Probing Majorana modes using entanglement measures and fractional - - PowerPoint PPT Presentation

Probing Majorana modes using entanglement measures and fractional ac Josephson effect

Moitri Maiti

Bogoliubov Laboratory of Theoretical Physics (BLTP), Joint Institute for Nuclear Physics (JINR) Novel Quantum States in Condensed Matter 20th November 2017

• Dynamics of unconventional Josephson junctions using RCSJ model

• presence of odd Shapiro steps 

• presence of additional steps in the devil’s staircase structure
• Entanglement measures in the Kitaev model on the honeycomb

lattice 


• qualitative behaviour of the entanglement entropy 

• Schmidt gap is dependent on the presence of gapless edge modes


Plan of the talk

Recent experiments detecting presence of Majorana modes

Recent proposal using one-dimensional nanowire

• Proximity induced effective p-wave pairing amplitude
• Main ingredients:


a) Strong spin-orbit (SO) coupling.
 b) spin polarization. 
 c) proximity induced superconductivity.

• Semiconductor nanowires

H(qz) = ~2q2

z

2meff + αˆ n · (σ × q)

strength of SO coupling direction 


• f SO coupling

✏(qz) = ~2q2

z

2meff ± ↵qz

Introduce in-plane magnetic field B 


• pens a gap at qz =0

✏(qz) = ~2q2

z

2meff ± p ↵2q2

z + B2 z

p µ2 + ∆2 < Bz

Condition for topological phase hosting 
 Majorana fermions:

Δ Δ

z z z

⭐ Mourik et. al Science (2012)

⭐ Das et. al Nat. Phys (2012) ⭐ Deng et. al Nano Lett. (2012) ⭐ Finck et. al PRL (2013)

Position of the zero bias peaks

Mid-gap zero-bias states using 1-d nanowire

⭐ Rokhinson et. al Nat. Phys. (2012)

Position of the zero-bias Majorana modes

Fractional ac Josephson effect: Doubling of Shapiro steps

⭐ Nadj-Perge et. al Science (2013)

Localised edge states on ferromagnetic atomic chains atop Pb superconductor

fractional Josephson effect

• Junctions with a layer of non-superconducting material sandwiched between

two superconducting layers.

• Systems with s-wave pairing amplitude, e.g Nb, Al, Pb etc

S S

IJ ∼ sin(φ)

✩ Josephson (1962)

Weak links

N S S

Tunnel junctions

B S S

superconducting current in the 
 absence of any external bias : Andreev bound states proximity effect Superconducting current is due to Superconducting current is due to

Josephson effect in conventional junctions

Systems with unconventional pairing amplitude:

g(k) = (kx + iky)/kF

B S S

φ = (φ1 ∼ φ2)

IJ ∼ sin(φ/2)

The current phase relation changes from to IJ ∼ sin(φ/2)

IJ ∼ sin(φ)

Doubling of the periodicity of the phase in the current-phase relation!

Josephson effect in unconventional junctions

∆(kF ) = ∆0g(kF ) exp(iφ)

g(kF ) : variation around the Fermi surface

φ

: global phase factor

Dynamics of unconventional Josephson junctions

Schematic representation of the junction

B S S

d V0

Phase ɸ is has time dependence in 
 presence of external radiation I + A sin ωt = IJ + C0 Φ0 2π d2φ dt2 + Φ0 2πR dφ dt

dφ dt = 2eV/h

Dissipation parameter

The resistively and capacitively shunted Josephson junction

Current-Voltage characteristics

β 7 1

(underdamped/overdamped)

V = n~ω/2e

• Shapiro steps

⭐ Shapiro (1963)

V = ω, 2ω, 3ω..

• Shapiro step structures are predicted to be different for Josephson junctions with

sin(ɸ) and sin(ɸ/2) current phase relations.

• 4π periodic Josephson effect or appearance of Shapiro steps at even multiples of

frequency of external radiation i.e

V = 2ω, 4ω, 6ω..

• Recent theoretical works in the overdamped regions for sin(ɸ/2) current-phase

relations using junctions of unconventional superconductors.

⭐ Domnguez et. al. PRB (2012) ⭐ Houzet et. al PRL (2013)

• Effect of including capacitance?

Shapiro steps in Josephson junction

D=0.4, A=20, β=0.2

dimensionless barrier strength

Appearance of both odd and even steps 
 in the current-voltage (I-V) characteristics. This is in contrast to the recent studies where only even steps are observed in the I-V characteristics. Even steps are enhanced compared to the 


• dd steps for a significant range of coupling 


~ the width of the odd steps are decreases 
 gradually in the resistive junctions.

