Transport through a mesoscopic Luttinger liquid with Rashba - - PowerPoint PPT Presentation

OUTLINE

Transport through a mesoscopic Luttinger liquid with Rashba spin-orbit coupling

Nicolas Pauget1 Mikhail Pletyukhov1 Vladimir Gritsev2

1Institut f¨

ur Theoretische Festk¨

• rperphysik

Universit¨ at Karlsruhe

2Condensed Matter Theory Group

Harvard University

2nd February 2006

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

OUTLINE

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

OUTLINE

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

OUTLINE

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Conductance of a quasi 1-dimensional ring

V VG, CG Φ

• B,

E

External parameters Gate voltage (VG),

chemical potential µ, Electric field E (Rashba SO-coupling), Charging energy EC.

Magnetic field ( B),

Magnetic flux Φ, Zeeman effect (negligible).

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Conductance of a quasi 1-dimensional ring

V VG, CG Φ

• B,

E

External parameters Gate voltage (VG),

chemical potential µ, Electric field E (Rashba SO-coupling), Charging energy EC.

Magnetic field ( B),

Magnetic flux Φ, Zeeman effect (negligible).

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Experiments

• J. Nitta et al., Physica E 12

(2002) 753-757 Measure of the magnetoresistance, InGaAs/InAlAs heterostructure. Contour plot: maxima (dark), minima (white).

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Experiments

• M. K¨
• nig et al.,

arXiv:cond-mat/0508396 Measure of the magnetoconductance, HgTe/HgCdTe heterostructure (strong Rashba coupling). Contour plot: maxima (yellow), minima (blue).

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Experiments

• M. K¨
• nig et al.,

arXiv:cond-mat/0508396 Measure of the magnetoconductance, HgTe/HgCdTe heterostructure (strong Rashba coupling). Contour plot: maxima (yellow), minima (blue).

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Experiments

• M. K¨
• nig et al.,

arXiv:cond-mat/0508396 Measure of the magnetoconductance, HgTe/HgCdTe heterostructure (strong Rashba coupling). Contour plot: maxima (yellow), minima (blue).

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Modulation of the conductance peaks

−2 −1 1 2 1 0.5

G/G0 Φ/Φ0

• B (

A)

+

E.A. Jagla et al., PRL 70, 5, 639 (1993)

• Y. Aharonov et al., PR 115, 3, 485 (1959)

Effect of the magnetic field Interferences: Aharonov-Bohm effect. Phase accumulated ( = 1): −e c Z

±

A · dx Flux quantum Φ0 = 2πc

e ,

Kinetic Hamiltonian: Hc = (k − kΦ)2 2m∗ , kΦ = Φ Φ0 2π L L: perimeter of the ring.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Modulation of the conductance peaks

EF E k kF

• B (

A)

+

E.A. Jagla et al., PRL 70, 5, 639 (1993)

• Y. Aharonov et al., PR 115, 3, 485 (1959)

Effect of the magnetic field Interferences: Aharonov-Bohm effect. Phase accumulated ( = 1): −e c Z

±

A · dx Flux quantum Φ0 = 2πc

e ,

Kinetic Hamiltonian: Hc = (k − kΦ)2 2m∗ , kΦ = Φ Φ0 2π L L: perimeter of the ring.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Modulation of the conductance peaks

kΦ EF E k kF

• B (

A)

+

E.A. Jagla et al., PRL 70, 5, 639 (1993)

• Y. Aharonov et al., PR 115, 3, 485 (1959)

Effect of the magnetic field Interferences: Aharonov-Bohm effect. Phase accumulated ( = 1): −e c Z

±

A · dx Flux quantum Φ0 = 2πc

e ,

Kinetic Hamiltonian: Hc = (k − kΦ)2 2m∗ , kΦ = Φ Φ0 2π L L: perimeter of the ring.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Modulation of the conductance peaks (2)

−2 −1 1 2 1 0.5

G/G0 zµ

2πvF L

Effect of the magnetic field Spectral flow, zµ =

L 2πkµ = (µ−vF kF )L 2π

+ 1

2,

Dispersion relation shifted vertically.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Modulation of the conductance peaks (2)

