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 TechniquesWe 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 tunnel-coupled 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

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.

H1 = p2 +V(x) + aE p sy - B × s

Spectrum for B = 0 Spectrum for B ¹ 0

p p

e e

sy > 0sy < 0

Single-mode nanowire tunnel-coupled to bulk superconductor

To get TS state:

Spectrum for B ¹ 0Adjust Fermi level to fall inside Zeeman gap at p =0. Two Fermi-points remain at, p = ± kF.

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 semi-infinite 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 Lwhere 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 nearly-degenerate ground states.

What is the energy splitting between ground states?

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 becomesδE ~ 1 / LK. K = κ NCH. (Not exponential, but decreases rapidly if NCH is large.)

Results:

What is the energy splitting between ground states?

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.

More Results:

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 finite-energy 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 Lwhere 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 Lwhere 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)

ConclusionsThe 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.