Fermionic Dipoles: Many-Body Physics ... - int.washington.edu · 2. Laughlin and Wigner crystal states in a rapidly rotating 2D dipolar Fermi gas In collaboration with: Ł. Dobrek,

Van der Waals - Zeeman Institute, University of Amsterdam

Fermionic Dipoles: Many-Body Physicswith Anisotropic Interactions

M.A. Baranov

1-26 August 2005, University of Washington, Seattle

Outline:

1. BCS pairing in a dipolar Fermi gas

2. Laughlin and Wigner crystal states ina rapidly rotating 2D dipolar Fermi gas

In collaboration with:

Ł. Dobrek, H. Fehrmann, M. Lewenstein, M. Mar’enko,Klaus Osterloh, Val. Rychkov, and G. Shlyapnikov


Dipolar Fermi gas

•Single-component gas of fermionic dipolar particles•All dipoles are oriented in the same direction (z-axis)

Hamiltonian

zr’

r−r ′

r

where is the dipole-dipole interaction

∫⎭⎬⎫

⎩⎨⎧

−+∆−= +

r trap rrVm

rH r

rrhr )()(2

)(2

ψµψ

∫ ++ −+',

)'()'()'()()(21

rr d rrrrVrrrr

rrrrrr ψψψψ

)cos31)(/()( 232rd rdrV r

r θ−=

Assumptions

Fnd ε<<2Gas: the mean dipole-dipole interaction is much less than the Fermi energy Fε

T ≈ F 2

2m 62n2/3Regime of quantum degeneracy:


Properties of the dipole-dipole interaction

• long range• couples different angular moments l• anisotropic (partially attractive)

Attractiveinteraction

Repulsive interaction

Possibility of BCS pairing at low temperatures?

05

40,10,1 2 <−=>====< dmlVml dπ !


Strength of the dipolar interaction

Scattering amplitude at low energies is energy independent(similar to a scattering with a short-rang potential) andcan be characterized by an effective scattering length

22

22hπ

mdad −=

Exampleso

d Aa 1450−=Polar molecules: 183 105.15.1, −×== DdND charge × cm

o

d AaDdNO 24,16.0, −==Atomic magnetic moments:

),106*(6, 2DdCr B−×== µµ

o

d Aa 5−=

a~10 1000ÅFor short-range interactions:1-26 August 2005, University of Washington, Seattle

BCS pairing:

Formation of pair-correlated states - Cooper pairs,characterized by an order parameter

Δ~⟨ ≠ 0

- the critical temperatureT ≤ Tcat temperatures

Consequences:

• single-particle excitations are gapped• Fermi system exhibits collective behaviorsimilar to Bose systems: superfluidity, superconductivity


What to expect for pairing with dipole-dipole interactions?(simple intuitive picture)

the order parameter


Critical temperature and the order parameter

can be found from the gap equation

The order parameter

p

Δp = Δp ′

p’

-p-p-p’

pVeff

or

∑∆=∆−lodd

ll pYpppp )ˆ()()(~)()( 0rrrr ψψ

3)2(')'(

)(22/)(tanh)',()(

h

rr

r

rrrr

πpdp

pETpEppVp eff ∆−=∆ ∫

where

cTThe gap equation has a nontrivial solution at temperatures below

the critical temperature

- excitation energy

- effective interaction

222 )2/()()( µ−+∆= mpppE rr

)',( ppVeffrr


Effective interaction

),',()'()',( ppVppVppV deffrrrrrr δ+−=

where

( )1cos33

4)'( '22 −=− −ppd dppV rr

rr θπ- direct dipole-dipoleinteraction

=

+ +

+ - many-bodycontributions

)',( ppV rrδ


1/)( <<− cc TTTFor temperatures close to the critical temperature:

the order parameter is small, ,cT<<∆

and the gap equation can be expanded in its powers:

3'

'

)2(')'(

2)2/tanh(

)',()(h

rrrrr

πξξ pdp

TppVp

p

peff ∆−=∆ ∫

,4

')'()',()(8

)3(7 322 π

ενπς ndnpnppV

T FFeffF

rrrr

∆− ∫

3/12 )6( np F πh= is the Fermi momentumwhere

322/)( hπεν FF mp= the density of states

3/222

)6(2

nmF πε h

=at the Fermi energy


Results for 1/)( <<− cc TTT PRA, 66, 013606 (2002)

The critical temperature

)12

exp(44.1 2ndT F

Fcπεε −=

Which values to expect?

