Correlation effects in superconducting quantum dot systems

V. Pokorny 1, M. Zonda 1, T. Novotny 1, V. Janis 2

(1) Faculty of Mathematics and Physics, Charles University in Prague

(2) Institute of Physics, Czech Academy of Sciences, Prague

collaboration: T. Domanski (UMCS Lublin, Poland)

pokornyv@karlov.mff.cuni.cz 11. 1. 2018, Uni Regensburg

Introduction

single-level quantum dot connected to two superconducting (sc) BCS leads:

I ideal system for studying the interplay of electronic correlations andsuperconducting order

I proximity effect: Cooper pairs from the superconducor leak into the quantumdot, opening a gap in the DoS (induced pairing)

I multiple Andreev reflections on the opposite interfaces give rise to discretesubgap states (Andreev bound states)

I Josephson current can flow between the superconducting electrodes

I presence of a third normal (non-sc) electrode populates the gap with finite DoS,gives control over the Kondo effect

I experimental and theoretical results show that system can undergo a quantumphase transition from spin-singlet (BCS or Kondo) ground state to spin-doubletground state: 0− π transition known from SFS junctions

I What does it mean “single-level”?level spacing: (like a particle in a box)

δE ∼500meV

L[nm]for short carbon nanotube

is the dominant energy scale

I long sc nanotubes/nanowires: different physics (e.g. Majorana fermions)

Introduction

single-level quantum dot connected to two superconducting (sc) BCS leads:

I ideal system for studying the interplay of electronic correlations andsuperconducting order

I proximity effect: Cooper pairs from the superconducor leak into the quantumdot, opening a gap in the DoS (induced pairing)

I multiple Andreev reflections on the opposite interfaces give rise to discretesubgap states (Andreev bound states)

I Josephson current can flow between the superconducting electrodes

I presence of a third normal (non-sc) electrode populates the gap with finite DoS,gives control over the Kondo effect

I experimental and theoretical results show that system can undergo a quantumphase transition from spin-singlet (BCS or Kondo) ground state to spin-doubletground state: 0− π transition known from SFS junctions

I What does it mean “single-level”?level spacing: (like a particle in a box)

δE ∼500meV

L[nm]for short carbon nanotube

is the dominant energy scale

I long sc nanotubes/nanowires: different physics (e.g. Majorana fermions)

Introduction

I theoretical description: single-impurity Anderson model (SIAM) + BCSI analytic solvers: Hartree-Fock, NCA, 2nd order PT...I heavy numerics: ED, NRG, fRG, QMCs...

I experimental realization:I carbon nanotubeI InAs/InSb nanowireI C60 molecule in break junctionI self-assembled SiGe quantum dots

PRB 88, 045101 (2013). Nature 442, 667 (2006).

I applications:I quantum computing: Andreev qubits: Science 349, 1199 (2015). (CNT)I single-molecule SQUIDs: Nat. Nano. 1, 53 (2006). (InAs nanowire)I Cooper pair splitters: Nature 461 960 (2009). (InAs nanowire)I . . .

Introduction Andreev bound states and the 0 − π transition

I electron with energy E < ∆ incidenton the QD-S interface penetrates intosc, creates a Cooper pair and isreflected back as a hole - Andreevreflection

I multiple Andreev reflections lead to aformation of Andreev bound stateswithin the sc gap

Sp

ectr

al f

un

ctio

n

Energy

Exp.: evolution of ABS with gate voltage

Nat. Nano. 9, 79 (2014).

I Crosing of ABS at the Fermi energymarks QPT from spin-singlet 0-phaseto spin-doublet π-phase.

I ABS are current-carrying states

I The Josephson current jumps frompositive to negative values at 0− πtransition - can be observed incurrent-phase relations.

