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Hierarchical Majoranas in a Programmable Nanowire Network

Christopher Mudry1 Zhi-Cheng Yang2 Claudio Chamon2 Thomas Iadecola3

1Paul Scherrer Institut, Switzerland; 2Boston University, USA, 3University of Maryland, USA

ETHZ, August 03 2018

• C. Mudry (PSI)

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Section

1

Main idea and result

2

A brief history

3

Realization with Majorana nanowires

4

Experimental considerations

5

Summary and Gedanken braiding experiment

6

Appendices

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Hierarchical Majoranas in a Programmable Nanowire Network 2 / 36

First hierarchy of quasi Majorana zero modes (QMZMs)

There are physical QMZMs:

Vg

|Ψ0(r)|2

Vg,1

Vg,2

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Second hierarchy of Majorana zero modes (MZMs)

There are logical (emergent) MZMs Vg + δVg r,α:

R(t)

δVg r,α(t) . .= V0 cos

• K+·sα+(K+−K−)·r+q arg
• r−R(t)
• ,

q = ±1.

• C. Mudry (PSI)

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Section

1

Main idea and result

2

A brief history

3

Realization with Majorana nanowires

4

Experimental considerations

5

Summary and Gedanken braiding experiment

6

Appendices

• C. Mudry (PSI)

Hierarchical Majoranas in a Programmable Nanowire Network 5 / 36

Exotic excitations in many-body quantum physics

When interactions are not too strong, excitations are “simple”:

◮ smooth fluctuations about the

magnetization (magnons, Bloch 1930),

◮ dressed electrons (Fermi-Liquid

quasiparticles, Landau 1957). When interactions are strong, excitations can be “exotic”:

◮ spinons carrying spin-1/2

quantum numbers in antiferromagnetic spin-1/2 chains (Faddeev and Takhtajan 1981),

◮ fractionally charged electrons in

polyacetylene (Jackiw and Rebbi 1976; Su, Schrieffer, and Heeger 1979).

RbMnFs, Windsor 1966

0.0 0.2 0.4 0.6 0.8 1.0

• 0.4
• 0.2

0.0 0.2 Binding Energy (eV) kx (Å-1)

• 0.4
• 0.2

0.0 0.2 kx (Å-1)

• 0.4
• 0.2

0.0 0.2 kx (Å-1) 0.0 0.2 0.4 0.6 0.8 1.0 x=1 x=0.6 x=0.5

Bix Pb1−x /Ag(111), Meier 2009 CuSO4:5D2O, Ronnow 2007

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Majoranas as “exotic” excitations: I

Starting from the pair of non-Hermitean operators c and c†

• beying the fermion algebra {

c, c†} . .= 1 we may identify the pair

• f Majorana operators

γ1 . .= ( c + c†) and γ2 . .= ( c − c†)/i obeying the Majorana algebra { γa, γb} = 2δab. We can always interpret an electron as being a bound state of two Majoranas. However, this interpreation is pertinent only if there are fermionic Hamiltonians whose “exotic” excitations are Majoranas! The Majoranas making up the electron are physically meaningfull iff they are:

◮ (fully) deconfined due to electron-electron interactions ◮ or (partially) deconfined due to defectuous backgrounds.

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Majoranas as “exotic” excitations: II

Moore and Read in 1991 propose that this is so in the fractional quantum Hall effect at the filling fraction 5/2. Read and Green in 2000 give a simpler interpretation for the Majoranas of Moore and Read as the zero modes bound to vortices of a two-dimensional type II chiral p-wave superconductor. Kitaev in 2001 give a one-dimensional version of Read and Green where the the role of the vortices is taken by domain walls (like in polyacetylene), each of which shall be called a Majorana nanowire. Fu and Kane in 2008 show that the vortices of an s-wave superconductor in contact with the surface of a three-dimensional toological insulator bound Majorana zero modes.

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Majoranas as “exotic” excitations: III

The Majorana zero modes (MZMs) bound to a vortex in a two-dimensional type II chiral p-wave superconductor obey non-Abelian statistics (Fröhlich 1988) when braided (Ivanov 2001). However, how does one braid a pair of superconducting vortices? An alternative physical platform for realizing MZMs that could be braided was proposed by several groups – Sau 2010, Alicea 2010, Lutchyn 2010, and Oreg 2010 – by building two-dimensional networks of Majorana nanowires. Hereto, there is a difficulty in that the braiding of these (physical) MZMs often violate adiabacity.

