qmc study of ultracold dipolar fermions in two dimensions · qmc study of ultracold dipolar...

QMC study ofultracold dipolar fermions in two dimensions

Natalia Matveeva, Stefano Giorgini

INO-CNR BEC center, University of Trento, Italy

Phys. Rev. Lett. 109, 200401, 2012.

Natalia Matveeva QMC study of ultracold dipolar fermions 1 / 26

Outline

1. Introduction: ultracold dipolar gases

2. The model: 2D dipolar fermions

3. Details of QMC calculations

4. Ground state properties: Fermi liquid, Wigner crystal and stripephases

⇒ Conclusion


Introduction:

ultracold dipolar gases


What are the ultracold quantum gases?

Cooling down an atomic ensamble

The first BEC: 1995

The first degenerate Fermigas: 1999.

Ground state of bosons andfermions in harmonic potential at

T = 0

Densities: 1012 − 1015cm−3

Temperatures: 10nK

Atom numbers: 10− 106


Why are the quantum gases interesting?

Ultracold atoms are clean and highly controllable systems:

tuning of scattering length, density, temperature, number of atoms

different dimensions: 1D, 2D, 3D

different spacial arrangments of atoms (square, triangular, honycomb lattices)

⇒ Possibility to study model hamiltonians for different condensed-matter

theories:

Bardeen-Cooper-Schrieffer theory of superconductivity

Hubbard model

ferromagnetism

etc ...


Contact and dipolar interactions

Atoms without a dipole moment: the s-wave scattering is a dominantprocess

Contact interaction potential: Ucont(r) = 4π~2am δ(r)

Reviews: Rev. Mod. Phys. 74, 463, 1999 (bosons), Rev. Mod. Phys. 80, 1215, 2008

(fermions).

Atoms with a dipole moment: not only the s-wave scattering

Udd(r) = Ucont + d2

r3 (1− 3 cos2 θ), characteristic length: r0 = md2

~2

Reviews: Rep. Prog. Phys. 72, 126401, 2009 (bosons), Chem. Rev. 112, 5012, 2012

(fermions).


Experimental realization of dipolarquantum gases

Atoms with large magnetic momentI Bosons: 52Cr (d = 0.054D), 168Er (d = 0.065D), 164Dy (d = 0.093D)

I Fermions: 53Cr, 161Dy

Phys. Rev. Lett. 108, 210401, 2012; Phys. Rev. Lett. 107, 190401, 2011; Phys. Rev. Lett. 108, 215301, 2012

Heteronuclear polar moleculesI Bosons: 41K87Rb (d = 0.56D), 87Rb133Cs

I Fermions: 40K87Rb, 23Na40K (d = 2.7D), 6Li133Cs (d = 5.5D)

Science 327, 853, 2010; Nature 464, 1324, 2010; Phys. Rev. Lett. 109, 085301, 2012.

Two-dimensional geometry supresses the chemical reaction rate for 40K87Rb


The model:

2D dipolar fermions


The model

H = ~2

2m

∑Ni=1 ∆i + d2

∑j<k

1|rj−rk|3

Dipole moments are aligned perpendicular to the plane of motion byan external field.

Purely repulsive 1/r3 interaction.

Purely two-dimensional system.

The goal is to investigate phase diagram at T = 0 by means of FNDMC.


Possible phases

Fermi liquid phaseElementary excitations of interacting fermions are described by almostindependent fermionic quasiparticles with the effective mass

Wigner crystalTriangular crystal which appears at large density (contrary to the Coulombinteraction case)

Stripe phase (stationary density modulations)2D dipolar Fermi gasStripe phase is predicted to appear at kF r0 ≈ 1.5− 6 based on differentmean-field approaches.2D homogeneous electron gas

I Microemulsion phases (stripes and bubbles) are predicted to appearbetween FL and WC (Spivak and Kivelson, PRB 70, 155114, 2004.)

I QMC study (Clark and Ceperley, PRL, 2009 ) did not find amicroemulsion phase


Details of FNDMC calculations.


