the imbalanced fermi gas at unitarity...the unitary fermi gas interacting system of two-component...
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The Imbalanced Fermi Gas at Unitarity
Olga Goulko
In collaboration with Matthew WingateBased on Phys.Rev.A 82, 053621 (2010)
DAMTP, University of Cambridge
Jefferson Lab Theory Seminar, 2 May 2011
Cold quantum gases
• High tunability• interaction strength• temperature• dimensionality• . . .
• Many analogues• atomic gases• neutron stars• quark-gluon plasma• . . .
The unitary Fermi gas
Interacting system of two-component fermions:Low-energy interactions are characterised by the scattering length a
∞ −∞0
1a
BEC regimestrongly bound
(bosonic) moleculesof two fermions
UNITARITYstrongly
interactingfermions
BCS regimepairs of fermionsweakly bound in
momentum space
What is interesting about unitarity?
• System is dilute (range of potential � interparticle distance)and strongly interacting (interparticle distance � scatteringlength) at the same time
• No length scales associated with interactions ⇒ universalbehaviour
• Only relevant parameters: temperature and density
• High-temperature superfluidity
neutron star Tc = 106K Tc = 10−5TF
high-Tc superconductor Tc = 102K Tc = 10−3TF
atomic Fermi gas Tc = 10−7K Tc = 10−1TF
• Experimental data available
Methods to study unitarity
Strong interactions ⇒ No small parameter for perturbation theory
No exact theory for Fermi gas at unitarity!
What to do?
• Approximate schemes (e.g. mean-field theory) involveuncontrolled approximations
• Numerical Methods=⇒ Good results for critical temperature and other quantities
Our project: Calculating the critical temperature of the imbalancedunitary Fermi gas with the Determinant Diagrammatic MonteCarlo (DDMC) algorithm [Burovski et al. cond-mat/0605350]
The Fermi-Hubbard model
Simplest lattice model for two-particle scattering
• Non-relativistic fermions
• Contact interaction between spin up and spin down
• On-site attraction U < 0 tuned to describe unitarity
• Grand canonical ensemble
• Finite 3D simple cubic lattice, periodic boundary conditions
• Continuum limit can be taken by extrapolation to zero density
H =∑k,σ
(εk − µσ)c†kσckσ + U∑x
c†x↑cx↑c†x↓cx↓,
where εk = 1m
∑3j=1(1− cos kj) is the discrete FT of −∇
2
2m .
Finite temperature formalism
Grand canonical partition function in imaginary time interactionpicture: Z = Tre−βH :
Z = 1 + + +− − ± . . .
Sign problem!
The diagrams of each order can be written as the product of twomatrix determinants [Rubtsov et al. cond-mat/0411344]
Z =∑p,Sp
(−U)p detA↑(Sp) detA↓(Sp),
where Sp is the vertex configuration and the matrix entries are free(finite temperature) propagators
Order parameter of the phase transition
Anomalous correlations in the superfluid phase:
⇒ Introduce pair annihilation/creation operators P and P†:
P(x, τ) = cx↑(τ)cx↓(τ) and P†(x′, τ ′) = c†x′↑(τ′)c†x′↓(τ
′)
At the critical point the correlation function
G2(xτ ; x′τ ′) =⟨TτP(x, τ)P†(x′, τ ′)
⟩=
1
ZTrTτP(x, τ)P†(x′, τ ′)e−βH
is proportional to |x− x′|−(1+η) as |x− x′| → ∞(in 3 spatial dimensions, where η ≈ 0.038 for U(1) universalityclass)
Order parameter of the phase transition
⇒ the rescaled integrated correlation function
R(L,T ) = L1+ηG2(xτ ; x′τ ′)
becomes independent of lattice size at the critical point
Finite-size corrections:
R(L,T ) = (f0 + f1(T − Tc)L1/νξ + . . .)︸ ︷︷ ︸universal scaling function
(1 + cL−ω + . . .)︸ ︷︷ ︸finite-size scaling
• Critical exponents for the U(1) universality class:νξ ≈ 0.67 and ω ≈ 0.8
• Non-universal constants to be determined:Tc , f0, f1, c (to first order)
Order parameter of the phase transition
Crossing of R(L,T ) curves for 2 lattice sizes Li , Lj :
R(Li ,Tij) = R(Lj ,Tij)⇒ Tij − Tc = const. · g(Li , Lj)
with
g(Li , Lj) =(Lj/Li )
ω − 1
L1νξ
+ω
j
(1− (Li/Lj)
1νξ
)+ cL
1νξ
j
(1− (Li/Lj)
1νξ−ω)
︸ ︷︷ ︸neglect?