Introduction description of the model

left lead quantum dot right lead

normal lead

I ε - on-site energy level

I U - on-site Coulomb interaction

I ∆α - superconducting gap (we assume ∆L = ∆R = ∆)

I Φα - order parameter phase

I Γα - tunneling rate (dot-lead coupling)

Observables depend only on the phase difference Φ = ΦL − ΦR between the sc leadsnot on the absolute values of the two phases.Result: gauge invariance under ΦL,R → ΦL,R + Φsh with important consequencesA. Kadlecova, M. Zonda, T. Novotny, Phys. Rev. B 95, 195114 (2017).

Introduction description of the model

left lead quantum dot right lead

normal lead

I ε - on-site energy level

I U - on-site Coulomb interaction

I ∆α - superconducting gap (we assume ∆L = ∆R = ∆)

I Φα - order parameter phase

I Γα - tunneling rate (dot-lead coupling)

Observables depend only on the phase difference Φ = ΦL − ΦR between the sc leadsnot on the absolute values of the two phases.Result: gauge invariance under ΦL,R → ΦL,R + Φsh with important consequencesA. Kadlecova, M. Zonda, T. Novotny, Phys. Rev. B 95, 195114 (2017).

Introduction the modified single-impurity Anderson model

H = Hdot +∑

α=R,L,N

(Hαlead +Hαc )

I quantum dot (single - level):

Hdot =∑σ

εσd†σdσ + Ud†↑d↑d

†↓d↓ εσ = ε+ σB

I leads:

Hαlead =∑kσ

εσ(k)c†α,kσcα,kσ−∆α

∑k

(e iΦαc†α,k↑c†α,−k↓+H.c.) α = R, L,N

I couplings:

Hαc = −tα∑kσ

(c†α,kσdσ + H.c.) Γα = 2πρα|tα|2 − tunneling rate

I reliable description of the system: numerically exact calculations (NRG, QMC)show good agreement with experiment (ABS frequencies, Josephson current)

Introduction the modified single-impurity Anderson model

H = Hdot +∑

α=R,L,N

(Hαlead +Hαc )

I quantum dot (single - level):

Hdot =∑σ

εσd†σdσ + Ud†↑d↑d

†↓d↓ εσ = ε+ σB

I leads:

Hαlead =∑kσ

εσ(k)c†α,kσcα,kσ−∆α

∑k

(e iΦαc†α,k↑c†α,−k↓+H.c.) α = R, L,N

I couplings:

Hαc = −tα∑kσ

(c†α,kσdσ + H.c.) Γα = 2πρα|tα|2 − tunneling rate

I reliable description of the system: numerically exact calculations (NRG, QMC)show good agreement with experiment (ABS frequencies, Josephson current)

Introduction the modified single-impurity Anderson model

H = Hdot +∑

α=R,L,N

(Hαlead +Hαc )

I quantum dot (single - level):

Hdot =∑σ

εσd†σdσ + Ud†↑d↑d

†↓d↓ εσ = ε+ σB

I leads:

Hαlead =∑kσ

εσ(k)c†α,kσcα,kσ−∆α

∑k

(e iΦαc†α,k↑c†α,−k↓+H.c.) α = R, L,N

I couplings:

Hαc = −tα∑kσ

(c†α,kσdσ + H.c.) Γα = 2πρα|tα|2 − tunneling rate

I reliable description of the system: numerically exact calculations (NRG, QMC)show good agreement with experiment (ABS frequencies, Josephson current)

Introduction the modified single-impurity Anderson model

H = Hdot +∑

α=R,L,N

(Hαlead +Hαc )

I quantum dot (single - level):

Hdot =∑σ

εσd†σdσ + Ud†↑d↑d

†↓d↓ εσ = ε+ σB

I leads:

Hαlead =∑kσ

εσ(k)c†α,kσcα,kσ−∆α

∑k

(e iΦαc†α,k↑c†α,−k↓+H.c.) α = R, L,N

I couplings:

Hαc = −tα∑kσ

(c†α,kσdσ + H.c.) Γα = 2πρα|tα|2 − tunneling rate

I reliable description of the system: numerically exact calculations (NRG, QMC)show good agreement with experiment (ABS frequencies, Josephson current)

