realized with
2× 2 Pauli matrices
1-d : α = σ2, β = σ1, p = 1i
ddx
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A further more profound deformation allows the mass term
m to depend on position.
What sort of dependence should we consider?
Surely a weak dependence will produce only insignificant changes
from the usual homogenous mass case; we are interested in a
significant dependence on position, which could significantly
alter the physical situation. Note that the gap 2 |m| dependsonly on the magnitude of m, and not on its sign. +m pro-
duces the same gap as -m. This suggest a deformation of
the mass term that interpolates between positive and nega-
tive values. In this way we are led to a Dirac equation in the
presence of a defect.
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(mass term position-dependent, soliton)
m→ ϕ (r)
[α · p+ β ϕ (r)]ψ = Eψ
Find:
continuum solutions E > 0, E < 0
AND isolated, normalizable E = 0 solution
“mid-gap” state is found by explicit calculation
is guaranteed by index theorems
CENTRAL QUESTION: in “vacuum” is mid-gap state empty
or filled, what is its charge?
UNEXPECTED ANSWER: charge Q = ± 12.
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Fractional Charge (Analytic Derivation)
Vacuum charge density:
ρ(r) =
0∫
−∞dE ρE (r) ρE = ψ
†EψE
renormalized charge in soliton background
Q =
∫
dr
0−∫
−∞dE
(
ρsE(r)− ρ0E(r))
Evaluation simple in the presence of
an energy reflection symmetry:
ρ-E = ρE (charge conjugation)
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Fractional Charge Calculation
Completeness:∞∫
−∞dE ψ
†E(r)ψE(r
′) = δ(r− r′)
⇒∞∫
−∞dE[ρsE(r)− ρ0E (r)] = 0
Conjugation (ρE = ρ-E) and zero mode ⇒0−∫
−∞dE
(
2ρsE(r)− 2ρ0E (r))
+ ψ†E=0 (r)ψE=0(r) = 0
0−∫
−∞dE
(
ρsE(r)− ρ0E(r))
= −1
2ψ†E=0(r)ψE=0(r)
Q = −1
2Any dimension!
Empty mid-gap state: Q = −12
Filled mid-gap state: Q = +12
Eigenvalue, not expectation value!
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Fractional Charge
(Second Quantized Description)
Expansion of quantum Fermi field in presence of defect &
zero mode
Ψ =∑
E>0
(bE ψsE + d
†Eψ
∗sE ) + aψE=0
Ψ† =∑
E>0
(b†E ψ
s∗E + dE ψ
sE) + a†ψE=0
a†a+ aa† = 1
How to realize on states:
a | + >=| − >, a† | + >= 0,
a | − >= 0, a† | − >=| + >
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Q =1
2
∫
d r(Ψ†Ψ−ΨΨ†)
=1
2
∑
E>0
(b†E bE + dE d
†E − bE b
†E − d
†E dE) +
1
2(a† a− a a†)
=∑
E>0
(b†E bE − d
†E dE) + a† a− 1
2
Q | − >= − 1
2| − >,
Q | + >= +1
2| + >
eigenvalue !
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One Dimensional Example (Polyacetylene)
Dirac equation with varying mass [σ2p+ σ1ϕ(x)]ψ = Eψ
E ≶ 0
0 − d
dx
ddx 0
+
0 ϕ(x)
ϕ(x) 0
ψuE
ψlE
= E
ψuE
ψlE
E = 0
0 − d
dx + ϕ(x)
ddx + ϕ(x) 0
ψu0
ψl0
= 0
ψu,l0 = N exp ∓ ∫ x dx′ ϕ(x′)
Normalizable provided ϕ(x) has kink profile.
Rebbi & RJ, PRD 13, 3398 (76)
Su, Schrieffer & Heeger, PRL 42, 1698 (79)
[high conductivity in polymers]
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Peierls’ Instability in Polyacetylene
O
B
A
Energetics of Polyacetylene Phonon Field
0
Energy density V (φ), as a function of a
constant phonon field φ. The symmetric
stationary point, φ = 0, is unstable. Sta-
ble vacua are at φ = + |φ0|, (A) andφ =
−|φ0|, (B).
Profiles of Phonon Field
A
S
S
B
The two constant fields, ± | φ0 |, cor-
respond to the two vacua (A and B).