EF E k kF 2πvF L Effect of the magnetic field Spectral flow, zµ =

L 2πkµ = (µ−vF kF )L 2π

+ 1

2,

Dispersion relation shifted vertically.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Modulation of the conductance peaks (2)

∆µ kµ EF E k kF

2πvF L

Effect of the magnetic field Spectral flow, zµ =

L 2πkµ = (µ−vF kF )L 2π

+ 1

2,

Dispersion relation shifted vertically.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Scattering Matrix Approach

−2 −1 1 2

zΦ

−2 −1 1 2

zµ

Contourplot of the conductance maxima (white), minima (black), zµ = (µ−vF kF )L

2π

+ 1

2,

zΦ = Φ

Φ0 ,

Equation of the peaks: zµ + zΦ = m + 1 2 zµ − zΦ = n + 1 2 with m, n ∈ Z.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Rashba spin-orbit coupling

3D = ⇒ 2D 2DEG − → Vertical confinement,

∂V ∂z

• z=0 = 0

− → Structure Inversion Asymmetry, H =

k2 2m∗ + αR

ez · ( σ × k) − → Relativistic correction. where αR = α0 + βVG.

V VG, CG Φ

• B,

E

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Rashba spin-orbit coupling

k E EF kF

n = 0 n = 1 n = 2

2D = ⇒ Quasi 1D Radial confinement (hard-wall potential), Dispersion relation: E± = (k−kΦ±kR)

2m∗

,

kR = π

L (

• 1 + ( αRm∗L

π

)2 − 1). Lowest band approximation: v1, v2 = vF

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Rashba spin-orbit coupling

k E EF kF+ kF−

n = 0 n = 1 n = 2

2D = ⇒ Quasi 1D Radial confinement (hard-wall potential), Dispersion relation: E± = (k−kΦ±kR)

2m∗

,

kR = π

L (

• 1 + ( αRm∗L

π

)2 − 1). Lowest band approximation: v1, v2 = vF

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Rashba spin-orbit coupling

n = 0

k E EF kF+ kF− v1 v2

2D = ⇒ Quasi 1D Radial confinement (hard-wall potential), Dispersion relation: E± = (k−kΦ±kR)

2m∗

,

kR = π

L (

• 1 + ( αRm∗L

π

)2 − 1). Lowest band approximation: v1, v2 = vF

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EXPERIMENTAL MOTIVATIONS CONDUCTANCE PEAKS EFFECT OF SPIN-ORBIT COUPLING

Scattering Matrix Approach with SO coupling

−2 −1 1 2

zΦ

−2 −1 1 2

zµ

A.G. Aronov et al., PRL 70, 3, 343 (1993) Contourplot of the conductance maxima (white), minima (black), zµ = (µ−vF kF )L

2π

+ 1

2,

zΦ =

Φ Φ0 ,

zR = 1

2(

p 1 + (αRmL/π)2−1), Equation of the peaks: zµ ± zΦ ± zR ∈ 1

2 + Z PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Hamiltonian

ψ(ϕ) cL(x) cR(x)

ϕ

tL tR Hring HL HR x

Hamiltonian of the system H = (HL + HR) + Hring + (HTL + HTR) HL =

• k1 εLk1c†

Lk1cLk1

HR =

• k2 εRk2c†

Rk2cRk2

HTL = tLc†

L(xL)ψ(π) + t∗ Lψ(π)†cL(xL)

HTR = tRc†

R(xR)ψ(0) + t∗ Rψ(0)†cR(xR)

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Hamiltonian

ψ(ϕ) cL(x) cR(x)

ϕ

tL tR Hring HL HR x

Hamiltonian of the system H = (HL + HR) + Hring + (HTL + HTR) HL =

• k1 εLk1c†

Lk1cLk1

HR =

• k2 εRk2c†

Rk2cRk2

HTL = tLc†

L(xL)ψ(π) + t∗ Lψ(π)†cL(xL)

HTR = tRc†

R(xR)ψ(0) + t∗ Rψ(0)†cR(xR)

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Hamiltonian

ψ(ϕ) cL(x) cR(x)

