Fractional fermion number and Cold Atoms · Fermion number ½ Consider two states Difference of two...
Transcript of Fractional fermion number and Cold Atoms · Fermion number ½ Consider two states Difference of two...
Fractional fermion number and Cold Atoms
Tommaso MacrìTommaso MacrìSissaSissa
February 27February 27thth 2009 2009
Plan of the talk
Fractional fermion number
The polyacetylene story
Recent application to Cold Atoms
BosonFermion mixtures and outlook
Conclusions
Fermion number ½
Consider two statesConsider two states
DifferenceDifference of two fermion numbers is of two fermion numbers is an an integerinteger!!
Assume Assume charge conjugation charge conjugation symmetrysymmetry
Fractional fermion number in QFT
Consider the Lagrangian:
Suppose that a classical solution exists (no fermions, not yet!)
Example:
Fermion zero modes
Static solution for the Dirac equationStatic solution for the Dirac equation
Properties of the solution:Properties of the solution:
StaticStatic
NomalizableNomalizable
Unique Unique
Zeroenergy eigensolution!Zeroenergy eigensolution!
Timedependent solutionTimedependent solution
Continuum spectrum
Quantization
From the analysis of the classical solution:
The operator a satisfy the anticommutation relations:
Sketch of the proof:
Postulate the existence of a soliton sector (orthogonal to the vacuum)
Study onesoliton matrix elements of the bosonic/fermionic field
Derive a set of equations for the components of the Fermi field and identify the fermion zeromode with the normalized classical solution
Heisenberg equationHeisenberg equation
Consequence: Fermion number ½ !
The conserved Ferminumber current The conserved Ferminumber current is:is:
Define the fermion number operator Define the fermion number operator in the soliton sectorin the soliton sector
The soliton state is doubly degerate The soliton state is doubly degerate with respect to the fermion zero with respect to the fermion zero mode!mode!
ComputeCompute
We get finally:We get finally:
Fractional fermion numberFractional fermion number
The polyacetylene story
Polyacetylene is a polymer consisting Polyacetylene is a polymer consisting of of parallel chains of carbon atomsparallel chains of carbon atoms
Electrons moving along the chainsElectrons moving along the chains
Hopping between chains suppressedHopping between chains suppressed
Onedimensional systemOnedimensional system
Lattice Hamiltonian:Lattice Hamiltonian:
Peierls instability
Diagonalize the HamiltonianDiagonalize the Hamiltonian
Experimentally: not uniform spacing!Experimentally: not uniform spacing!
Reduced Brilloiun zoneReduced Brilloiun zone
Degeneration at the Fermi levelDegeneration at the Fermi level
Quasiparticle spectrum develops a Quasiparticle spectrum develops a gap!gap!
Ground state energyGround state energy
Two degenerate ground statesTwo degenerate ground states
Discrete model QFT
Solitons and link with fractionization? Solitons and link with fractionization?
Model in the continuumModel in the continuum
If If uunn is the scalar field, how to obtain a is the scalar field, how to obtain a Dirac equation in nonrelativistic QM...Dirac equation in nonrelativistic QM...
For electrons in the vicinity of Fermi level:For electrons in the vicinity of Fermi level:
Schroedinger equation
Study the effective potential for the model Study the effective potential for the model (see e.g. A. Zee book)(see e.g. A. Zee book)
DiscreteDiscrete Z Z22 symmetry is symmetry is brokenbroken
Study the system as in Jackiw & Rebbi. Study the system as in Jackiw & Rebbi. Fractional charge is measuredFractional charge is measured!!
Ultracold Bosons vs Fermions
Quantum statistics effectsQuantum statistics effects
Bosons: condensation
Fermions: quantum degeneracy
Different temperature scalesDifferent temperature scales
Bosons: condensation temperatureBosons: condensation temperature
Fermions: Fermi temperatureFermions: Fermi temperature
Evidence for quantum degeneracyEvidence for quantum degeneracy(De Marco, Papp and Jin, 2001)
Ultracold Diluted Fermi Systems
Some numbers:Some numbers:
Number of particles: 1.000 100.000Number of particles: 1.000 100.000
Temperature: 100 nK – 1Temperature: 100 nK – 1 K K
Density of particles: 10Density of particles: 101111 particles / cm particles / cm3 3 normal metal: 10normal metal: 102323 particles / cm particles / cm33
Dilution parameter: n |aDilution parameter: n |a33| | ~~ 10 1033
What do these numbers tell us?What do these numbers tell us?
Diluted particles interactingDiluted particles interacting relevance of two body physics!relevance of two body physics!
Moreover: Moreover: possibility of tuning the interaction!possibility of tuning the interaction!
