liquid and crystal phase of dipolar fermions in two dimensionsbqmc.upc.edu/pdfs/doc854.pdf · 2019....

Liquid and crystal phase of

dipolar fermions in two dimensions

Natalia Matveeva and Stefano Giorgini

Dipartimento di Fisica, Universita di Trento and INO-CNR BEC Center, I-38123 Povo, Italy

arXiv:1206.3904, accepted to PRL

Natalia Matveeva and Stefano Giorgini (Dipartimento di Fisica, Universita di Trento and INO-CNR BEC Center, I-38123 Povo, Italy)Liquid and crystal phase of dipolar fermions in two dimensions 1 / 24

Outline

1. Introduction: dipolar ultracold gas

2. The experimental realization

3. Fixed-node diffusion Monte-Carlo method

4. Two-dimensional dipolar Fermi systemI Fermi liquid phaseI Wigner crystal phaseI Quantum phase transitionI Absence of stripe phaseI Correlation functions

⇒ Conclusion and outlook


Ultracold atomsCooling down of an atomic

ensamble

The first BEC: 1995

The first degenerate Fermigas: 2000

Ground state of bosons andfermions in harmonic potential at

T = 0

Densities: up to 1015cm−3

Temperatures: less than 1µK

Atom numbers: 102 − 106


Dipolar interactionProperties

Dipolar potential: Vd =~d1~d2R2−3(~d1

~R)(~d2~R)

R5 ∼ 1R3

Anisotropic (partially repulsive, partially attractive) character

Long-range compared with contact interaction

Relatively easy to tune: by changing the external fields or confiningpotentials

Experimental realization

Atoms with large magnetic momentum

Polar molecules


Experimental realizationBEC of atoms with large magnetic momentum52Cr , µ = 6µB ; 168Er , µ = 7µB ; 164Dy , µ = 10µB .

Polar molecules40K 87Rb, dmax = 0.566D, degenerate gas of ground state molecules, T/TF = 1.4,d = 0.2D, Nature 464, 1324, 2010.23Na40K , dmax = 2.7D, Bose-Fermi mixture: PRA 85, 051602(R), 2012.23Na6Li , dmax = 0.6D, Feshbach molecules, arXiv:1205.5304v2.


Two-dimensional system of dipolar fermions in a bulk

Fig. Fermionic dipoles confined to the xy -plane. The dipolesare aligned in the xz-plane and form an angle θ0 with the

z-axis. Taken from PRA 84, 063633, (2011).

Fig. Phase diagram of the quasi-2D dipolar Fermi gas atT = 0, g = kF r0. SF is a superfluid, NFL is a novel

Fermi-Liquid, DW is density waves, LWI is a long wave-lengthinstability. Taken from PRA 84, 063633, (2011).

Novel Fermi liquid:Z.-K. Lu and G.V. Shlyapnikov, PRA 85 023614, (2012)J.P. Kestner and S. Das Sarma, PRA 82, 033608, (2010)

Stripe (CDW) phase:Y. Yamaguchi, T. Sogo, T. Ito and T. Miyakawa, PRA 82, 013643, (2010)M. Babadi and E. Demler, PRB, 84, 235124, (2011)M.M. Parish and F.M. Marchetti, PRL, 108, 145304, (2012)

Wigner crystal:Bosons: G.E. Astracharchik, J. Boronat, I.L. Kurbakov and Yu. E. Lozovik, PRL 98, 060405, (2007)Bosons: C. Mora, O. Parcolet and X. Waintal , PRB 76, 064511, (2007)

Superfluid phase:G.M. Bruun and E. Taylor, PRL 101, 245301, (2008)L.M. Sieberer and M.A. Baranov, PRA 84, 063633, (2011)J. Levinsen, N.R. Cooper and G.V. Shlyapnikov, PRA 84, 013603, (2011)


Fixed-node diffusion Monte CarloImaginary time Schrodinger equation:

∂φ(R, τ)

∂τ= D∇2φ(R, τ) + (ET − V (R))φ(R, τ).

Distribution function: f (R, τ) = ψT (R)φ(R, τ).