ϕ

tL tR Hring HL HR x

Hamiltonian of the system H = (HL + HR) + Hring + (HTL + HTR) HL =

• k1 εLk1c†

Lk1cLk1

HR =

• k2 εRk2c†

Rk2cRk2

HTL = tLc†

L(xL)ψ(π) + t∗ Lψ(π)†cL(xL)

HTR = tRc†

R(xR)ψ(0) + t∗ Rψ(0)†cR(xR)

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Hamiltonian

ψ(ϕ) cL(x) cR(x)

ϕ

tL tR Hring HL HR x

Hamiltonian of the system H = (HL + HR) + Hring + (HTL + HTR) HL =

• k1 εLk1c†

Lk1cLk1

HR =

• k2 εRk2c†

Rk2cRk2

HTL = tLc†

L(xL)ψ(π) + t∗ Lψ(π)†cL(xL)

HTR = tRc†

R(xR)ψ(0) + t∗ Rψ(0)†cR(xR)

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Kubo formula

• J. M. Kinaret et al., PRB, 57, 7, 3777 (1998)

Kubo formula for conductance G(Ω) = IL(t) V(t) = −1 Ω t

−∞

dt′e−iΩ(t′−t)[ˆ IL(t),ˆ IR(t′)] Current operators: IL = i[tLc†

L(xL)ψ(π) − t∗ Lψ(π)†cL(xL)]

IR = i[tRc†

R(xR)ψ(0) − t∗ Rψ(0)†cR(xR)]

ˆ IL(t) and ˆ IR(t′) are written in the Heisenberg representation. = ⇒ Static limit (Ω → 0)

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Kubo formula

• J. M. Kinaret et al., PRB, 57, 7, 3777 (1998)

Kubo formula for conductance G(Ω) = IL(t) V(t) = −1 Ω t

−∞

dt′e−iΩ(t′−t)[ˆ IL(t),ˆ IR(t′)] Current operators: IL = i[tLc†

L(xL)ψ(π) − t∗ Lψ(π)†cL(xL)]

IR = i[tRc†

R(xR)ψ(0) − t∗ Rψ(0)†cR(xR)]

ˆ IL(t) and ˆ IR(t′) are written in the Heisenberg representation. = ⇒ Static limit (Ω → 0)

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Expansion of the current-current correlator (Keldysh)

Current-current correlator (t > t′) [ˆ IL(t),ˆ IR(t′)] = −2iImTtˆ IL(t)ˆ IR(t′)

• Ttei

R t

−∞ HT (t′′)dt′′IL(t)Tte−i

R t

t′ HT (t′′)dt′′IR(t′)Tte−i

R t′

−∞ HT (t′′)dt′′

HT = HTL + HTR

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Expansion of the current-current correlator (Keldysh)

Current-current correlator (t > t′) [ˆ IL(t),ˆ IR(t′)] = −2iImTtˆ IL(t)ˆ IR(t′)

• Ttei

R t

−∞ HT (t′′)dt′′IL(t)Tte−i

R t

t′ HT (t′′)dt′′IR(t′)Tte−i

R t′

−∞ HT (t′′)dt′′

HT = HTL + HTR

−∞ t′ t

= ⇒ Perturbation theory in HT

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Expansion of the current-current correlator (Keldysh)

Current-current correlator (t > t′) [ˆ IL(t),ˆ IR(t′)] = −2iImTtˆ IL(t)ˆ IR(t′)

• Ttei

R t

−∞ HT (t′′)dt′′IL(t)Tte−i

R t

t′ HT (t′′)dt′′IR(t′)Tte−i

R t′

−∞ HT (t′′)dt′′

HT = HTL + HTR

−∞ t′ t

= ⇒ Perturbation theory in HT

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Expansion of the current-current correlator (Keldysh)

Current-current correlator (t > t′) [ˆ IL(t),ˆ IR(t′)] = −2iImTtˆ IL(t)ˆ IR(t′)

• Ttei

R t

−∞ HT (t′′)dt′′IL(t)Tte−i

R t

t′ HT (t′′)dt′′IR(t′)Tte−i

R t′

−∞ HT (t′′)dt′′

HT = HTL + HTR

−∞ t′ t

= ⇒ Perturbation theory in HT

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Expansion of the current-current correlator (Keldysh)