Methods Nambu Green function and Hartree-Fock approximation

I Nambu spinor: Nambu Green function:

Ψ(τ) =

(d↑(τ)

d†↓(τ)

)G(τ) = −

⟨Tτ [Ψ(τ)Ψ†(0)]

⟩I 2× 2 matrix with normal (diagonal) and anomalous (off-diagonal) components

G = −(〈d↑d

†↑〉 〈d↑d↓〉

〈d†↓d†↑〉 〈d

†↓d↓〉

)≡(Gn Ga

G∗a G∗n

)=

I equilibrium physics: Matsubara frequencies ωn = (2n + 1)π/β

Hartree - Fock approximation (spin - symmetric)

I static mean-field, simplest way how to study Coulomb interaction effects

I static self-energies: ΣHF = U〈Gn〉, SHF = U〈Ga〉I self-consistent equations:

ΣHF =U

β

∑n

Gn(iωn,ΣHF ,SHF ), SHF =

U

β

∑n

Ga(iωn,ΣHF ,SHF )

I straightforward continuation to the real axis → Gn(ω + i0), Ga(ω + i0)(important for calculating ABS frequencies)

Methods Nambu Green function and Hartree-Fock approximation

I Nambu spinor: Nambu Green function:

Ψ(τ) =

(d↑(τ)

d†↓(τ)

)G(τ) = −

⟨Tτ [Ψ(τ)Ψ†(0)]

⟩I 2× 2 matrix with normal (diagonal) and anomalous (off-diagonal) components

G = −(〈d↑d

†↑〉 〈d↑d↓〉

〈d†↓d†↑〉 〈d

†↓d↓〉

)≡(Gn Ga

G∗a G∗n

)=

I equilibrium physics: Matsubara frequencies ωn = (2n + 1)π/β

Hartree - Fock approximation (spin - symmetric)

I static mean-field, simplest way how to study Coulomb interaction effects

I static self-energies: ΣHF = U〈Gn〉, SHF = U〈Ga〉I self-consistent equations:

ΣHF =U

β

∑n

Gn(iωn,ΣHF ,SHF ), SHF =

U

β

∑n

Ga(iωn,ΣHF ,SHF )

I straightforward continuation to the real axis → Gn(ω + i0), Ga(ω + i0)(important for calculating ABS frequencies)

Methods second-order perturbation theory

I first step beyond Hartree-Fock approximation

I dynamical self-energy: Σ(iωn) = ΣHF + Σ(2)(iωn), input: HF propagators

Σ = − −

S = − −

I interacting Green function: matrix Dyson equation

G−1(iωn) = G−1U=0(iωn)− Σ(iωn)

I static self-consistency between input and output Green functions:

ΣHF =U

β

∑n

Gn[iωn; Σ(iωn),S(iωn)] SHF =U

β

∑n

Ga[iωn; Σ(iωn),S(iωn)]

I Josephson current: Spectral functions:

J =4

β

∑n

Im[Ga(iωn)S(0)(iωn)

]ρn/a(ω) = −

1

πImGn/a(ω + i0)

Results two-terminal setup: 2nd order PT vs. NRG

two-terminal setup: ΓN = 0

ABS frequencies: current-phase relation:

0

0.1

0.2

0.3

0 0.2 0.4 0.6 0.8 1

U = 8∆

U = 4∆

Γ = 2∆, ε = -U/2

ω /

∆

Φ / π

NRG

2nd PT, U = 4∆

U = 8∆

-0.2

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1

Jose

ph

son

cu

rren

t

Φ / π

NRG

2nd PT, U = 4∆

U = 8∆

I NRG data calculated using NRG Lubliana code (nrgljubljana.ijs.si)

I good agreement with NRG in the 0-phase

I we can extract the position of the 0− π transition QCP from ABS frequencies orJosephson current (both experimentally measurable)

I fails completely to describe the π-phase

I breakdown of the perturbation technique: double-degenerate ground state,violation of the of the Gell-Mann - Low theorem

Results two-terminal setup: 2nd order PT vs. NRG

I NRG: ∼ hours on a cluster node

I 2nd order PT: ∼ minutes on a laptop

I Fundamental theorem of scientificcomputing: Nobody cares how fastyou can calculate a wrong answer.

