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Dissertation ETH Number 21384 Numerical simulations of bosons and fermions in three dimensional optical lattices Ping Nang MA Ph.D. oral examination (Sept. 27, 2013) Main Collaborators: Prof. Lode POLLET (LMU, Munich), Dr. Sebastiano PILATI

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Numerical simulations of bosons and fermions in three dimensional optical lattices

Ping Nang MA Ph.D. oral examination (Sept. 27, 2013)

Dissertation ETH Number 21384 Supervisor: Prof. Matthias TROYER Main Collaborators: Prof. Lode POLLET (LMU, Munich), Dr. Sebastiano PILATI (ICTP, Italy)

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Contents

  • 1. Optical lattice - introduction
  • 2. Magnetism in optical lattices

Density Functional Theory Reference:

  • 3. Thermometry in optical lattices

Fluctuation-dissipation thermometry Wing thermometry Reference:

  • 4. Directed worm algorithm (QMC) - optional
  • 5. Conclusion/Outlook
  • P. N. Ma, S. Pilati, M. Troyer, and X. Dai,

Density functional theory for atomic Fermi gases, Nature Phys. 8, 601 (2012)

  • P. N. Ma, L. Pollet, and M. Troyer,

Measuring the equation of state of trapped ultracold bosonic systems in an optical lattice with in-situ density imaging ,

  • Phys. Rev. A. 82, 033627 (2010)

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Optical lattices

Cartoon illustration

  • - setup by 3 orthogonal pairs of laser beams.

V (x) = X

xi=x,y,z

V0 sin2(kxi)

Short-range Van der Waals interaction

Lattice separation: Lattice strength: V0

nvironmen ng d = λ

2,

tial of an

  • r k = 2π

λ = π d [

be confined by

Lattice wavevector: s-wave scattering length: a

  • - setup by 3 orthogonal pairs of laser beams.

Immanuel Bloch’s laboratory, Max Planck Institute

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Optical lattices

  • - Gaussian laser beams induce trapping of atoms.

V0 exp ✓ −2r2 w2 ◆

trapping envelope:

  • - Current experiments:

Atoms are around the vicinity of the center.

~ harmonic trapping: VT(x) = VTx2 ~ waist effects are minimal:

Bosons in an optical lattice. QMC-DWA simulation. U/t = 8.11, T/t = 1.00, N = 280,000, wo = 150μm (Red) Minimal effect seen due to waist corrections.

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Optical lattices

For deep lattices, or large V0:

ˆ H = t X

hi,ji

ˆ b†

iˆ

bj + U 2 X

i

ni(ni 1) X

i

(µ VTx2

i )ni

  • 1. bosons in an optical lattice
  • 2. fermions in an optical lattice

ˆ H = t X

hi,ji,σ

ˆ c†

iσˆ

cjσ + U X

i

ni"ni#

1) X

i

(µ VTx2

i )ni

>>> import numpy; >>> import pyalps.dwa; >>> >>> V0 = numpy.array([8.805, 8. ,

  • 8. ]); #lattice strength [Er]

>>> wlen = numpy.array([765., 843., 843.]); #laser wavelength [nm] >>> a = 101; #s−wave scattering length [bohr radius] >>> m = 86.99; #mass [a.m.u.] >>> L = 160; #lattice of size L^3 >>> >>> band = pyalps.dwa.bandstructure(V0, wlen, a, m, L); >>> >>> band Optical lattice: ================ V0 [Er] = 8.805 8 8 lamda [nm] = 765 843 843 Er2nK = 188.086 154.89 154.89 L = 160 g = 5.51132 Band structure: =============== t [nK] : 4.77257 4.77051 4.77051 U [nK] : 38.7027 U/t : 8.1094 8.1129 8.1129

(boson Hubbard model) (Hubbard model)

Easy and convenient conversion within ALPS Python:

hopping

  • nsite interaction

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Temperature U/J Superfluid Mott Insulator QCP Normal Tc quantum critical region

Large t: superfluid BEC

Phase diagram (V=0)