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Section

1

Main idea and result

2

A brief history

3

Realization with Majorana nanowires

4

Experimental considerations

5

Summary and Gedanken braiding experiment

6

Appendices

• C. Mudry (PSI)

Hierarchical Majoranas in a Programmable Nanowire Network 10 / 36

Elementary building block

The elementary building block is a nanowire which at low temperatures supports a topological superconducting gap ∆nw, i.e., the nanowire hosts a pair of QMZMs at its endpoints when superconducting. We shall call such a nanowire a “Majorana nanowire.”

Vg

|Ψ0(r)|2

Vg,1

Vg,2

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Honeycomb lattice made of Majorana nanowires

Imagine that all nearest-neighbor bonds of the honeycomb lattice are realized by identical Majorana nanowires

Vg

|Ψ0(r)|2

Vg,1

Vg,2

There are then two energy scales in the problem: a hybridization U and a hopping amplitude t.

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Vertices of the honeycomb lattice are Y junctions of Majorana nanowires

The QMZMs are depicted as green dots. Effectively, there are three flavors of QMZMs on each lattice site. We label the operators creating QMZMs by γα

S,j, where α = x, y, z denotes the bond to which the

QMZM belongs, while S = A, B denotes the sublattices, and j is the label for the lattice sites. B B B A

γx

A,j

γy

A,j

γz

A,j

γx

B,j+sx

γy

B,j+sy

γz

B,j+sz

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The trimer limit (U = 0, t = 0) is defined by

U U

• Htrimer .

.=

• S=A,B
• j∈ΛS

iU

• γx

S,j

γy

S,j +

γy

S,j

γz

S,j +

γz

S,j

γx

S,j

• where the honeycomb lattice Λ is made of two interpenetrating

triangular sublattices ΛA and ΛB, while we impose the Majorana algebra

• γα

S,j,

γα′

S′,j′

• = 2δα,α′δS,S′δj,j′,
• γα†

S,j =

γα

S,j.

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The trimer Hamiltonian is the sum over S = A, B and j ∈ ΛS of the pairwise commuting operators iU

• γx

S,j

γy

S,j +

γy

S,j

γz

S,j +

γz

S,j

γx

S,j

• .

As each one of these operators has the three single-particle eigenvalues − √ 3 U, 0, + √ 3 U, with the Majorana zero mode ˆ η . .= 1 √ 3

• γx

S,j +

γy

S,j +

γz

S,j

• ,

it supports three doubly-degenerate flat bands with the single-particle energies − √ 3 U, 0, + √ 3 U, respectively.

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The dimer limit (U = 0, t = 0) is defined by

t

• Hdimer .

.=

• j∈ΛA
• α=x,y,z

it γα

A,j

γα

B,j+sα,

where sα are the unit vectors connecting the three sites in ΛB that are nearest-neighbor to a site in ΛA. The dimer Hamiltonian supports two triply-degenerate flat bands with the single-particle energies −|t| and +|t|, respectively. These single-particle energies correspond to the fermionic state ˆ cα†

j |0 .

.= 1 2

• γα

A,j − i

γα

B,j+sα

• |0,

ˆ cα

j |0 .

.= 0, being empty or occupied, respectively. There is no zero mode in the dimer limit.

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Reversal of time

Time reversal is defined by the rules i → −i,

• γα

A,j → +

γα

A,j,

• γα

B,j+sα → −

γα

B,j+sα.

The motivation for this definition is that we would like to interpret ˆ cα

A,j .

.= 1 2

• γα

A,j + i

γα

B,j+sα

• as a fermion operator localized on the directed bond

j ∈ ΛA, j + sα ∈ ΛB of the honeycomb lattices that is left invariant by the operation of time reversal. One verifies that

• Hdimer → +

Hdimer,

• Htrimer → −

Htrimer. Although Htrimer is odd under time reversal, the zero-energy flat band transforms trivially, whereas the finite-energy bands are interchanged.