I need to set limits of applicability of2nd order PT

phase diagram in U −∆ plane (Φ = 0)

0.01

0.1

1

0 2 4 6 8 10 12 14

∆ /

Γ

U / Γ

sc 2nd PT

NRG

0.01

0.1

1

0 2 4 6 8 10 12 14

0 - phase(singlet)

π - phase(doublet)BCS singlet

Kondo singlet

BC

S-K

ondo c

ross

over

Andreev bound states

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

ω /

∆

U / Γ

2nd, ∆ / Γ = 0.94

0.16

0.04

0.016

NRG

Induced gap µ = U〈d†d†〉

0

0.2

0.4

0.6

0 2 4 6 8 10 12 14

Indu

ced g

ap

U / Γ

2nd, ∆ / Γ = 0.940.16

0.016NRG

Results two-terminal setup: 2nd order PT vs. experiment

R. Delagrange et al, PRB 91, 241401(R) (2015).

0− π transition:

0

0.2

0.4

0.6

0.8

1

0.7 0.8 0.9 1 1.1 1.2 1.3

Φ /

π

ε [meV]

∆ = 0.17 meVΓR + ΓL = 0.44 meVΓR = 4ΓLU = 3.2 meV +/- 10%

experiment

shifted CT-INT

2nd, U = 3.44 meV

two-terminal setup: quantum dot with scleads embedded in asymmetric squid setup:

I doped Si substrate + SiO2 layer

I quantum dot: carbon nanotube

I Al electrodes (TC = 1.2K) withPd/Nb coating

I T = 50mK

parameters:

I ∆ = 0.17 meV

I U = 3.2 meV ± 10%

I ΓL + ΓR = 0.44 meV, ΓL/ΓR = 4

I Φ = (2e/~)BS

Good fit of exp. data obtained by varyingU within the error bars.

Methods three-terminal setup

three-terminal heterostructure (two superconducting and one normal lead):

I NRG is ineffective in this setup, except Φ = 0 case (three-channel problem)

I normal coupling ΓN smears out the phase boundary and mixes 0 and π phases:2nd order PT becomes unreliable (cannot describe the π-phase)

I normal coupling also populates the gap with non-zero density of states, leading tothe onset of the Kondo effect

test: NRG vs 2nd order PT at Φ = 0:

(ΓS = ΓL + ΓR )

I We need a robust method to solve the modified SIAM, capable to describeI superconducting pairingI both 0 and π phaseI Kondo behavior

Methods three-terminal setup

three-terminal heterostructure (two superconducting and one normal lead):

I NRG is ineffective in this setup, except Φ = 0 case (three-channel problem)

I normal coupling ΓN smears out the phase boundary and mixes 0 and π phases:2nd order PT becomes unreliable (cannot describe the π-phase)

I normal coupling also populates the gap with non-zero density of states, leading tothe onset of the Kondo effect

test: NRG vs 2nd order PT at Φ = 0:

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

∆ = ΓS, Φ = 0

T = 0

Ind

uce

d g

ap

U / ΓS

NRG, ΓN / ΓS = 0

0.010.1

1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

∆ = ΓS, Φ = 0

T = 0

U / ΓS

2nd PT, ΓN / ΓS = 0

0.010.1

1

(ΓS = ΓL + ΓR )

I We need a robust method to solve the modified SIAM, capable to describeI superconducting pairingI both 0 and π phaseI Kondo behavior

Methods three-terminal setup

three-terminal heterostructure (two superconducting and one normal lead):

I NRG is ineffective in this setup, except Φ = 0 case (three-channel problem)

