Renormalization Group Approach from Ultra Cold Atoms to the Hot QGP
Yukawa Institute for Theoretical Physics, Kyoto, Japan

One-dimensional Fermi gases:
Density-matrix renormalization group studyof ground state properties and dynamics
30 August 2011
Masaki TEZUKA (Kyoto University)

Plan of the talk• Introduction– Low-dimensional cold atom systems– Density-matrix renormalization group• Harmonically trapped imbalanced system– Larkin-Ovchinnikov state• Optical lattice with disorder– Proving superfluidity by dynamics• Collision dynamics– Non-classical reflectance• Summary

PRL 100, 110403 (2008)
New J. Phys 12, 055029 (2010)

PRA 82, 043613 (2010)arXiv: 1107.0774

Collaborators• Masahito Ueda (University of Tokyo)– Harmonically trapped imbalanced system

PRL 100, 110403 (2008)
New J. Phys 12, 055029 (2010)• Antonio M. García-García (Cambridge University)– Optical lattice with disorder
PRA 82, 043613 (2010)• Jun’ichi Ozaki (Kyoto University)• Norio Kawakami (Kyoto University)– Collision dynamics

arXiv: 1107.0774

Low-dimensional cold atom systems
single atom
detection

2D hologram

2D Bose-Einstein condensates (BEC) released and collideBerezinskii-Kosterlitz-Thouless (BKT) transition is observed
Hadzibabic et al.: Nature 441, 1118 (2006)
WS Bakr et al.: Nature 462, 74 (2009)Low-dimensional cold atom systems
Quasi-1D system in optical lattices

see e.g. I. Bloch, Nature Phys. (2005)

2D lattice of
1D systems

1D Bosons: absence of thermalization
Kinoshita et al.:Nature 440, 900 (2006)

Speckle potential for bosons
Billy et al.: Nature
453, 891 (2008)“1D” : Strong radial confinement
Radial level separation >> energy scale of axial motion(Fermi energy in fermionic system)• Low-dimensional but not necessarily solvable• How to simulate numerically without uncontrollable
approximations?

Density-Matrix Renormalization Group(DMRG)
environment block

System S = B + s + s’ + B’

ijj 'i '  cijj 'i '  i  j  j ' i 'ijj 'i 'Partial density matrix for the left block L (k≡(ij))

 kk '   kj 'i ' kj 'i '   ckj 'i 'ck*' j 'i '
i' j'

i' j'

Has all information on L = B + s when S is in the target state |Ψ>Diagonalize ρ : states with larger eigenvalues are more importantcf. In NRG (numerical renormalization group) the lowest energy states are kept

Reduce the dimension to mof the Hilbert space for L= “renormalization”

S. R. White: PRL (1992) PRB (1993)
Reviews:
Schollwöck: Ann. Phys. 326, 96 (2011),
RMP 77, 259 (2005)
Hallberg: Adv. Phys. 55, 477 (2006)

Infinite system DMRG
Calculate the DM andreduce the dimension

Add two sites
at the center

Iterate until the desired system size is reachedFinite system DMRG
reduce the dimension to m

prepared in previous steps

we know thetransfer matrix

wavefunction in new basis can be predicted

One of the blocks can be made longer iteratively

Finite system DMRG
predicted wavefunction iteratively updated

m can be increased during the process for higher precision

Iterate until physical quantities (e.g. energy) converge

Application of DMRG:
Low-dimensional quantum systems
DMRG: variational method
Error in ground state energy is positive,and decrease as # of states m is increased
1D Heisenberg model (White: PRB 48, 10345 (1993))
10-1

H   Si  Si 1iConjecture:
gapped for spin N∈N,
gapless for spin N+1/2(Haldane PRL 1983)

10-9
0

25

50

75

100

~ 150 papers per year!