DMC estimate of energy:

EDMC =

∫dREL(R)f (R, τ →∞)∫

dRf (R, τ →∞),EL(R) =

HψT (R)

ψT (R).

Requirement: f (R, τ) ≥ 0!

Bosons:

f (R, τ →∞) = ψT (R)ψ0(R),

where ψ0(R) is the exact ground state ⇒ EDMC is the exact ground state energy.

Fermions - fixed-node approximation:

f (R, τ →∞) = ψT (R)ψFN(R),

where ψFN(R) is an approximation to exact ground-state wave function,

⇒ EDMC is an upper bound to the exact eigenenergy.


The model

Fig. 2D system of one-component fermions with dipolar

momentum oriented perpendicular to the plane

The goal: phase diagram(FL, WC and stripe phase)

at T = 0 by means of FN DMC

Hamiltonian:

H = ~2

2m

∑Ni=1 ∆i + d2

∑j<k

1|~rj−~rk |3 .

The units:characteristic length of dipole-dipole force r0 = md2/~2;the strength of the dipolar interaction: kF r0 or dimensionless density nr2

0 ;

the units of the energy: EHF = N εF2 (1 + 128

45πkF r0) or ~2

mr20

(nr20 )3/2

Trial wave function:

ΨT (~r1,~rN) = ∆S ·ΨJ ,

where ∆S is the Slater determinant, ΨJ is the Jastrow part.


Fermi liquid phase: definition and trial wave function

Concept of Landau’s Fermi liquid theory

elementary exitations of interacting fermions are described by almostindependent fermionic quasiparticles with the effective mass

state of Fermi liquid is simply described by quasiparticles distribution

Trial wave function

ΨT (~r1,~rN) = ∆SΨJ

Slater determinant: ∆S = {φi (~rj)},where φi (~rj) = exp{ı~qi~rj} are plane waves in a square box,

Jastrow: ΨJ =∏N

j<k f (|~rj −~rk |),

for r < R it is the solution of two-body scattering problem,for r > R f (~r) ∼ exp(−const/r).


Calculation of the potential energySummation 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 .

Estimation of potential energy at the exampleof a system containing 3 particles with

Lcut = L.

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.5 1 2 5 7 10 20 30 40

E/E

HF

m

ΣΣ+Etail

0.4936

0.4938

0.494

0.4942

0.4944

0.4946

0.5 1 2 5 7 10 20 30

The dependence of the direct sum ofpotential energy Σ (the red points) on cutoffm (Lcut = mL); the black points on the main

figure and on the inset show Σ + Etail .


Finite size scaling

Periodic boundary conditions (PBC)

kα = (2π/L)(nxα, nyα)

Twist averaged boundary conditions (TABC)C. Lin, F.H. Zong and D.M. Ceperley, PRE, 64 016702, (2001).

kα(θ) = (2π/L)(nxα + θx , nyα + θy )

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*

TABCEnergy in thermodynamical limit

N = 25, 29, 37, 49, 61, 69, 81

PBC: EN = ETL + mm∗

∆TPBCN + a

N

TABC: EN = ETL + ∆TTABCN + a

N


Optimization of DMC parameters: time step and numberof walkers

29.735

29.74

29.745

29.75

29.755

29.76

29.765

0.01 0.1

E/N

dt

5.199

5.1995

5.2

5.2005

5.201

5.2015

5.202

5.2025

1e-06 1e-05 0.0001

E/N

dt

Dependence of DMC energy on the time step dt. The left figure corresponds to the density nr20 = 0.02 (kF r0 = 0.5) and the

right figure corresponds to the density nr20 = 32 (kF r0 = 20). The energy is in units of ~2

mr20

(nr20 )3/2.

29.735 29.74

29.745 29.75

29.755 29.76

29.765 29.77

29.775 29.78

0 0.01 0.02 0.03 0.04 0.05

E/N

1/N

5.2

5.201

5.202

5.203

5.204

5.205

5.206

5.207

0 0.01 0.02 0.03 0.04 0.05

E/N

1/N

Dependence of DMC energy on the number of walkers N. The left figure corresponds to the density nr20 = 0.02 (kF r0 = 0.5)

and the right figure corresponds to the density nr20 = 32 (kF r0 = 20). The energy is in units of ~2

mr20

(nr20 )3/2.