U t /

1 = n

2 = n

= n

U / µ

Large U: incompressible Mott- insulator at Integer filling

Quantum phase transition varying U/t

Phase diagram for homogeneous systems:

  • M. Fisher et al, PRB 40, 546 (1989)

Bosons in an optical lattice

ˆ H = t X

hi,ji

ˆ b†

iˆ

bj + U 2 X

i

ni(ni 1) X

i

(µ VTx2

i )ni

Quantitative validation:

a b c

2hk

Experiment QMC

13.6 nK 18.8 nK 26.5 nK 30.7 nK 43.6 nK 11.9 nK 19.1 nK 26.5 nK 31.8 nK 47.7 nK

r c At zero temperature At finite temperature

  • S. Trotsky, L. Pollet et al, Nature Phys. 6, 998-1004 (2010)
  • 1. on time-of-flight (tof) images:
  • 2. on density profiles:

experiment QMC

  • S. Trotsky, L. Pollet et al, Nature Phys. 6, 998-1004 (2010)
  • S. Fang, et al, PRA 83, 031605 (2011)

V0/ER = 8 , U/t = 8.11, N = 280,000

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Fermions in an optical lattice

ˆ H = t X

hi,ji,σ

ˆ c†

iσˆ

cjσ + U X

i

ni"ni#

1) X

i

(µ VTx2

i )ni

Quantum Monte Carlo -- negative sign problem

  • M. Troyer, U-J. Weise, PRL 94, 170201 (2005)

A =

  • c A(c)p(c)
  • c p(c)

=

  • c A(c)s(c)|p(c)| /

c |p(c)|

  • c s(c)|p(c)| /

c |p(c)|

≡ As′ s′ .

∆s s =

  • (s2 − s2) /M

s =

  • 1 − s2

√ Ms ∼ eβN∆f √ M .

~ scales exponentially with 1) inverse temperature β, and 2) system size N.

Therefore, phase diagram for fermions is not entirely clear in general At half-filling, the Hubbard model exhibits antiferromagnetic ground state.

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich 1947: Birth of the transistor

Magnetism

Time Line 585 B.C.

“ ... loadstone attracts iron because it has a soul.” — Thales of Miletus, ∼ 585 B.C.

1887

Classical electromagnetism (Maxwell’s equations):

1926

Quantum Mechanics (Schrodinger equation):

i~ ∂ ∂tψ(x, t) = ✓ ~2 2mr2 + V (x, t) ◆ ψ(x, t)

2009 1965

1928: Quantum mechanics appiled to solids Birth of Heisenberg model

Kohn-Sham DFT:

R ✓ ~2 2mr2 + V (r) + V HXC

σ

(⇢↑, ⇢↓; r) ◆ σ

n = ✏σ n σ n

Stoner ferromagnetism experimentally detected in (ultracold) gases.

  • 1. many-body ➝ effective single-body quantum problem
  • 2. largely successful in electronic structure problems
  • 3. inadequate to explain (strong) magnetism

Magnetism in gases Magnetism in solids

ˆ H = X

k,σ

✏kˆ c†

kˆ

ck + 1 2 U N X

k1k2 q6=0

ˆ c†

k1+q "ˆ

c†

k2q #ˆ

ck2 #ˆ ck1 "

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

P kF a

0.2 0.4 0.6 0.8 1 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Magnetism in (ultracold) gases

Cartoon illustration

VT(x) = VTx2

Trapping potential Ultracold atomic gas density

Short-range Van der Waals interaction

Interaction strength: kFa

Stoner ferromagnetism:

ˆ H = X

k,σ

✏kˆ c†

kˆ

ck + 1 2 U N X

k1k2 q6=0

ˆ c†

k1+q "ˆ

c†

k2q #ˆ

ck2 #ˆ ck1 "

  • - first observed experimentally in an

ultracold 6Li gaseous cloud in 2009.

G-B. Jo, et al, Itinerant ferromagnetism in a Fermi gas of ultracold atoms, Science 325, 5947 (2009).

