Budapest, 12-10-2015 Niccolò Traverso Ziani

Wigner oscillations and fractional Wigner oscillations in Luttinger liquids

Genova,

Sassetti's group:

Fabio Cavaliere,Giacomo Dolcetto

Wuerzburg,

Trauzettel's group:

Francois Crépin

Introduction:● Wigner crystallization● Luttinger liquid

Introduction:

Normal systems:● Density vs interaction● Density vs temperature

● Wigner crystallization● Luttinger liquid

Introduction:● Wigner crystallization● Luttinger liquid

Normal systems:● Density vs interaction● Density vs temperature

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

WIGNER CRYSTAL

Electron liquid

Usually electrons in solids are in a liquid state

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

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

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

3D

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

3D

2D

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

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).

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

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

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

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

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

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 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

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

ELECTRON DENSITY

A one dimensional (strongly) interacting quantum dot

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).

ELECTRON DENSITY

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

ZERO T DENSITY VS INTERACTION

ZERO T DENSITY VS INTERACTION

ZERO T DENSITY VS INTERACTION

ZERO T DENSITY VS INTERACTION

ZERO T DENSITY VS INTERACTION

Does it work?

Soeffig et al.PRB 79, 195114 (2009)

LUTTINGER PARAMETERSHow to get at least a reasonable range?

LUTTINGER PARAMETERSHow to get at least a reasonable range?

LUTTINGER PARAMETERSHow to get at least a reasonable range?

LUTTINGER PARAMETERSHow to get at least a reasonable range?

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

At low energy

DENSITY VS TEMPERATURE

DENSITY VS TEMPERATURE

SPIN EXCITED STATES CAN BE POPULATED BEFORE THAN CHARGED ONES

Ground

Low energy

DENSITY VS TEMPERATURE

4kF Wigner

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

2kF Friedel

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

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

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

NUMERICAL EVIDENCE:TWO ELECTRONS

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

NUMERICAL EVIDENCE:TWO ELECTRONS

NUMERICAL EVIDENCE:TWO ELECTRONS

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

Summary

Ground

Low energy

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

SOC

2DTI=helical liquid=spin momentum locking

2DTI=helical liquid=spin momentum locking

CdTe

CdTe

HgTe

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

2DTI=helical liquid=spin momentum locking

CdTe

CdTe

HgTe

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

Non interacting!

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

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

Technical aspects

Technical aspects

P odd

Technical aspects

P odd

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

Technical aspects

Friedel and Wigner oscillations

P odd

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

Technical aspects

Friedel and Wigner oscillations

P odd

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

Technical aspects

Friedel and Wigner oscillations

No spin momentum locking

P odd

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

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)

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)

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)

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

Model

● 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)

Semiclassical solution

NTZ, Crépin,Trauzettel, arXiv:1504.07143

5 particles10 peaks..

Charge ½?No decay,Let's includeQuantumfluctuations

Haldane's expansion in hLL

Haldane's expansion in hLL

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

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=

+

=

Fractional Wigner oscillation + anisotropic spin

Physical picture● Two particle backscattering involves..two particles

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)

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

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.

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).

Summary

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)

TRANSPORT PROPERTIES

LOCAL TRANSPORT

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

TUNNEL COUPLINGDot Luttinger liquid

Leads and STM Fermi gases

Connection TunnelingHamiltonian

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

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

TUNNEL COUPLINGChemical potential No first order correction; difficult

to employ as a probe

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

Linear conductance

Linear conductance

Linear conductance

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

TEMPERATURE CONDUCTANCE

20-21

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

g=0.5

g=0.5

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

g=0.5

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

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

● 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

● 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 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

WHAT ELSE?● Local capacitive coupling

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

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

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

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

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

Quantum transport and nanodevices

● 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

LUTTINGER PARAMETERS

How to get at least a reasonable range?

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

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

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

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

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

Local transport (STM) detection of vibrons in CNTs

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

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

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

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

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

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

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

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

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

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

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

MFM tip

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

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

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)

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