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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS2348

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Supplementary material for “Density functional theory for atomic Fermi gases”

Ping Nang Ma,1 Sebastiano Pilati,1, 2 Xi Dai,3 and Matthias Troyer1

1Theoretische Physik, ETH Zurich, CH-8093 Zurich, Switzerland2The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy

3Institute of Physics, Chinese Academy of Science, Beijing China

I. FORMALISM OF KOHN-SHAM DFT FOR

FERMIONIC OPTICAL LATTICES

A. The Kohn-Sham equations

For N interacting spin-half fermions in an isotropic3D optical lattice V (r) with cubic symmetry, the groundstate energy and density can be obtained by minimizingthe total energy

E =

∫dr [T0 [ρ↑, ρ↓; r] + V (r) (ρ↑ (r) + ρ↓ (r))

+ �HXC [ρ↑, ρ↓; r] ] (1)

with respect to the 3D spin densities

ρσ (r) =

∫Drj δ (r− r1)

∣∣∣∣∣∣∏

σ=↑,↓

det [φσn (rj)]

∣∣∣∣∣∣

2

=

occ∑nσ

|φσn (r)|2 .

(2)

Here φσn (r) are normalized single-quasiparticle spin-

orbitals filled to the Fermi level, T0 [ρ↑, ρ↓; r] representsthe kinetic term and �HXC [ρ↑, ρ↓; r] represents the inter-action energy derived in the next section.

Constrained minimization leads to the coupled Kohn-Sham eigenvalue equations

HσKS φ

σn = �n φ

σn (3)

where the Kohn-Sham Hamiltonian HσKS is given by

HσKS = −

h2

2m∇2 + V eff

σ (ρ↑, ρ↓; r) (4)

and the effective potential V effσ (ρ↑, ρ↓; r) is

V effσ (ρ↑, ρ↓; r) = V (r) +

∂

∂ρσ[�HXC (ρ↑, ρ↓; r)] . (5)

B. Translational symmetry and Bloch’s theorem

In a periodic optical lattice, i.e. V (r+ d) = V (r), weshall make use of Bloch’s theorem to write the solutionsas

φσnk (r) = e2πik·ruσ

nk (r) , (6)

with periodic Bloch orbitals uσnk (r). The wavevectors k

run over the first Brillouin zone of the reciprocal lattice.Working in the units of lattice spacing d = λ

2 and recoil

energy ER = h2

2m

(2πλ

)2(h is the reduced Planck con-

stant and m the atomic mass) the coupled Kohn-Shamequations become[1

π2(−i∇+ 2πk)2 + V eff

σ (ρ↑, ρ↓; r)

]uσnk (r) = �σnku

σnk (r) .

(7)They must be solved self-consistently with the groundstate densities

ρσ (r) =∑nk

|uσnk (r)|

2 Θ (μ− �σnk) (8)

where Θ(· · · ) is the Heaviside function. The total groundstate energy is calculated from the set of all quasiparticleenergies �σnk as

E =∑nkσ

�σnkΘ (μ− �σnk) (9)

C. Plane Wave Basis

Expanding the Bloch orbitals in a plane wave basis

uσnk (r) =

∑G

cσnk (G) exp (2πiG · r) , (10)

turns the Kohn-Sham equation (7) into a coupled set ofmatrix eigenvalue equations

4 (G+ k)2 cσnk (G) +∑G′

V effG−G′cσnk (G

�) = �σnkcσnk (G)

(11)for reciprocal vectors G, in particularly Gα =0,± 1

Mα

,± 2Mα

, . . . for a simple cubic lattice. As a re-mark, the effective potential possesses both translationaland inversion symmetry, and therefore its Fourier com-ponents

V effG =

1

M3

∫

unitcell

V eff (r) exp (−2πiG · r) dr (12)

must be real and related by V effG

= V eff−G

.

D. Oh point group symmetry

The band structure of our cubic lattice possesses an Oh

point group symmetry, in particularly the inversion (J)and reflection (σh, σd) symmetries, that give rise to thefollowing 48-fold degeneracy for the quasiparticle energies�σnk:

Density functional theory for atomic Fermi gases

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1. J-symmetry:

�σn(kx,ky,kz)= �σn(−kx,−ky,−kz)

(13)

2. σh-symmetry:

�σn(kx,ky,kz)= �σn(±kx,±ky,±kz)

(14)

3. σd-symmetry:

�σn(kx,ky,kz)= �σn(ky,kx,kz)

(15)

�σn(kx,ky,kz)= �σn(kx,kz ,ky)

(16)

�σn(kx,ky,kz)= �σn(kz ,ky,kx)

(17)

Recognizing this reduces computational effort greatly.