  • - phase diagram
  • S. Pilati, G. Bertaina, S. Giorgini, and M. Troyer,

Itinerant ferromagnetism of a repulsive atomic Fermi gas: a quantum Monte Carlo study

  • Phys. Rev. Lett. 105, 030405 (2010).

P = ρ↑ − ρ↓ ρ↑ + ρ↓

FM PM

setup an

  • ptical lattice

Q: Magnetism?

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Magnetism in optical lattices

0.1 0.2 0.3

a / d

0.5 1 1.5 2

n

0.2 0.4 0.6 0.8 1

KS-LSDA V0 = 0.5 ER (d) 0.1 0.2 0.3

a / d

0.5 1 1.5 2

n

0.2 0.4 0.6 0.8 1

KS-LSDA V0 = 2.0 ER (b) 0.1 0.2 0.3

a / d

0.5 1 1.5 2

n

0.2 0.4 0.6 0.8 1

KS-LSDA V0 = 4.0 ER (c)

  • 1

1 2 3 4 5 6 1 2 3 4 5

E / N [ER] V0 [ER]

n = 0.5 a/d = 0.04

KS-LSDA FN-DMC

0.6 1 1.4 1.8

  • 0.4 -0.2

0.2 0.4

n(x,0,0) x / d V0 = 2 ER

P = ρ↑ − ρ↓ ρ↑ + ρ↓

Ferromagnetism is enhanced by optical lattice (band structure effects):

✓ ~2 2mr2 + V (r) + V HXC

σ

(⇢↑, ⇢↓; r) ◆ σ

n = ✏σ n σ n

Kohn-Sham density functional theory (KS-DFT):

V HXC

σ

(⇢↑, ⇢↓; r) =

  • ⇢σ(r) [✏HXC(⇢↑, ⇢↓; r)]
  • - KS-DFT is exact only with exact h.x.c. potential:
  • - Local density approximation (LDA) to h.x.c. potential

(See appendix B in thesis.)

Q: How valid is KS-DFT? A: KS-DFT works (quantitatively) for shallow optical lattice

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Magnetism in optical lattices

M Γ X R M

  • 2

2 4

ε [ER]

M Γ X R M

  • 2

2 4

ε [ER]

M Γ X R M

  • 2

2 4

ε [ER]

M Γ X R M

  • 2

2 4

ε [ER]

M Γ X R M

  • 2

2 4

ε [ER]

M Γ X R M

  • 2

2 4

ε [ER] 0.1 0.2 0.3

a / d

0.5 1 1.5 2

n

0.2 0.4 0.6 0.8 1

KS-LSDA V0 = 2.0 ER (b) 0.1 0.2 0.3

a / d

0.5 1 1.5 2

n

0.2 0.4 0.6 0.8 1

KS-LSDA V0 = 4.0 ER (c)

a/d = 4% a/d = 8% a/d = 16%

KS-DFT:

  •  1

⇡2 (ir + 2⇡k)2 + V eff

σ

(⇢", ⇢#; r)

  • uσ

nk (r) = ✏σ nkuσ nk (r)

simple cubic (sc) lattice:

b1 = ˆ x , b2 = ˆ y , b3 = ˆ z

Notation k-point Γ (0, 0, 0) X (1/

2, 0, 0)

M (1/

2, 1/ 2, 0)

R (1/

2, 1/ 2, 1/ 2)

Conclusion:

Band structure effects stablizes ferromagnetism.

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Magnetism in optical lattices

Antiferromagnetism:

0.1 0.2 0.3

a / d

1 2 3 4 5

V0 [ER]

  • 1
  • 0.5

0.5 1

ρd3 = 1

L Γ X W L

1 2 3 4 5

ε [ER] ∆SDW

fcc lattice:

Notation k-point Γ (0, 0, 0) X (1/

2, 0, 0)

W (1/

4, 1/ 2, 0)

L (1/

4, 1/ 4, 1/ 4)

(due to symmetry reduction) in the limit towards Hubbard model V0/ER = 4 , a/d = 8% :

  • - qualitatively correct (AFM ground state) towards the

Hubbard limit.

