Topological Superconductivity in One
Dimension and Quasi-One Dimension
Bertrand I. Halperin
Harvard University
Conference on Topological Insulators and Mathematical
Science
Harvard University, September, 2014
References
Jay D. Sau, B. I. Halperin, K. Flensberg, and S. Das Sarma,
Number-conserving theory for topologically protected
degeneracy in one-dimensional fermions, Phys. Rev. B 84,
144509 (2011)
(see also Fidkowski et al, Phys. Rev. B 84, 195436 (2011).
Gilad Ben-Shach, Arbel Haim, Ian Appelbaum, Yuval Oreg,
Amir Yacoby, and B. I. Halperin, Detecting Majoranas in 1D
wires by charge sensing, arXiv: 1406.5172
Background
• Models of 1D superconductors can exist in a topological state which
gives rise to zero-energy localized Majorana states at the ends of an
infinitely long wire.
• Models generally describe a nanowire where superconductivity is
induced from the outside, by proximity coupling to an infinite external
three-dimensional superconductor
•
Bulk superconductor has an energy gap for single-electron excitations,
and has long-range superconducting order. The superconducting phase
can be well defined.
• Effect on the nanowire is described by creation and annihilation
operators for Cooper pairs. Charge in the nanowire is only conserved
modulo 2. Hamiltonian has a Z2 gauge symmetry rather than U(1).
Issues
• What happens if the “bulk” superconductor is actually very thin?
• What happens if the superconductivity arises from interactions within
the nanowire itself, so there is no external superconductor?
• Charge is conserved absolutely, not just modulo 2.
• System of finite length, total charge is conserved. For a system of
infinite length, charge is conserved locally. The system will not have
true long-range superconducting order, even at T=0, so the
superconducting phase is not well defined.
• Note: We will confine ourselves to T=0. We will emphasize
conceptual issues rather than practical ones.
• To discuss what happens, we need to formulate the questions properly.
Questions depend on the geometry.
Geometries
• Infinite 1D wire
• Semi-infinite wire – behavior near end
• Long but finite wire
• Collections of coupled wires
Mathematical Techniques
We use Luttinger liquid theory, bosonization, and
renormalization group methods to derive results. I give here
only key results and some physical arguments.
Definitions
• 1D wire – nanowire with a finite number of conducting
channels, long compared to width.
• Proximity induced superconductivity – nanowire tunnelcoupled to a bulk superconductor with an energy gap.
Cooper pairs can tunnel between nanowire and
superconductor, but not single electrons
• Isolated superconducting nanowire – nanowire with
charge-conservation and finite number of channels, with
attractive short range interactions that favor
superconductivity.
Related question
• For a finite nanowire connected to an infinite bulk
superconductor charge in the wire is not conserved and is
not quantized. However, number-parity is conserved.
• For topological superconductor, the ground-state number
parity can change as one varies parameters, such as the
chemical potential, applied magnetic field, or wire length,
as the state of the Majorana end modes changes between
occupied and empty.
• Is there a change in the mean electron density associated
with such an event? If so, where is it located?
Experimental realization: Proximity induced
topological superconductivity.
Semiconductor nanowire with strong spin orbit coupling
and applied magnetic (Zeeman) field coupled to s-wave
bulk superconductor.
Candidate structure: Use InAs or InSb nanowire for
semiconductor.
TS state occurs if the Fermi level of the nanowire is
adjusted so that an odd-number of spin-split energy bands
are occupied at Fermi energy. (Assumes superconducting
gap is small compared to spin splitting.)
Single-mode InSb nanowire with strong
Zeeman field
H1 = p2 +V(x) + aE p sy - B  s
Orientation of wire (w) chosen in x-direction. Rashba E-field,
perpendicular w, chosen in z-direction. Magnetic field B
should not be parallel to the y axis. s is the the electron spin.
Spectrum for B  0
Spectrum for B = 0
e
e
sy < 0
sy > 0
p
p
Single-mode nanowire tunnel-coupled to bulk
superconductor
To get TS state:
Adjust Fermi level to fall
inside Zeeman gap at p =0.
Two Fermi-points remain
at, p =  kF.
Spectrum for B  0
EF
-kF
kF
Infinite 1D wire
Key measurement. Single-electron tunneling into middle of
the wire, with applied voltage V. Differential conductance
measures energy spectrum G(E) of one-electron Green’s
function at E=eV.
Proximity induced superconductivity gives an energy gap Δ for
all excitations. G(E) = 0 for E < Δ.
Isolated superconducting wire has phonon modes with ω -> 0
at long wavelengths. However, G(E) can still have an energy
gap, even for weak attractive interactions, provided that
number of channels NCH is >1. (We shall only be interested in
case NCH >1.) Energy gap in G(E) means that topologically
distinct phases may be defined.
Semi-infinite wire
Key measurement. Single-electron tunneling at a point x, at or
near the end of the wire.
Proximity induced superconductivity gives a delta-function at
zero energy, in the TS phase: G(E) = f(x) δ(E) , where f(x) falls
off exponentially with distance from the end of the wire.
Isolated superconducting wire has no delta-function
contribution. Spectrum in TS phase has form G(E) ~ f(x) E-1+γ,
with γ proportional to 1/NCH. For large NCH, E-1+γ approaches a
delta function.
Specifically: γ = 1/K, where K is the Luttinger liquid parameter,
determined by the compressibility and the phonon velocity.
(K ~ NCH .)