II. ENERGY FUNCTIONAL FOR REPULSIVE

FERMI GASES

The Hartree and exchange-correlation energy function�HXC (ρ↑, ρ↓) is defined as:

�HXC (ρ↑, ρ↓) = e (ρ↑, ρ↓)−3

5ρ↑EF↑ −

3

5ρ↓EF↓; (18)

the first term on the right-hand side e (ρ↑, ρ↓) is the en-ergy density of a homogeneous repulsive Fermi gas, whilethe second (third) term is the energy of the spin-↑ (spin-↓) component of a noninteracting Fermi gas (EF↑(↓) =(hkF↑(↓)

)2/2m is the Fermi energy of the spin-↑ (spin-↓)

particles and kF↑(↓) =(6π2ρ↑(↓)

)1/3is the Fermi momen-

tum).To determine the zero-temperature equation of state ofthe homogeneous Fermi gas we use the Fixed-Node Dif-fusion Monte Carlo (FN-DMC) method. This quantumMonte Carlo technique has been employed in severalstudies of the ground-state properties of resonantly in-teracting Fermi gases with balanced [1, 2] as well as im-balanced [3–5] populations of the two components.

Despite the fact that – to circumvent the sign problem– one has to introduce the fixed-node constraint, mean-ing that the ground-state wave function is forced to havethe same nodal surface as a trial wave function, FN-DMChas proven to be extremely accurate. It provides a rig-orous upper bound for the ground-state energy, whichis exact if the nodes of the trial wave function coincidewith those of the exact ground state. Predictions forthe ground-state energies obtained with this techniquehave been benchmarked against experimental results forthe low temperature equations of state of both normaland superfluid atomic gases on the attractive branch ofFeshbach resonances [6, 7]. For the repulsive branch –similarly to the normal phase of the attractive branch –

1.1

1.2

1.3

1.4

1.5

1.6

0 0.2 0.4 0.6 0.8 1

e [3

/5 ρ

E F]

P

kFa = 0.9kFa = 0.82kFa = 0.75kFa = 0.65kFa = 0.5kFa = 0.25

FIG. 1: (color on-line). Energy per volume as a functionof the polarization for different values of the interaction pa-rameter kFa. Squares are Monte Carlo data, solid (black)lines the global energy function equation (20), dashed anddot-dashed lines represent the low-P and large-P expansions,equations (22) and (25), respectively.

the trial wave function employed is in the Jastrow-Slaterform [8]:

ψT ({ri}, {ri�}) =∏i,i�

f(|ri − ri� |)D↑(N↑)D↓(N↓) ; (19)

here, i (i�) labels the N↑ spin-↑ (N↓ spin-↓) particles.The positive-definite Jastrow correlation term f(x) is ob-tained from the solution of the two-body scattering prob-lem in free space. It does not affect the nodal surfaceand is introduced to reduce Monte Carlo fluctuations.

D↑(N↑) = det[φ↑j (ri)

]N↑×N↑

is the spin-↑ Slater deter-

minant of the single-particle orbitals φ↑j (ri) (D↓(N↓) is

the spin-↓ Slater determinant). The index j (j�) labelsthe N↑ (N↓) lowest-energy eigenstates, which solve equa-tion (3) in the non-interacting case a = 0 (a is the s-wave scattering length). For the homogeneous system(V (r) = 0), the eigenstates φσ

j are plane waves in a 3Dbox with periodic boundary conditions.

In the following we report an accurate and simpleparametrization of the equation of state of the homo-geneous repulsive Fermi gas. The bare quantum MonteCarlo data are reported in section IV. The function e de-fined below gives the ground-state energy per volume asa function of the interaction parameter kF a = (3π2ρ)1/3a(where ρ = ρ↑ + ρ↓ is the total density) and the polar-ization P = (ρ↑ − ρ↑)/(ρ↑ + ρ↑). The global function einterpolates between the functions e<, which gives theenergy in the small polarization regime, and e>, givingthe energy at large polarization. The interpolation makesuse of the damping function fdamp. The general form ofthe equation of state is the following:

e (kF a, P ) = [1− fdamp(P )] e< (kFa, P ) +

fdamp(P ) e> (kF a, P ) . (20)

3

The damping function is defined as:

fdamp(P ) =1

2tanh [2πCdamp1 (P − Cdamp2)] +

1

2; (21)

the coefficient Cdamp1 = 1.8 determines the steepness,and Cdamp2 = 0.5 is the center of the transition region.In figure 1 we show the equation of state (20), the smalland large polarization limits, together with the MonteCarlo data.