I normal coupling ΓN smears out the phase boundary and mixes 0 and π phases:2nd order PT becomes unreliable (cannot describe the π-phase)

I normal coupling also populates the gap with non-zero density of states, leading tothe onset of the Kondo effect

test: NRG vs 2nd order PT at Φ = 0:

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

∆ = ΓS, Φ = 0

T = 0

Ind

uce

d g

ap

U / ΓS

NRG, ΓN / ΓS = 0

0.010.1

1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

∆ = ΓS, Φ = 0

T = 0

U / ΓS

2nd PT, ΓN / ΓS = 0

0.010.1

1

(ΓS = ΓL + ΓR )

I We need a robust method to solve the modified SIAM, capable to describeI superconducting pairingI both 0 and π phaseI Kondo behavior

Methods three-terminal setup: CT-HYB

continuous-time quantum Monte Carlo [1]:

I in experiment, U > Γ, so we chose the strong-coupling, hybridization-expansionCT-QMC solver implemented in the package (ipht.cea.fr/triqs) [2,3]

I include superconducting pairing:particle-hole transformation in the σ = ↓ segment [4]:

d†↑ → d†↑ , d†↓ → d↓, c†α,k,↑ → c†α,k,↑, c†α,k,↓ → cα,−k,↓, α = L,R,N

I it maps the sc SIAM model on normal SIAM with attractive interaction:

εσ → σεσ , εσ(k)→ σεσ(k), tα → σtα, U → −U

I as εσ = ε+ σB, it maps the magnetic field B on the local energy ε and vice versa

I gauge invariance under ΦL,R → ΦL,R + Φsh used to keep the hybridizations real

I no fermionic sign problem

I finite temperatures only

I Green function is calculated in imaginary time, no direct access to spectralfunctions/ABS frequencies, only the Josephson current, filling and the inducedgap can be obtained directly

(1) E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Rev. Mod. Phys. 83, 349 (2011).(2) O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, Comput. Phys. Commun. 196, 398 (2015).(3) P. Seth, I. Krivenko, M. Ferrero, and O. Parcollet, Comput. Phys. Commun. 200, 74 (2016).(4) D. J. Luitz and F. F. Assaad, Phys. Rev. B 81 024509, (2010).

Methods three-terminal setup: CT-HYB

continuous-time quantum Monte Carlo [1]:

I in experiment, U > Γ, so we chose the strong-coupling, hybridization-expansionCT-QMC solver implemented in the package (ipht.cea.fr/triqs) [2,3]

I include superconducting pairing:particle-hole transformation in the σ = ↓ segment [4]:

d†↑ → d†↑ , d†↓ → d↓, c†α,k,↑ → c†α,k,↑, c†α,k,↓ → cα,−k,↓, α = L,R,N

I it maps the sc SIAM model on normal SIAM with attractive interaction:

εσ → σεσ , εσ(k)→ σεσ(k), tα → σtα, U → −U

I as εσ = ε+ σB, it maps the magnetic field B on the local energy ε and vice versa

I gauge invariance under ΦL,R → ΦL,R + Φsh used to keep the hybridizations real

I no fermionic sign problem

I finite temperatures only

I Green function is calculated in imaginary time, no direct access to spectralfunctions/ABS frequencies, only the Josephson current, filling and the inducedgap can be obtained directly

(1) E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Rev. Mod. Phys. 83, 349 (2011).(2) O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, Comput. Phys. Commun. 196, 398 (2015).(3) P. Seth, I. Krivenko, M. Ferrero, and O. Parcollet, Comput. Phys. Commun. 200, 74 (2016).(4) D. J. Luitz and F. F. Assaad, Phys. Rev. B 81 024509, (2010).

