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
Plan of the talk 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
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 • ~ 2 q 2 z + α ˆ H ( q z ) = n · ( σ × q ) Δ Δ 2 m eff direction strength of SO coupling z z z of SO coupling ~ 2 q 2 z ✏ ( q z ) = ± ↵ q z 2 m eff Condition for topological phase hosting Introduce in-plane magnetic field B Majorana fermions: opens a gap at q z =0 p µ 2 + ∆ 2 < B z ~ 2 q 2 z p ✏ ( q z ) = ↵ 2 q 2 z + B 2 ± z 2 m eff
Mid-gap zero-bias states using 1-d nanowire Position of the zero bias peaks ⭐ � Mourik et. al Science (2012) ⭐ � Das et. al Nat. Phys (2012) ⭐ � Deng et. al Nano Lett. (2012) ⭐ � Finck et. al PRL (2013)
Fractional ac Josephson effect: Doubling of Shapiro steps ⭐ � Rokhinson et. al Nat. Phys. (2012) Position of the zero-bias Majorana modes
Localised edge states on ferromagnetic atomic chains atop Pb superconductor ⭐ � Nadj-Perge et. al Science (2013)
fractional Josephson effect
Josephson effect in conventional junctions • 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 superconducting current in the S S I J ∼ sin( φ ) absence of any external bias : ✩ Josephson (1962) Weak links Tunnel junctions S S S S N B Superconducting current is due to Superconducting current is due to Andreev bound states proximity effect
Josephson effect in unconventional junctions g ( k F ) : variation around the Fermi surface ∆ ( k F ) = ∆ 0 g ( k F ) exp( i φ ) φ : global phase factor Systems with unconventional pairing amplitude: g ( k ) = ( k x + ik y ) /k F S S I J ∼ sin( φ / 2) B φ = ( φ 1 ∼ φ 2 ) I J ∼ sin( φ ) to I J ∼ sin( φ / 2) The current phase relation changes from Doubling of the periodicity of the phase in the current-phase relation!
Dynamics of unconventional Josephson junctions
The resistively and capacitively shunted Josephson junction d d 2 φ d φ Φ 0 dt 2 + Φ 0 S B S I + A sin ω t = I J + C 0 2 π 2 π R dt Phase ɸ is has time dependence in V 0 presence of external radiation d φ dt = 2 eV/h Schematic representation of the junction Dissipation parameter Current-Voltage characteristics β 7 1 (underdamped/overdamped) • Shapiro steps V = n ~ ω / 2 e V = ω , 2 ω , 3 ω .. ⭐ Shapiro (1963)
Shapiro steps in Josephson junction • Shapiro step structures are predicted to be different for Josephson junctions with sin( ɸ ) and sin( ɸ /2) current phase relations. • 4 π periodic Josephson effect or appearance of Shapiro steps at even multiples of frequency of external radiation i.e V = 2 ω , 4 ω , 6 ω .. • 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?
Appearance of odd Shapiro steps! ⭐ PRB 92, 224501 (2015) D=0.4, A=20, β =0.2 Appearance of both odd and even steps in the current-voltage (I-V) characteristics. dimensionless barrier strength 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 odd steps for a significant range of coupling ~ the width of the odd steps are decreases : Width of Shapiro steps gradually in the resistive junctions.