: Width of Shapiro steps

Appearance of odd Shapiro steps!

⭐ PRB 92, 224501 (2015)

Appearance of odd Shapiro steps!

C-V characteristics variation with frequency of external radiation:

¨ φn + β ˙ φn = fn(t) + In

f0 = A sin(ωt) f1 = − sin(φ0/2)

Perturbative analysis

h ˙ φn>0i = 0

= X

n

✏nn, I = X

n

✏nIn

⭐ Kornev et. al, J Phys. Conf. Ser., 43, 1105 (2006)

φ0(t) = φ

0 + I0t/β + A

ωγ sin(ωt + α0)

• In first order,

α0 = arccos(ω/γ), γ = p β2 + ω2

I(0)

s

∼ sin(φ0(t)/2) = Im

∞

X

n=−∞

Jn(A/2γω)e(i[I0/(2β)+nω]t+nα0+φ

0/2)

∆Ieven

s

= 2Jn( A 2ω p β2 + ω2 )

~ contribution from the harmonics Condition for Shapiro steps

I0 = 2|n|ωβ

• In the regime β𝜕, 𝜕, A >> 1, perturbative analysis of the non-linear term.
• I0 ~ applied current, In>0 ~ determined from
• For n < 2

where

~ contribution from the sub-harmonics

∆Iodd

s

= X

n>m

Jn(

A 2ω√ β2+ω2 )J2m+1−n( A 2ω√ β2+ω2 )

2(β2 + (2m + 1 − 2n)2ω2/4)

• n=1

φ1 =

1

X

n=1

Jn(x)(γnωn)1 cos(ωnt + nα0 + δ0 + nφ0/2)

I(1)

s

∼ 1 2φ1(t) cos(φ0(t)/2) = X

n1,n2

Jn1(x)Jn2(x)(4γn1ωn1)1 × h sin([ωn1 + ωn2]t + [n1 + n2](α0 + φ0/2) + δn1) + sin([ωn1 − ωn2]t + [n1 − n2](α0 + φ0/2) + δn1) i

Condition for Shapiro steps

I0 = |n1 + n2|ωβ

(n1 + n2) = 2m + 1

η = ∆Ieven

s

∆Iodd

s ωn = I0/(2β) + nω, δn = arccos(ωn/γn), γn = p ω2

n + β2

0.5 1

β

3 6 9

η

P-P Simulation P-P Theory S-S Simulation

α0 = 6.09, α1 = 0.31 α0 = 5.98, α1 = 0.32

Simulation: Theory:

Plot of the ratio of the step width η as a function of dissipation parameter β:

η = α0 exp(α1β2)

• For p-wave junction η has exponential dependence on the junction capacitance C0 ~

presence of odd Shapiro steps do not signify absence of Majorana fermions.

• This provides a universal phase sensitive signature for the presence of Majorana

fermions.

Devil’s staircase structure:

I V

0.575 0.58 0.585 0.59 0.595 2.7 2.8 2.9 3

p-p s-s

11/2 17/3 23/4

...

16/3 40/7 52/9 N-2/n N=6 28/5 29/5 p-p - A=0.6; s-s - A =0.8; ω=0.5 64/11 N-1/n, N=6

I V

0.64 0.66 0.68 0.7 0.72 3 3.1 3.2 3.3 3.4 3.5

p-p s-s

31/5 25/4 N+2/n N=6

...

44/7 20/3 7/1 p-p - A=0.77, s-s - A =0.9; ω=0.5 13/2 32/5

...

19/3 N+1/n, N=6

V = (N ± 2/n)ω

V = (N ± 1/n)ω

s-wave junctions p-wave junctions

Experimental proposal:

• Measurement of 𝜃 as a function of β in the RCSJ model ~ exponential dependence of 𝜃

with β.

• Additional steps in the CV-characterstics for Josephson junctions hosting Majorana fermions

⭐ Kulikov et.al, accepted for publication in

JETP (2017)

The step structures of the 4! periodic current prevails!

IJ = p (D) sin(φ/2) + sin(φ)

V = (N ± 2/n)ω

Summary - I

• Unconventional Josephson junctions Majorana quasiparticles subjected to external radiation ~

phase sensitive detectors.

• The current-voltage characteristics of junctions with p-wave pairing symmetry shows presence
• f both odd and even steps in the Shapiro step structures. The origin of the odd Shapiro steps in

the current-voltage characteristics are essentially of different origin and is shown to exist due to the sub-harmonics.

• Presence of additional step sequences in the Devil- staircase structure.