Current-current correlator (t > t′) [ˆ IL(t),ˆ IR(t′)] = −2iImTtˆ IL(t)ˆ IR(t′)

• Ttei

R t

−∞ HT (t′′)dt′′IL(t)Tte−i

R t

t′ HT (t′′)dt′′IR(t′)Tte−i

R t′

−∞ HT (t′′)dt′′

HT = HTL + HTR

−∞ t′ t

= ⇒ Perturbation theory in HT

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Expansion in the tunneling elements

−2 −1 1 2 1 0.5

G/G0 Φ/Φ0

Infinite order

−2 −1 1 2 1 0.5

G/G0 Φ/Φ0

2nd order Perturbative expansion Summation over all orders possible without interactions, Summation difficult with interactions, Position of the peaks, → Second order.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Expansion in the tunneling elements

Expansion in the tunneling elements The first non-vanishing term is of the order |tLtR|2. Θ(t − t′)[IL(t), IR(t′)] = −2i× + t

−∞

dt1 t′

−∞

dt′

1[[IL(t), HT(t1)], [HT(t′ 1), IR(t′)]]

− t

−∞

dt1 t1

−∞

dt2[[[IL(t), HT(t1)], HT(t2)], IR(t′)] − t′

−∞

dt′

1

t′

1

−∞

dt′

2[IL(t), [HT(t′ 2), [HT(t′ 1), IR(t′)]]]

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Factorisation of the correlators

Factorisation of the correlators:

• IL
• c†

L(xL, t)ψ(π, t)

• HL
• ψ†(π, t1)cL(xL, t1)
• HR
• c†

R(xR, t′)ψ(0, t′)

• IR
• ψ†(0, t′

1)cR(xR, t′ 1)

• ≡ −

B B @ic†

L(xL, t)cL(xL, t1)

| {z }

G<

L (t1−t)

1 C C A B B @ic†

R(xR, t′)cR(xR, t′ 1)

| {z }

G<

R (t2−t′)

1 C C A ψ(π, t)ψ†(π, t1)ψ(0, t′)ψ†(0, t′

1)

| {z }

4-particle correlator PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

General result for the linear response

Two contributions G(Ω) = e2ΓRΓL

−∞ −∞

dt1dt2 e−iΩt1(1 − e−iΩt2) 2iΩβ sinh[πt2/β] ×Re{[[ψ(π, 0), ψ†(π, 0)], ψ(0, t1)], ψ†(0, t1 + t2)} +e2ΓRΓL

−∞ −∞ −∞

dt1dt2dt3 1 2β sinh[πt1/β] sin Ωt3 − sin Ωt2 2Ωβ sinh[π(t3 − t2)/β] ×Im

• Θ(t1 > t2 > t3){[{ψ(π, 0), ψ†(π, t1)}, ψ†(0, t2)], ψ(0, t3)} + · · ·
• where ΓR/L = 2πνR/L|tR/L|2 are the scattering rates

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

General result for the linear response

Two contributions G(Ω) = e2ΓRΓL

−∞ −∞

dt1dt2 e−iΩt1(1 − e−iΩt2) 2iΩβ sinh[πt2/β] ×Re{[[ψ(π, 0), ψ†(π, 0)], ψ(0, t1)], ψ†(0, t1 + t2)} +e2ΓRΓL

−∞ −∞ −∞

dt1dt2dt3 1 2β sinh[πt1/β] sin Ωt3 − sin Ωt2 2Ωβ sinh[π(t3 − t2)/β] ×Im

• Θ(t1 > t2 > t3){[{ψ(π, 0), ψ†(π, t1)}, ψ†(0, t2)], ψ(0, t3)} + · · ·
• where ΓR/L = 2πνR/L|tR/L|2 are the scattering rates

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Bosonisation in the non-interacting case

k E EF kF+ kF− −kF+ −kF− (r+) (r−) (l+) (l−)

Bosonisation Ansatz ψασ(ϕ) ≡ Fασ √ L e±i(Nασ−zασ)ϕe−i

√ 2πφ†

ασ(ϕ)e−i

√ 2πφασ(ϕ)

where φασ(ϕ) =

i √ 2π

+∞

n=1 e±inϕ √n bn ασ

α ∈ {l, r} and σ ∈ {+, −}.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Bosonization in the non-interacting case

