thermal metal in topological superconductors. -...
TRANSCRIPT
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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