Methods three-terminal setup: CT-HYB

continuous-time quantum Monte Carlo [1]:

I in experiment, U > Γ, so we chose the strong-coupling, hybridization-expansionCT-QMC solver implemented in the package (ipht.cea.fr/triqs) [2,3]

I include superconducting pairing:particle-hole transformation in the σ = ↓ segment [4]:

d†↑ → d†↑ , d†↓ → d↓, c†α,k,↑ → c†α,k,↑, c†α,k,↓ → cα,−k,↓, α = L,R,N

I it maps the sc SIAM model on normal SIAM with attractive interaction:

εσ → σεσ , εσ(k)→ σεσ(k), tα → σtα, U → −U

I as εσ = ε+ σB, it maps the magnetic field B on the local energy ε and vice versa

I gauge invariance under ΦL,R → ΦL,R + Φsh used to keep the hybridizations real

I no fermionic sign problem

I finite temperatures only

I Green function is calculated in imaginary time, no direct access to spectralfunctions/ABS frequencies, only the Josephson current, filling and the inducedgap can be obtained directly

(1) E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Rev. Mod. Phys. 83, 349 (2011).(2) O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, Comput. Phys. Commun. 196, 398 (2015).(3) P. Seth, I. Krivenko, M. Ferrero, and O. Parcollet, Comput. Phys. Commun. 200, 74 (2016).(4) D. J. Luitz and F. F. Assaad, Phys. Rev. B 81 024509, (2010).

Methods three-terminal setup: CT-HYB

continuous-time quantum Monte Carlo [1]:

I in experiment, U > Γ, so we chose the strong-coupling, hybridization-expansionCT-QMC solver implemented in the package (ipht.cea.fr/triqs) [2,3]

I include superconducting pairing:particle-hole transformation in the σ = ↓ segment [4]:

d†↑ → d†↑ , d†↓ → d↓, c†α,k,↑ → c†α,k,↑, c†α,k,↓ → cα,−k,↓, α = L,R,N

I it maps the sc SIAM model on normal SIAM with attractive interaction:

εσ → σεσ , εσ(k)→ σεσ(k), tα → σtα, U → −U

I as εσ = ε+ σB, it maps the magnetic field B on the local energy ε and vice versa

I gauge invariance under ΦL,R → ΦL,R + Φsh used to keep the hybridizations real

I no fermionic sign problem

I finite temperatures only

I Green function is calculated in imaginary time, no direct access to spectralfunctions/ABS frequencies, only the Josephson current, filling and the inducedgap can be obtained directly

(1) E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Rev. Mod. Phys. 83, 349 (2011).(2) O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, Comput. Phys. Commun. 196, 398 (2015).(3) P. Seth, I. Krivenko, M. Ferrero, and O. Parcollet, Comput. Phys. Commun. 200, 74 (2016).(4) D. J. Luitz and F. F. Assaad, Phys. Rev. B 81 024509, (2010).

Results three-terminal setup: CT-HYB

Induced gap µ =U〈d†d†〉: test against NRG for Φ = 0

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

∆ = ΓS, Φ = 0

T = 0

µ

U / ΓS

NRG, ΓN / ΓS = 0

0.01

0.1

1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

∆ = ΓS, Φ = 0

T = 0.025ΓS

µ

U / ΓS

CT-HYB, ΓN / ΓS = 0

0.01

0.1

1

Josephson current: finite ΓN vs. finite temperature

I Normal coupling ΓN and temperature have qualitatively similar effect on the system:vanishing of the π-phase (spin-doublet)

Results three-terminal setup: CT-HYB

Induced gap µ =U〈d†d†〉: test against NRG for Φ = 0

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

∆ = ΓS, Φ = 0

T = 0

µ

U / ΓS

NRG, ΓN / ΓS = 0

0.01

0.1

1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

∆ = ΓS, Φ = 0

T = 0.025ΓS

µ

U / ΓS

CT-HYB, ΓN / ΓS = 0

0.01

0.1

1

Josephson current: finite ΓN vs. finite temperature

-0.1

0

0.1

0.2

0.3

0 0.2 0.4 0.6 0.8 1

Ud = 3ΓS, ∆ = ΓS

T = 0.025ΓS

Jose

phso

n c

urr

ent

Φ / π

NRG (T = 0), ΓN = 0CT-HYB, ΓN = 0

0.010.1

1

-0.1

0

0.1

0.2

0 0.2 0.4 0.6 0.8 1

Ud = 3ΓS, ∆ = ΓS

ΓN = 0.01ΓS

Φ / π

CT-HYB, T / ΓS = 0.025

0.050.10.2

I Normal coupling ΓN and temperature have qualitatively similar effect on the system:vanishing of the π-phase (spin-doublet)