Bosonization Ansatz ψασ(ϕ) ≡ Fασ √ L e±i(Nασ−zασ)ϕe−i

√ 2πφ†

ασ(ϕ)e−i

√ 2πφασ(ϕ)

where φασ(ϕ) =

i √ 2π

+∞

n=1 e±inϕ √n bn ασ

Klein factors Fασ (α, β ∈ {l, r}) Fermionic-like operator

{Fασ, Fα′σ′} = {F †

ασ, F † α′σ′} = 0 (if (ασ) = (α′σ′)),

{Fασ, F †

α′σ′} = 2δα,α′δσ,σ′

Commutation relations:

[Fασ, Nα′σ′] = δα,α′δσ,σ′Fασ, [F †

ασ, Nα′σ′] = −δα,α′δσ,σ′F † ασ

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Bosonization in the non-interacting case

Bosonization Ansatz ψασ(ϕ) ≡ Fασ √ L e±i(Nασ−zασ)ϕe−i

√ 2πφ†

ασ(ϕ)e−i

√ 2πφασ(ϕ)

where φασ(ϕ) =

i √ 2π

+∞

n=1 e±inϕ √n bn ασ

Topological part Nασ: number operator for (ασ), zασ =

L 2πkασ: Fermi momentum for (ασ) (dimensionless).

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Bosonization in the non-interacting case

Bosonization Ansatz ψασ(ϕ) ≡ Fασ √ L e±i(Nασ−zασ)ϕe−i

√ 2πφ†

ασ(ϕ)e−i

√ 2πφασ(ϕ)

where φασ(ϕ) =

i √ 2π

+∞

n=1 e±inϕ √n bn ασ

Bosonic part bn

ασ, bn† ασ: bosonic operators (related to the density

• perators),

Commutation relations: [bn

ασ, bn α′σ′] = [bn† ασ, bn† α′σ′] = 0,

[bn

ασ, bn† α′σ′] = δα,α′δσ,σ′

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Bosonized Hamiltonian Hring = H0 + Hb

Bosonic Hamiltonian (non-interacting) Hb = 2πvF L

• α,σ

+∞

• n=1

nbn†

ασbn ασ

Zero-mode Hamiltonian (non-interacting) H0 = 2πvF L

• α,α′
• σ,σ′

aασ,α′σ′NασNα′σ′

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Bosonized topological Hamiltonian H0

Topological Hamiltonian (non-interacting)

H0 = πvF

8L

• (Nc − 4zµ)2 + (Jc − 4zΦ)2 + N2

s + (Js − 4zR)2

Linear transformation        Nc = Nr+ + Nl+ + Nr− + Nl− Jc = Nr+ − Nl+ + Nr− − Nl− Ns = Nr+ + Nl+ − Nr− − Nl− Js = Nr+ − Nl+ − Nr− + Nl− zµ = (µ−vF kF )L

2π

+ 1

2,

zΦ = Φ

Φ0 ,

zR = 1

2(

• 1 + (αRmL/π)2 − 1)

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Bosonized Hamiltonian with interactions

Bosonic Hamiltonian Hb =

2πvF L

• σ∈{+,−}

+∞

n=1 n

• bn†

rσbn rσ + bn† lσ bn lσ

• H0

=

πvF 8L

• (Nc − 4zµ)2 + (Jc − 4zΦ)2 + N2

s + (Js − 4zR)2

gc, gs 1 dn

1c, dn 2c, dn 1s and dn 2s are obtained by linear canonical transformation

from the operators bn

r+, bn l+, bn r− and bn l− (depends on gc and gs).

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Bosonized Hamiltonian with interactions

Bosonic Hamiltonian Hb =

2πvF L

• η∈{c,s}

vη vF

+∞

n=1 n

• dn†

1ηdn 1η + dn† 2ηdn 2η

• H0

=

πvF 8L

• νc(Nc − 4zµ)2 + λc(Jc − 4zΦ)2 + νsN2

s + λs(Js − 4zR)2

gc, gs 1 dn

1c, dn 2c, dn 1s and dn 2s are obtained by linear canonical transformation

from the operators bn

r+, bn l+, bn r− and bn l− (depends on gc and gs).

