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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