Tunneling spectrum at the end of a semiinfinite TS wire
Physical Reason
Why the semi-infinite isolated superconducting wire has no
delta-function contribution for tunneling into the end:
Injecting an electron into a Majorana mode at the end of the
wire will be accompanied by creation of low-energy phonons.
In 1D, all spectral density is pushed into multiphonon
sidebands, no spectral weight is left in zero-energy Majorana
mode.
Finite Proximity-Coupled TS Wire
Majorana modes at the wire ends lead to two low-lying energy states of
different number parity. For proximity coupled wire, energy splitting falls
off exponentially with length. (Will also oscillate in sign.)
For a single mode wire,
δE ~ E0 e—L/ξ cos kF L
where ξ is the induced superconducting coherence length.
δE
L
Reason for exponential fall-off
For L >> ξ , quantum fluctuations in electron number due
to tunneling of Cooper pairs in and out of the nanowire are
large compared to unity. Then ground states with even and
odd electron number can have the same average electron
number and become indistinguishable by any local
measurement.
Example: Even parity state may be a superposition of N = 96, 98, 100,
102, 104.
Odd parity state can be a superposition of N= 97, 99, 101, 103.
Both have mean number <N>=100.
Number fluctuations do not occur if electron number is conserved.
Quantum states of different parity have different electron number.
Energy splittings for a finite isolated wire
If electron number is conserved, we have, for a long wire
E(N) ~ L ε(ρ), where ρ = N/L .
In TS state we don’t care whether N is even or odd. Then, if
energy is minimized at N=N0 , we have
E(N0±1) - E(N0) ~ (1/L) d2ε/dρ2 .
Difference between lowest energy even and odd parity states
is generically of order 1/L.
Tunneling into end of a finite isolated wire
Tunneling into ground state with one more electron increases
the ground-state energy by order 1/L .
Emitted low energy phonons also have energy of order 1/L.
Tunneling spectrum has series of delta-functions with
energies of order 1/L.
Weight in the lowest energy state falls off as a power law with
increasing L.
Group of coupled wires
Three wires with odd number of modes, attractive interactions, in TS state.
Majorana modes (MF) at ends of each wire.
Wires are tunnel-coupled in region of overlap. (Blue region)
Lengths of blue region and dangling ends all large compared to ξ.
Total electron number is fixed but pairs can tunnel between wires. For fixed
total electron number, six Majorana end-modes lead to 23-1= 4 nearlydegenerate ground states.
What is the energy splitting between
ground states?
Results:
If tunneling coupling varies adiabatically at ends of blue region,
and there is no backscattering due to impurities, the energy
splitting can be exponentially small.
With backscattering due to non-adiabaticity or to impurities
near the ends of the blue region, energy splitting becomes
δE ~ 1 / LK. K = κ NCH. (Not exponential, but decreases
rapidly if NCH is large.)
What is the energy splitting between
ground states?
More Results:
If there are impurities in the middle of blue region but not at the
ends, and the tunneling coupling varies adiabatically at ends of
blue region, the energy splitting can still be exponentially small,
provided that the number of coupled nanowires is odd.
If the number of coupled wires is even, then backscattering
impurities inside the blue region will lead to power law
dependence on L.
Reason for difference between even and
odd wire number.
If there is an even number of electron modes in the blue region,
impurity back scattering can lead to quantum coherent 2π phase
slips in the superconducting order parameter, which lead to
energy splitting of the Majorana modes.
If the blue region contains an odd number of modes only 4π
(double) phase slips are allowed, which do not lead to such
energy splitting. (2π phase slips are accompanied by a finiteenergy electronic excitation in the TS phase.)
Charging events in a tunnel-coupled wire
• For a finite nanowire connected to an infinite bulk
superconductor, the ground-state number parity can change
as one varies parameters, such as the chemical potential,
applied magnetic field, or wire length.
• Is there a change in the mean electron density associated
with such an event? If so, where is the charge located?
Charging events in a finite wire
We shall consider the case of a single mode nanowire with non-interacting
electrons, coupled to a bulk superconductor. Recall, in TS state:
Energy splitting between even and odd parity states has the form
δE = E0 e—L/ξ cos kF L
where ξ is the induced superconducting coherence length. Parity jumps
occur when kFL = (n + ½) π.
δE
L
Charging events in a finite wire
We shall consider the case of a single mode nanowire with non-interacting
electrons, coupled to a bulk superconductor. Recall, in TS state:
Energy splitting between even and odd parity states has the form
δE = E0 e—L/ξ cos kF L
where ξ is the induced superconducting coherence length. Parity jumps
occur when kFL = (n + ½) π.
kF depends on chemical potential μ.
Difference in mean electron number between even and odd parity states is
equal to d (δE) / dμ ~ e—L/ξ . Charge jump is small, but not zero.
Charging events in a finite wire
Results: Parity jumps are accompanied by charge jumps of order e- L/ξ .
Although the Majorana wavefunctions are exponentially localized at the
two wire ends, the charge density is uniformly spread along the length of
the wire, with modulation at frequency 2kF: δρ(x) ~ e- L/ξ sin2 kF x
Charge density is
proportional to product of
Majorana wavefunctions at
the two ends of the wire.
δρ(x)
Conclusions
The topological superconductor state can exist in a
quasi-one-dimensional wire that is not coupled to a bulk
superconductor, if there are attractive interactions in the
wire.
Properties of the TS state are modified by the effects of
charge conservation, in ways that depend on the
geometry.
Most interesting is the case of several coupled wires,
where spitting between ground states will generally fall
off as a power of the length scale L, rather than
exponentially. The power is proportional to the number
of channels.