c can take values of O(1)⇒ perform non-linear fit to 4 parametersinstead
Order parameter of the phase transition
Example: fit of the rescaled integrated correlator R(L,T )
0.130.14
0.15T
10
12
14
16L
0.04
0.06
0.08
R
(data taken at 4 different temperatures and 4 different lattice sizes)
Diagrammatic Monte Carlo
Burovski et al. cond-mat/0605350:
• sampling via a Monte Carlo Markov chain process
• the configuration space is extended → worm vertices
• physical picture: at lowdensities multi-ladder diagramsdominate
• updates designed to favourprolonging existing vertexchains
The worm updates
Updates of the regular 4-point vertices: adding/removing a4-point vertex (changes the diagram order)
• Diagonal version: add or remove a random vertex
• Alternative using worm: move the P(x, τ) vertex to anotherposition and insert a 4-point vertex at its old position.⇒ choose new coordinates of P very close to its initialcoordinates⇒ the removal update always attempts to remove the nearestneighbour of P
AutocorrelationsThe original worm algorithm achieved high acceptance ratios, butat the cost of strongly autocorrelated results:
40
60
80
100
120
140
160
0 20000 40000 60000 80000 100000
40
60
80
100
120
140
160
0 20000 40000 60000 80000 100000
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 10000 20000 30000 40000 50000 60000 70000
Rela
tive E
rror
Block Size
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 10000 20000 30000 40000 50000 60000 70000
Rela
tive E
rror
Block Size
Worm updates Diagonal updates
Alternative updates
Alternative set of updates: both weak autocorrelations and highacceptance rates [Goulko and Wingate, arXiv:0910.3909].
• Choose a random 4-point vertex from the configuration (willact as a worm for this step).
• Addition: add another 4-point vertex on the same lattice siteand in some time interval around the worm.
• Removal: remove the nearest neighbour of the worm vertex
This setup still prolongs existing vertex chains, but autocorrelationsare reduced since the worm changes with every update.
Alternative updates
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 20 40 60 80 100 120 140
Rela
tive E
rror
Block Size
Comparison between diagonal setup (red circles) and alternativeworm setup (blue squares) at low filling factor
The balanced Fermi gas
An interacting system with equal number of spin up and spin downfermions (µ↑ = µ↓)
The imbalanced Fermi gas
Interactions are suppressed in presence of an imbalance (µ↑ 6= µ↓)
The imbalanced Fermi gas
Thermal probability distribution:
ρ(Sp) =1
Z(−U)p detA↑(Sp) detA↓(Sp)
Sign problem: µ↑ 6= µ↓ ⇒ detA↑ detA↓ 6= | detA|2
Sign quenched method: write ρ(Sp) = |ρ(Sp)|sign(Sp) and use|ρ(Sp)| as the new probability distribution
〈X 〉ρ =
∑X (Sp)ρ(Sp)∑
ρ(Sp)=
∑X (Sp)|ρ(Sp)|sign(Sp)∑ |ρ(Sp)|sign(Sp)
=〈X sign〉|ρ|〈sign〉|ρ|
Problems if 〈sign〉 ≈ 0
But for the unitary Fermi gas 〈sign〉|ρ| ≈ 1 for a wide range of ∆µ
The imbalanced Fermi gas
Schematic plot of the sign near the critical point:
0.00 0.05 0.10 0.15 0.20
DΜ
¶F
0.2
0.4
0.6
0.8
1.0
<sign>
Results
Relationship between ∆µ/εF = |µ↑ − µ↓|/εF and δν/ν =|ν↑−ν↓|ν↑+ν↓
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 0.05 0.1 0.15 0.2 0.25
∆ν/ν
∆µ/εF
Results: the critical temperature
The critical temperature using only balanced data:
0.05
0.1
0.15
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Tc/ε
F
ν1/3
Tc(ν = 0) = 0.173(6)εF
ν → 0 corresponds to the continuum limit
Surface fits for the imbalanced gas
Surface fit of a physical observable X as a function of filling factorν1/3 and imbalance h = ∆µ/εF :
• At fixed imbalance X is a linear function of ν1/3, with slopeα(X )(h).
• X (h) and α(X )(h) viewed as functions h can be Taylorexpanded.
• Due to symmetry in h all odd powers in the expansion of X (h)and α(X )(h) have to vanish.
Hence the fitted function takes the form
X (ν, h) = X (h) + α(X )(h)ν1/3
We will expand the functions X (h) and α(X )(h) to 2nd order in h.