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 0

0.2

0.4

0.6

0.8

e 0 χ-1

kF a

FIG. 2: (color on-line). Energy per volume (circles, left axis)and inverse magnetic susceptibility (squares, right axis) of theunpolarized Fermi gas. Units are the ideal Fermi gas values,3/5ρEF and χ0 = 3ρ/(2EF ), respectively. Solid lines arethe fitting functions (23) in red and (24) in green, while thedashed lines correspond to second order perturbation theory.

At small population imbalance, the equation of stateis quadratic in the polarization. Accordingly, e< can becast in the following form:

e< (kFa, P ) =3

5ρEF

[e0 (kFa) +

5

9P 2χ−1 (kF a)

], (22)

where the energy of the unpolarized gas (in units of35ρEF , where EF = (kF h)

2/2m is the Fermi energy) isgiven by:

e0 (kFa) = 1 + CE1kF a+ CE2(kFa)2 +

CE3(kF a)3 + CE4(kF a)

4, (23)

while χ−1 is the inverse of the magnetic susceptibil-ity divided by the ideal gas result 3ρ/(2EF ), which weparametrize as:

χ−1 (kFa) = 1−Cχ1kFa−Cχ2(kFa)2−Cχ3 (kFa)

3 . (24)

The coefficients CE1 = 0.3536, CE2 = 0.1855,Cχ1 = 0.6366 and Cχ2 = 0.2911 have been deter-mined using second order perturbation theory [9–13],while we obtain CE3 = 0.307(7), CE4 = −0.115(8) andCχ3 = 0.56(1) from a best-fit to the Monte Carlo resultswith zero or small population imbalance (in the range

0

0.4

0.8

1.2

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1

1.05

1.1

1.15

1.2

1.25

A m*

kF↑ a

FIG. 3: (color on-line). Chemical potential at zero concen-tration (circles, left axis) and effective mass (squares, rightaxis) of the repulsive polaron. Units are 3/5EF↑ and bareatomic mass, respectively. Solid lines are the fitting functions(26) in red and (27) in green. The dashed line is the chemicalpotential in second order perturbation theory.

kFa ≤ 1 and P ≤ 0.5). The functions e0(kF a) andχ−1(kF a) are shown in figure 2.

At large population imbalance the behavior of a normalFermi gas is well described by the Landau-PomeranchukHamiltonian [4, 8, 14]. In this approach, the minoritycomponent is regarded as a gas of weakly interactingquasi-particles, the so-called Fermi polarons. The energyper volume takes the following form:

e> (kF↑a, x) =3

5ρ↑EF↑

[1 +A (kF↑a)x

+x5/3

m∗ (kF↑a)+ F (kF↑a)x

2

]; (25)

where x = (1 − P )/(1 + P ) is the concentration ofthe minority component, while the spin-up Fermi mo-mentum and energy are given by the relations kF↑ =

kF (2/(1 + x))1/3 and EF↑ = EF (2/(1 + x))2/3. Thefunction A(kF↑a) gives the polaron chemical potential(devided by 3/5EF↑) at zero concentration and can beparametrized as follows:

A (kF↑a) =5

3

[CA1kF↑a+ CA2 (kF↑a)

2+ CA3 (kF↑a)

3];

(26)second order perturbation theory [14–16] gives the firsttwo coefficients CA1 = 0.4244 and CA2 = 0.2026, whileCA3 = 0.105(2) results from a best-fit to Monte Carlodata for the energy of a single spin-down impurity im-mersed in the Fermi sea of spin-up particles. The po-laron effective mass m∗ is extracted from the disper-sion relation of an impurity with finite momentum. Weparametrize the interaction parameter dependence of this

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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS23482

1. J-symmetry:

�σn(kx,ky,kz)= �σn(−kx,−ky,−kz)

(13)

2. σh-symmetry:

�σn(kx,ky,kz)= �σn(±kx,±ky,±kz)

(14)

3. σd-symmetry:

�σn(kx,ky,kz)= �σn(ky,kx,kz)

(15)

�σn(kx,ky,kz)= �σn(kx,kz ,ky)

(16)

�σn(kx,ky,kz)= �σn(kz ,ky,kx)

(17)

Recognizing this reduces computational effort greatly.

