thermal metal in topological superconductors. -...

Thermal metal in topological superconductors.

J. Tworzydło

University of Warsaw, Poland

in collaboration with Leiden University, The NetherlandsCosma Fulga, Anton Akhmerov, Benjamin Beri, Carlo Beenakker

Nano-CTM Network MeetingCargese, Corsica, 26 October 2012

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Relevant papers

I.C. Fulga, A.R. Akhmerov, J. Tworzydlo, B. Beri, C.W.J. BeenakkerPhys. Rev. B 86, 054505 (2012)M. V. Medvedyeva, J. Tworzydło, and C. W. J. BeenakkerPhys. Rev. B 81, 214203 (2010)M. Wimmer, A.R. Akhmerov, M.V. Medvedyeva, J. Tworzydlo,and C.W.J. BeenakkerPhys. Rev. Lett. 105, 046803 (2010)

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Plan

1 Introduction: topological superconductors

2 Thermal metal

3 Thermal metal with time reversal symmetry

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Plan

1 Introduction: topological superconductors

2 Thermal metal

3 Thermal metal with time reversal symmetry

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SuperconductorBCS condensate absorbs charge −→

excitations only carry heat current

“partihole” excitations above the gaptransport inhibited for kBT < ∆

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Thermal QHE

imagine an edge quasiparticle excitation bulk superconductor actsas heat insulatorthermal conductance quantumg0 = π2k2

BT/3hSchwab et al. Nature ’00, Saito talk

temperature imbalance givesJQ = κxyδTκxy = g0/2 quantizedas in QH Sendhil, Fisher PRB ’00

... topologically protected in p-wave chiral SC (TRS broken)

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Proposals

surface film(A-phase in 3He)

Volovik JETP Lett. ’99,Kopnin & Salomaa PRB ’91

Read & Moore ’91pairing state at ν = 5/2

Read & Green PRB ’00exp. Willett et al. PRB ’10exp. Heiblum ’12

p-wave superconductorSr2RuO4

exp. J. Jang et al. Science ’11

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Revived interest

interface topological insulator/superconductor Fu & Kane PRL ’08

Nb-Bi2Te3: proximity & TI

exp. M. Veldhorst et al. Nature Mat. ’11

CuxBi2Se3: SC from intercalated TI

exp. Y.S. Hor et al. PRL ’10

expected helical edge statesthermal analog of Quantum Spin Hall Effect Ryu, Moore, Ludwig ’10

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BdG equation for p-wave superconductor

Bogoliubov-de-Gennes equation(p2

2m + U − µ ∆p

∆∗p −(

p2

2m + U − µ) )( u

v

)= E

(uv

)for p-wave superconductorwith an order parameter ∆p = v∆ · (px − ipy )

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BdG equation for p-wave superconductor

Bogoliubov-de-Gennes equation(p2

2m + U − µ ∆p

∆∗p −(

p2

2m + U − µ) )( u

v

)= E

(uv

)for p-wave superconductorwith an order parameter ∆p = v∆ · (px − ipy )

Dirac quasiparticlesH = (pxτx + pyτy ) + (M + 1

2p2)τz

mass M = (U − µ)/mv2∆

e-h symmetry: τxH∗τx = −Hsystem in D symmetry class (TRS broken)J.Tworzydło (Warsaw & Leiden) - Thermal metal - 9 / 19

Majorana-Shockley state

electrostatic line defect δU[mv2∆]

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Majorana-Shockley state

electrostatic line defect δU[mv2∆]

MS state amplitude

Wimmer et al. ’10

gap closes upon varying δUe-h symmetrypins the state at E = 0protected MS states format the ends

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Plan

1 Introduction: topological superconductors

2 Thermal metal

3 Thermal metal with time reversal symmetry

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Collective properties of Majoranas

numerical study of quasiparticle localizationquantity studied: thermal conductivity κ/g0

random mass disorder: δUstaggered-fermion lattice discretization in 2D

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Phase diagram

∗ free fermions (or Ising)RG attractive fixed point

(A) Insulator-InsulatorQHE–like transition

(B) Metal-Insulatorpercolation of Majorana fermions

(C) tricritical point

similar: Cho-Fisher model (1997), recent study Kagalovsky & Nemirovsky PRB ’10;

Lauman, Ludwig, Huse, Trebst arXiv 1106.6265; Kraus, Stern New J. Phys. ’11

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Scaling at M-I transition

metallic one-parameterscaling curvelogarithmic conductivity ∝ 1

πln L

(predicted in Senthil & Fisher ’00)

M-I scaling exponentν ′ = 1.02± 0.06compare: I-I exponent ν = 1critical κc/g0 = 0.41± 0.01

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Plan

1 Introduction: topological superconductors

2 Thermal metal

3 Thermal metal with time reversal symmetry

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Helical superconductor in class DIII

Spin dependent BdG equationH = (pxτxσZ + pyτyσ0) + (M + 1

2p2)τzσ0 + K τyσy

extra TRS symmetric copy→ σyH∗σy = Hcoupling K (lowest order in p)

mass M = (U − µ)/mv2∆

helical edge states present for M < 0

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Phase diagram

metallic phase develops alreadyat small disordergenerically no I-I transition at K 6= 0

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Metal-insulator transition in class DIII

clear signature ofa thermal metalcritical conductivityκc/g0 = 0.74± 0.02multi-paramter scaling fitwith sub-leading correctionsνDIII = 2.06± 0.05

raw (not rescaled) data up to L = 200, W = 800network model used instead of discretization

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Summary

Majorana states form on electorstatic defectsin p-wave chiral superconductorboth chiral and helical Majorana modes percolateto form a thermal metalnew exponent for 2D M-I transition νDIII ≈ 2.0I-I not found in DIII class

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