  • - AFM phase can be deduced indirectly by probing ΔSDW
  • - Cooling towards GS is an experimental challenge!

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Magnetism in optical lattices

0.1 0.2 0.3

a / d

0.5 1 1.5 2

n

0.2 0.4 0.6 0.8 1

KS-LSDA V0 = 4.0 ER (c) 0.1 0.2 0.3

a / d

0.5 1 1.5 2

n

0.2 0.4 0.6 0.8 1

HK-LSDA V0 = 4.0 ER (d)

  • 1

1 2 3 4 5

E [ER] ρ(E) [a.u.]

Inadequacy of Hohenberg-Kohn DFT (HK-DFT):

Phase diagram: Density-of-states:

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Thermometry

0oC

water mercury thermometer

In-situ density images:

non-destructive measurement

Fluorescence experiment: a single measurement of atom distribution of bosons in an optical lattice. Single-site resolution

  • J. F. Sherson, et al, Nature 467, 68 (2010)

destructive measurement

Unable to distinguish a doublon

We can then collect a timeseries of density measurements, thereby able to evaluate density-related observables, for instance:

  • 1. average density , i.e. <n(r)>
  • 2. density correlations , i.e. <n(r)n(r’)> - <n(r)> <n(r’)>

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Fluctuation-Dissipation Thermometry

  • Q. Zhou, T
  • L. Ho,

Universal thermometry for quantum simulation,

  • Phys. Rev. Lett 106, 225301 (2011)

Fluctuation-dissipation theorem:

∂ n( r ) ∂µ = β n( r )N − n( r ) N

[ ] Local Density Approximation n(r; T, µ) = no(µ(r), T ), µ

, µ(r) = µ − V (r)

, where

( V(r) = 1/2 Mω2r2 )

dµ(r) dr = −Mω 2r

δ n(r) δµ(r) = − 1 Mω 2r ∂ n(r) ∂r

(Thereby, we have and .)

Universal Thermometry:

− kB Mω 2r ∂ n( r ) ∂r × T = n( r )N − n( r ) N L(r) × T = R(r)

  • r :
  • 0.05 0.1 0.15 0.2 0.25

Ρ 0.005 0.015 0.025 0.020 0.010 Ρ

1200 non-interacting fermions, 2D optical lattice, T/t = 0.1 125 000 interacting bosons, 3D optical lattice, T/t = 1.0

20 independent measurements

Uncontrolled statistical noise!

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

  • Rξ(ρ)=
  • dρ{n(ρ)n(ρ) − n(ρ)n(ρ)}θ(ξ − |ρ − ρ|),
  • - density-density correlation length
  • - unknown in reality
  • - yet can be “obtained” by slowly

enlarging the “window size”

=2 N(r) n(r) r

Graphical illustration of ξ = 2

ight n(ρ) =

  • dzn(r), a

correlations,

coordinate r(ρ,φ,z), is the Hea

while ρ(ρ,φ) viside step function.

where the 3D density is integrated along the line-of-sight:

Window-sizing:

L(ρ) = − 1 Mω 2ρ ∂ n(ρ) ∂ρ

Fluctuation-Dissipation Thermometry

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

0.5 1 1.5 2 L(ρ) 0.5 1 1.5 2 Rξ(ρ) (units of t)

True temperature ξ = 0

U/t = 10 , T/t = 1.0

0.5 1 1.5 2 L(ρ) 0.5 1 1.5 2 Rξ(ρ) (units of t)

True temperature ξ = 1

0.5 1 1.5 2 L(ρ) 0.5 1 1.5 2 Rξ(ρ) (units of t)

True temperature ξ = 2

2 L(ρ) 0.5 1 1.5 2 L(ρ) 0.5 1 1.5 2 Rξ(ρ) (units of t)

True temperature ξ = 3

0.5 1 1.5 2 L(ρ) 0.5 1 1.5 2 Rξ(ρ) (units of t)