31210 −> cmn3ND nKTc 100>for at densities


The order parameter

at the Fermi surface

at arbitrary momentum

z

pθ

)(/15.2)( 0 nTTTnp ccFrr φ−=∆

)cos2

sin(2)(0 nn rr θπφ =

∫ ∆−=∆π

π4

')'()'(6

)(2 ndnpnppVp FFd

rrrrr


Physical consequences

/21. Anisotropic order parameter with line of zeros at(similar to the polar phase of andsome heavy-fermion compounds)

3He

As a result:

2. Anisotropic gap in the spectrum of single-particle excitations

3. Anisotropic collective excitations2~ TC4. Specific heat


Pairing in a trap (the role of a confining potential)

Hamiltonian:

∫⎭⎬⎫

⎩⎨⎧

−+∆−= +

r trap rrVm

rH r

rrhr )()(2

)(2

ψµψ

∫ ++ −+',

)'()'()'()()(21

rr d rrrrVrrrr

rrrrrr ψψψψ

])([2

)( 22222 zyxmrV ztrap ωωρ ++=rwhere

z/1/2Important parameter: - aspect ratio


The order parameter

),()()()(),( 212121 rRrrrrVrr drrrrrrrr ∆=−=∆ ψψ

2/)( 21 rrR rrr+= 21 rrr rrr −=andwhere

The (linearized) gap equation

with the kernel

i i − where and

).()()(2

2

rrrVm trap

rrrhννν ψεψ =

⎭⎬⎫

⎩⎨⎧

+∆−

∫ ∆−=∆',')( )','()',';,()(),(

rReffd rRrRrRKrVrR rrrrrrrrrrr

),()()()()(2

)2

tanh()2

tanh().;,( 4

*3

*21

, 21

21

4321 2121

21

rrrrTTrrrrK rrrrrrrrνννν

νν

ψψψψξξ

ξξ

∑ +

+=


1. The case cz T<<ρωω ,

,)6( 3/10

2npF πh=

and, as a function of

The quantity ∫ ∆−=∆ rdRrrpiRp rrrrrrr ),()exp(),(~

of on the scaleRr

,/~0 cF mTphξ

is a slow varying function

pr , it varies

0non a scale of the order of where is the gas

density in the trap center.

Ansatz for the order parameter

( ),2/)cos(sin)(~),)((~nF RRnRp r

rrrrθπ×∆=∆

where is the local Fermi momentum.))((2)( RVmRp trapF

rr−= µ


)(~ Rr

∆Equation on

)(~24

1)(

48)3(7

, 22

2

2 Rnd

RVf

Tv

ziF

F

trapRi

c

Fi

rr

∆⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛ ++∇⎟⎟

⎠

⎞⎜⎜⎝

⎛− ∑ =ρ

πεεπ

ζ

)(~ln RTTtrap

c

c r∆=

2

61π

+=zf2

31πρ −=fwhere and


Results for cz T<<ρωω ,

1.1. The critical temperature in the trap:

)24

1(48

)3(722 ndTT

TT F

cc

ctrap

c πεπ

ςω+−=

−

,61312 3/42

3/22

⎭⎬⎫

⎩⎨⎧

++−× − λπ

λπ

( ) 3/12zωωω ρ= ρωωλ /z=where and


3/4ωλω =z

3/2−= ωλω ρFor a given number of particles in a trap

and

)(24

1 20

λπεω tdnTT

TT F

cc

ctrap

c +−=−

λ

0.694

0.384

t λ( )

2.50.2 λ0 0.5 1 1.5 2 2.5

0.3

0.4

0.5

0.6

)(λt

The optimal aspect ratio

8.0≈optλ


1.2. The order parameter in the trap

has a gaussian form:

,22

exp~)( 2

2

2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−−∆

z

z

lR

lR

Rρ

ρr

with 4/12203/155.0

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛≈

cF

F

Tdnvl ω

εωλρ

4/12203/268.0

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛≈ −

cF

Fz T

dnvl ωεω

λ


Comparison of different scales

ω/FTF vR = size of the cloud

size of the pairing regionnear

cT2/1)/(~ cTF TRl ω

TFccF RTTv )/(/~0 ωξ = coherence length

TFRl <<<<0ξ


, z Tc , z 2. The case (but )

For a given dipolar interaction strengththere is critical aspect ratio c above which the BCS pairingdoes not take place

λ

Γ

1.2

1.0

0.8

0.60.5 1.0 1.5 2.0 2.5 3.0

−1

N=105

N=2*10

N=106

6

no BCS

BCS

PRL, 92, 250403 (2004)

Fdn πε/)0(36 2=Γ


The order parameter at criticality:

0 0.2 0.4 0.6 0.8 1

0

0.5

1

∆∆ z

ρ

λ = 2.2−1

Δ R

zRz

, R

Ri - the size of the cloud in the i-th direction


Resume:

The BCS pairing in a polarized dipolar Fermi gas is

anisotropic and very sensitive to confining potential


Laughlin and Wigner crystal states inrapidly rotating 2D dipolar Fermi gas

Motivation:

Is it possible to model Fractional Quantum Hall and Wigner crystal states with ultracold (Fermi) gases?