125

S=1 Haldane gap
detected
An improved method
gives 0.4104792485(4)(Ueda and Kusakabe:
PRB 84, 054446 (2011))

Eric Jeckelmann, April 16, 2010
http://www.itp.uni-hannover.de/~jeckelm/
dmrg/paper_stat5.pdf

Application of DMRG:
Low-dimensional quantum systems
Fractional Quantum Hall systems(Landau levels: effectively 1D)

Mixtures of bosons and fermions Spin-polarized half-filled fermions + bosons
Mering andFleischhauser:
PRA 81, 011603(R) (2010)
Phase separation
between CDW and
Mott insulator

 Correlated electrons + phonons

Shibata and Yoshioka: PRL 86, 5755 (2001)

Phase diagram of Hubbard-Holstein model
Tezuka, Arita and Aoki: PRB 76, 155114 (2007)

Also, quantum chemistry, 2D & 3D classical systems, …

Two-component Fermi gas
Neutral atoms: bosons or fermions

6Li

depending on parity of A + Z(nucleon number + electron number)

Atom: fine structure, hyper fine structure
electron spin S,
orbital degrees of freedom,
nuclear spin I
Gupta et al.: Science 300, 1723 (2003)

Fermions in two hyperfine states:
Loss due to three-body collisions is rare  Pauli principle(pseudo-) spin population preserved Fermi gas with fixed spin imbalance can be realizedFeshbach resonance
A (highly excited) molecular state close to EB=0 can modify the scattering length
Closed channel:
Quantum numbers differ
from initial state of

Finite overlap of
wavefunctions

When the open and close channels differ in magnetic moment, we can utilize

Rapid change of scattering length as function of magnetic field B
energyclosed channel
open channel
1/(kFa)>0

Feshbach resonance:energyOne bound (=molecular)state has binding energy = 0

closed channel magnetic field
open channel
1/(kFa)<0

internuclear distance

1/(kFa)=0

internuclear distance interaction (including sign) can be controlled

BFor equal number of atoms,

BEC-BCS crossover
BEC : Bosons might break into fermions at energy Δ, but Δ is not correlated with Tc
BCS-type condensate : Pairing gap Δ is in proportion to Tc(density determines Tc)

BEC of diatomic molecules : smoothly connected with BCS condensate?
Theory: Eagles (1969), Leggett (1980), Nozières and Schmidt-Rink (1985), …

Experiment for the same number of Fermi atoms in two of the hyperfine states:
40K

BCS-like superfluid
BEC of molecules

Greiner et al.: PRL (2003, 2004) Nature (2003)

 What happens when the numbersof two spins are not equal?

1) Harmonically trapped imbalanced system
Motivation: condensation of population imbalanced fermions in elongated traps
Zwierlein et al.(MIT): Science 311, 492 (2006)
More recent experiment:
Nascimbène et al. (ENS): Nature 463, 1057 (2010)
Partridge et al.(Rice): Science 311, 503 (2006)
 Discrepancy in cloud shape and maximum imbalance P for condensate
P=(N↑- N↓)/(N↑+ N↓)

Q. What happens when the atom trap is essentially 1D?
s-wave scatt. length ( g 1 )Kinetic energy << level separation of the radial trap
Olshanii: PRL 1998

2D optical lattice  array of 1D traps realized(Esslinger group, ETH Zürich; Hulet group, Rice)I. Bloch: Nature Phys. (2005)

(Situation in 2006-2008)

3D: Controversy over experiments
MIT

Rice

5

Trap aspect ratio λ

50

~107

Number of 6Li atoms N~105~equipotential surface

Density distibution

significant deformation

 



1



1/α
Axial dir.

Shin et al.:PRL 97, 030401(2006)

PCC=75-80%

Radial dir.

 Partridge et al.: PRL 97, 190407 (2006)

Upper limit for imbalance PCC>95%
P for condensation
P=(N↑- N↓)/(N↑+ N↓)

What happens in 1D?

: majority

: minority(two hyperfine states of 6Li)

Pair with finite momentum:
FF and LO states

Fulde-Ferrell
state

Larkin-Ovchinnikov
state

Δ～exp(iqx)

Δ～cos(qx)Different Fermi
momentum forup and down spins
Pairs with non-zero
momentum condense
Density difference
LO : density difference
oscillates withwave number 2q

Order parameter
Machida and Nakanishi: PRB 30, 122 (1984)

