119
Budapest, 12-10-2015 Niccolò Traverso Ziani Wigner oscillations and fractional Wigner oscillations in Luttinger liquids

Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

  • Upload
    others

  • View
    27

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Budapest, 12-10-2015 Niccolò Traverso Ziani

Wigner oscillations and fractional Wigner oscillations in Luttinger liquids

Page 2: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Genova,

Sassetti's group:

Fabio Cavaliere,Giacomo Dolcetto

Wuerzburg,

Trauzettel's group:

Francois Crépin

Page 3: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Introduction:● Wigner crystallization● Luttinger liquid

Page 4: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Introduction:

Normal systems:● Density vs interaction● Density vs temperature

● Wigner crystallization● Luttinger liquid

Page 5: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Introduction:● Wigner crystallization● Luttinger liquid

Normal systems:● Density vs interaction● Density vs temperature

2D TIs:● Why is it interesting● Fractional Wigner oscillations

Page 6: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

WIGNER CRYSTAL

Electron liquid

Usually electrons in solids are in a liquid state

Page 7: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

WIGNER CRYSTAL

Electron liquid

Wigner crystal

Usually electrons in solids are in a liquid state

When the average inter-particle distance a isincreased (the density is lowered), electronscan crystallize

Page 8: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

WIGNER CRYSTAL

Electron liquid

Wigner crystal

Usually electrons in solids are in a liquid state

When the average inter-particle distance a isincreased (the density is lowered), electronscan crystallize Formal explanation → Jellium model

Simple idea:Competition between Kinetic energy T and Interaction U

P≈ ħ/a ; T≈ ħ²/(m a²) ;U≈e²/aPotential energy dominates for

a> ħ²/(m e²)

G. F. Giuliani and G. Vignale, Theory of the electron liquid

Page 9: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

E. Wigner, Phys. Rev. 46, 1002 (1943).R. S. Crandal, Phys. Lett. A 7, 404 (1971).

3D

Page 10: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

E. Wigner, Phys. Rev. 46, 1002 (1943).R. S. Crandal, Phys. Lett. A 7, 404 (1971).

3D

2D

Page 11: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

E. Wigner, Phys. Rev. 46, 1002 (1943).R. S. Crandal, Phys. Lett. A 7, 404 (1971).M. Bockrath, et al., Nature Phys 4, 314 (2008).S. Pecker,, et al., Nature Phys. 9, 576 (2013).

3D

1D

2D

1D

Page 12: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

3D

1D

2D

1D

E. Wigner, Phys. Rev. 46, 1002 (1943).R. S. Crandal, Phys. Lett. A 7, 404 (1971).M. Bockrath, et al., Nature Phys 4, 314 (2008).S. Pecker,, et al., Nature Phys. 9, 576 (2013).

Page 13: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER LIQUID FOR WIRESIn 1D the Fermi liquid picture breaks down Luttinger liquid

Haldane, Phys. Rev. Lett 47 (1981) 1840T. Giamarchi, Quantum Physics in one dimension

Page 14: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER LIQUID FOR WIRES

The low energy excitations of 1D systems of interacting electrons are free bosons (with linear dispersion relation).

In 1D the Fermi liquid picture breaks down Luttinger liquid

Page 15: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER LIQUID FOR WIRES

The low energy excitations of 1D systems of interacting electrons are free bosons (with linear dispersion relation).

In 1D the Fermi liquid picture breaks down Luttinger liquid

Page 16: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER LIQUID FOR WIRES

The low energy excitations of 1D systems of interacting electrons are free bosons (with linear dispersion relation).

In 1D the Fermi liquid picture breaks down Luttinger liquid

Page 17: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER LIQUID FOR WIRES

The low energy excitations of 1D systems of interacting electrons are free bosons (with linear dispersion relation).

In 1D the Fermi liquid picture breaks down Luttinger liquid

More on velocities will come later

Page 18: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER LIQUID FOR WIRES

The low energy excitations of 1D systems of interacting electrons are free bosons (with linear dispersion relation).