True temperature ξ = 5

0.5 1 1.5 2 L(ρ) 0.5 1 1.5 2 Rξ(ρ) (units of t)

True temperature ξ = ∞

Systematic error arises due to lack

  • f correlations

Statistical noise arises with increasing ξ

1 2 3 5 ∞

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Number of shots System ξ = 3 ξ = ∞ U/t = 10, T/t = 1 20 O(104) U/t = 10, T/t = 3 14 O(104) U/t = 50, T/t = 1 21 O(104) U/t = 50, T/t = 3 12 O(104)

  • - The simple trick of “window-sizing” leads to orders-of-magnitude improvement!
  • - Statistical noise is drastically reduced.
  • - Fluctuation-dissipation thermometry will remain a feasible tool so long as the

density-density correlation length remains short !

Number of independent shots required to estimate the temperature within 5% accuracy

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

At a lower temperature, a larger window is needed due to increasing correlation length :

0.5 1 1.5 2 L(ρ) 0.25 0.5 0.75 1 R5(ρ) (units of t) True temperature ξ = 5

U/t = 10, T/t = 0.5 100 shots

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

U/t = 10 , T/t = 1.0 U/t = 50 , T/t = 1.0

20 40 60 80 ρ 0.5 1 1.5 2 L(ρ) single-site resolution 2-site resolution 3-site resolution 4-site resolution 5-site resolution 20 40 60 80 ρ 0.5 1 1.5 2 L(ρ) single-site resolution 2-site resolution 3-site resolution 4-site resolution 5-site resolution 0.5 1 1.5 2 L(ρ) 0.5 1 1.5 2 2.5 3 R3(ρ) (units of t) single-site resolution 2-site resolution 3-site resolution 4-site resolution 5-site resolution True temperature 0.5 1 1.5 2 L(ρ) 0.5 1 1.5 2 2.5 3 R3(ρ) (units of t) single-site resolution 2-site resolution 3-site resolution 4-site resolution 5-site resolution true temperature

Lower Resolution

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Fluorescence experiments can only detect parity densities

ρ 0.5 1 1.5 2 L(ρ) 1 2 3 4 Rξ(ρ) (units of t)

True temperature ξ = 3

ρ 0.5 1 1.5 2 L(ρ) 1 2 3 4 Rξ(ρ) (units of t)

True temperature ξ = 3

U/t = 10 , T/t = 1.0 U/t = 50 , T/t = 1.0

  • normal measurements
  • parity measurements

Can only trust mesurements with densities <~ 0.4 ...

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

  • n(0)

i

  • =

1 Z(0)

i

  • {ni}

nie−β(Di−µini),

HTE0 :

Z(0)

i

=

  • {ni}

e−β(Di−µini), Di = U 2 ni(ni − 1).

  • - only valid deep in the

normal phase

ni =

  • n(0)

i

  • +
  • i,j

(−βt)2 Z(0)

i Z(0) j

×  

(−,+)

  • {ni,nj }
  • δni +

χδ

ij

δ

ij

  • ninδ

je−β(Di+Dj −µini−µjnj)δ ij

+

(+,−)

  • {ni,nj }
  • δni −

χδ

ji

δ

ji

  • nδ

i nje−β( Di+Dj−µini−µjnj)δ ji

  (13)

δ

ij =

1 − eβγ δ

ij

  • βγ δ

ij

  • (βγji) −

1 − eβ

  • γ δ

ij+γji

  • βγ δ

ij + βγji

  • (βγji),

we also introduce ,

χδ

ij =

eβγ δ

ij

  • βγ δ

ij

  • (βγji) +

1 − eβ

  • γ δ

ij+γji

  • βγ δ

ij + βγji

  • (βγji)2

− (1 − eβγ δ

ij )

  • βγ δ

ij − βγji

  • βγ δ

ij

2(βγji)2 ,

HTE2 :

  • To fit the wings of 2D cross sectional density

profile against HTE2 (or better) density.