Trial wave function

ΨT (r1, . . . , rN) = ∆SΨJ

Jastrow: ΨJ =∏N

j<k f (|rj − rk|),for r < R it is the s-wave solution of a two-bodyscattering problem,

for r > R f (r) ∼ exp(−const/r). 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

f(r)

r/(L/2)

kFr0=0.5kFr0=20

Slater determinant:

Fermi liquid phase ∆S = det[e ıqirj ]qi = 2π/L(nx , ny ) are PBC wave vectors in a square box

Wigner crystal phase ∆S = det[e−α(ri−Rm)2]

Rm are the lattice points of the triangular lattice, α is the variational parameter.

Stripe phase (pattern of equaly spaced 1D stripes along y -direction)

∆S = det[e ikαxxi−ξ(yi−ym)2]

ym denotes the y coordinate of the m-th stripe,kαx = 2πnαx/Lx are the PBC wave vectors in the x-direction,

ξ is a variational parameter.


Two-dimensional pair-distribution function

0 2 4 6 8 10 12

kF x

0

2

4

6

8

10

12

kF y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12

kF x

0

2

4

6

8

10

12

kF y

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12

kF x

0

2

4

6

8

10

12

kF y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Finite size errors

Potential energy

Summation in real space over all replicas of the simulation box:

Vdd = 12

∑nx ,ny

(∑Ni=1

∑Nj=1

d2

|ri−(rj+nL)|3

).

Simpler formula:

〈V 〉 = Σ + Etail,

Σ denotes Vdd calculated with the constarin |ri − rj − R| ≤ Lcut ,Etail (m)

N = 12

∫∞mLx

d2

r3 g(r)2πr dr .

Kinetic energy

Shell errors for a noninteracting Fermi gas: ∆TN = TTH − TN

Shell errors for a Fermi liquid: mm∗

∆TN

No shell errors for a Wigner crystal

Negligible shell errors for 1D stripes


Finite size scaling: Fermi liquid and Wigner crystal phases

Fermi liquid phase

0.9832

0.9834

0.9836

0.9838

0.984

0.9842

0.9844

0.9846

0.9848

0.985

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

E/E

HF

1/N

PBC - mPBC - m*

TABC

N = 25, 29, 37, 49, 61, 81, 89

Periodic boundary conditions(PBC)EN = E + m

m∗∆TPBC

N + aN

→ we can extract the effective massof a quasiparticle

Twist-averaged boundary conditions(TABC) give the same E as PBC

Wigner crystal phase

0.5632

0.5634

0.5636

0.5638

0.564

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

E/E

HF

1/N

N = 30, 56, 90 (almoust squaresimulation box)

EN = E + bN


Finite size scaling: stripe phase

The overall density is fixed

The square simulation box

The simulations are performed for 25 (5× 5), 49 (7× 7) and 81(9× 9) particles (we need an odd number of particles per stripe inorder to have filled 1D shells)

→ The linear dependence of the energy on the number of particles:EN = E + c

N

0.608

0.609

0.61

0.611

0.612

0.613

0.614

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

E/E

HF

1/N


Main results.


The equation of state and quantum phase transition

0.5

0.6

0.7

0.8

0.9

1

0.1 1 10

E/E

HF

kFr0

-0.002 0

0.002 0.004 0.006

20 30 40 50 60 70

∆E/E

HF

kFr0

EHF = εF2

(1 + 12845π

kF r0)

Red circles: EFL

Green triangles: EWC

Inset, blue circles: EWC − EFL

Inset, black circles:EST − EFL

Red line: Epe

Purple solid line: Ecl

At weak interaction we find good agreement withEpe = EHF + (NεF /8)(kF r0)2 log(1.43kF r0) fromLu and Shlyapnikov (2012)

At strong interaction the WC energy approaches

the energy of a purely classical crystal (purple

dashed line) corrected by the zero-point motion

of phonons from C. Mora et al (2007):

Ecl = N εF2

kF r04

(1.597 + 2.871√

kF r0

).

Quantum phase transition between FLand WC happens at kF r0 = 25(3).

The region of phase coexistence isvery small δkF r0 ≈ 0.01.

The stripe phase is never energeticallyfavorable.


Pair-distribution function

g(r) = 1n2 〈Ψ+(~s)Ψ+(~s +~r)Ψ(~s +~r)Ψ(~s)〉

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 2 4 6 8 10 12

g(r)

kF r

ideal gaskF r0 =1.10kF r0 =6.14kF r0 =20.0

crystal, kF r0 =61.4

The short-range repulsion increases by increasing the interaction strength.The shell structure starts to appear on approaching the freezing density.