Appearance of odd Shapiro steps! C-V characteristics variation with frequency of external radiation:
Perturbative analysis In the regime β 𝜕 , 𝜕 , A >> 1, perturbative analysis of the non-linear term. • X X ✏ n � n , I = ✏ n I n � = ⭐ Kornev et. al, J Phys. Conf. Ser., 43, 1105 (2006) n n h ˙ I 0 ~ applied current, I n>0 ~ determined from φ n> 0 i = 0 • φ n + β ˙ ¨ φ n = f n ( t ) + I n For n < 2 • f 0 = A sin( ω t ) where f 1 = − sin( φ 0 / 2) 0 + I 0 t/ β + A α 0 = arccos( ω / γ ) , In first order, φ 0 ( t ) = φ ωγ sin( ω t + α 0 ) • β 2 + ω 2 p γ = I (0) ∼ sin( φ 0 ( t ) / 2) s Condition for Shapiro steps ∞ 0 / 2) X J n (A / 2 γω )e (i[I 0 / (2 β )+n ω ]t+n α 0 + φ = Im I 0 = 2 | n | ωβ n= −∞ A = 2 J n ( β 2 + ω 2 ) ~ contribution from the harmonics ∆ I even s p 2 ω
• n=1 1 J n ( x )( γ n ω n ) � 1 cos( ω n t + n α 0 + δ 0 + n φ 0 / 2) X φ 1 = n = �1 p ω n = I 0 / (2 β ) + n ω , δ n = arccos( ω n / γ n ) , γ n = ω 2 n + β 2 ∼ 1 I (1) 2 φ 1 ( t ) cos( φ 0 ( t ) / 2) s X J n 1 ( x ) J n 2 ( x )(4 γ n 1 ω n 1 ) � 1 = n 1 ,n 2 h i sin([ ω n 1 + ω n 2 ] t + [ n 1 + n 2 ]( α 0 + φ 0 / 2) + δ n 1 ) + sin([ ω n 1 − ω n 2 ] t + [ n 1 − n 2 ]( α 0 + φ 0 / 2) + δ n 1 ) × Condition for Shapiro steps I 0 = | n 1 + n 2 | ωβ ( n 1 + n 2 ) = 2 m + 1 A A J n ( β 2 + ω 2 ) J 2 m +1 − n ( β 2 + ω 2 ) 2 ω √ 2 ω √ X ∆ I odd ~ contribution from the sub-harmonics = 2( β 2 + (2 m + 1 − 2 n ) 2 ω 2 / 4) s n>m η = ∆ I even s ∆ I odd s
Plot of the ratio of the step width η as a function of dissipation parameter β : η = α 0 exp( α 1 β 2 ) 9 6 P-P Simulation η P-P Theory Simulation: S-S Simulation 3 α 0 = 6 . 09 , α 1 = 0 . 31 0 0.5 1 Theory: β α 0 = 5 . 98 , α 1 = 0 . 32 • For p-wave junction η has exponential dependence on the junction capacitance C 0 ~ 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: 3 7/1 3.5 N-2/n p-p s-s N+1/n, N=6 s-s N=6 p-p ... 3.4 64/11 2.9 p-p - A=0.6; 29/5 p-p - A=0.77, 52/9 V 23/4 s-s - A =0.8; 20/3 s-s - A =0.9; 40/7 V ω =0.5 3.3 17/3 ω =0.5 13/2 2.8 28/5 32/5 3.2 N-1/n, 11/2 16/3 19/3 N=6 44/7 25/4 ... 31/5 3.1 2.7 ... N+2/n N=6 0.575 0.58 0.585 0.59 0.595 3 I 0.64 0.66 0.68 0.7 0.72 I s-wave junctions V = ( N ± 1 /n ) ω p-wave junctions V = ( N ± 2 /n ) ω 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 p I J = ( D ) sin( φ / 2) + sin( φ ) JETP (2017) The step structures of the 4 ! periodic current prevails! 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 • of 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. • 7/1 3.5 9 N+1/n, s-s N=6 p-p 3.4 p-p - A=0.77, 6 V P-P Simulation 20/3 s-s - A =0.9; η P-P Theory 3.3 ω =0.5 S-S Simulation 13/2 3 32/5 3.2 19/3 44/7 25/4 0 31/5 ... 3.1 ... N+2/n 0.5 1 N=6 3 β 0.64 0.66 0.68 0.7 0.72 I
entanglement in the Kitaev model
Kitaev model A two dimensional quantum spin model which is exactly solvable X H = − ( α = x, y, z ) J α jk σ α j σ α k <j,k> α x-x Dimensionless coupling constant J α y-y σ k 𝛽 component of Pauli matrices z-z α k , b y { b x k , b z Following Kitaev’s prescription, we introduce a set of four Majorana fermions: k , c k } H = i X ˜ J α jk ˆ u jk c j c k u jk = ib α jk b α jk ˆ 2 j k <j,k> α link operators defined on a given link <jk> ⭐ Kitaev, 2006 Ann. Phys.
Kitaev model J z =1 • Gapped quantum phases robust to any small (local) perturbation A quasiparticle excitations which obey fractional statistics B topological entanglement entropy 𝜹 : leading A A order correction to the universal area law J x =1 J y =1 Gapped phase A Gapless phase B
Entanglement entropy B S A = S A,F + S A,G − ln 2 S A,F Contribution from the Majorana fermions A S A,G Contribution from the Z 2 gauge field topological entanglement entropy ln 2 ⭐ Yao et. al. 2010 PRL S A,F = − Tr[ ρ A,F ln ρ A,F ] ⇣ k = ( exp ( ✏ k ) + 1) − 1 reduced density matrix with eigenvalues ϵ k ⭐ Peschel (2002) JPhys A: Math Gen. N A 1 + ζ i ln 1 + ζ i + 1 − ζ i ln 1 − ζ i X S A,F = 2 2 2 2 i =1
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