Results CT-HYB: spectral functions

I in CT-HYB, Green function is calculated in imaginary time, no direct access tospectral functions/ABS frequencies

I analytic continuation −iτ → t of noisy data is a well-known ill-defined problem

G(τ) =

∫dωK(τ, ω)ρ(ω) K(τ, ω) =

e−τω

1 + e−βω

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 10 20 30 40

Gn(τ

)

Imaginary time

-0.006

-0.004

-0.002

0

15 20 25

G11G22

0 β/2 β

Imaginary time

-10

-5

0

5

10

Rea

l fr

equen

cy 0

0.2

0.4

0.6

0.8

1

10-38

10-24

10-12

10-4

I solvers:I Pade approximants (very unstable) [1]I maximum-entropy method (cannot resolve sharp features, needs covariance data) [2]I stochastic sampling - Mishchenko method (computationally very demanding) [3]

(1) H. J. Vidberg and J. W. Serene, J. Low Temp. Phys. 29, 179 (1977).(2) M. Jarrell and J. E. Gubernatis, Phys. Rep. 269, 133 (1996).(3) A. S. Mishchenko, N. V. Prokof’ev, A. Sakamoto, and B. V. Svistunov, Phys. Rev. B 62, 6317 (2000).

Results CT-HYB: spectral functions

Experiment:J.-D. Pillet et al., Phys. Rev. B, 88045101 (2013).

I two-terminal setup, CNT with Al leads

I ∆ = 150µeV, T=35mK

I U = 13.3∆, Γ = 0.9∆

I spectral function measured via STM

I good agreement between experimentand NRG

Spectral function from CT-HYB:stochastic sampling method

NRG

2ndPT

0 2 4 6 8 10

Gate voltage

-1.5

-1

-0.5

0

0.5

1

1.5

En

erg

y

V. Pokorny, M. Zonda, arXiv:1001.2700 (2017).

I T = 44mK (0.025∆)

I good agreement with T = 0 NRGresults in 0-phase

I good agreement aroung QCP

I ABS from CT-HYB shifted w.r.t.NRG in π-phase

Other possible applications

multiple-quantum dot systems:

R. Zitko et al, Phys. Rev. Lett. 105, 116803 (2010).R. Zitko, Phys. Rev. B 91, 165116 (2015).D. Sherman et al., Nat. Nanotech. 12, 212 (2017).Z. Su et al., Nat. Commun. 8, 585 (2017).

Su et al., Nat. Commun. 8, 585 (2017).

InSb nanowire with NbTiN leads∆ ≈ 400µeV , U = 1− 2meV

I singlet, doublet and triplet Andreev molecular states

I superexchange effects (tunable by interdot coupling tLR)

I fingerprints of Majorana bound states (?)

I interesting for applications in topological quantum computing

I model is solvable by the TRIQS CT-HYB implementation

Other possible applications

multiple-quantum dot systems:

R. Zitko et al, Phys. Rev. Lett. 105, 116803 (2010).R. Zitko, Phys. Rev. B 91, 165116 (2015).D. Sherman et al., Nat. Nanotech. 12, 212 (2017).Z. Su et al., Nat. Commun. 8, 585 (2017).

Su et al., Nat. Commun. 8, 585 (2017).

InSb nanowire with NbTiN leads∆ ≈ 400µeV , U = 1− 2meV

I singlet, doublet and triplet Andreev molecular states

I superexchange effects (tunable by interdot coupling tLR)

I fingerprints of Majorana bound states (?)