In 1D the Fermi liquid picture breaks down Luttinger liquid

More on velocities will come later

No strong SOC

For repulsive interactions

Page 19: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER LIQUID FOR WIRES

The low energy excitations of 1D systems of interacting electrons are free bosons (with linear dispersion relation).

In 1D the Fermi liquid picture breaks down Luttinger liquid

More on velocities will come later

No strong SOC

For repulsive interactions

Luttinger liquid is very general,

Parameters are model dependent

Page 20: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER LIQUID FOR WIRES

The low energy excitations of 1D systems of interacting electrons are free bosons (with linear dispersion relation).

In 1D the Fermi liquid picture breaks down Luttinger liquid

The electron operator can be expressed in terms of bosons

G. F. Giuliani and G. Vignale, Theory of the electron liquidGiamarchi, Quantum Physics in one dimension

Page 21: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

ELECTRON DENSITY

A one dimensional (strongly) interacting quantum dot

Page 22: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

ELECTRON DENSITY

A one dimensional (strongly) interacting quantum dot

Friedel oscillations (2kF, due to finite size effects)

Wigner oscillations (4kF, due to finite size effects)

12 electrons, no interaction 12 electrons, strong interaction

We expect a competition between Friedel and Wigner oscillations, with Wigner oscillations dominating for strong interaction (low density).

Page 23: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

ELECTRON DENSITY

Page 24: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

ELECTRON DENSITY

Flat

2kF

4kF

Friedel

Wigner

Mantelli et al. J. Phys: Condens Matter 24 (2012) 432202Traverso et al. New J. Phys. 15 (2013) 063002

Page 25: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

ZERO T DENSITY VS INTERACTION

Page 26: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

ZERO T DENSITY VS INTERACTION

Page 27: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

ZERO T DENSITY VS INTERACTION

Page 28: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

ZERO T DENSITY VS INTERACTION

Page 29: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

ZERO T DENSITY VS INTERACTION

Page 30: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Does it work?

Soeffig et al.PRB 79, 195114 (2009)

Page 31: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER PARAMETERSHow to get at least a reasonable range?

Page 32: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER PARAMETERSHow to get at least a reasonable range?

Page 33: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER PARAMETERSHow to get at least a reasonable range?

Page 34: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER PARAMETERSHow to get at least a reasonable range?

Fiete et al. Phys. Rev. B 73 (2005) 165104

At low energy

Page 35: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

DENSITY VS TEMPERATURE

Page 36: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

DENSITY VS TEMPERATURE

SPIN EXCITED STATES CAN BE POPULATED BEFORE THAN CHARGED ONES

Ground

Low energy

Page 37: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

DENSITY VS TEMPERATURE

4kF Wigner

Traverso et al. New J. Phys. 15 (2013) 063002

2kF Friedel

Page 38: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

DENSITY VS TEMPERATURE

4kF Wigner

Enhancement of the visibility of Wigner correlations

Traverso et al. New J. Phys. 15 (2013) 063002

12 electrons

2kF Friedel

Page 39: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

DENSITY VS TEMPERATURE2kF

4kF

Friedel

Wigner

This is in accordance with the limiting case of spin incoherent LL

Traverso et al. EuroPhys. Lett. 102 (2013) 47006Fiete et al. Phys. Rev. B 73 (2005) 165104

Page 40: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Cavaliere,NTZ, Sassetti J. Phys.: Condens. Matter 26, 505301 (2014)

NUMERICAL EVIDENCE:TWO ELECTRONS

Page 41: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Cavaliere,NTZ, Sassetti J. Phys.: Condens. Matter 26, 505301 (2014)

NUMERICAL EVIDENCE:TWO ELECTRONS

Page 42: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

NUMERICAL EVIDENCE:TWO ELECTRONS

Cavaliere,NTZ, Sassetti J. Phys.: Condens. Matter 26, 505301 (2014)