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Hamiltonian for the network of nanowires

The pair of particle-hole symmetric bands with the lowest energy in magnitudes for ( Htrimer + Hdimer)/t when U/t = 10 with U > t > 0 displays a Haldane gap at the corners of the Brillouin zone ΩBZ (depicted in light blue) [the magnitude of the Haldane gap follows from ε±(K+) = ε±(K−) ≈ ±

t2 2 √ 3 U + O(t4/U3)]:

• 0.5

2 3 0.5 2 1

• 1
• 2
• 2
• 3
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A Kekulé dimerization competes with the Haldane gap:

If we add the perturbation δ Hdimer . .= i

• j∈ΛA
• α=x,y,z

δtj,α γα

A,j

γα

B,j+sα,

δtj,α . .= ∆ eiK+·sα eiG·rj +c.c., G . .= K+−K−, where the Kekulé amplitude is defined by ∆ . .= ∆0 eiϕ, ∆0 . .= |∆|, and ϕ ∈ [0, 2π), we lower the space-group symmetry to C B A B C A B C A C C A B B

ΩK

BZ

Γ

K+

K−

p

ΩBZ

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If so, the two lowest particle-symmetric single-particle bands are (Hou 2007; Ryu 2009)

• 0.4
• 0.2

0.2 1 0.4 1

• 1
• 1
• 0.4
• 0.2

0.2 1 0.4 1

• 1
• 1

Haldane gap critical point Kekule gap

(a) (b) (c) (d)

• 0.4
• 0.2

0.2 1 0.4 1

• 1
• 1

0.02 0.04 0.06 0.08 0.1 0.02 0.04 0.06

Gap

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A logical Majorana zero mode (MZM) is bound to the Kekulé vortex (Hou 2007) δtr,α . .= ∆0 cos (K+ · sα + G · r + q arg (r)) , q = ±1:

(b) (a)

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Section

1

Main idea and result

2

A brief history

3

Realization with Majorana nanowires

4

Experimental considerations

5

Summary and Gedanken braiding experiment

6

Appendices

• C. Mudry (PSI)

Hierarchical Majoranas in a Programmable Nanowire Network 22 / 36

Estimate of U

The relevant energy scales in a Majorana nanowire of length a are:

Vg

|Ψ0(r)|2

Vg,1

Vg,2

∆nw . .=

g µB |B| 2

−

• ∆2

sc + V 2 g ≈ U > 0

the external magnetic field |B| needed to break time-reversal symmetry, the proximity-induced superconducting gap |∆sc| needed to break charge conservation, the chemical potential |Vg| of the wire in proximity of the s-wave superconducting substrate, the Rashba energy scale vF,nw needed to break spin-rotation symmetry that enters through the Fermi velocity vF,nw of the nanowire.

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Estimate of t

Physical QMZMs are bound to the end points of this Majorana nanowire if and only if

gµB |Bz| 2 >

• ∆2

sc + V 2 g .

The decay length for a physical QMZM bound to the end points

• f a Majorana nanowire is

ξphysical = vF,nw ∆nw . It follows that the overlap between two physical QMZMs is then approximately given by t ∼ vF,nw a κ e−κ, κ . .= a ∆nw vF,nw , when measured in units of energy.

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Estimate of ∆0 and ξlogical

If we compute the leading order change δ∆nw resulting from Vg → Vg + δVg, we find the estimate ∆0 t ∼ δt t ≈ κ − 1 κ a ξsc V 2

g /∆2 sc

• 1 + V 2

g /∆2 sc

δVg Vg for the Kekulé gap measured in units of t. The decay length of a logical MZM is then ξlogical . .= t δt a. For an InSb/Al Majorana wire (Lutchyn 2018), ∆sc ∼ 0.2 meV, vF,nw ∼ 0.2 − 1.0 eV × , ξsc ∼ 100 − 500 nm so that for a Majorana nanowires of length a ∼ 1µm and with κ ≈ 2, a ξsc ∼ 2 − 10, ∆0 t ≈ 1 2 a ξsc V 2

g /∆2 sc

• 1 + V 2

g /∆2 sc

δVg Vg .

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The prefactor in front of δVg/Vg on the right-hand side can be chosen to be of order one by choosing the ratio V 2

g /∆2 sc so as to compensate

the factor a/(2ξsc) ∼ 1.0 − 5.0. (The corresponding bias Vg should thus be of roughly the same order as ∆sc.) If so, the ratio ∆0/t ≈ δVg/Vg.

Consequently, by using modulations with δVg of the same order as Vg,

• ne can make the Kekulé gap of the order of t, and hence the size of

the logical MZMs as small as the length scale of the wire size a.

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Section

1

Main idea and result

2

A brief history

3

Realization with Majorana nanowires

4

Experimental considerations

5

Summary and Gedanken braiding experiment

6

Appendices

• C. Mudry (PSI)

Hierarchical Majoranas in a Programmable Nanowire Network 27 / 36

There follows the support of the logical MZM in

R(t)

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What could be measured?