0.5 1

β

3 6 9

η

P-P Simulation P-P Theory S-S Simulation

I V

0.64 0.66 0.68 0.7 0.72 3 3.1 3.2 3.3 3.4 3.5

p-p s-s

31/5 25/4 N+2/n N=6

...

44/7 20/3 7/1 p-p - A=0.77, s-s - A =0.9; ω=0.5 13/2 32/5

...

19/3 N+1/n, N=6

entanglement in the Kitaev model

Kitaev model

x-x y-y z-z

⭐Kitaev, 2006 Ann. Phys.

H = − X

<j,k>α

Jαjkσα

j σα k

(α = x, y, z)

Jα σk

α

Dimensionless coupling constant 𝛽 component of Pauli matrices A two dimensional quantum spin model which is exactly solvable

˜ H = i 2 X

<j,k>α

Jαjkˆ ujkcjck

Following Kitaev’s prescription, we introduce a set of four Majorana fermions:

ˆ ujk = ibαjk

j

bαjk

k

link operators defined on a given link <jk>

{bx

k, by k, bz k, ck}

Kitaev model

• Gapped quantum phases robust to any small (local) perturbation 



 quasiparticle excitations which obey fractional statistics 
 
 topological entanglement entropy 𝜹 : leading 


• rder correction to the universal area law

B A A A Jx=1 Jy=1 Jz=1

A B Gapped phase Gapless phase

SA = SA,F + SA,G − ln 2

Contribution from the Majorana fermions Contribution from the Z2 gauge field topological entanglement entropy

SA,F SA,G ln 2

A B

⭐Yao et. al. 2010 PRL

Entanglement entropy

⇣k = (exp(✏k) + 1)−1

SA,F = −Tr[ρA,F ln ρA,F ]

SA,F =

NA

X

i=1

1 + ζi 2 ln 1 + ζi 2 + 1 − ζi 2 ln 1 − ζi 2

⭐ Peschel (2002) JPhys A: Math Gen.

reduced density matrix with eigenvalues ϵk

A B

Entanglement spectra

⇣k = (exp(✏k) + 1)−1

⭐ Li et. al. 2008 PRL

Eigenvalues of the reduced density matrix Entanglement (Schmidt) gap

∆A = −(ln ΓM − ln ΓM 0)

largest eigenvalue second largest eigenvalue

∆A = ln (1 + |ζ|min) (1 − |ζ|min)

Γ =

NA

Y

i=1

(1 + ζi) 2 (1 − ζi) 2

Square/rectangular region

Half-region

Nx: Number of sites along the x-axis Ny: Number of sites along the y-axis

Sub-systems we consider:

• Impose periodic boundary conditions
• Numerically analyse the entanglement entropy 


and the entanglement spectrum and corroborate with perturbative analysis.

⭐ PRB 94, 045421 (2016)

Results

Half-region

(a) Plot of entanglement entropy as function of coupling strength Jz

B A A A Jx=1 Jy=1 Jz=1

A B Gapped phase Gapless phase

(b) Plot of derivative of the correlation functions (obtained analytically) 
 as function of coupling strength Jz

Jy/Jx = 0.3

• Ny coupled one-dimensional chains
• Aspect ratio Ny /Nx =10

Prominent cusps at Jz=Jx+Jy : Transition from gapless to gapped phase; Non-monotonic behaviour in the gapless region.

Oscillations in the two- point correlation function corresponding to those in entanglement entropy

Czz = Re[Jxeikx + Jyeiky + Jz] |Jxeikx + Jyeiky + Jz|

Half-region

B A A A Jx=1 Jy=1 Jz=1

A B Gapped phase Gapless phase

Jy/Jx = 0.3

(a) Plot of entanglement gap as function of coupling strength Jz

Entanglement gap is finite in the gapped phase and zero in the gapless phase. However, it still remains zero even in the large Jz limit!

Presence of zero energy edge modes in the gapless and in the 
 large Jz limit

⭐ Thakurati et. al 2014 PRB

Square/rectangular region

• Ny coupled one-dimensional chains
• Aspect ratio Ny /Nx =10

B A A A Jx=1 Jy=1 Jz=1

A B Gapped phase Gapless phase

Jy/Jx = 0.3

Gapless

Gapless

(a) Plot of entanglement entropy as function of coupling strength Jz The qualitative behaviour of the entanglement entropy (non- monotonic/monotonic) in the gapless region depends on system parameters (transverse coupling, geometry).

Jy = Jx

(b) Plot of nearest-neighbour correlation functions with varying Jz

Open: Jy/Jx=1 Solid: Jy/Jx=0.3

Non-monotonic behaviour of different correlation functions - ratio of the x- and z-bonds depends on system size.