νc = vc

vF (1/gc + 2EcL/πvF),

λc = vc

vF · gc.

νs = vs

vF · 1/gs,

λs = vs

vF · gs.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Bosonized Hamiltonian with interactions

Bosonic Hamiltonian Hb =

2πvF L

• η∈{c,s}

vη vF

+∞

n=1 n

• dn†

1ηdn 1η + dn† 2ηdn 2η

• H0

=

πvF 8L

• νc(Nc − 4zµ)2 + λc(Jc − 4zΦ)2 + νsN2

s + λs(Js − 4zR)2

gc, gs 1 dn

1c, dn 2c, dn 1s and dn 2s are obtained by linear canonical transformation

from the operators bn

r+, bn l+, bn r− and bn l− (depends on gc and gs).

zµ = Ec(CGVG − 2N0) + µ/2 − 2πvF/4L πvF/Lgc + 2Ec

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Topological/bosonic separation

Hring = H0 + Hb ψασ(ϕ, t) = Fασ(ϕ, t)Bασ(ϕ, t) Disentanglement topological/bosonic part

ψr+(0, 0)ψ†

l+(0, 0)ψl+(ϕ, t1)ψ† r+(ϕ, t2)

= Fr+(0, 0)F †

l+(0, 0)Fl+(ϕ, t1)F † r+(ϕ, t2)F

× Br+(0, 0)B†

l+(0, 0)Bl+(ϕ, t1)B† r+(ϕ, t2)B PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Bosonic sector

Disentanglement of the bosonic part Br+(0, 0)B†

l+(0, 0)Bl+(ϕ, t1)B† r+ϕ, t2)B

Idea of the calculation Use of Wick’s theorem (good approximation):

Br+(0, 0)B†

l+(0, 0)Bl+(ϕ, t1)B† r+(ϕ, t2)B

≈ Br+(0, 0)B†

r+(ϕ, t2)B

× B†

l+(0, 0)Bl+(ϕ, t1)B

Periodicity in space and REAL time ⇒ Fourier transform

B†

l+(0, 0)Bl+(ϕ, t1)b = +∞

X

l1=0

gl1eiωl1t1, Br+(0, 0)B†

r+(ϕ, t2)b = +∞

X

l2=0

gl2eiωl2t2

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY KUBO FORMULA KELDYSH BOSONISATION

Topological sectors

Topological part Fr+(0, 0)F†

l+(0, 0)Fl+(ϕ, t1)F† r+(ϕ, t2)F

Idea of calculation Typical time evolution:

Fl+(ϕ, t1) = Fl+e−i(Nl+−zl+)ϕe−i

πvF 2L [νcNc−λcJc+νsNs−λsJs−(νc−λc+νs−λs)]t1

[Fασ, Nα′σ′] = δα,α′δσ,σ′Fασ, [F †

ασ, Nα′σ′] = −δα,α′δσ,σ′F † ασ

F †

l+Fl+ = 1, F † r+Fr+ = 1

Averaging: O = Tr[Z −1e−βH0O], Selecion rules:

1

Nc ± Ns and Jc ± Js are simultaneaouly even or odd,

2

Nc ± Ns + Jc ± Js = · · · , −4, 0, 4, · · ·

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Summary of the different parameters

zµ = Ec(CGVG−2N0)+µ/2−2πvF /4L

πvF /Lgc+2Ec

, zΦ = Φ

Φ0 ,

zR = 1

2(

• 1 + ((α0+βVG)mL

π

)2 − 1), gc, gs, vc, vs.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Effect of interactions

−2 −1 1 2

zΦ

−2 −1 1 2

zµ Parameters gc = gs = 1, vc = vs = 1, zR = 0, EC = 0. νc = νs = 1 λc = λs = 1

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Effect of interactions

−2 −1 1 2

zΦ

−2 −1 1 2

zµ Parameters gc = 0.8, gs = 1, vc = vs = 1, zR = 0, EC = 0. νc = 1.25, νs = 1 λc = 0.8, λs = 0