I interesting for applications in topological quantum computing

I model is solvable by the TRIQS CT-HYB implementation

Conclusions

second order of perturbation theory:

I fast, simple, charge conserving (if ∆L = ∆R) and thermodynamicallyconsistent approximation, does not break spin symmetry, direct access to spectralfunctions/ABS frequencies

I gives reliable results for 0-phase properties (spectral/transport) and phaseboundaries in realistic range of parameters (outside the Kondo region)

I present formulation not applicable to the π-phase - double degenerate groundstate - violation of the Gell-Mann - Low theorem

I not reliable in the three-terminal setup due to the mixing of 0 and π phases

hybridization-expansion CT-QMC:

I based on the TRIQS CT-HYB solver (ipht.cea.fr/triqs)

I works well in both the two-terminal and the three-terminal setups

I able to describe all phases including the π-phase and the Kondo regime

I works only for finite temperatures

I formulated in imaginary time: no direct access to spectral functions/ABSfrequencies without analytic continuation

Conclusions

second order of perturbation theory:

I fast, simple, charge conserving (if ∆L = ∆R) and thermodynamicallyconsistent approximation, does not break spin symmetry, direct access to spectralfunctions/ABS frequencies

I gives reliable results for 0-phase properties (spectral/transport) and phaseboundaries in realistic range of parameters (outside the Kondo region)

I present formulation not applicable to the π-phase - double degenerate groundstate - violation of the Gell-Mann - Low theorem

I not reliable in the three-terminal setup due to the mixing of 0 and π phases

hybridization-expansion CT-QMC:

I based on the TRIQS CT-HYB solver (ipht.cea.fr/triqs)

I works well in both the two-terminal and the three-terminal setups

I able to describe all phases including the π-phase and the Kondo regime

I works only for finite temperatures

I formulated in imaginary time: no direct access to spectral functions/ABSfrequencies without analytic continuation

Conclusions

acknowledgments:

I Czech Science Foundation grant 15-14259S

I National Grid Infrastructure MetaCentrum

I IT4Innovations National Supercomputing Center

reading:I M. Zonda, V. Pokorny, V. Janis, and T. Novotny, Sci. Rep. 5, 8821 (2015). (2ndPT)

I M. Zonda, V. Pokorny, V. Janis, and T. Novotny, PRB 93, 024523 (2016). (2ndPT)

I T. Domanski, M. Zonda, V. Pokorny, G. Gorski, V. Janis, and T. Novotny, PRB 95, 045104(2017). (3-terminal, QMC)

I V. Pokorny, M. Zonda, arXiv:1001.2700 (2017). (2-terminal, QMC)

I www.fzu.cz/∼pokornyv/pages/research (more resources, including this presentation)

codes:I 2nd order PT solver: github.com/pokornyv/SQUAD

I CT-HYB code (using TRIQS solver): github.com/pokornyv/SQUAD-CTHYB

Thank you for your attention.

Conclusions

acknowledgments:

I Czech Science Foundation grant 15-14259S

I National Grid Infrastructure MetaCentrum

I IT4Innovations National Supercomputing Center

reading:I M. Zonda, V. Pokorny, V. Janis, and T. Novotny, Sci. Rep. 5, 8821 (2015). (2ndPT)

I M. Zonda, V. Pokorny, V. Janis, and T. Novotny, PRB 93, 024523 (2016). (2ndPT)

I T. Domanski, M. Zonda, V. Pokorny, G. Gorski, V. Janis, and T. Novotny, PRB 95, 045104(2017). (3-terminal, QMC)

I V. Pokorny, M. Zonda, arXiv:1001.2700 (2017). (2-terminal, QMC)

I www.fzu.cz/∼pokornyv/pages/research (more resources, including this presentation)

codes:I 2nd order PT solver: github.com/pokornyv/SQUAD

I CT-HYB code (using TRIQS solver): github.com/pokornyv/SQUAD-CTHYB

Thank you for your attention.