Discretization of the space Apply DMRG
U(r) = gδ(r)Kinetic energy

Confining potential (r )Ar

K (k ) 2 k 2 / 2m

2

U=g/a

t 2 / 2ma2

Site potential L 1 i Aa 2  i 

2



2

L-site Hubbard model

K. E. dispersion

K (k )2t (1  cos ka)  ta 2 k 2aH  tcf. optical lattice systems
λ=2a
2V0

 cˆ cˆ  cˆ cˆ  U  nˆ nˆ   nˆ†i
i , j , ,

j

†
j i

i i

i

i ,

i i

DMRG simulation of continuous systemwith the lattice introduced

Density difference

Smaller lattice spacingcontinuum limit approached
40 atoms
L: number of sites

Normalized real-space coordinate
Density difference: shows oscillation incommensurate with latticePair correlation and density distribution
Pair correlation


†
(N)
ˆ
ˆ
ˆ


 0( N ) cˆ†
c
c
c



z
0
i z j 
i , i , j , j ,imbalance parameterN  NP N  N
positive

negative

M. Tezuka and M. Ueda,
PRL 100, 110403 (2008)up

down differencePair correlation and density distribution
Pair correlation


†
(N)
ˆ
ˆ
ˆ


 0( N ) cˆ†
c
c
c



z
0
i z j 
i , i , j , j ,imbalance parameterN  NP N  N
positive

negative

M. Tezuka and M. Ueda,
PRL 100, 110403 (2008)up

down differencePair correlation and density distribution
Pair correlation


†
(N)
ˆ
ˆ
ˆ


 0( N ) cˆ†
c
c
c



z
0
i z j 
i , i , j , j ,imbalance parameterN  NP N  N
positive

negative

M. Tezuka and M. Ueda,
PRL 100, 110403 (2008)up

down differencePair correlation and density distribution
Pair correlation


†
(N)
ˆ
ˆ
ˆ


 0( N ) cˆ†
c
c
c



z
0
i z j 
i , i , j , j ,imbalance parameterN  NP N  N
positive

negative

M. Tezuka and M. Ueda,
PRL 100, 110403 (2008)up

down differencePair correlation and density distribution
Pair correlation


†
(N)
ˆ
ˆ
ˆ


 0( N ) cˆ†
c
c
c



z
0
i z j 
i , i , j , j ,imbalance parameterN  NP N  N
positive

negative

M. Tezuka and M. Ueda,
PRL 100, 110403 (2008)up

down differencePair correlation and density distribution
Pair correlation


†
(N)
ˆ
ˆ
ˆ


 0( N ) cˆ†
c
c
c



z
0
i z j 
i , i , j , j ,imbalance parameterN  NP N  N
positive

negative

M. Tezuka and M. Ueda,
PRL 100, 110403 (2008)up

down differencePair correlation and density distribution
Pair correlation


†
(N)
ˆ
ˆ
ˆ


 0( N ) cˆ†
c
c
c



z
0
i z j 
i , i , j , j ,imbalance parameterN  NP N  N
positive

negative

M. Tezuka and M. Ueda,
PRL 100, 110403 (2008)up

down differenceCondensate? – two-body density matrix
N Fermions in M states:
Maximum possible eigenvalue = N(M-N+2)/M ~ N(C.N. Yang, RMP 1962)
 Measure of pair condensation
L4 matrix elementsfor L-site chain

Diagonalize to obtain eigenvalue distribution
Eigenfunction: state occupied by (↑, ↓) pairs

cf. Condensate fraction for Bose gas  one-body DM

Eigenvalue

Largest eigenvalue

Eigenvalue distribution
Almost irrelevant of
total number of atoms

Non-int. atoms
P

Kinks observed

Eigenstate number

M. Tezuka and M. Ueda:
PRL 100, 110403 (2008)

 Most of minority atoms contribute to quasi-condensate

LO condensate at trap center
Pair correlation:
periodic sign change

P=0.1

P=0.3

Population difference: M. Tezuka and M. Ueda:
constant + oscillation PRL 100, 110403 (2008)
kF  n in 1D

Phase diagram
P=0.1

No up-only
regionP=0.4

Up-only;
Phase sep.