II. ENERGY FUNCTIONAL FOR REPULSIVE

FERMI GASES

The Hartree and exchange-correlation energy function�HXC (ρ↑, ρ↓) is defined as:

�HXC (ρ↑, ρ↓) = e (ρ↑, ρ↓)−3

5ρ↑EF↑ −

3

5ρ↓EF↓; (18)

the first term on the right-hand side e (ρ↑, ρ↓) is the en-ergy density of a homogeneous repulsive Fermi gas, whilethe second (third) term is the energy of the spin-↑ (spin-↓) component of a noninteracting Fermi gas (EF↑(↓) =(hkF↑(↓)

)2/2m is the Fermi energy of the spin-↑ (spin-↓)

particles and kF↑(↓) =(6π2ρ↑(↓)

)1/3is the Fermi momen-

tum).To determine the zero-temperature equation of state ofthe homogeneous Fermi gas we use the Fixed-Node Dif-fusion Monte Carlo (FN-DMC) method. This quantumMonte Carlo technique has been employed in severalstudies of the ground-state properties of resonantly in-teracting Fermi gases with balanced [1, 2] as well as im-balanced [3–5] populations of the two components.

Despite the fact that – to circumvent the sign problem– one has to introduce the fixed-node constraint, mean-ing that the ground-state wave function is forced to havethe same nodal surface as a trial wave function, FN-DMChas proven to be extremely accurate. It provides a rig-orous upper bound for the ground-state energy, whichis exact if the nodes of the trial wave function coincidewith those of the exact ground state. Predictions forthe ground-state energies obtained with this techniquehave been benchmarked against experimental results forthe low temperature equations of state of both normaland superfluid atomic gases on the attractive branch ofFeshbach resonances [6, 7]. For the repulsive branch –similarly to the normal phase of the attractive branch –

1.1

1.2

1.3

1.4

1.5

1.6

0 0.2 0.4 0.6 0.8 1

e [3

/5 ρ

E F]

P

kFa = 0.9kFa = 0.82kFa = 0.75kFa = 0.65kFa = 0.5kFa = 0.25

FIG. 1: (color on-line). Energy per volume as a functionof the polarization for different values of the interaction pa-rameter kFa. Squares are Monte Carlo data, solid (black)lines the global energy function equation (20), dashed anddot-dashed lines represent the low-P and large-P expansions,equations (22) and (25), respectively.

the trial wave function employed is in the Jastrow-Slaterform [8]:

ψT ({ri}, {ri�}) =∏i,i�

f(|ri − ri� |)D↑(N↑)D↓(N↓) ; (19)

here, i (i�) labels the N↑ spin-↑ (N↓ spin-↓) particles.The positive-definite Jastrow correlation term f(x) is ob-tained from the solution of the two-body scattering prob-lem in free space. It does not affect the nodal surfaceand is introduced to reduce Monte Carlo fluctuations.

D↑(N↑) = det[φ↑j (ri)

]N↑×N↑

is the spin-↑ Slater deter-

minant of the single-particle orbitals φ↑j (ri) (D↓(N↓) is

the spin-↓ Slater determinant). The index j (j�) labelsthe N↑ (N↓) lowest-energy eigenstates, which solve equa-tion (3) in the non-interacting case a = 0 (a is the s-wave scattering length). For the homogeneous system(V (r) = 0), the eigenstates φσ

j are plane waves in a 3Dbox with periodic boundary conditions.