Page 43: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Summary

Ground

Low energy

Page 44: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

There are systems when one can hardly think that spin can loose itsrole

SOC

2DTI=helical liquid=spin momentum locking

Page 45: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

2DTI=helical liquid=spin momentum locking

CdTe

CdTe

HgTe

[Qi and Zhang, Rev. Mod. Phys. 83, 1057 (2010)]

Page 46: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

2DTI=helical liquid=spin momentum locking

CdTe

CdTe

HgTe

[Qi and Zhang, Rev. Mod. Phys. 83, 1057 (2010)]

Non interacting!

Page 47: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Du's group: PRL 115, 136804 (2015), PRL 107, 136603 (2011)

InAs/GaSb, v=10^4m/s K=0.22

Page 48: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Technical aspects

Page 49: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Technical aspects

P odd

Page 50: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Technical aspects

P odd

Safi, Schulz PRB 59 3040 (1999)Schulz PRL 47 1840 (1981)

Page 51: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Technical aspects

Friedel and Wigner oscillations

P odd

Safi, Schulz PRB 59 3040 (1999)Schulz PRL 47 1840 (1981)

Page 52: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Technical aspects

Friedel and Wigner oscillations

P odd

Safi, Schulz PRB 59 3040 (1999)Schulz PRL 47 1840 (1981)

Page 53: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Technical aspects

Friedel and Wigner oscillations

No spin momentum locking

P odd

Safi, Schulz PRB 59 3040 (1999)Schulz PRL 47 1840 (1981)

Page 54: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Technical aspects

Friedel and Wigner oscillations

No spin momentum locking

No protection from backscattering

P odd

Safi, Schulz PRB 59 3040 (1999)Schulz PRL 47 1840 (1981)

Page 55: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Technical aspects

Friedel and Wigner oscillations

No spin momentum locking

Forbidden

P odd

No protection from backscattering

Safi, Schulz PRB 59 3040 (1999)Schulz PRL 47 1840 (1981)

Page 56: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Technical aspects

Friedel and Wigner oscillations

No spin momentum locking

No protection from backscattering

Forbidden

Forbidden

P odd

Spin momentum locking does notlike charge oscillations.

Safi, Schulz PRB 59 3040 (1999)Schulz PRL 47 1840 (1981)

Page 57: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Technical aspects

Friedel and Wigner oscillations

No spin momentum locking

No protection from backscattering

Forbidden

Forbidden

Spin momentum locking does notlike charge oscillations.

Compromise betweenSOC and interactions

Safi, Schulz PRB 59 3040 (1999)Schulz PRL 47 1840 (1981)

P odd

Page 58: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Model

Page 59: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

● It is similar to the umklapp term which leads to Wigner oscillations

● It is compatible with time reversal symmetry● It can emerge in generic helical liquids● It is relevant for g<1/2

Model

Orth, et al. PRB 91, 081406(R) (2015)

Page 60: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Semiclassical solution

NTZ, Crépin,Trauzettel, arXiv:1504.07143

5 particles10 peaks..

Charge ½?No decay,Let's includeQuantumfluctuations

Page 61: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Haldane's expansion in hLL

Page 62: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Haldane's expansion in hLL

Fractional Charge oscillations:4Qx is the wavevector of the fractional oscillations

Page 63: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Haldane's expansion in hLL

Strongly anisotropic spinCorrelations: ¼,-3/4,5/4,..