The kink-soliton fields, ±φs, interpolate
between the vacua and represent domain
walls.
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Polyacetylene realization of (1− d) “Dirac” equation
Kinetic term: linearization at Fermi level p
Potential term:
Peierls’ instability
energy profile seen
by phonon field
0m
B A
unstable
m
0
V (ϕ)
ϕ0
configurations
of phonon
field ϕS
S
mϕ(x)
+|m|
-|m|
ϕ(x)
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Fractional Charge Schematic
B
A
2S
S S
Two soliton state carries one fewer link
relative to no-soliton vacuum A.
Separate solitons to ∞ ⇒split quantum numbers of link ⇒fermion number fractionalization!
16
(2− d) ⇒ Fractional charge in quantum Hall effect inspired by soli-
tons result, but different mechanism.
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Graphene Realization of (2-d) Dirac Equation
Graphene hexagonal lattice
Sublattices
Kinetic term:
Linearization at Fermi level
2 “Dirac points” per sublattice (conduction and valence bands
meet)
⇒ 4× 4 Dirac Hamiltonian in 2 spatial dimensions
Wallace,PR 71, 662 (47); Semenoff,PRL53, 2449 (84);
Gaim & Novoselov, Nobel Prize (10)
Potential term: ? Speculate: “Kekule’ distortion”
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Dirac Hamiltonian
H = ψ†α · pψ︸ ︷︷ ︸
+ψ†β [ϕre − i ϕim γ5]︸ ︷︷ ︸ψ (4×4)
��
��
��
���✠
ϕ ≡ ϕre+ i ϕim❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❥
α =(
σ 00 −σ
), β =
(0 II 0
), γ5 =
(I 00 −I
), p = 1
i∇
(All vectors are 2-dimensional)
linearization Kekule distortion
at Fermi level ϕ constant: ϕ0 ⇒ mass gap
ϕ soliton/vortex: ϕs = |ϕ(r)|eiθ, ϕ(0) = 0, |ϕ(∞)| = ϕ0
⇒ mid-gap state
Conclusion : Q = ±12 (existence proof!)
Hou, Chamon & Mudry, PRL 98, 186809 (07) [cond-mat/0609740]
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Majorana Equation
electrically charged particles:
particle is different from anti-particle
created by complex field
electrically neutral particles:
particle can be identified with its anti-particle
created by real fielde.g. neutral pion (S = 0)
photon (S = 1)graviton (S = 2)
⟩
all bosons
Majorana fermion = neutral fermion
Majorana Matrix Equation
(α · p+ βm)Ψ = i ∂∂tΨ
Ψ real (neutral excitations)
p = 1i ∇ imaginary
α real = α∗β imaginary = −β∗
}
Majorana Representation
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Majorana Representation
α1M =
(
0 σ1
σ1 0
)
α2M =
(
I 00 −I
)
α3M =
(
0 σ3
σ3 0
)
βM =
(
0 σ2
σ2 0
)
Ψ∗M = ΨM
Majorana in arbitrary representation
Cα∗C−1 = α, Cβ∗C−1 = −β CΨ∗ = Ψ
e.g. Weyl α =
(
σ 00 −σ
)
β =
(
0 II 0
)
C =
(
0 −iσ2
iσ2 0
)
(
σ · p m
m −σ · p
) (
ψχ
)
= i∂
∂t
(
ψχ
)
CΨ∗ = Ψ ⇒ χ = iσ2ψ∗
σ · pψ+ iσ2mψ∗ = i∂
∂tψ (2× 2)
ψ mixes with ψ∗
NB C = I in Majorana representation
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Majorana Equation (2 component)
σ · pψ+ iσ2mψ∗ = i∂
∂tψ
NB. Dirac mass term: preserves quantum numbers (charge, particle number)
Majorana mass term: does not preserve any quantum numbers
⇒ no distinction between particle and anti-particlesince there are no conserved quantities to tellthem apart, particle is its own anti-particle
Dirac field operator
Ψ =∑
E>0
(
aE e−iEtΨE + b†E e
iEtCΨ∗E
)
Majorana field operator
Ψ =∑
E>0
(
aE e−iEtΨE + a†E e
iEtCΨ∗E
)
anti-particle operators (b, b†) have disappeared
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Are there Majorana fermions in Nature?
neutrinos?