(c) Plot of entanglement gap with varying Jz

Jy = Jx ◼ 12 x 12 △ 12 x 18 ⨀ 18 x 12 ▲ 18 x 18

• 18 x 30

☐ 30 x 18

• Non-monotonic behaviour of entanglement gap within the gapless region.
• Localised gapless edge modes for small and large Jz values with small extensions in the bulk.

0.1, 2.6x10

−3

0.3, 6.17x10

−4

0.69, 1.28x10

−4

1.37, 9.88x10

−6

1.77, 1.11x10 0.79, 6.85x10

−6 −6

Ny Nx

(c) Plot of edge states with varying Jz (with the value of Schmidt gap)

Perturbative analysis: small Jz limit (weakly coupled chain limit)

H = H0 + H0

• Ny coupled one-dimensional chains with periodic boundary condition

H0 =

Ny

X

m

Hm

0 ,

Hm

0 =

X

n

⇣ iJxcm

n,acm n,b + iJycm n,bcm n+1,a

⌘

• Unperturbed Hamiltonian of the mth one-dimensional chain.
• Interchain coupling :

H0 = X

m

H0

m,m+1,

H0

m,m+1 =

X

n

iJzcm

n,acm1 n,b

• Diagonalising we get:

Hm

0 =

X |✏k| ⇣ ↵m†

k ↵m k − m† k m k

⌘ , Hm,m+1

p

= X

k≥0

Jzeiθk 4 ⇣ ↵m†

k ↵m−1 k

− ↵m†

k m−1 k

+ m†

k ↵m−1 k

− m†

k m−1 k

⌘ + h.c ✓ cm

k,a

cm

k,b

◆ = 1 √ 2 ✓ ieiθk ieiθk 1 −1 ◆

θk = tan−1 ✓ Jy sin k Jx + Jy cos k ◆

, ✏k = |Jx + Jyeik|

• Ground state of the system:

|G1i = |Gi ⇣ 1 Ny X

k

J2

z

64 1 ✏2

k

⌘

• X

m

(1)Ny−mJz 8✏k ⇣ e−iθk|0, 0; m 1i|1, 1; mi|G : m, m 1i eiθk|0, 0; m + 1i|1, 1; mi|G : m + 1, mi ⌘

|Gi =

Ny

Y

m=1

Y

k

βm†

k |0i

for the unperturbed Hamiltonian ground state of the mth chain

• 1st order corrections:
• Eigenvalues of the reduced density matrix

1 = 0 + X

k

(N 0 − 1)J2

z

32✏2

k

⇣ 0 − J2

z

64✏2

k

⌘1 2 = J2

z

64(Jx − Jy)2 0 = 1 − Ny X

k

J2

z

32✏2

k

Perturbative analysis: small Jz limit (weakly coupled chain limit)

Perturbative analysis: Large Jz limit

• Isolated z-bonds

H = X

n

Jzicn,1cn,2 + X

n

Jαicn,1cn+δα,2, α = x, y

perturbation: hopping of the Majorana fermions between nearest neighbour dimers

• 1st order corrections to the

ground state of the system:

|G1i = ✓ 1 ˜ N J2 8J2

z

◆ |Oi + X

<i,i+δα>

Jα 2Jz |1n, 1n+δαi

unperturbed ground state filled states at nth nearest neighbour dimers.

ψ ~ linear combination of c

fermions

• Calculation of the reduced density matrix: ρE

Perturbative analysis: Large Jz limit

ρE = ✓ 1 N J2 4J2

z

◆ |OihO| + X

i

⇣ J2 4J2

z

| ˜ O, 1iih ˜ O, 1i|+ X

i,j

 J2 4J2

z

| ˜ O, 1i, 1jih ˜ O, 1i, 1j| + J 2Jz |Oih ˜ O, 1i, 1j| ⌘

⇒ Nz degenerate values ⇒ Vanishing Schmidt gap

• The extensive study of the entanglement entropy and spectra for the vortex free ground state
• f the Kitaev model.
• For the half-region, entanglement gap is found to be finite in small coupling limit, while it

was shown to be zero in the large coupling limit. Presence of gapless edge modes were attributed.

• For the square/rectangular block, non-monotonic behaviour of entanglement entropy

attributed to the competition between the correlation functions in different kind of bonds.

Summary - II

Collaborators

• K. Sengupta (IACS, India)
• Y. M Shukrinov (BLTP, JINR)
• K. M Kulikov (BLTP, JINR)
• S. Mandal (Institute of Physics, India)
• V. K Varma (ICTP, Italy)
• PRB 92 224501 (2015).
• PRB 94, 045421 (2016).