  • Expand the partition function exact up to (βt)2

Wing Thermometry

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

  • - Should the entire system be in the

normal phase, no more than a single measurement is needed to estimate the temperature accurately!

  • - Whenever possible, HTE2 is the

choice of preference for thermometry.

10 20 30 40 50 r 0.5 1 1.5 2 n(r) U/t=50, T/t=3.0 (single shot) 2

nd order (fit)

normal region (LDA) everywhere

Wing Thermometry

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

10 20 30 40 50 60 r 0.5 1 1.5 2 n(r) U/t=10.0, T/t=3.0

th order (true)

2

nd order (true)

normal region superfluid region Tc/t (U/t=10.0, <n>=0.42) ~ 3.0 (LDA) (LDA)

  • HTE0 works poorly.
  • HTE2 captures the entire normal phase.
  • HTE2 works better with larger normal region.

HTE0 / HTE2 Theory in a Picture Fitting density “wings” to HTE2

10 20 30 40 50 r 0.5 1 1.5 n(r) normal region superfluid region (LDA) (LDA)

  • The wings of 2D cross sectional density

profile is fitted to HTE2 density.

Wing Thermometry

Thursday, September 26, 13

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Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Directed worm algorithm

Feynmann perturbation (boson Hubbard model):

ˆ H0 = U 2 X

i

ni(ni − 1) − X

i

µini ˆ V = t X

hi,ji

ˆ b†

iˆ

bj ,

Z =

∞

X

m=0

X

i1···im

e−✏1 Z d⌧1 · · · Z ⌧m−1 d⌧m

  • e−⌧1✏1 Vi1i2 e⌧1✏2

· · ·

  • e−⌧m✏m Vimi1 e⌧m✏1

(2.9)

re ˆ H0|ii = ✏i|ii

d Vij = hi|ˆ V |ji.

Z = X

C

Z(C) X

  • r simply:

Worldlines configuration

: |i1i = |0, 1, 0, 1, 3, 0, 1, 0, 2, 0i : |i2i = |0, 0, 1, 1, 3, 0, 1, 0, 2, 0i : |i3i = |0, 0, 1, 1, 3, 0, 0, 1, 2, 0i : |i4i = |0, 0, 2, 0, 3, 0, 0, 1, 2, 0i : |i5i = |0, 1, 1, 0, 3, 0, 0, 1, 2, 0i : |i6i = |0, 1, 1, 0, 3, 0, 1, 0, 2, 0i : |i7i = |0, 1, 0, 1, 3, 0, 1, 0, 2, 0i : |i8i = |1, 0, 0, 1, 3, 0, 1, 0, 2, 0i

In this example:

Vi1i2 = 1 , Vi2i3 = 1 , Vi3i4 = 1 , Vi4i5 = p 2 Vi5i6 = p 2 , Vi6i7 = 1 , Vi7i8 = 1 , Vi8i1 = 1

O N : N(C) = 8 E0 : E0(C) = 16

Thursday, September 26, 13

slide-26
SLIDE 26

Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Directed worm algorithm

Z(X) = 4βL Z(Y ) = n

P(X → Y ) = 1 4βL P(Y → X) = 1 n .

Creation/ Annihilation of worms

  • L. Pollet, K.
  • V. Houcke, S. M. A. Rombouts,

Engineering local optimality in QMC algorithms,

  • J. Comp. Phys. 225/2, 2249-2266 (2007)

Pacceptance (X → Y) = 1 Pacceptance (Y → X) = 1 Move is globally balanced.

Thursday, September 26, 13

slide-27
SLIDE 27

Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Directed worm algorithm

  • L. Pollet, K.
  • V. Houcke, S. M. A. Rombouts,

Engineering local optimality in QMC algorithms,

  • J. Comp. Phys. 225/2, 2249-2266 (2007)

P(X1 ! X2) = ✏pe✏p⌧p0p

P(X1 → X2) = Z ∞

⌧vp

✏p e−✏p⌧p0p d⌧p0p P(X1 → X2) = e−✏p⌧vp

Movement of worms:

  • 1. not halted
  • 2. halted

Move is NOT globally balanced.

