Rembert Duine

Arnaud Koetsier

Immanuel Bloch

Henk Stoof

2

Motivation

• Can use ultracold fermionic atoms in an optical lattice to map out the Hubbard Model.

• High-Tc SC: Electrons in a doped 2D lattice conjectured to undergo d-wave superconducting transition — not understood in the context of the Hubbard model.

• Also interesting for studying many aspects of quantum magnetism including frustration effects.

• Advantage of using cold atoms: Experimentalists can easily vary:

‣ lattice dimensionality/symmetry‣ interaction strength‣ lattice impurities‣ etc…

‣ doping‣ density

3

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,↑

4

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.

5

What is the Néel State?

• The Néel State is the antiferromagnetic ground state of the Fermi-Hubbard Model at half filling (i.e. 1 particle per site), in the limit (tight binding limit):

• 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

U À t

6

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

How to reach the Néel state: 1

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

SFG = NkBπ2 T

TF

V =1

2mω2r2

Trapping potential:

7

2. 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: 2

8

3. When the temperature becomes cold enough, the fermions antialign. The entropy remains constant throughout.

How to reach the Néel state: 3

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

We need to know the entropy of the Néel state in the lattice.

To reach the Néel state:

SFG(Tini) = SLat(T ≤ Tc)

9

• Consider half filling, when we are deep in the Mott phase.

• Then, at low temperatures , the Hubbard model reduces to the Heisenberg model:

• Entropy of Néel state: perform mean-field analysis with as the mean field

Heisenberg Model

Usual spin-½ operator:

Superexchange constant (describes virtual hops):

hni

H =J

2

Xhjki

Sj · Sk

U À t

kBT ¿ U

S = 12σ

J =4t2

U

H ' J

2

Xhiji

½(−1)inSj + (−1)jnSi − Jn2

¾

10

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

Entropy of the Néel State

• Landau free energy:

Self-Consistency

Entropy

Heating

Cooling

S = −N ∂fL(hni)∂T

∂fL(n)

∂n

¯n=hni

= 0

hni

Lattice depth,6ER

Tc = 0.036 TF

kBTc = 3J/2

Mott

fL(n) =Jz

2n2 − 1

βln

∙cosh(

β|n|Jz2

)

¸− 1βln(2).

Née

l

11

Mean-Field Theory not accurate enough

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

No temperature dependence above Tc

is the correct limit.

Incorrect low temperature behavior

• Entropy exponentially suppressed.

• Model neglects spin wave excitations which dominate near T=0. They lead to power law suppression.

NkB ln(2) T →∞

12

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:

13

2-Site Mean-Field Theory: Entropy

Successes:

•Correct temperature dependence at high temperatures.

Initial temperature to reach the Néel state is lower

Shortcomings:•Incorrect critical exponent and universal amplitude ratio

•Incorrect low temperature behavior: spin waves ( )absentT/T F

S/Nk B

0 0.02 0.04 0.06 0.080

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1-site2-sitetrap

• Comparison with 1-site theory:

ωk ∝ |k|

14

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/kBFrom quantum Monte-Carlo [Staudt et al. ’00]:

15

Achieving the Néel state in an Optical Lattice

Correct high-T, low-T and critical behavior of entropy in the latticeFinal temperature in the lattice can be found for all Tini < TF

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

l

16

Concluding Remarks

• Néel state is reached with cold atoms by adiabatically ramping up an optical lattice. Corrections to mean-field theory reduce the initial temperature by ~20% but it remains experimentally accesible.

• Initial temperature close to limit of what is experimentally viable, accurate determination of the critical temperature is therefore crucial; fluctuations important.

• In principle our results form a lower bound to since we underestimate the entropy of the edge states of the Mott insulator.

Future research:• d=2 case: Start with d=3 Néel state then

decrease tunneling in one direction.• Doped lattices: population imbalance• Impurity scattering: introduce atoms of different species in the lattice

Insight into high-Tc SC

Tini

17

Maximum Number of Particles

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

18

2-Site Mean-Field Theory: 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 theory:

• Depletion at zero temperature due to quantum fluctuations

Tc ' 1.44kBJ

• Lowering of Tc:

19

Entropy, 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

20

Entropy, 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

21

Entropy, T<Tc: Coefficients

• 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

22

[MFT Details 1: Path Integral for Spins]

• For the unit vector:

• we define spin-coherent states obeying

• The partition function of Heisenberg model in terms of these states is a path integral for spins:

• is the vector potential of a monopole.

Ω =

⎛⎝sin θ cosφsin θ sinφcos θ

⎞⎠hΩ|S|Ωi = ~SΩ

Z =

ZDΩ(τ) exp

½−1~

Z ~β

0

dτ

∙Xj

i~SA(Ωj(τ))·∂Ω

∂τ+Xjk

Jjk2S2Ωj(τ)·Ωk(τ)

¸¾

|Ωi =mS=SXmS=−S

q2SCS+mS

eimSφ cosS+mS (θ/2) sinS−mS (θ/2)|S,mSi

A(Ωj(τ)) = ∇×Ωj(τ))

23

[MFT Details 2: Hubbard-Stratonovich]

• Hubbard-Stratonovich transformation to auxiliary field whichis on average related to the spins as

Result:

Where is the effective magnetic field at site j due to

all the other spins.

hmj(τ )i = hSΩj(τ)i

mj(τ)

Bj = −Xk

Jjkmk(τ)

Z =

ZDΩDm exp

(−1~

Z ~β

0

dτXj

"−Xk

Jjk2mj(τ)mk(τ)

− SBj ·Ωj(τ) + i~SA(Ωj(τ)) ·∂Ωj(τ)

∂τ

#)

24

[MFT Details 3: Free Energy]

To study antiferromagnets occurring when U>0,

• Introduce the staggered or “Néel” order parameter

• Consider nearest-neighbor interactions only: , etc.

• Next, integrate out the fields. For S= ½, this yieldsΩj(τ)

mj = (−1)jn

Jhjki = J

ZAFM 'ZDn exp

½− βNJzn2

2+N ln

∙2 cosh

β|nJz|2

¸¾≡ZDn e−βNfL