Strongly interacting fermionsin optical lattices

Arnaud Koetsier

PhD thesis talk

2

Outline

• Topics covered in this talk:Chapter 2: introduction to Hubbard and Heisenberg modelsChapter 3: cooling into the Néel stateChapter 4: imbalanced antiferromagnets in optical lattices.

• Topics in the thesis but not covered in this talk:Chapter 1: Introduction (mainly historical)Chapter 6: BEC-BCS crossover for fermions in an optical latticeChaoter 7: Analogy of the BEC-BCS crossover for bosonsJuicy details

3

Introduction

• Fermions in an optical lattice• Described by the Hubbard model• Realised experimentally [Esslinger ’05], fermionic Mott insulator recently seen [Esslinger ’08, Bloch ’08]• There is currently a race to create the Néel state

— How to achieve the Néel state in an optical lattice?

• Imbalanced Fermi gases• Experimentally realised [Ketterle ’06, Hulet ’06]• High relevance to other areas of physics (particle physics, neutron stars, etc.)

— How does imbalance affect the Néel state?

4

Fermi-Hubbard Model

Sums depend on:Filling NDimensionality (d=3)

On-site interaction: U Tunneling: t

Consider nearest-neighbor tunneling only.

The positive-U (repulsive) Fermi-Hubbard Model, relevant to High-Tc SC

H = −tPσ

Phjj0i

c†j,σcj0,σ + UPjc†j,↑c

†j,↓cj,↓cj,↑

5

Quantum Phases of the Fermi-Hubbard ModelFi

lling

Frac

tion

0

0.5

1

Mott Insulator (need large U)

Band Insulator

Conductor

Conductor

Conductor

• Positive U (repulsive on-site interaction):

• Negative U: Pairing occurs — BEC/BCS superfluid at all fillings.

6

• At half filling, when and we are deep in the Mott phase.hopping is energetically supressedonly spin degrees of freedom remain (no transport)

• Integrate out the hopping fluctuations, then the Hubbard model reduces to the simpler Heisenberg model:

U À t

Mott insulator: Heisenberg Model

kBT ¿ U

J =4t2

U

Szi =1

2

³c†i,↑ci,↑ − c†i,↓ci,↓

´S+i =c

†i,↑ci,↓

S−i =c†i,↓ci,↑

H =J

2

Xhjki

Sj · Sk

Spin ½ operators: S = 12σ

Superexchange constant (virtual hops):

7

Néel State

• The Néel state is the antiferromagnetic ground state for

• Néel order parameter measures amount of “anti-alignment”:

• Below some critical temperature Tc, we enter the Néel state and becomes non-zero.

0 Tc0

0.5

T

⟨n⟩

0 ≤ h|n|i ≤ 0.5

h|n|i

h|n|i

nj = (−1)jhSji

J > 0

8

Start with trapped 2-component fermi gas of cold atoms. The entropy is:

How to reach the Néel state: Step 1

Total number of particles: NFermi temperature in the harmonic trap: kBTF = (3N )1/3~ω

SFG = NkBπ2 T

TF

V =1

2mω2r2

Trapping potential:

9

Adiabatically turn on the optical lattice. We enter the Mott phase: 1 particle per site.

Entropy remains constant: temperature changes!

How to reach the Néel state: Step 2

10

How to reach the Néel state: Step 3

Prepare the system so that the initial entropy in the trap equals the final entropy below Tc in the lattice:

What is the entropy of the Néel state in the lattice?

To reach the Néel state:

SFG(Tini) = SLat(T ≤ Tc)

Final temperature is below Tc : we are in the Néel state. The entropy remained constant throughout.

11

• Mott insulator:

• Perform a mean-field analysis about the equilibrium value of the staggered magnetisation :

• Landau free energy:

hni

Mean-field theory in the lattice

H ' J

2

Xhiji

½(−1)inSj + (−1)jnSi − Jn2

¾

H =J

2

Xhjki

Sj · Sk

fL(n) =Jz

2n2 − 1

βln

∙cosh(

β|n|Jz2

)

¸− 1βln(2).