In the following we report an accurate and simpleparametrization of the equation of state of the homo-geneous repulsive Fermi gas. The bare quantum MonteCarlo data are reported in section IV. The function e de-fined below gives the ground-state energy per volume asa function of the interaction parameter kF a = (3π2ρ)1/3a(where ρ = ρ↑ + ρ↓ is the total density) and the polar-ization P = (ρ↑ − ρ↑)/(ρ↑ + ρ↑). The global function einterpolates between the functions e<, which gives theenergy in the small polarization regime, and e>, givingthe energy at large polarization. The interpolation makesuse of the damping function fdamp. The general form ofthe equation of state is the following:

e (kF a, P ) = [1− fdamp(P )] e< (kFa, P ) +

fdamp(P ) e> (kF a, P ) . (20)

3

The damping function is defined as:

fdamp(P ) =1

2tanh [2πCdamp1 (P − Cdamp2)] +

1

2; (21)

the coefficient Cdamp1 = 1.8 determines the steepness,and Cdamp2 = 0.5 is the center of the transition region.In figure 1 we show the equation of state (20), the smalland large polarization limits, together with the MonteCarlo data.

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 0

0.2

0.4

0.6

0.8

e 0 χ-1

kF a

FIG. 2: (color on-line). Energy per volume (circles, left axis)and inverse magnetic susceptibility (squares, right axis) of theunpolarized Fermi gas. Units are the ideal Fermi gas values,3/5ρEF and χ0 = 3ρ/(2EF ), respectively. Solid lines arethe fitting functions (23) in red and (24) in green, while thedashed lines correspond to second order perturbation theory.

At small population imbalance, the equation of stateis quadratic in the polarization. Accordingly, e< can becast in the following form:

e< (kFa, P ) =3

5ρEF

[e0 (kFa) +

5

9P 2χ−1 (kF a)

], (22)

where the energy of the unpolarized gas (in units of35ρEF , where EF = (kF h)

2/2m is the Fermi energy) isgiven by:

e0 (kFa) = 1 + CE1kF a+ CE2(kFa)2 +

CE3(kF a)3 + CE4(kF a)

4, (23)

while χ−1 is the inverse of the magnetic susceptibil-ity divided by the ideal gas result 3ρ/(2EF ), which weparametrize as:

χ−1 (kFa) = 1−Cχ1kFa−Cχ2(kFa)2−Cχ3 (kFa)

3 . (24)

The coefficients CE1 = 0.3536, CE2 = 0.1855,Cχ1 = 0.6366 and Cχ2 = 0.2911 have been deter-mined using second order perturbation theory [9–13],while we obtain CE3 = 0.307(7), CE4 = −0.115(8) andCχ3 = 0.56(1) from a best-fit to the Monte Carlo resultswith zero or small population imbalance (in the range

0

0.4

0.8

1.2

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1

1.05

1.1

1.15

1.2

1.25

A m*

kF↑ a

FIG. 3: (color on-line). Chemical potential at zero concen-tration (circles, left axis) and effective mass (squares, rightaxis) of the repulsive polaron. Units are 3/5EF↑ and bareatomic mass, respectively. Solid lines are the fitting functions(26) in red and (27) in green. The dashed line is the chemicalpotential in second order perturbation theory.

kFa ≤ 1 and P ≤ 0.5). The functions e0(kF a) andχ−1(kF a) are shown in figure 2.

At large population imbalance the behavior of a normalFermi gas is well described by the Landau-PomeranchukHamiltonian [4, 8, 14]. In this approach, the minoritycomponent is regarded as a gas of weakly interactingquasi-particles, the so-called Fermi polarons. The energyper volume takes the following form:

e> (kF↑a, x) =3

5ρ↑EF↑

[1 +A (kF↑a)x

+x5/3

m∗ (kF↑a)+ F (kF↑a)x

2

]; (25)

where x = (1 − P )/(1 + P ) is the concentration ofthe minority component, while the spin-up Fermi mo-mentum and energy are given by the relations kF↑ =

kF (2/(1 + x))1/3 and EF↑ = EF (2/(1 + x))2/3. Thefunction A(kF↑a) gives the polaron chemical potential(devided by 3/5EF↑) at zero concentration and can beparametrized as follows:

A (kF↑a) =5

3

[CA1kF↑a+ CA2 (kF↑a)

2+ CA3 (kF↑a)

3];

(26)second order perturbation theory [14–16] gives the firsttwo coefficients CA1 = 0.4244 and CA2 = 0.2026, whileCA3 = 0.105(2) results from a best-fit to Monte Carlodata for the energy of a single spin-down impurity im-mersed in the Fermi sea of spin-up particles. The po-laron effective mass m∗ is extracted from the disper-sion relation of an impurity with finite momentum. Weparametrize the interaction parameter dependence of this