The existence of the “logical" MZMs can be probed via scanning tunneling microscopy (STM), where they manifest themselves as zero-bias peaks in the tunneling differential conductance. For a system with 2N “logical" MZMs, each pair of MZMs constitutes a fermionic state that can be either empty or filled. The fermion parity (even or odd, respectively) of each pair then specifies the state of a qubit. Thus, the dimension of the Hilbert space spanned by the quantum states of these qubits grows as 2N−1 once the total fermion parity of the 2N MZMs has been fixed. To verify that braiding the “logical" MZMs acts in the desired way, one needs a means of measuring the fermion parity of any pair of MZMs. If we exploit the fact that the “logical" MZMs can be moved adiabatically by adjusting the array of gate voltages, bringing a pair of “logical" MZMs together by merging two Kekulé vortices effectively “fuses" the two MZMs. To determine whether the pair of MZMs were in an even- or odd-fermion-parity state, one can measure – with scanning single-electron transistor microscopy (SSETM) – the local charge distribution in the vicinity of the fused pair.

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Section

1

Main idea and result

2

A brief history

3

Realization with Majorana nanowires

4

Experimental considerations

5

Summary and Gedanken braiding experiment

6

Appendices

• C. Mudry (PSI)

Hierarchical Majoranas in a Programmable Nanowire Network 30 / 36

Logical MZM as zero mode of an effective Dirac Hamiltonian with a Kekulé vortex ∆vtx(r) . .= ∆0(r) ei(ϕ+nθ) with r = |r|

• cos θ

sin θ T and ϕ ∈ [0, 2π):

The pair of particle-hole symmetric bands with the lowest energies is encoded by

• HKek(r) .

.=     2i∂z ∆vtx(r) 2i∂¯

z

∆vtx(r) ∆vtx(r) −2i∂z ∆vtx(r) −2i∂¯

z

    , z = x + iy. When n = −1, a normalizable zero mode is supported on sublattice A and given by ΨA,0(r) = N         e+i( π

4 + ϕ 2 ) e

−

r

• dr′ ∆0(r′)

e−i( π

4 + ϕ 2 ) e

−

r

• dr′ ∆0(r′)

        , ϕ ∈ [0, 2π). There follows the “logical” MZM operator

• γA .

.=

• d2r
• uA(r)

aA,+(r) + uA(r) aA,−(r)

• =

γ†

A

as aA,+(r) = a†

A,−(r).

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Braiding of vortices

Interchanging two vortices is a commutative operation.

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Braiding of logical MZMs

Interchanging two logical MZMs always implies that one and only

• ne string is crossed by one and only one MZM.

The single-particle wave function of the logical MZM that crosses the string changes by the phase factor exp(i2π/2) = −1.

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Braiding of logical MZMs: definition

t 2 3 4 5 6 7 8 1 y x t position string at infinity i i+1 Ti Representation

Interchange Ti of two logical MZMs is Ti :

• γj
• →
• γj
• ,
• γj → Ti (

γj) . .=      +γi+1, j = i, −γi, j = i + 1, + γj, j = i, i + 1.

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Braiding of logical MZMs: representation

Let i = 1, · · · , 2n index 2n Kekulé vortices. The interchange of two Kekulé vortices – such that their strings never cross the remaining 2(n − 1) Kekulé vortices – is represented through conjugation by

• τ(Ti ) .

.= exp π 4 γi+1 γi

• =

1 √ 2

• 1 +

γi+1 γi

• .

They realize a 2n-dimensional representation of the braiding group B2n, the group generated by the interchanges Ti with i, j = 1, · · · , 2n modulo the relations Ti Tj = Tj Ti with |i − j| > 1 Ti Ti+1 Ti = Ti+1 Ti Ti+1

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Hierarchical Majoranas in a Programmable Nanowire Network 35 / 36

Ti Tj = Tj Ti if |i − j| > 1 Ti Ti+1 = Ti+1 Ti Ti Ti+1 Ti = Ti+1 Ti Ti+1 =

j j+1

...

i+1 i i+1 i

...

i i+1 i i+1 j j+1 j j+1 j j+1

=

i i i+1 i+2 i+2 i i+2 i+1 i+2 i i+1 i+1

=

i i+2 i+1 i i+2 i+1 i i+2 i+1 i i+2 i+1

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