Fractional Charge oscillations:4Qx is the wavevector of the fractional oscillations

R=L=

Page 64: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

+

=

Fractional Wigner oscillation + anisotropic spin

Page 65: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Physical picture● Two particle backscattering involves..two particles

Page 66: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Physical picture● Two particle backscattering involves..two particles● When two particle backscattering dominates, with a Landau-like

approach, I expect it to be a one particle backscattering of a new field (see refermionization)

Page 67: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Physical picture● Two particle backscattering involves..two particles● When two particle backscattering dominates, with a Landau-like

approach, I expect it to be a one particle backscattering of a new field (see refermionization)

● The momentum transfer is 4k_F, so each quasiparticle carries 2k_F instead of k_F

Page 68: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Physical picture● Two particle backscattering involves..two particles● When two particle backscattering dominates, with a Landau-like

approach, I expect it to be a one particle backscattering of a new field (see refermionization)

● The momentum transfer is 4k_F, so each quasiparticle carries 2k_F instead of k_F

● the number of quasiparticles is twice as big as the number of electrons, their charge is ½.

Page 69: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Physical picture● Two particle backscattering involves..two particles● When two particle backscattering dominates, with a Landau-like

approach, I expect it to be a one particle backscattering of a new field (see refermionization)

● The momentum transfer is 4k_F, so each quasiparticle carries 2k_F instead of k_F

● the number of quasiparticles is twice as big as the number of electrons, their charge is ½.

● Fractional quasiparticles in strongly interacting 2DTI had already been proposed at the interface with superconductors

Orth, et al. PRB 91, 081406(R) (2015)F. Zhang and Kane, PRL 113, 036401 (2014).

Page 70: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually
Page 71: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Summary

Page 72: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually
Page 73: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

increasing interaction

G. Dolcetto, NTZ, et al., Phys. Rev. B 87, 235423 (2013)G. Dolcetto, NTZ, et al., Phys. Status Solidi RRL 7, 1059 (2013)

Page 74: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

TRANSPORT PROPERTIES

Page 75: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LOCAL TRANSPORT

● Provide a direct method for detecting the Wigner molecule● We consider sequential tunneling regime

Page 76: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

TUNNEL COUPLINGDot Luttinger liquid

Leads and STM Fermi gases

Connection TunnelingHamiltonian

Traverso et al. New J. Phys. 15 (2013) 063002

Page 77: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

TUNNEL COUPLINGDot Luttinger liquid

Leads and STM Fermi gases

Connection TunnelingHamiltonian

No coupling to the electron density, but “unusual” tunneling Hamiltonian

When an electron is added in the dot, one electron with opposite spin is reflected

Traverso et al. New J. Phys. 15 (2013) 063002

Page 78: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

TUNNEL COUPLINGChemical potential No first order correction; difficult

to employ as a probe

Page 79: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

TUNNEL COUPLING

Linear conductanceSecond order perturbation theory:Second order in the tunneling Hamiltonian

Leads Bosonic excitations

THERMAL DISTRIBUTIONDefinite number ofelectrons in each channel

Chemical potential No first order correction; difficult to employ as a probe

Page 80: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Linear conductance

Page 81: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Linear conductance

Page 82: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Linear conductance

Page 83: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LOW TEMPERATURE CONDUCTANCE

Even when the system shows Wigner oscillations in the density2-3 3-4

20-21 STM tunneling experiments are not able to detect the Wigner molecule?

g=0.5

Page 84: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

TEMPERATURE CONDUCTANCE

20-21

No! Only the linear conductance is much more sensitive to temperature than the density

g=0.5

Page 85: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

g=0.5

Page 86: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Tunneling density of states

A. Secchi et al. Phys. Rev. B 85 (2012) 121410 (2012)

In black the electron density of N electrons at T=0

In red the linear conductance peak for N-1 N at T=0

When T is raised the number of peaks of the linear conductance equals the number of peaks of the electron density

2-3 electrons

Page 87: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

g=0.5

Page 88: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Numerical results

A. Secchi et al. Phys. Rev. B 85 (2012) 121410 (2012)

In black the electron density of N electrons at T=0

In red the linear conductance peak for N-1 N at T=0

When T is raised the number of peaks of the linear conductance equals the number of peaks of the electron density

2-3 electrons

Page 89: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Numerical results

In black the electron density of N electrons at T=0

When T is raised the number of peaks of the linear conductance equals the number of peaks of the electron density