Entropy:

S = −N ∂fL(hni)∂T

Self-Consistency:

∂fL(n)

∂n

¯̄̄̄n=hni

= 0hni

12

Mean-field theory results

• Entropy in the lattice:

kBTc = 3J/2

0.02 0.04 0.06 0.080.0

0.2

0.4

0.6

ln(2)

T/TF

S/Nk B

Lattice EntropyTrap Entropy

Mott

Née

l

Heating

Cooling

Lattice depth,6ER

Tc = 0.036 TF

13

• Improve on “single-site” mean-field theory by including the interaction between two sites exactly:

• Incorporate correct low-temperature and critical behaviour

Improved 2-site mean-field theory

0.02 0.04 0.06 0.080.0

0.2

0.4

0.6

ln(2)

T/TF

S/Nk B

Lattice, MFTLattice, fluc.Trap

Mott

Née

lExample:

atoms with a lattice depth of :

Tc = 0.012 TF

40K

8ER

Tini = 0.059 TF

14

• Until now, • What happens if — spin population imbalance? • This gives rise to an overal magnetization

N↑ 6= N↓

• Add a constraint to the Heisenberg model that enforces

Heisenberg Model with imbalance

Effective magnetic field (Lagrange multiplier):

H =J

2

Xhjki

Sj · Sk −Xi

B · (Si −m)

m = (0, 0,mz)

mz = SN↑ −N↓N↑ +N↓

(fermions: )S = 12

N↑ = N↓ = N/2

hSi =m

B

15

• ground state is antiferromagnetic (Néel state)Two sublattices:

• Linearize the Hamiltonian

• Magnetization:

• Néel order parameter:

• Obtain the on-site free energy subject to the constraint (eliminates )

J > 0⇒

Mean field analysis

A, B

m =hSAi+ hSBi

2

n =hSAi− hSBi

2

SA(B)i = hSA(B)i i+ δS

A(B)i

f(n,m;B)

∇Bf = 0 B

B

A B

A

A

B

AB

B

B

A

A

16

Phase Diagram in three dimensions

0 Tc0

0.5

T

⟨n⟩

Add imbalance0.0

0.2

0.4mz0 0.3 0.6

0.91.2kBT�J

0.0

0.2

0.4n

mz

k B T

/J

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

m

nIsing:

m

n

Canted:

n 6= 0

n = 0

17

Spin waves (magnons)

• Spin dynamics can be found from:

• Imbalance splits the degeneracy:

−π2

π

2kd

h̄ω/Jz

00

0.1

0.2

0.3

0.4

0.5No imbalance: Doubly degenerate antiferromagnetic dispersion

Antiferromagneticmagnons: ω ∝ |k|

ω ∝ k2Ferromagneticmagnons:

dS

dt=i

~[H,S]

Gap:(Larmorprecessionof n)

18

Long-wavelength dynamics: NLσM

• Dynamics are summarised a non-linear sigma model with an action

• The equilibrium value of is found from the Landau free energy:

• NLσM admits spin waves but also topologically stable excitations in the local staggered magnetisation .

S[n(x, t)] =

Zdt

Zdx

dD

½1

4Jzn2

µ~∂n(x, t)

∂t− 2Jzm× n(x, t)

¶2− Jd

2

2[∇n(x, t)]2

¾

F [n(x),m] =

Zdx

dD

½Jd2

2[∇n(x)]2 + f [n(x),m]

¾

• lattice spacing:• number of nearest neighbours:• local staggered magnetization:

d = λ/2z = 2Dn(x, t)

n(x, t)

n(x, t)

19

• The topological excitaitons are vortices; Néel vector has an out-of-easy-plane component in the core

• In two dimensions, these are merons:• Spin texture of a meron:

• Ansatz:

• Merons characterised by:Pontryagin index ±½VorticityCore size λ

Topological excitations

n =

⎛⎝ pn2 − [nz(r)]2 cosφ

nvpn2 − [nz(r)]2 sinφ

nz(r)

⎞⎠nv = 1

nv = −1nv = ±1

nz(r) =n

[(r/λ)2 + 1]2

20

Meron size

• Core size λ of meron found by plugging the spin texture into and minimizing (below Tc):

• At low temperatures, the energy of a single meron diverges logarithmically with the system area A as

merons must be created in pairs.