Equation of state at small interaction

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

E/N

(EF/2

)

kF r0

EmcFit of Emc

EpeEHF

1

1.1

1.2

1.3

1.4

1.5

0 0.1 0.2 0.3 0.4 0.5

Dependence of FL energy in thermodynamical limit on strength of dipolar interaction kF r0.

The result of perturbation theoryZ.K. Lu and G.V. Shlyapnikov, PRA, 85, 023614, (2012):

Epe

N=

~2k2F

4m(1 +

128

45πkF r0 +

1

4(kF r0)2ln(ukF r0)), u = 1.43.

→ Epe is quantatively accurate up to kF r0 ≈ 0.5Natalia Matveeva and Stefano Giorgini (Dipartimento di Fisica, Universita di Trento and INO-CNR BEC Center, I-38123 Povo, Italy)Liquid and crystal phase of dipolar fermions in two dimensions 13 / 24

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

Fig. Dependence of the effective mass m/m∗ on strength ofdipolar interaction kF r0 at the wealky interacting regime.

Black points are QMC results, dashed blue line is theperturbative expantion up to the first order, red line is the

perturbative expantion up to the second order.

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

Fig. Effective mass and renormalization factor in the liquidphase as a function of the interaction strength. The line

corresponds to the perturbation expansion ( m∗m

)pe . Inset:Momentum distribution corresponding to kF r0 = 20 for

different system sizes.

Perturbatively calculated effective mass:Z.K. Lu and G.V. Shlyapnikov, PRA, 85, 023614, (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

Renormalization factor Z is extracted from the jump of the momentumdistribution at k = kF


Wigner crystal phase: trial wave function and finite size scaling


ψT (r1, ..., rN) =∏i<j

f (rij) det[e−α(~ri−~Rm)2

],

f (r) is the same as for FL,~Rm are the lattice points of the triangular lattice,

α is the variational parameter.

Fig. Left: the pseudo-elementary cell oftriangular lattice which contains two atoms,ay = 3

√ax . Right: the example of triangular

lattice for 30 particles, black circles are positions

of lattice points ~Rm .

Optimization of α

4.797 4.798 4.799

4.8 4.801 4.802 4.803 4.804 4.805 4.806 4.807

200 220 240 260 280 300 320 340 360 380

E/N

α

Fig. Dependence of variational energy on the value ofvariational parameter α for N = 30 at nr2

0 = 96

(kF r0 = 34.73). The energy is in units of ~2

mr20

(nr20 )3/2.

Finite size scaling

5.832

5.834

5.836

5.838

5.84

5.842

5.844

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

E/N

1/N

Fig. Finite size scaling of WC crystal energy for 30, 56 and90 particles at nr2

0 = 128 (kF r0 = 40.11). The energy is in

units of ~2

mr20

(nr20 )3/2.


Stripe phase: definition and trial wave functionStripe phase is characterized by stationary density modulations.2D Coulomb gas

Microemulsion phases (stripes and bubbles) are predicted to appear between FLand WC (B.Spivak and S.A. Kivelson)

Carefull MC study (B. K. Clark et al ) did not find a microemulsion phase

2D dipolar Fermi gas

Density wave instability, which breaks both rotational and translational symmetry,is predicted based on different mean-field like approaches (Y. Yamaguchi et al, M.Babadi and E. Demler, M.M. Parish and F.M. Marchetti)

Such stripe phase is expected to occupy large region on phase diagram stratingfrom kF r0 ≈ 1.5− 6.


ψT (r1, ..., rN) =∏i<j

f (rij) det[e ikαx xi−γ(yi−ym)2

],

f (r) is the same as for FL,ym denotes the y coordinate of the m-th stripe,

kαx = 2πnαx/Lx are the PBC wave vectors in the

x-direction, γ is a variational parameter.Fig. Pattern of equally spaced stripes

along y direction.