+ equal-population

1D: LDA (local density approximation) results
Exact solution for system without trap
(Yang’s generalied Bethe ansatz  Gaudin’s integral equation)

Trap depth

Small P

Large PFFLO

Hu, Liu and Drummond, PRL 98, 070403 (2007)

DensityOrso, PRL 98, 070402 (2007)

Chemical potential difference Consistent with our DMRG results

1D Experiment (Rice group)
Density at central tube reconstructed

Liao et al.: Nature 467, 567 (Sep 2010)
P=0.015

P=0.055

P=0.10P=0.33Majority
Minority

Difference
T/TF ~ 0.15 (fit to thermodynamic Bethe ansatz with LDA)

P=0.05

Low enough for condensation?

Our data at T=0(interaction strength
NOT tuned)

P=0.10P=0.35

Conclusion: (1)Population imbalance + harmonic trap

Attraction strength• Pairing?  LO (quasi-) condensate• Phase separation?  Yes (LO at center)• Upper limit in imbalance P for
condensation?  not observed

M. Tezuka and M. Ueda:
PRL 100, 110403 (2008); NJP 12, 055029 (2010)see also:
Feiguin and Heidrich-Meisner: PRB 76, 220508R (2007); PRL 102, 076403 (2009) (Ladder);Lüscher, Noack, and Läuchli: PRA 78, 013637 (2008); Batrouni et al.: PRL 100, 116405 (2008) (QMC)
Machida, Yamada, Okumura, Ohashi, and Matsumoto: PRA 77, 053614 (2008);
Rizzi et al.: PRB 77, 245105 (2008); Machida et al.: PRB 78, 235117 (2008)

2) Optical lattice with disorder
Disorder:enhance?
suppress?

Interaction:
Superconductivity
Superfluidity

pairing force?
other phase?

Population imbalance= “magnetic field” : FFLO
Now widely controllable in cold Fermi atomic gases• Optical lattice  laser standing wave
– Bichromatic lattice [Roati et al.: Nature 453, 895 (2008)]• Holographic potential imprinting in 2D– M. Greiner’s group *Gillen et al.: PRA 2009; Bakr et al.: Nature 2009]Experimental realization• Optical lattice  laser standing wave
– Bichromatic lattice• G. Roati et al., Nature 453, 895 (2008) (Firenze-Rome-Trento)• Another possibility: holographic potential imprinting in 2D• M. Greiner’s group– J. I. Gillen et al., PRA 80, 021602(R) (2009)– W.S. Bakr et al., Nature 462, 74 (2009)

Our motivation: what happens in 1D?
• Quantum fluctuation suppresses
true long range order (even for
T=0)• Finite system : can havecondensate (superfluid)• Is coherence length O(systemsize)?

Can be studied withnumerically exact
low-energy methods(Here we use DMRG)

Existing results• Speckle potential (Gaussian random)– All eigenstates exponentially localized• L. Sanchez-Palencia et al.: PRL 98, 210401(2007)• A.M. García-García and E. Cuevas: PRB 79, 073104 (2009)• Fibonacci potential ABAABABA ...
–J. Vidal et al.: PRL 83, 3908 (1999), PRB 65, 014201 (2001); K. Hida: PRL 86, 1331 (2001)– Critical irrespective of the strength of λ• Bichromatic potential

V n   cos2n   “Aubry-Andre model”– non-interacting : metal-insulator transition at λ=2J (J: hopping) Numerical studies of interacting systems

 Bose Hubbard (DMRG): Deng et al.: PRA 78, 013625 (2008); Roux et al.:
PRA 78, 023628 (2008)
 Spinless Fermions : Chaves and Satija (ED, PRB 55, 14076 (1997)); Schuster
et al. (PBC DMRG, PRB 65, 115114 (2002))

Quasiperiodic potential
Hopping J

Interaction U

Laser standing
waves

Disorderamplitude 2λJU2λModeled by a single-band Hubbard
model with site level modification

Formulation: Hubbard model +quasiperiodic potential J: unit of energy(=1)
U: negative for
attractive interaction

V n    cos2n   ,   Fk / Fk 1 

F0  F1  1, Fk 2  Fk 1  Fk 

 5 1/ 2, L  F  1• Ratio of consective Fibonacci numbers  golden ratio(=irrational number) as k∞• (Nς, L) = (10,90), (26, 234), (42, 378) : ν=2/9• Non-interacting case: all eigenstates become critial at λ=2Jk 1