Fractional quantum Hall states

Cold gases analogs:Ingredients of the FQHE:

- (quasi)2D system - (quasi)2D trapped gas

- Trap rotation(critical rotation)

- Long-range dipole-dipoleinteraction

- speckle radiation from arotating diffractive mask

- Strong Magnetic field(Landau levels)

- Long-range Coulomb interaction(strongly correlated states)

- Quenched disorder(robustness - plateaus)


System

N dipolar fermions polarizedin the z-direction in a(quasi)2D rotating harmonic trap

Possible experimental realizations

1. Close-to-critical rotation of the harmonic trap(JILA, 2003)

2. Critical rotation + extra confining potential(LKB ENS, 2003)


Hamiltonian (in the rotational frame)

H ∑j1N pj

2

2m Vtraprj − Lz Vdipole

rj x j,yj 0 0ezwhere

V trapr m02r2/2 - trapping potential

Vdrk − rl d2 /|rk − rl |3Vdipole ∑kl Vdrk − rl with

- dipole-dipole interaction


Hamiltonian (continued)

H ∑j1N

HLandau

12m p j − m0rj

2

HΔ

0 − Lz Vdipole

→ 0For critical rotations

HΔ HLandau,Vdipole

and

H → H0 HLandau Vdipole


HLandauProperties of

HLandau describes the motion in a constant perpendicularmagnetic filed with the cyclotron frequency c 20

HLandauSpectrum of

n cn 1/2Consists of equidistant (Landau) levels

l0 /mcDegeneracy of every level g 1/2l02 , where

nfFor a given (area) density of fermions

nf/gThe filling fraction gives the number of occupied levels


The lowest Landau level regime

For the eigenstates of have the formHLandau 1

zj Pz1 , . . . , zNexp −∑j1N |zj|2 /4

Pz1 , . . . , zNwhere is an antisymmetric polynomial of all

zj x j iyjparticle coordinates


HLandau VdipoleLaughlin variational wave function for

1/3The trial wave function for the ground state at filling fraction

L Nklzk − zl3 exp −∑j1

N |zj |2 /4

Why this function?

1. Exact for short-range potentials

2. The overlap with the numerically found exact ground state forsmall number of electrons is close to 1

3. Expected also to be good for “intermediate-range” dipole-dipoleinteractions


Excitations (quasi-holes)

Quasi-hole at 0

qh zj , 0 N 0j1

Nzj − 0 L

Excitations energy

[ ]∫ −−=∆ ),(),()( 210212122

12 zzgzzgzzVzdzd qhdε

gqhwhere and are the pair correlation functions for the groundg0

and for the quasi-hole states, respectively


Pair correlation functions [MacDonald, Girvin, 1986]

g0z1,z2 2

221 − e−|z1−z2|2/2 − 2∑odd j

Cj

4 jj!|z1 − z2 |2je−|z1−z2|2/4

gqhz1,z2 2

22

j1,21 − e−|zj |2/2

−e−|z1 |2|z2 |2/2 ez1z2∗/2 − 1 2 2∑ oddj

Cj

4jj!∑k0 |Fj,kz1,z2 |

4 kk!

Fj,kz1 , z2 z1z2

2 ∑r,sj,k j

rks

−1rz1rsz2

jk−rs

rs1jk1−rswhere

1/3 : C1 1 C3 −1/2and for


Results PRL, 94 (2005)

d 1Debye and 0 103Hz

30

2

)019.0927.0(ld

±=∆ε

For

02 ωε h<∆

ε∆<∆HThe requirement

put a constraint on Nand on 0 − /0


Wigner crystal state in 2D dipolar gas

R i

Dipolar moments are polarized perpendicular to the plane

E1 d2∑Ri≠01

Ri3Classical energy per particle

a 23

n−1/2favors triangular lattice with the lattice spacing

321 /0.11 adE =and


Dynamics of the lattice (phonons)

For the displacements uR i in the α-direction

m d2

dt2 uR i −∑j, ,ijuR j

where

,ii ∂2

∂Ri∂Ri∑Rij≠0 VdR ij

,i≠j − ∂2

∂Ri∂RiVdR ij

- components of the dynamical matrix


Phonon dispersion curves

k xk y

k d2

ma 5

a

− a


Existence of the dipolar crystal state

Lindemann criterion: crystal is stable if

u i−u i12

a 2 ≤ c

where cwith anharmonicity effects taken into account

should be defined from microscopic calculations

( for different 2D crystals)c 0. 03 0. 08


Crystal in a non-rotating dipolar gas at T=0

/ma2D

D d 2

ma 5where is the characteristic (Debye) frequency

a adc2 ad~ md2

2 ≤ cStability condition requires

i.e. large densities (contrary to electron gas!)

E1 FIn this regime - strongly interacting system


Crystal in a rotating dipolar gas at T=0

/ma2c c D( )

Stability condition

requires small densities

/2 c

n 1l0

2 c

c 0. 08(?) crystal state is stable for 1/6For


Resume:

Interesting subject for studies:

Fractional quantum Hall states versus Wigner crystalin rotating dipolar Fermi (Bose) gas


Conclusions:

Dipolar Fermi gases exhibit veryinteresting many-body physics


Documents

Fermionic Dipoles: Many-Body Physics ... - int.washington.edu · 2. Laughlin and Wigner crystal states in a rapidly rotating 2D dipolar Fermi gas In collaboration with: Ł. Dobrek,