Results: the critical temperature
Surface fit of the critical temperature versus ν1/3 and h:
0.0 0.2 0.4 0.6
Υ1�3
0.0
0.1
0.2
DΜ�¶F
0.05
0.10
0.15
Tc�¶F
Data is consistent with Tc(ν = 0) = 0.171(5)εF , independent of h.
Results: the critical temperature
Lower bounds for the deviation of Tc from the balanced limit:
0.00 0.05 0.10 0.15 0.20 0.25
DΜ�¶F
0.14
0.15
0.16
0.17
Tc�¶F
lower bound: Tc(h)− Tc(0) > −0.5εFh2,
with additional assumption: Tc(h)− Tc(0) > −0.04εFh2
Results: the critical temperature
Comparison with other numerical studies and experiment:
• Crossings• Burovski, Prokof’ev, Svistunov, Troyer (DDMC) 0.152(7)• Burovski, Kozik, Prokof’ev, Svistunov, Troyer 0.152(9)• Bulgac, Drut, Magierski 0.15(1)
• Full fit• Abe, Seki 0.189(12)• Goulko, Wingate (DDMC) 0.171(5)
• Experiment• Nascimbene, Navon, Jiang, Chevy, Salomon 0.157(15)• Horikoshi, Nakajima, Ueda, Mukaiyama 0.17(1)
Results: the chemical potential
The average chem. pot. projected onto the (ν1/3 − µ) plane:
0.2
0.25
0.3
0.35
0.4
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
µ/ε
F
ν1/3
µ(ν = 0) = 0.429(7)εF
ν → 0 corresponds to the continuum limit
Results: the energy per particle
The energy per particle using only balanced data:
0.25
0.3
0.35
0.4
0.45
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
E/E
FG
ν1/3
E(ν = 0) = 0.46(2)EFG
ν → 0 corresponds to the continuum limit; EFG = (3/5)NεF
Results: the energy per particleSurface fit of the energy per particle versus ν1/3 and h:
0.0 0.2 0.4 0.6 0.8
Υ1�3
0.00
0.050.10
0.150.20
DΜ�¶F
0.3
0.4
0.5
0.6
E�EFG
Results: the contact density
The contact can be interpreted as a measure for the local pairdensity [Braaten, arXiv:1008.2922].
Definition contact [Werner and Castin, arXiv:1001.0774]:
C = m2g0Eint,
where g0 is the physical coupling constant.
The contact density is C = C/V and has units ε2F .
This was the first numerical calculation of the contact density atfinite temperature [Goulko and Wingate, arXiv:1011.0312]
Results: the contact density
The contact density using only balanced data:
0.08
0.085
0.09
0.095
0.1
0.105
0.11
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
C/ε
F2
ν1/3
C(ν = 0) = 0.1102(11)εF2
ν → 0 corresponds to the continuum limit
Results: the contact densitySurface fit of the contact density versus ν1/3 and h:
0.0 0.2 0.4 0.6 0.8
Υ1�3
0.0
0.1
0.2
DΜ�¶F
0.09
0.10
0.11
C�¶F2
Outlook: temperatures beyond Tc
Problem: fixing a physical isotherm for T 6= Tc
Setting the scale: set lattice spacing b to be independent oftemperature ⇒ b = b(µ)
⇒ ν(µ,T )
ν(µ,Tc)=
n(T )
n(Tc)
(b(µ,T )
b(µ,Tc)
)3
=n(T )
n(Tc)
If the fix the lattice temperature ratio T (µ)/Tc(µ) we will movealong an isotherm
Outlook: temperatures beyond Tc
Works for T/Tc ≤ 4 for sufficiently small µ:
0
2
4
6
8
10
0 2 4 6 8 10
ν(µ
,T)/ν(µ
,Tc)
T/Tc
µ=0.4µ=0.5µ=0.7
Outlook: temperatures beyond Tc
Preliminary results: temperature dependence of the chemicalpotential
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
µ/ε
F
ν1/3
T/Tc=3
T/Tc=2
T/Tc=1.5
T/Tc=1
Outlook: temperatures beyond Tc
Preliminary results: temperature dependence of the contact
0.075
0.1
0.125
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
C/ε
F
2
ν1/3
T/Tc=3
T/Tc=2
T/Tc=1.5
T/Tc=1
Conclusions
• Lattice Field Theory is a useful tool for studying stronglyinteracting systems in condensed matter physics
• The DDMC algorithm can be applied to study the phasetransition of the unitary Fermi gas
• Imbalanced case with the sign quenched method
• Results for Tc/εF , µ/εF , E/EFG and C/ε2F for equal andunequal number of fermions in the two spin components
• Temperatures beyond Tc accessible
Thank you!