One-electron level scheme (non-interacting)
λ=1

λ=2

site number isite number i

λ=3site number i

611 sites; 1st, 10th, 306th, 611th eigenstate wavefunction

λ=10

λ=3
λ=1

Fermions localize for λ>2 for |U|=0(Energy spectrum is fractal and changes smoothly
as λ is increased)

Interaction strength
Negative U: |interaction energy| linearly increase as |U| is increased
Positive U: it has a peak because double occupancy is suppressed as Ularge

t=1(band width=4)|U|=6 : strongly interacting|U|=1 : weakly interacting

band widthHow to detect pairing and delocalization?
PairingOn-site pair correlation function:easy to calculate with DMRG
Depends on the site potentials of the site pair†ˆi, r   cˆ†ccˆ cˆ
i  r ,  i  r ,  i , i ,Ps 
i, r 
Averaged equal-time pair structure factorri
Sum of pair correlation for all lengths
Increasing function of L
 average over sites
cf. Hurt et al.: PRB 72, 144513 (2005);
Mondaini et al.: PRB 78, 174519 (2008)

if decay of correlation is slow

Delocalization
Phase sensitivity: requires (anti-)periodic condition [see e.g. Schuster et al.: PRB 65, 115114 (2002) ]Hard to calculate within DMRG (not open BC) in large systems (OK for small systems)






2
ˆ
ˆ
I E    ni N 1, N 1  ni N , N  i

Inverse participation ratio (IPR)
Add 2 atoms  How uniformly is the population increase distributed?
Compare between different system sizes

1

The case without disorder (λ=0)
Pair structure factor

Inverse participation ratio

indicator of global (quasi long-range) superfluidity

indicator of atom delocalization

Both increase with |U|, and system size L

Weak |U|

Tezuka and García-García:
PRA 82, 043613 (2010)

Pair structure factor

U=-1 :
Quasi long-range
pairing disappears(λp~0.95) before
localization (λc~1.00)

Inverse participation ratio

Pairing enhanced
by disorderLarger |U|
Pair structure factor

U=-6 :
Quasi long-range
pairing disappears
at localization(λc~0.30)

Inverse participation ratio

Tezuka and García-García:
PRA 82, 043613 (2010)

Tezuka and Garcia-Garcia: PRA 82, 043613 (2010)

Spin gap
Minimum energy to break a pair by spin flipping s  E0 n  1, n  1  E0 n, n

Continues to increase even after λc

Delocalized, superconducting pairs
below sc-metal transition

Localized, tightly boundpairs at large λM. Tezuka and A. M. Garcia-Garcia: PRA 82, 043613 (2010)

Schematic phase diagram• Effect of coexisting disorder (bichromatic potential)
and short-range attractive interaction– Studied for 1D fermionic atoms on optical latticeλ/J2Disorder• For strong attraction (|U|≫J),
pairing decreases as disorder λ isincreased, and localizes at ~insulating transition λc

Insulator, not superfluid
spin, charge gaps

Metal without superfluidity• For weaker attraction (|U|~J),
pairing has a peak as a function ofenhanceddisorder λ, but localizes before λc
Global superfluidity

Attraction|U|/JWhat about dynamics?
• Many experiments observe the dynamics of the atomic clouds
after release from a trap
Bosons: E. Lucioni et al. (LENS, Florence): PRL 106, 230403 (2011)
Subdiffusion observed in bichromatic lattice (3D)

V(x) = V1cos2(k1x) + V2cos2(k2x), k1=2π/(1064.4nm), k2=2π/(859.6nm)
50 thousand 39K atoms, almost spherical trap switched off at t=0
Initially a=280a0 (repulsive), λ~3J (localized)  tuned to final value within 10 ms
λ=4.9J

J/h = 300Hz

α=0.5: normal diffusionwidth ς: square root of
the second moment
Eint=2.3J

α=0.5: normal diffusion
J/h = 180Hz

Eint=1.8J

Simulation (1D DNLSE)
Larcher et al. PRA 80, 053606

λ=5.3(4)J

Eint=0 What happens for interacting fermions?