Fractional fermion numbers in Cold Atoms: a recent proposal
1D system of 1D system of two fermionic speciestwo fermionic species in in optical latticeoptical lattice
HoppingHopping of atoms beween the sites due to coherent em field
Neglect swave scattering
Dimerized latticeDimerized lattice
Proced as in the polyacetyleneProced as in the polyacetylene
Diagonalize the hamiltonian and find a Diagonalize the hamiltonian and find a gapped energy spectrumgapped energy spectrum
Defect in the lattice
Introduce a defect in the patternIntroduce a defect in the pattern
Break the symmetry of the dimerized Break the symmetry of the dimerized latticelattice
Properties of the field Properties of the field ::
Reproduce aReproduce a phasekink phasekink
Specific form is not important: Specific form is not important: asymptotic asymptotic behaviour!behaviour!
Fractional fermion number:Fractional fermion number:Finite latticeFinite latticeExpectation valueExpectation value vs vs Quantum numberQuantum number!!
Study Study fluctuations of fermion number fluctuations of fermion number (see(see R.Rajaraman & J.S. Bell, Phys. R.Rajaraman & J.S. Bell, Phys. Lett.B 1982Lett.B 1982))
Continuum model and arbitrary fermion numbers
Studying the model in the continuumStudying the model in the continuum
Generalization of the polyacetylene Generalization of the polyacetylene lagrangianlagrangian
No chargeconjugation symmetry!No chargeconjugation symmetry!
Define a 2component spinorDefine a 2component spinor
Get a Dirac Hamiltonian for a spinor Get a Dirac Hamiltonian for a spinor coupled to a bosonic field coupled to a bosonic field
The constants are defined as:
All the constants can be tuned All the constants can be tuned experimentally!experimentally!
= m = 0 charge conjugation sym. = m = 0 charge conjugation sym. preserved preserved N = 1/2 N = 1/2
= m = m 0 arbitrary fractional fermion number 0 arbitrary fractional fermion number
Fermion number for arbitrary J. Goldstone & F. Wilczek, Phys. Rev. Lett. 47, 1981
Conclusions
In certain QFT fractional fermion number can arise from the quantization In certain QFT fractional fermion number can arise from the quantization of the soliton sector in presence of a Fermi zero mode.of the soliton sector in presence of a Fermi zero mode.
Experimental observation of fractional charge in linear chains of Experimental observation of fractional charge in linear chains of polyacetylenepolyacetylene
Recent proposal for measuring fractional fermion numbers in a Cold Atom Recent proposal for measuring fractional fermion numbers in a Cold Atom system with coupling of two spin species with electromagnetic fieldssystem with coupling of two spin species with electromagnetic fields
Fermion – Bosons mixtures and the chance to get fractional fermion Fermion – Bosons mixtures and the chance to get fractional fermion numbersnumbers
Are there systems of fermion interacting with bosons in a cold atom Are there systems of fermion interacting with bosons in a cold atom mixture giving rise to the phenomenon of Fractionalization?mixture giving rise to the phenomenon of Fractionalization?
References
Fractional fermion number in QFTFractional fermion number in QFT
R. Jackiw and C. Rebbi, Phys. Rev. R. Jackiw and C. Rebbi, Phys. Rev. DD 13, 3398 (1976) 13, 3398 (1976)
R. Rajaraman and J.S. Bell, Physics Letters B Vol. 116, Issue 5 (1982)R. Rajaraman and J.S. Bell, Physics Letters B Vol. 116, Issue 5 (1982)
A. J. Niemi and G. W. Semenoff, Physics Reports Vol. 135 , Issue 3 (1986) A. J. Niemi and G. W. Semenoff, Physics Reports Vol. 135 , Issue 3 (1986)
R. Jackiw, arxiv:mathph/0503039v1 (2005) R. Jackiw, arxiv:mathph/0503039v1 (2005)
The polyacetylene storyThe polyacetylene story
W. P. Su, J. R. Schrieffer, A. J. Heeger, Phys. Rev. W. P. Su, J. R. Schrieffer, A. J. Heeger, Phys. Rev. BB 22, 2099 (1980) 22, 2099 (1980)
R. Jackiw and G. W. Semenoff, Phys. Rev. Lett. 50, 439 (1983)R. Jackiw and G. W. Semenoff, Phys. Rev. Lett. 50, 439 (1983)
Fractional fermion number in cold atomsFractional fermion number in cold atoms
J. Ruostekoski, G. V. Dunne, J. Javanainen, Phys. Rev. Lett. J. Ruostekoski, G. V. Dunne, J. Javanainen, Phys. Rev. Lett. 8888, 18041 (2002), 18041 (2002)
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J. Ruostekoski, J. Javanainen, G. V. Dunne, Phys. Rev. A J. Ruostekoski, J. Javanainen, G. V. Dunne, Phys. Rev. A 7777 013603 (2008) 013603 (2008)