0 0.1 0.20.3

0.4mz0.3

0.60.9

1.21.5

kBT�J0

2

4

6Λ

d

F [n(x),m]

Jn2π

2ln

A

πλ2

mz

k B T

/J

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

Meronspresent

Meron core size

21

Meron pairs

Low temperatures:A pair of merons with opposite vorticity, has a finite energy since the deformation of the spin texture cancels at infinity:

Higher temperatures:Entropy contributions overcome the divergent energy of a single meronThe system can lower its free energy through the proliferation of single merons

22

Kosterlitz-Thouless transition

• The unbinding of meron pairs in 2D signals a KT transition. Thisdrives down Tc compared with MFT:

• New Tc obtained by analogy to an anisotropic O(3) model (Monte Carlo results: [Klomfass et al, Nucl. Phys. B360, 264 (1991)] )

mzk B

T/J

0 0.05 0.10

0.02

0.04

0.06

n 6= 0

mz

k B T

/J

0 0.2 0.40

0.2

0.4

0.6

0.8

1

KT transition

MFT in 2D

23

Experimental feasibility

• Experimental realisation:Néel state in optical lattice: adiabatic coolingImbalance: drive spin transitions with RF field

• Observation of Néel stateCorrelations in atom shot noiseBragg reflection (also probes spin waves)

• Observation of KT transitionInterference experiment [Hadzibabic et al. Nature 441, 1118 (2006)]In situ imaging [Gericke et al. Nat. Phys. 4, 949 (2008); Würtz et al.arXiv:0903.4837]

24

Conclusion

• Néel state appears to be is experimentally achievable (just) by adiabatically ramping up the optical lattice – accurate determination of initial T is crucial.

• The imbalanced antiferrromagnet is a rich systemspin-canting below Tcferro- and antiferromagnetic properties, topological excitationsin 2D, KT transition significantly lowers Tc compared to MFT resultsmodels quantum magnetism, bilayers, etc.possible application to topological quantum computation and information: merons possess an internal Ising degree of freedom associated toPontryagin index

• Future work:incorporate equilibrium in the NLσM actiongradient of n gives rise to a magnetization (can possibly be used to manipulate topological excitations)effect of imbalance on initial temperature needed to achieving the canted Néel state by adiabatic cooling

25

• On-site free energy:

where• Constraint equation:

• Critical temperature:

• Effective magnetic field below the critical temperature:

Free energy and Tc for imbalanced AFM

f(n,m;B) =Jz

2(n2 −m2) +m ·B

− 12kBT ln

∙4 cosh

µ |BA|2kBT

¶cosh

µ |BB |2kBT

¶¸BA (B) = B− Jzm± Jzn

B = 2Jzm

m =1

4

∙BA|BA|

tanh

µ |BA|2kBT

¶+BB|BB |

tanh

µ |BB |2kBT

¶¸

Tc =Jzmz

2kBarctanh(2mz)

26

Finding KT transition: Anisotropic O(3) model

• Dimensionless free energy of the anisotropic O(3) model [Klomfass et al, Nucl. Phys. B360, 264 (1991)] :

• KT transition:

• Numerical fit:

βf3 = −β3Xhi,ji

Si · Sj + γ3Xi

(Szi )2

γ3(β3) =β3

β3 − 1.06exp[−5.6(β3 − 1.085)].