Traverso et al. EuroPhys. Lett. 102 (2013) 47006

6-7 electrons 2-3 electrons

Page 90: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

● We inspected the density of a strongly interacting LL and we found a temperature enhancement of Wigner correlations

● We also inspected the density density correlation functions finding similar results

Page 91: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

● We inspected the density of a strongly interacting LL and we found a temperature enhancement of Wigner correlations

● We also inspected the density density correlation functions finding similar results

● Transport properties tunnel coupling

We demonstrated that local transport properties are effective in the detection of the Wigner molecule

Page 92: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

● We inspected the density of a strongly interacting LL and we found a temperature enhancement of Wigner correlations

● We also inspected the density density correlation functions finding similar results

● Transport properties tunnel coupling

We demonstrated that local transport properties are effective in the detection of the Wigner molecule

We showed how the emergence of a spin incoherent regime occurs

Page 93: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

WHAT ELSE?● Local capacitive coupling

Page 94: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

WHAT ELSE?● Local capacitive coupling● CNT as NEMS/molecular sensors

Page 95: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

WHAT ELSE?● Local capacitive coupling● CNT as NEMS/molecular sensors● Vibrons do not strongly affect the physics

Page 96: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

WHAT ELSE?● Local capacitive coupling● CNT as NEMS/molecular sensors● Vibrons do not strongly affect the physics● Spin incoherent Luttinger liquid

Page 97: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

WHAT ELSE?● Local capacitive coupling● CNT as NEMS/molecular sensors● Vibrons do not strongly affect the physics● Spin incoherent Luttinger liquid● Spin oscillations in 2D Topological Insulators

Page 98: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

N. Traverso Ziani, G. Piovano, F. Cavaliere, and M. Sassetti, Electrical probe for mechanical vibrations in suspended carbon nanotubesPhys. Rev. B 84, 155423 (2011)

N. Traverso Ziani, F. Cavaliere, G. Piovano, and M. Sassetti, Temperature dependence of transport properties in a suspended carbon nanotube , Phys. Scr. T151, 014041 (2011)

F. Remaggi, N. Traverso Ziani, G. Dolcetto, F. Cavaliere, and M. Sassetti, Carbon nanotube sensor for vibrating molecules, New J. Phys. 15, 083016 (2013)

N. Traverso Ziani, F. Cavaliere, and M. Sassetti, Signatures of Wigner correlations in the conductance of a one-dimensional quantum dot coupled to an AFM tip, Phys. Rev. B 86, 125451 (2012)

N. Traverso Ziani, F. Cavaliere, and M. Sassetti, Temperature-induced emergence of Wigner correlations in a STM-probed one-dimensional quantum dot , New J. Phys. 15, 063002 (2013)

N. Traverso Ziani, F. Cavaliere, and M. Sassetti, Theory of the STM detection of Wigner molecules in spin-incoherent CNTs, Europhys. Lett. 102, 47006 (2013)

N. Traverso Ziani, F. Cavaliere, E. Mariani, and M. Sassetti, Interaction and temperature effects on the pair correlation function of a strongly interacting 1D quantum dot, Physica E 54, 295 (2013)

N. Traverso Ziani, G. Dolcetto, F. Cavaliere, and M. Sassetti, Wigner oscillations in strongly correlated CNT quantum dots, to appear in EPJ plus (2014)

N. Traverso Ziani, liquid theory of the 1D Wigner crystal: static and transport properties, to appear in the Proceedings of RTG 1570 workshop (2014)

N. Traverso Ziani, F. Cavaliere, and M. Sassetti, Probing Wigner correlations in a suspended carbon nanotube, J. Phys.: condens. matter 25, 342201 (2013)

G. Dolcetto, N. Traverso Ziani, M. Biggio, F. Cavaliere, and M. Sassetti, Coulomb blockade microscopy of spin-density oscillations and fractional charge in quantum spin Hall dots, Phys. Rev. B 87, 235423 (2013)