– recent development in neutrino physics
experimental observation of neutrino oscillations ⇒• neutrinos have mass (< 0.1eV)
• lepton number is not conserved separately for each
flavor.
⇒ they could be Majorana fermions
Hypothetical Majorana fermions:
• supersymmetry – supersymmetric partners of
photon, neutral Higgs boson, etc. are necessarily
Majorana fermions
• cosmology – dark matter candidates
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Majorana fermions in superconductor in contact with a topologicalinsulator
superconductor
proximity effects ⇒ Cooper pairs
tunnel through to the surface of TI
topological insulator
Hamiltonian density for the model:
H = ψ∗ (σ · 1i∇− µ) ψ+
1
2(△ψ∗ i σ2ψ∗ + h.c.)
ψ =
(
ψ↑
ψ↓
)
,σ = (σ1, σ2),
µ is chemical potential, △ is the order parameter
△ may be constant:△ = △0j
or take vortex profile: △(r) = v(r)eiθ, v(0) = 0, v(∞) = △0.
Equation of motion: i ∂t ψ = (σ · p− µ) ψ+△ i σ2ψ∗
In the absence of µ, and with constant △ , the above system is a (2+1)-dimensional version of the (3+1)-dimensional, two component Majoranaequation!
⇒ governs chargeless spin 12fermions with Majorana mass |△|.
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Zero Mode
In the presence of a single vortex order parameter ∆(r) = v(r)eiθ thereexists a zero-energy (static) isolated mode(Fu & Kane, PRL 100, 096407 (08); Rossi & RJ NPB 190, 681 (81)
)
ψ0 = N
(
J0(µr) exp {−iπ/4− V (r)}J1(µr) exp {i(θ+ π/4)− V (r)}
)
N real constant, V ′(r) = v(r)
Majorana field expansion:Ψ = ................ + aΨ0
E 6=0 modes
where zero mode operator a satisfies
{a, a†} = 1, a† = a ⇒ a2 = 1/2
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[Chamon, Nishida, Pi, Santos & RJ; PRB 81, 224515 (10)]
(i) Two 1-dimensional realizations: take vacuum state to be eigenstateof a, with possible eigenvalue ±1/
√2.
a |0±〉 = ± 1√2
|0±〉
There are two ground states |0+〉 and |0−〉. Two towers of states
are constructed by repeated application of a†E. No operator connectsthe two towers.
Fermion parity is broken because a is a fermionic operator. Like inspontaneous breaking, a vacuum |0+〉 or |0−〉 must be chosen, andno tunneling connects to the other ground state.
(ii) One 2-dimensional realization: vacuum doubly degenerate |1〉 , |2〉,and a connects the two vacua.
a |1〉 = 1√2
|2〉
a |2〉 = 1√2
|1〉
Two towers of states are constructed by repeated application of a†E.a connects the towers. Fermion parity is preserved.
We shall assume that fermion parity is preserved, and adopt secondpossibility
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• Curious fact in (1-d)
total L for scalar kink ⊕ fermions
L = 12∂µΦ ∂µΦ+ µ
22Φ2 − λ
8
2Φ4 + iΨ γµ ∂µΨ− gΦΨΨ
L possesses SUSY for g = λ,Ψ Majorana
Center anomaly in SUSY algebra ⇒ fermion parity can be absent.
[Losev, Shifman & Vainshtein, PLB 522, 327 (01)
Any relevance for condensed matter?
Semenoff & Sodano, EJTP 10, 57 (08)]
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Multiple Vortices
With N vortices, governed by operators a1, a2, . . . , aN that satisfy
{ai, aj} = 2 δij (Clifford algebra)
one can show that one needs
N = 2N
2 states for even N
and N = 2N+1
2 states for odd N
N = 1 N = 2 σ1 or σ2 (not σ3) (2× 2)
N = 2 N = 2 σ1 and σ2 (not σ3) (2× 2)
N = 3 N = 4 α1 α2 α3 or β (not diagonal) (4× 4)
N = 4 N = 4 α1 α2 α3 and β (not diagonal) (4× 4)
etc.
Clifford algebra, with a restriction: use for ai Pauli, Dirac, . . . , matrices
excluding diagonal one since it would correspond to diagonalizing a mode
operator and would produce fermion parity violation [Pi & RJ, PRB 85,
033102 (12)]
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