The existence of long-range ordering can be seen from the oscillating

behaviour of g(r) at large r .


Static structure factor

S(k) = 1 + n∫dre ik·r[g(r)− 1]

NS(k) = 〈ρkρ−k〉 = 〈∑

i ,j eik·(ri−rj )〉

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5k/kF

ideal gas kF r0 =1.10 kF r0 =6.14 kF r0 =20.0

7.8 8

8.2crystal, kF r0 =34.7

The direct estimator exhibits a more pronounced peak compared to thesmoother Fourier transform.

This peak appears at the wave vector corresponding to the lowest non-zeroreciprocal lattice vector of the triangular lattice.


Effective mass and renormalization factor

0.55 0.6

0.65 0.7

0.75 0.8

0.85 0.9

0.95 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

m*/

m

kF r0

(m*/m)MC(m*/m)1st(m*/m)2nd

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20

Ren

orm

aliz

atio

n fa

ctor

Z a

nd m

*/m

kFr0

Zm*/m

0 0.2 0.4 0.6 0.8

0 0.5 1 1.5 2

n(k)

k/kF

N=29N=49N=61

Perturbatively calculated effective mass:Lu and Shlyapnikov, (2012)

(m∗m

)pe = 1/(1 +4

3πkF r0 + 0.25(kF r0)2 ln(0.65kF r0)).

Effective mass extracted from the fit of FNDMC energy:

EN = ETL +m

m∗∆TPBC

N +a

N

There is no dependence of momentum distribution on number of particles.


Main results

The phase diagram of a 2D single-component homogeneous dipolarFermi gas at T = 0 was investigated by means of FNDMC.

The important characteristics of Fermi liquid such as the effectivemass of quasiparticles and the renormalization factor were found.

Quantum phase transition between Fermi liquid and Wigner crystalphase occurs at kF r0 = 25(3).

Stripe phase is never energetically favorable.

The extention of this work:”The impurity problem in a bilayer system of dipoles”, arXiv:1306.5588v1.


Thank you for your attention!.

Recent review on QMC study of ultracold gases:L. Pollet, Rep. Prog. Phys. 75, 094501, 2012.


Optimization of DMC parameters: time step and numberof walkers

Dependence of DMC energy on the time step dt.

0.919 0.9191 0.9192 0.9193 0.9194 0.9195 0.9196 0.9197 0.9198 0.9199

0.92

0 0.02 0.04 0.06 0.08 0.1

E/E

HF

dt

kF r0 = 0.5

0.48865

0.4887

0.48875

0.4888

0.48885

0.4889

0.48895

0.489

0 2e-05 4e-05 6e-05 8e-05 0.0001

E/E

HF

dt

kF r0 = 20

Dependence of DMC energy on the number of walkers Nw .

0.919

0.9192

0.9194

0.9196

0.9198

0.92

0.9202

0.9204

0 0.01 0.02 0.03 0.04 0.05

E/E

HF

1/Nw

kF r0 = 0.5

0.4888

0.4889

0.489

0.4891

0.4892

0.4893

0.4894

0 0.01 0.02 0.03 0.04 0.05

E/E

HF

1/Nw

kF r0 = 20


Stripe phase: optimization of the stripe separation

The distance between two stripes a = |ym+1 − ym|

The overall density in the simulation box with area S = Lx × Ly isfixed

The distance a is changed when the ratio ρ = Lx/Ly changes

0.61

0.615

0.62

0.625

0.63

0.635

0.64

0.645

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

E/E

HF

a/asq

→ The optimal value is a = asq, where asq is the stripe separation for asquare box.


Momentum distribution

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

n(k)

k/kF

kF r0 =1.10kF r0 =6.14kF r0 =20.0

crystal, kF r0 =34.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2

n(k)

k/kF

N=29N=49N=61

FL: the discontinuity of n(k) at k = kF decreases with the increase of kF r0,but always stays finite

WC: n(k) does not have a discontinuity

Momentum distribution does not depend on the system size


qmc study of ultracold dipolar fermions in two dimensions · qmc study of ultracold dipolar...

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