Does the phase depend on filling?
What do we see?
If the phase diagram issensitive to the filling

Density decrease after release may
induce (de)localization

Intuitively,

But we checked

The density is decreased as theatoms flow to the outer side of
the system; density of states at
the Fermi surface changes(not monotonously)

The ground state phase diagram:does not depend strongly up tofilling ~ 0.31 (per spin per site)
λ=10
λ=3
λ=1

Small change of densityof states does not affectλloc strongly
Gap at ~31% filling

One parameter scaling theory
Abrahams et al.: PRL 42, 673 (1979)
see also Garcia-Garcia and Wang: PRL 100, 070603 (2008)

L : system size, δ : (one-particle) mean level spacing
Dimensionless conductanceg(L) = ET / δ
ET : Thouless energy( related to typical time for particle to travel L )--- Metal-insulator transition : g(L) = gc --Insulator : g(L) ∝ exp(-L/ξ) →0<x2(t)> ∝ tα
Motion slowed down,~ L2/α time to propagate L,
ET-1 ∝ t -2/α, α<1(d>2)D Normal metal : g(L) ∝ Ld-2→∞∵ ET ∝ L-2, δ ∝ L-d

Multifractal spectrum withHausdorff dimension dH δ ∝ L-d/dH

MIT should occur at α = 2dH/d = 2dH(has been checked for the non-interacting case;
Artuso et al.: PRL 68, 3826 (1992);
Piechon et al.: PRL 76, 4372 (1996))

Localization length ξ:
should diverge as |λ-λc|-ν(ν=1 at U=0)

Near metal―insulator transition
Disordered 1D system, U < 0
Diffusion <x2(t)> ∝ tα

|U| 0intermediate |U||U| ∞

α~1?

α~2
ballistic motion

One parameter scalingdH~1brownian motion
Hausdorff dimension
dH~0.5of the spectrum dH see e.g. Artuso et al.:α = 2dH at MIT

PRL 68, 3826 (1992)

Localization length
close to transitionξ ∝ |λ – λc|-νν~1

Our conjectureν = 1/(2dH) = 1/α?

Not fractal

ν~1/2
mean-field like;similar to Cayley tree Let us numerically check by studying the dynamics

Setup (1)• Optical lattice + incommensurate potential (Aubry-Andrémodel)• On-site attractive interaction• Initially trapped in a harmonic
potential• Remove the harmonic potential but keep theincommensurate potential on: what happens?
 Study by time-dependent DMRG

Finite system DMRG

wavefunction constructed from the previous step

Iterate until physical quantities (e.g. energy) converge

Time-dependent DMRGWhite and Feiguin: PRL 93, 076401 (2004)
Application of exp(-iτHi,j) is almost exact
if Hi,j only affects neighboring sites i, j
T/τ finite system iterations to reach time T

Hˆ  Hˆ 1, 2  Hˆ 2,3    Hˆ L  2, L 1  Hˆ L 1, L
exp( iHˆ ) exp( iHˆ

Suzuki-Trotter decompotision
L / 2 , L / 2 1/ 2)  exp( iHˆ 2,3 / 2) exp( iHˆ 1, 2 )exp( iHˆ 2,3 / 2)  exp( iHˆ L  2, L 1 / 2) exp( iHˆ L 1, L )exp( iHˆ L  2, L 1 / 2) exp( iHˆ L / 2, L / 21 / 2)  O( 3 )exp( iHˆ L / 2, L / 21 / 2)exp( iHˆ L / 21, L / 2 2 / 2)

Operator applied to the wavefunction at each step
exp( iHˆ L 1, L )exp( iHˆ L2, L1 / 2)exp( iHˆ 1, 2 )exp( iHˆ 2,3 / 2)exp( iHˆ L / 2, L / 21 / 2)

Time-dependent DMRG:
other schemes
Cazalilla and Marston: PRL 88, 256403 (2002)
Luo et al.: PRL 91, 049701 (2003)
Vidal: PRL 93, 040502 (2004)
Feiguin and White: PRB 72, 020404 (2005)
Dutta and Ramasesha: PRB 82, 035115 (2010)
Time-evolving block decimation (TEBD) has
also been applied to cold atom systems
e.g. Macroscopic quantum tunneling between different supercurrent states in a ring
Danshita and Polkovnikov: PRB 82, 094304 (2010)

