Achieving the Neel state in an optical lattice
-
Upload
utrecht-university -
Category
Education
-
view
656 -
download
1
Transcript of Achieving the Neel state in an optical lattice
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