τp’p ~ Exp(εp)

  • 1. QMC-DWA assigns:
  • 2. It either gets halted or not.

⇒ Choose a exponential random number.

  • 3. Memoryless

Thursday, September 26, 13

slide-28
SLIDE 28

Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Directed worm algorithm

1) Inserting vertex, 2) deleting vertex, 3) relinking vertex, or 4) worm bounce:

P(X, Y1, Y2 ! X) = ✏v hiv+|ˆ b†

j|ivi

P(X, Y1, Y2 ! Y1) = t hiv+|ˆ b†

i|ivihiv|ˆ

biˆ b†

j|ivi

P(X, Y1, Y2 ! Y2) = t hiv+|ˆ b†

k|ivihiv|ˆ

bkˆ b†

j|ivi

  • 1. QMC-DWA assigns:
  • 2. The following moves are globally balanced:
  • a. unhalted move +

insert vertex/ bounce worm

  • b. halted move +

delete/relink vertex/ bounce worm

  • L. Pollet, K.
  • V. Houcke, S. M. A. Rombouts,

Engineering local optimality in QMC algorithms,

  • J. Comp. Phys. 225/2, 2249-2266 (2007)

Thursday, September 26, 13

slide-29
SLIDE 29

Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

Directed worm algorithm

  • L. Pollet, K.
  • V. Houcke, S. M. A. Rombouts,

Engineering local optimality in QMC algorithms,

  • J. Comp. Phys. 225/2, 2249-2266 (2007)

Crossing vertex:

Z(X) = e✏p⌧phiv|ˆ b†|ipie✏v⌧p ⇥ e✏v⌧vhi0

v|ˆ

b†|ivie✏0

v⌧v

Z(Y ) = e✏p⌧vhiv|ˆ b†|ipie✏v⌧v ⇥ e✏v⌧ +

v hi0

v|ˆ

b†|ivie✏0

v⌧ + v

Z(Y ) Z(X) = e✏p⌧vp e✏v⌧vp

! ! P(X ! Y ) = e✏p⌧vp P(Y ! X) = e✏v⌧vp

  • 1. It gets halted by like-vertex.
  • 2. It crosses the vertex with

acceptance probability 1

Move is globally balanced.

Thursday, September 26, 13

slide-30
SLIDE 30

Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

ALPS-2.2: QMC-DWA

Tutorial example: https://alps.comp-phys.org/mediawiki/index.php/ALPS_2_Tutorials:DWA-02_Density_Profile

Implementing QMC-DWA is easy and convenient within ALPS Python!

  • 1. Setup the parameters:
  • 2. Run simulation:
  • 3. Evaluate results:

4. Visualize:

Thursday, September 26, 13

slide-31
SLIDE 31

Numerical simulations of bosons and fermions in three dimensional optical lattices.

Ping Nang MA - pingnang@phys.ethz.ch September 27, 2013 PhD oral examination, ETH Zurich

  • 1. Magnetism in optical lattices

Density Functional Theory (for shallow fermionic optical lattices) Magnetism is stabilized by lattice bandstructure effects Phase diagram, ferromagnetism/ antiferromagnetism, SDW gap as indirect probe

  • 2. Thermometry in optical lattices

Fluctuation-dissipation thermometry -- feasible via window sizing Wing thermometry -- HTE2 valid entirely in normal region.

  • 3. QMC-DWA implementation in ALPS-2.2

easy and convenient within ALPS Python

  • P. N. Ma, S. Pilati, M. Troyer, and X. Dai,

Density functional theory for atomic Fermi gases, Nature Phys. 8, 601 (2012)

  • P. N. Ma, L. Pollet, and M. Troyer,

Measuring the equation of state of trapped ultracold bosonic systems in an optical lattice with in-situ density imaging ,

  • Phys. Rev. A. 82, 033627 (2010)

Conclusion/Outlook

Thursday, September 26, 13