1.0 1.2 1.4 1.6 1.8 2.00.0

0.2

0.4

0.6

0.8

1.0

b3

g 3êH1+g

3L

27

Finding KT transition: Analogy with the anisotropic O(3) model

• Landau free energy:

0.0 0.1 0.2 0.3 0.4 0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

m

gHm,bLê

J

βF =− βJ

2

XhI,ji

ni · nj + βXi

f(m,ni,β)

'− βJn2

2

XhI,ji

Si · Sj + βn2γ(m,β)Xi

(Szi )2

β3 =Jβn2

2

γ3 =2β3Jγ(m,β)

Mapping of our model toAnisotropic O(3) model:

Numerical fit parameter

0.02

1/Jβ =

0.2

0.4

0.60.8

Tc

28

2-Site Mean-Field Theory

1 2

• First Term: Treats interactions between two neighboring sites exactly,

• Second Term: Treats interactions between other neighbors within mean-field theory

H = JS1 · S2 + J(z − 1)|n|(Sz1 − Sz2) + J(z − 1)n2

• Improve on standard mean-field approach by including 2 sites exactly:

29

2-site mean-field theory: Néel order parameter

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

k BT/J⟨n⟩

2-site1-site

Comparison with 1-site mean-field theory:

• Depletion at zero temperature due to quantum fluctuations

Tc ' 1.44kBJ

• Lowering of Tc:

30

Entropy in the lattice: Three temperature regimes

• Low T: entropy of magnon gas

• High T: 2-site mean field theory result

• Intermediate T: non-analytic critical behaviour

Where, from renormalization group theory [Zinn-Justin]

S(T À Tc) = NkB

∙ln(2)− 3J2

64k2BT2

¸

S(T ¿ Tc) = NkB4π2

45

µkBT

2√3Jhni

¶3

d = 3, ν = 0.63, A+/A− ' 0.54

t =T − TcTc

→ 0±S(T = Tc) = S(Tc)±A±|t|dν−1

Tc = 0.957J/kB•Tc: from quantum Monte-Carlo [Staudt et al. ’00]:

31

Entropy in the lattice, T>Tc

First term: Critical behavior

Other terms: To retrieve correct high-T limit of 2-site theory. → Found by expanding critical term and subtracting all terms of lower order than in T than high-T expression, which is .

•Result:

•Function with the correct properties above Tc:

S(T ≥ Tc)NkB

' α1

∙µT − TcT

¶κ− 1 + κTc

T

¸+ ln(2)

α1 =3J2

(32κ(κ − 1)k2BT 2c )κ = 3ν − 1 ' 0.89

∼ 1/T 2

32

Entropy in the lattice, T<Tc

First term and last term: Critical behavior and continuousinterpolation with T>Tc result.

Other terms: Retrieve low-T behavior of magons, again found by expanding critical term and subtracting all terms of lower order than .

• Function for with the correct properties below Tc:

S(T ≤ Tc)NkB

= −α2∙µ

Tc − TTc

¶κ− 1 + κ

T

Tc− κ(κ− 1)

2

T 2

T 2c

¸+ β0

T 3

T 3c+ β1

T 4

T 4c

T 3

33

Entropy in the lattice: Coefficients for T<Tc

• Result:

α1 =3J2

(32κ(κ− 1)k2BT 2c )κ = 3ν − 1 ' 0.89

(same as high-T expression):

α2 =6

(κ− 1)(κ− 2)(κ− 3)

µ4π2k3BT

3c

135√3J3− α1(κ− 1) + β1 − ln(2)

¶β0 =

κ

(κ− 3)

µ4π2k3BT

3c

45√3κJ3

+ α1(κ− 1)− β1 + ln(2)

¶β1 = ln 2− J2

6(A+/A− + 1) + κ(κ− 5)64κk2BT

2c A

+/A−− 4π

2k3BT3c

135√3J3

34

Maximum number of particles in the trap

For smooth traps, tunneling is not site-dependent, overfilling leads to double occupancy:

The trap limits the number of particles to avoid double occupancy:

Destroys Mott-insulator state in the centre!

N ≤ Nmax =4π

3

µ8U

mω2λ2

¶3/2Example:

atoms with a lattice depth ofand

40K

λ = 755 nm

8ER

⇒ Nmax ' 3 × 106