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SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS23484

effective mass (in units of the bare atomic mass) as:

m∗ (kF↑a) = 1 + Cm∗1 (kF↑a)Cm∗2 (27)

where Cm∗1 = 0.0807(50) and Cm∗2 = 1.59(15). Thefunctions A(kF↑a) andm∗(kF↑a) are shown in fig. 3. Thelast term in equation (25) describes the interactions be-tween polarons. To determine the coefficient F we fitthe Monte Carlo data for a highly imbalanced Fermi gas(x <

∼ 0.5) using the Landau-Pomeranchuck functional,where we insert equations (26) and (27). We assumethat F varies with the square of the interaction param-eter F (kF↑a) = CF (kF↑a)

2, and obtain the coefficient

CF = 0.419(4).To facilitate the implementation of the functional pre-sented in this section we summarize the values of all ofthe coefficients in the following table:

Cdamp1 = 1.8 Cχ1 = 0.6366 Cm∗1 = 0.0807Cdamp2 = 0.5 Cχ2 = 0.2911 Cm∗2 = 1.59CE1 = 0.3536 Cχ3 = 0.56 CF = 0.419CE2 = 0.1855 CA1 = 0.4244CE3 = 0.307 CA2 = 0.2026CE4 = −0.115 CA3 = 0.105

III. QUANTUM MONTE CARLO ALGORITHM

FOR FERMIONS IN OPTICAL LATTICES

To test the accuracy of DFT in optical lattices,we extend our previous Diffusion Monte Carlo studyof the homogeneous system [8]. To simulate Fermigases is optical lattices, we employ the trial wavefunction (19) using the Bloch states (6) (obtained bysolving the equation (3) with a = 0 but finite V (r)) assingle-particle orbitals. The Bloch waves are expandedin a plane-wave basis, as in equation (10), using up to133 states. Differently from the Monte Carlo techniquesfor the single-band Fermi-Hubbard model, which is – inprinciple – reliable only for deep optical lattices, thiscontinuous-space Monte Carlo method allows to simulatealso moderate and shallow lattices. It is analogous toa recent bosonic Monte Carlo method based on theground-state Path-Integral Monte Carlo algorithm,which has been used to perform a continuous-spacesimulation of the superfluid-to-insulator transition ofhard-sphere bosons in optical lattices, going beyond thesingle-band approximation [17].

IV. QUANTUM MONTE CARLO DATA USED

TO EXTRACT THE FUNCTIONAL

Here we summarize the quantum Monte Carlo dataused to obtain the Hartree and exchange-correlation en-ergy function �HXC (ρ↑, ρ↓). Most of the data providedhere have been obtained as part of the work done inreference [8]. Table I contains the data for the polaron

kF↑a A [3/5EF↑] m∗/m0.25 0.208 (1) 1.018 (10)0.40 0.353 (3) 1.019 (6)0.50 0.468 (2) 1.025 (4)0.60 0.591 (2) 1.034 (6)0.70 0.733 (3) 1.047 (9)0.80 0.880 (2) 1.056 (14)0.90 1.038 (3) 1.075 (12)1.00 1.222 (4) 1.069 (10)1.10 1.415 (4) 1.112 (8)1.20 1.631 (4) 1.118 (55)1.25 1.731 (7) 1.083 (16)1.30 1.845 (6) 1.104 (36)1.40 2.073 (5) 1.131 (24)

TABLE I: (a) Polaron chemical potential at zero concentra-tion A, (b) Polaron effective mass m∗/m, vs. interactionstrength kF↑a.

kF a e [3/5ρEF ]0.125 1.0482 (1)0.25 1.1049 (2)0.4 1.1887 (2)0.5 1.2544 (3)0.6 1.3296 (4)0.65 1.3713 (8)0.7 1.4153 (5)0.75 1.4622 (7)0.8 1.5104 (6)0.85 1.5648 (4)0.9 1.6195 (6)1 1.7300 (7)1.1 1.8330 (8)

TABLE II: Energy per volume e vs. interaction strength kFaat P = 0, ie. unpolarized gas.

chemical potential and effective mass shown in figure 3,while table II reports the energy density of the unpolar-ized gas (see figure 2) and table III the energy density ofthe (partially) polarized gas (see figure 1).

[1] Astrakharchik, G. E., Boronat, J., Casulleras, J. &Giorgini, S. Equation of state of a Fermi gas in the BEC-BCS crossover: A quantum Monte Carlo study. Phys.

