8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Luther-Emery phase and atomic-density waves in a
trapped fermion gas
Marco Polini(NEST-CNR-INFM and Scuola Normale Superiore)
Collaborators:Rosario Fazio, Xianlong Gao, Matteo Rizzi, and Mario Tosi (Italy)
Vivaldo Campo Jr. and Klaus Capelle (Brasil)
Jairo Sinova and Allan MacDonald (Texas)
Pisa (Pisa (TuscanyTuscany,, ItalyItaly))
October 2006October 2006
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Outline
!Introduction and motivationsUltracold atoms and optical lattices
Why are cold gases interesting?
Bloch Oscillations, Vortices, Tonks-Girardeau limit,
Quantum Phase Transitions
!Rotating optical lattices, effective magnetic fields, and frustration
!A reminder of density-functional theory
(the Hohenberg-Kohn theorem and the Kohn-Sham mapping)
!One-dimensional two-component attractive fermions on a lattice
(…and very brief intro to the Luther-Emery liquid)
!Spin-pairing and atomic-density waves in the presence of confinement
!Conclusions and Future Perspectives
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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thanks to Markus Greiner
Optical lattices: artificial crystals of light for cold atoms
I. Bloch, Nature Physics 1, 23 (2005)
M.P.A. Fisher et al., Phys. Rev. B 40, 546 (1989)
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cold cesium atoms87
Rb BEC
40K Fermi gas
Bloch oscillations in cold atom systems
M.B. Dahan et al., Phys. Rev. Lett. 76, 4508 (1996) B.P. Anderson and M.A. Kasevich, Science 282, 1686 (1998)
G. Roati et al., Phys. Rev. Lett. 92, 230402 (2004)
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Superfluid-to-Mott insulator quantum phase transition
M. Greiner et al., Nature 415, 39 (2002)
D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998)
complete tunability of interactions
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Vortices and Abrikosov vortex arrays
K.W. Madison et al., Phys. Rev. Lett. 84, 806 (2000)
16 32 80 130J.R. Abo-Shaeer et al., Science 292, 476 (2001)
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Superfluidity in spin polarized gases
M.W. Zwierlein et al., Science 311, 492 (2006)
Polarization increases in this direction
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Phase separation and exotic
superfluid states (FFLO, et cetera)
Y. Shin et al., Phys. Rev. Lett. 97, 030401 (2006)
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Outline
!Rotating optical lattices, effective magnetic fields, and frustration
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Rotating optical lattices and effective magnetic fields
z
!
d
M. Polini et al., Laser Physics 14, 603 (2004)
C. Wu et al., Phys. Rev. A 69, 043609 (2004)Other ways:
D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003)
E.J. Mueller, Phys. Rev. A 70, 041603 (2004)
A.S. Sorensen et al., Phys. Rev. Lett. 94, 086803 (2005)
R. Fazio and H. van der Zant, Phys. Rep. 355, 235 (2001)
rotating hologram
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Fully Frustrated Cold Bosons: U(1) and Ising order
M. Polini, R. Fazio, A.H. MacDonald, and M.P. Tosi, Phys. Rev. Lett. 95, 010401 (2005)
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Outline
!A reminder of density-functional theory
(the Hohenberg-Kohn theorem and the Kohn-Sham mapping)
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Density-functional theory: The Hohenberg-Kohn theorem
3. The GS total energy functional can be written as
“universaluniversal”
2. The GS density minimizesminimizes the total energy functional
1. The GS expectation value of every observable
is a unique functionalfunctional of the GS density
“basicbasic variablevariable”
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Density-functional theory: The Kohn-Sham mapping
For any interacting system there exists a local single-particle potential
such that the exact GS density of the interacting system
equals the GS density of the auxiliary noninteracting system
we need to approximate “ONLY” the XC potential!
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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The local density approximation for the xc potential
n1
n2
V xc taken from
the homogeneous
electron liquid
at that density
V xc taken from
the homogeneous
electron liquidat that density
approximated, exactly known (e.g. 1D),
or known from QMC
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Outline
!Spin-pairing and atomic-density waves in the presence of confinement
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Density-functional theory for trapped Fermi gases
Two-component Fermi gases in a 1D optical lattice +trapping potential
exactly solvable by Bethe-Ansatz E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20, 1445 (1968)
U
=
t
t
t
t
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Generic phase diagram for a 1D quantum fluid
Charge gapless
Spin gapless(Luttinger liquid)
Charge gapless
Spin gapful(Luther-Emery liquid)
T. Giamarchi, Quantum Physics in One Dimension (Clarendon Press, Oxford, 2004)
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Spin-density and charge-density waves
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Outline
!Spin-pairing and atomic-density waves in the presence of confinement
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Tendency to spin pairing
thermodynamic limit in the presence of an harmonic trapK. Damle et al., Europhys. Lett. 36, 7 (1996)
N N+2
N+1 N+1
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Ground-state site occupation for a 1D
attractive Fermi gases
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Ground-state site occupation in a strong
harmonic potential
filled symbols LDA
x DMRG data
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Crossover from weak to strong coupling
(emergence of atomic-density waves…)
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Crossover from weak to strong coupling
(…and their disappearance)
8/3/2019 Marco Polini- Luther-Emery phase and atomic-density waves in a trapped fermion gas
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Observability of the atomic-density waves
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Conclusions
References:
Phys. Rev. Lett. 95, 010401 (2005)
Phys. Rev. A 73, 033609 (2006)
Phys. Rev. B 73, 165120 (2006)
Phys. Rev. B 73, 161103 (R) (2006)
cond-mat/0609346
1. Ultracold atomic gases in low-dimensional geometries are of momentous experimental
and theoretical interest (quest for the FFLO superfluid state…)
2. Quoting J.I. Cirac and P. Zoller, Science 301, 176 (2003),
"in the strong interaction regime, atomic experiments may help us to understand several
physical phenomena that have been predicted or observed in solid-state systems"
3. Many cold atom systems constitute, already at this time, an ideal, highly-tunable, and
controllable laboratory realization of many one-dimensional exactly-solvable modelsof condensed matter physics
4. Density-functional theory and density-matrix renormalization-group techniques are ideal
theoretical tools to study the interplay between interactions and inhomogeous
external potentials in one-dimensional systems of interacting fermions