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Effect of interactions

−0.5 0.5

zΦ

−0.5 0.5

zµ Parameters gc = 0.8, gs = 1, vc = vs = 1, zR = 0, EC = 0. νc = 1.25, νs = 1 λc = 0.8, λs = 0

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Effect of interactions

−0.5 0.5

zΦ

−0.5 0.5

zµ 1 − g (1 − g)/2

Parameters gc = 0.8, gs = 1, vc = vs = 1, zR = 0, EC = 0. νc = 1.25, νs = 1 λc = 0.8, λs = 0

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Effect of Rashba spin-orbit coupling

−2 −1 1 2

zΦ

−2 −1 1 2

zµ Parameters gc = gs = 1, vc = vs = 1, zR = 0.3, EC = 0. νc = νs = 1 λc = λs = 1

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Effect of Rashba spin-orbit coupling

−0.5 0.5

zΦ

−0.5 0.5

zµ 2zR Parameters gc = gs = 1, vc = vs = 1, zR = 0.3, EC = 0. νc = νs = 1 λc = λs = 1

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Effect of Rashba spin-orbit coupling

−0.5 0.5

zΦ

−0.5 0.5

zµ Parameters gc = 0.8, gs = 1, vc = vs = 1, zR = 0.3, EC = 0. νc = 1.25, νs = 1 λc = 0.8, λs = 1

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Outline

1

Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling

2

Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique

3

Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Effect of Rashba spin-orbit coupling

−0.5 0.5

zΦ

−0.5 0.5

zµ Parameters gc = gs = 1, zR = 0.3, EC = 0. νc = 1, νs = 1 λc = λs = 1

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Effect of Rashba spin-orbit coupling

−0.5 0.5

zΦ

−0.5 0.5

zµ Parameters gc = gs = 1, zR = 0.3, EC = 0.5 πvF

2L .

νc = 1.5, νs = 1 λc = λs = 1

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Effect of Rashba spin-orbit coupling

−0.5 0.5

zΦ

−0.5 0.5

zµ Parameters gc = gs = 1, zR = 0.3, EC = 3 πvF

2L .

νc = 4, νs = 1 λc = λs = 1

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Spin-charge separation

−0.5 0.5

zΦ

−0.5 0.5

zµ Parameters gc = 0.7, gs = 1, vc = 0.8, vs = 1, zR = 0, EC = 0.4 πvF

2L .

νc = 1.46, νs = 1 λc = 0.56, λs = 0.7

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY EFFECT OF INTERACTIONS INTRODUCTION OF RASHBA SPIN-ORBIT COUPLING INTRODUCTION OF CHARGING ENERGY

Gate voltage vs. Magnetic flux

−2 −1 1 2

zΦ

−2 −1 1 2

VG Parameters gc = 0.7, gs = 1, vc = vs = 1, zµ and zR: functions of VG, EC = 0.4 πvF

2L .

νc = 1.82, νs = 1 λc = 0.7, λs = 1

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY

Summary

Summary Static limit of the conductance in a mesoscopic ring, modelled by a Luttinger liquid, with Rashba SO-coupling. Straightforward technique

Linear response, Real-time approach, diagramatic technique, Exact consideration of interactions in the Tomonoga-Luttinger model,

Control of the position of the conductance peaks for several parameters (flux, chemical potential, Rashba, charging energy, interactions, Zeeman). Outlook Study of the interacting term in the linear response.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY

Summary

Summary Static limit of the conductance in a mesoscopic ring, modelled by a Luttinger liquid, with Rashba SO-coupling. Straightforward technique

Linear response, Real-time approach, diagramatic technique, Exact consideration of interactions in the Tomonoga-Luttinger model,

Control of the position of the conductance peaks for several parameters (flux, chemical potential, Rashba, charging energy, interactions, Zeeman). Outlook Study of the interacting term in the linear response.

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID

TUNNELING CONDUCTANCE LINEAR RESPONSE ANALYTICAL RESULTS SUMMARY

Outlook

• T. Bergsten et al., arXiv:cond-mat/0512264

PAUGET, PLETYUKHOV, GRITSEV TRANSPORT THROUGH A MESOSCOPIC LUTTINGER LIQUID