Rev. Lett. 93, 200404 (2004).[2] Chang, S.-Y., Pandharipande, V. R., Carlson, J. &

Schmidt, K. E. Quantum Monte Carlo studies of su-

perfluid Fermi gases. Phys. Rev. A 70, 043602 (2004).[3] Carlson, J. & Reddy, S. Asymmetric two-component

fermion systems in strong coupling. Phys. Rev. Lett. 95,060401 (2005).

[4] Lobo, C., Recati, A., Giorgini, S. & Stringari, S. Normalstate of a polarized Fermi gas at unitarity. Phys. Rev.

5

Lett. 97, 200403 (2006).[5] Pilati, S. & Giorgini, S. Phase separation in a polarized

Fermi gas at zero temperature. Phys. Rev. Lett. 100,030401 (2008).

[6] Navon, N., Nascimbene, S., Chevy, F. & Salomon, C.The Equation of State of a Low-Temperature Fermi Gaswith Tunable Interactions. Science 328, 729–732 (2010).

[7] Nascimbene, S. et al. Fermi-Liquid Behavior of the Nor-mal Phase of a Strongly Interacting Gas of Cold Atoms.Phys. Rev. Lett. 106, 215303 (2011).

[8] Pilati, S., Bertaina, G., Giorgini, S. & Troyer, M. Itin-erant Ferromagnetism of a Repulsive Atomic Fermi Gas:A Quantum Monte Carlo Study. Phys. Rev. Lett. 105,030405 (2010).

[9] Huang, K. & Yang, C. N. Quantum-mechanical many-body problem with hard-sphere interaction. Phys. Rev.

105, 767–775 (1957).[10] Lee, T. D. & Yang, C. N. Many-body problem in quan-

tum mechanics and quantum statistical mechanics. Phys.Rev. 105, 1119–1120 (1957).

[11] Duine, R. A. & MacDonald, A. H. Itinerant Ferromag-netism in an Ultracold Atom Fermi Gas. Phys. Rev. Lett.95, 230403 (2005).

[12] Recati, A. & Stringari, S. Spin fluctuations, susceptibil-ity, and the dipole oscillation of a nearly ferromagneticfermi gas. Phys. Rev. Lett. 106, 080402 (2011).

[13] Kanno, S. Criterion for the ferromagnetism of hardsphere fermi liquid. ii. Progress of theoretical physics 44,813–815 (1970).

[14] Mora, C. & Chevy, F. Normal Phase of an ImbalancedFermi Gas. Phys. Rev. Lett. 104, 230402 (2010).

[15] Cui, X. & Zhai, H. Stability of a fully magnetized ferro-magnetic state in repulsively interacting ultracold Fermigases. Phys. Rev. A 81, 041602 (2010).

[16] Recati, A. Private communication (2010).[17] Pilati, S. & Troyer, M. Bosonic superfluid-insulator tran-

sition in continuous space. Phys. Rev. Lett. 108, 155301(2012).

P e [3/5ρEF ]kF a=0.25 kF a=0.50 kFa=0.60 kF a=0.65 kF a=0.70 kF a=0.75 kF a=0.80 kF a=0.82 kF a=0.85 kFa=0.90 kF a=1.00