Stripe phase: optimization of the stripe separationTrial wave function:

ψT (r1, ..., rN) =∏i<j


],

the gaussian localization parameter γ is roughly the same as for WC

Optimization of the stripe separation a = |ym+1 − ym|

the overall density in the simulation box with area S = Lx × Ly is fixed

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

6.5

6.55

6.6

6.65

6.7

6.75

6.8

6.85

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

E/N

a/asq

Fig. Dependence of VMC energy on stripes separation a at nr20 = 32 (kF r0 = 20.053). Energy is in units of ~2

mr20

(nr20 )3/2.

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


Stripe phase: finite size scalingTrial wave function:

ψT (r1, ..., rN) =∏i<j


],

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 in order to have filled 1D shells)

→ The perfect linear dependence of the energy on the number of particles!

6.47

6.48

6.49

6.5

6.51

6.52

6.53

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

E/N

1/N

Fig. Finite scaling for stripes. Red points are FN DMC energies, line is a linear fit and black square is the extrapolated energy in

thermodynamical limit. Energy is in units of ~2

mr20

(nr20 )3/2. Error bars are smaller than symbol size.


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

Fig. Equation of state of the liquid and solidphase in units of the Hartree-Fock energy

EHF =εF2

(1 + 12845π

kF r0). Circles refer to theliquid and triangles to the solid. Inset: Energy

difference between the solid and the liquid (bluecircles) and between the stripe phase and the

liquid (black circles). The blue solid line isobtained from a best fit to the equation of state

of the liquid and solid phase. Error bars aresmaller than the symbols size and are

comparable in the three phases.

At weak interaction we find good agreement withEpe = EHF + (NεF /8)(kF r0)2 log(1.43kF r0) fromZ.-K. Lu and G.V. Shlyapnikov (red dashed line)

At strong interaction the WC energy approaches

the energy of a purelly classical crystal (purple

dashed line) corrected by the zero-point motion

of phonons from C. Mora et al:

Ecl = N εF2

kF r04

(1.597 + 2.871√

kF r0

)(the purple

solid line).

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

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

The stripe phase is never energeticallyfavorable!


Pair correlation function and static structure factor

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

Fig. Pair correlation function in the liquid and in the crystalphase. The pair correlation function of the non-interacting

gas is also shown.

The short-range repulsion increases by increasing theinteraction strength.

The shell structure starts to appear on approachingthe freezing density.

The existence of long-range ordering can be seenfrom the oscillating behaviour of g(r) at large r .

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

Fig. Static structure factor in the liquid (for 49 particles) andin the crystal (for 56 particles) phase. In the liquid phase,

solid lines correspond to the Fourier transform of g(r) whilesymbols correspond to the direct calculation of S(k). The

static structure factor of the non-interacting gas is alsoshown.

The direct estimator exhibits a more pronouncedpeak compared to the smoother Fourier transform.

This peak appears at the wave vector correspondingto the lowest non-zero reciprocal lattice vector of thetriangular lattice.


Two dimensional pair correlation function g(x , y)

0 2 4 6 8 10 12kF x

0

2

4

6

8

10

12

k F y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Fig. g(x, y) for FL (N = 49) at kF r0 = 20.

0 2 4 6 8 10 12kF x

0

2

4

6

8

10

12

k F y

0

0.5

1

1.5

2

2.5

Fig. g(x, y) for WC (N = 56) at kF r0 = 34.7.

0 2 4 6 8 10 12kF x

0

2

4

6

8

10

12

k F y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Fig. g(x, y) for stripes (N = 49) at kF r0 = 20.


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


Fig. Momentum distribution for liquid (for 49 particles) andfor crystal (for 56 particles) at different kF r0.

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

Fig. Momentum distribution for FL for different system sizeat kF r0 = 20.053

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 any discontinuity

Momentum distribution does not depend on the system size!


One-body density matrix

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

f(r)

kF r

ideal gaskF r0 =6.14kF r0 =20.0


Fig. One-body density matrix for FL (49 particles) and for WC (56 particles).


Conclusion and outlook

The phase diagramm 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.

Perspectives:

to add the coupling to a second layer,

to consider the effect of a tilting angle, making the interaction in the2D plane anisotropic.

Thank you for your attention!


liquid and crystal phase of dipolar fermions in two dimensionsbqmc.upc.edu/pdfs/doc854.pdf · 2019....

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