G. Dolcetto, N. Traverso Ziani, M. Biggio, F. Cavaliere, and M. Sassetti, Spin textures of strongly correlated spin Hall quantum dots, Phys. Status Solidi RRL 7, 1059 (2013)

CNT

W IGNER

T I

Page 99: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Quantum transport and nanodevices

Page 100: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

● We inspected the density of a strongly interacting LL and we found a temperature enhancement of Wigner correlations

● We also inspected the density density correlation functions finding similar results

● Transport properties capacitive coupling tunnel coupling

We demonstrated that local transport properties are effective in the detection of the Wigner molecule

We showed how the emergence of a spin incoherent regime occurs

Page 101: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER PARAMETERS

How to get at least a reasonable range?

Matveev et al. Phys. Rev. B 76 (2007) 155440

Page 102: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER PARAMETERS

How to get at least a reasonable range?

High energy

Fiete et al. Phys. Rev. B 73 (2005) 165104

Spin incoherent Luttinger liquid

Page 103: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

LUTTINGER PARAMETERS

How to get at least a reasonable range?

High energy

Fiete et al. Phys. Rev. B 73 (2005) 165104

Spin incoherent Luttinger liquid

Page 104: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

How to get at least a reasonable range?

High energySpin incoherent Luttinger liquid

LUTTINGER PARAMETERS

One single interacting fermion (holon)

Matveev et al. Phys. Rev. B 76 (2007) 155440

Page 105: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

How to get at least a reasonable range?

High energySpin incoherent Luttinger liquid

LUTTINGER PARAMETERS

One single interacting fermion (holon)

Matveev et al. Phys. Rev. B 76 (2007) 155440

Page 106: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Local transport (STM) detection of vibrons in CNTs

Traverso et al. Phys. Rev. B 84 (2011) 155423Traverso et al. Phys. Script. T151 (2012) 014041

Page 107: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Local transport (STM) detection of vibrons in CNTs

Traverso et al. Phys. Rev. B 84 (2011) 155423Traverso et al. Phys. Script. T151 (2012) 014041

Stretching mode

Page 108: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Local transport (STM) detection of vibrons in CNTs

Traverso et al. Phys. Rev. B 84 (2011) 155423Traverso et al. Phys. Script. T151 (2012) 014041

p=1 p=2

Stretching mode

Page 109: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Local transport for detecting molecules on CNTs

Remaggi, Traverso et al. accepted by New J. Phys. (2013)

When there is charge transfer Semiconducting CNT;Shift of chemical potential

For weakly coupled molecules Coupling between librational modes and electron density

Page 110: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Spin oscillations in quantum spin Hall quantum dots defined by magnetic barriers

Page 111: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Spin oscillations in quantum spin Hall quantum dots defined by magnetic barriers

Dolcetto, Traverso, et al. Phys. Rev. B 87 (2013) 235423

Page 112: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Spin oscillations in quantum spin Hall quantum dots defined by magnetic barriers

Page 113: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Spin oscillations in quantum spin Hall quantum dots defined by magnetic barriers

Page 114: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Spin oscillations in quantum spin Hall quantum dots defined by magnetic barriers

Page 115: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Spin oscillations in quantum spin Hall quantum dots defined by magnetic barriers

MFM tip

Page 116: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Spin oscillations in quantum spin Hall quantum dots defined by magnetic barriers

Probability density of finding an electron with spin up along x at distance x from an electron with spin up

Page 117: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Spin oscillations in quantum spin Hall quantum dots defined by magnetic barriers

Probability density of finding an electron with spin up along x at distance x from an electron with spin up

Page 118: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Spin oscillations in quantum spin Hall quantum dots defined by magnetic barriers

Traverso, Dolcetto, in preparation

Probability density of finding an electron with spin up along x at distance x from an electron with spin up

Intercalation of spin up and spin down (each electron has both)

Page 119: Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

THANK YOU FOR YOUR ATTENTION

and come to Genova

zpn5c8