Applications of time-dependent DMRG
on cold atom systems
Propagating 1D density waves of bosons in opticallattice
Kollath et al.: PRA 71, 053606 (2005)

Spin-charge separation
in S=1/2 Hubbard model
Kollath et al.: PRL 95, 176401 (2005)

Spin-charge separationin two-component BoseHubbard model
Kleine et al.: PRA 77, 013607 (2008)

Applications of time-dependent DMRG
on cold atom systems

Andreev-like reflection in bosons
Daley et al.: PRL 100, 110404 (2008)

Optical superlattice quenched
Yamamoto et al.: JPSJ 78, 123002 (2009)Setup (1)• Optical lattice + incommensurate potential(Aubry-André model)• On-site attractive interaction• Initially trapped in a harmonic
potential• Remove the harmonic potential but keep theincommensurate potential on: what happens?
 Study by time-dep. DMRG for Hubbard modelNon-interacting case

Smooth diffusion until bouncing backat (artificial) system boundaryλ=0.5 < λc(U=-1)

Atoms quickly flow to both sides

λ=1.0 ~ λc(U=-1)

Significantly slower diffusionλ=1.5 > λc(U=-1)

Atoms move only locallySetup (2)• Optical lattice + incommensurate potential(Aubry-André model)• On-site attractive interaction• Initially trapped in a box
potential• Remove the box potential but keep theincommensurate potential on: what happens?
 Study by time-dep. DMRG for Hubbard modelCheck: energy is nearly preserved
160 sites, 12+12 fermions, Δt = 0.01
U=-1

-0.1

0
Energy change

Energy change

0

U=-6

-0.01

t=100
(104 steps)

t=100

Suppressed motion for λ > λc
200 sites, 18 up and 18 down fermions, U=-1

λc ~ 1.0
λ=0.6

λ=0.9

λ=1.1

J. Ozaki, M. Tezuka, and N. Kawakami: arXiv: 1107.0774

3) Collision dynamics
Elongated two-component Fermi gas:
up and down spins released from separate
traps to collide (“Little Fermi Collider”)

cf. 1D Bosons: absence of thermalization
Kinoshita et al.:Nature 440, 900 (2006)

Spin diffusion constant minimumat Feshbach resonance;
converges as T0
Motion ofcenter of
mass

1sec
“Universal spin transport in a stronglyinteracting Fermi gas”
Sommer et al.: Nature 472 201 (2011)

What happens at 1D?

Motivation
What kind of many-bodyeffects are observed during asingle collision between two
one-dimensional fermion
clusters?
• A spin-dependent harmonic trap
• Quenched to a shared potential• The fermions collide at thetrap center

Model: Hubbard model
Method: time-dependent DMRGU/J = 0.80

Example(7+7 atoms, ~55% reflectance)Weak interaction: most of atoms are
not reflectedparticle reflectance for n + n atoms Rn: R1 ∝ u2 (u → 0)

Quasi-classical model
Quasi-classical model:
a series of one-to-one collisions between
two types of independent classical particles(i) u → 0 :n2 times of independent spin-up
and spin-down collisions⇒ Rqcn = n2 R1/n = nR1
Consistent with the simulation

Strong interaction: most of atoms arereflected backTransmittance for n + n atoms Tn : T1 ∝ u-2 (u → ∞) , Rn + Tn = 1

Quasi-classical model
Quasi-classical model:
a series of one-to-one collisions between
two types of independent classical particles(ii) u → ∞ :n atoms collide successively
against n atoms ⇒ Tqcn = nT1/n = T1
Inconsistent with the simulation!

What is going on?
The work is in progress.

Summary• Application of DMRG for static and dynamic behaviorof Fermi cold atom gases in 1D– Population-imbalanced gas in harmonic trap: FFLO-like
PRL 100, 110403 (2008); New J. Phys 12, 055029 (2010)
condensate– Quasiperiodic disorder• Can enhance condensation for weak attraction PRA 82, 043613 (2010)• Trap-release dynamics close to metal-insulator transition:
anomalous diffusion observed
arXiv: 1107.0774– Collision of spin clusters: more atoms pass through thanquasi-classically expected = emergent many-body behavior• There is much more to explore with powerful
numerical methods!