0.000 1.1049 (2) 1.2544 (3) 1.3296 (4) 1.3713 (8) 1.4153 (5) 1.4622 (7) 1.5105 (6) 1.5324 (5) 1.5648 (4) 1.6195 (6) 1.7300 (7)0.061 1.1069 (3) 1.2544 (5) 1.3288 (5) 1.3704 (8) 1.4151 (3) 1.4622 (9) 1.5105 (4) 1.5310 (6) 1.5620 (6) 1.6158 (7) 1.7269 (7)0.121 1.1118 (3) 1.2579 (4) 1.3315 (3) 1.3734 (8) 1.4167 (5) 1.4634 (4) 1.5107 (3) 1.5299 (5) 1.5619 (5) 1.6152 (6) 1.7257 (7)0.182 1.1199 (3) 1.2637 (4) 1.3351 (4) 1.3767 (9) 1.4188 (5) 1.4659 (6) 1.5113 (5) 1.5315 (4) 1.5610 (5) 1.6152 (6) 1.7248 (6)0.242 1.1318 (3) 1.2722 (3) 1.3417 (4) 1.3805 (6) 1.4224 (6) 1.4662 (9) 1.5121 (4) 1.5318 (6) 1.5603 (7) 1.6110 (3) 1.7191 (6)0.303 1.1467 (1) 1.2811 (4) 1.3486 (4) 1.3868 (7) 1.4253 (6) 1.4678 (9) 1.5117 (4) 1.5299 (4) 1.5586 (7) 1.6065 (6) 1.7074 (9)0.364 1.1653 (3) 1.2934 (4) 1.3580 (4) 1.3925 (7) 1.4297 (5) 1.4723 (6) 1.5125 (4) 1.5305 (4) 1.5562 (9) 1.6026 (7) 1.6963 (9)0.424 1.1867 (3) 1.3069 (10) 1.3690 (3) 1.4028 (7) 1.4370 (4) 1.4747 (6) 1.5145 (3) 1.5307 (5) 1.5557 (7) 1.5986 (5) 1.6827 (9)0.485 1.2118 (3) 1.3248 (3) 1.3810 (3) 1.4125 (4) 1.4446 (4) 1.4796 (7) 1.5143 (3) 1.5292 (6) 1.5516 (5) 1.5888 (9) 1.6678 (10)0.545 1.2412 (2) 1.3446 (3) 1.3955 (6) 1.4234 (8) 1.4528 (4) 1.4845 (7) 1.5160 (4) 1.5298 (4) 1.5504 (8) 1.5861 (4) 1.6507 (10)0.606 1.2734 (1) 1.3662 (3) 1.4122 (4) 1.4379 (4) 1.4642 (4) 1.4926 (7) 1.5215 (3) 1.5341 (6) 1.5525 (5) 1.5823 (3) 1.6475 (10)0.667 1.3099 (3) 1.3916 (3) 1.4321 (4) 1.4538 (5) 1.4766 (4) 1.5029 (4) 1.5276 (2) 1.5387 (4) 1.5554 (5) 1.5825 (5) 1.6379 (8)0.727 1.3502 (2) 1.4192 (3) 1.4536 (3) 1.4713 (7) 1.4924 (3) 1.5114 (4) 1.5349 (3) 1.5432 (3) 1.5573 (4) 1.5817 (4) 1.6303 (7)0.788 1.3947 (1) 1.4499 (2) 1.4771 (3) 1.4921 (5) 1.5076 (3) 1.5243 (4) 1.5426 (2) 1.5493 (4) 1.5608 (4) 1.5808 (4) 1.6208 (5)0.848 1.4427 (2) 1.4839 (2) 1.5040 (3) 1.5143 (6) 1.5263 (4) 1.5384 (5) 1.5516 (3) 1.5572 (3) 1.5660 (3) 1.5792 (2) 1.6089 (4)0.909 1.4962 (1) 1.5217 (2) 1.5340 (2) 1.5409 (4) 1.5482 (3) 1.5560 (2) 1.5634 (1) 1.5664 (3) 1.5718 (3) 1.5804 (3) 1.5986 (3)0.970 1.5552 (1) 1.5641 (1) 1.5683 (1) 1.5706 (2) 1.5730 (1) 1.5756 (1) 1.5784 (1) 1.5794 (1) 1.5810 (1) 1.5838 (1) 1.5905 (2)1.000 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0)

TABLE III: Energy per volume e vs. polarization P at various interaction strengths kF a.

© 2012 Macmillan Publishers Limited. All rights reserved.

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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS23484

effective mass (in units of the bare atomic mass) as:

m∗ (kF↑a) = 1 + Cm∗1 (kF↑a)Cm∗2 (27)

where Cm∗1 = 0.0807(50) and Cm∗2 = 1.59(15). Thefunctions A(kF↑a) andm∗(kF↑a) are shown in fig. 3. Thelast term in equation (25) describes the interactions be-tween polarons. To determine the coefficient F we fitthe Monte Carlo data for a highly imbalanced Fermi gas(x <

∼ 0.5) using the Landau-Pomeranchuck functional,where we insert equations (26) and (27). We assumethat F varies with the square of the interaction param-eter F (kF↑a) = CF (kF↑a)

2, and obtain the coefficient

CF = 0.419(4).To facilitate the implementation of the functional pre-sented in this section we summarize the values of all ofthe coefficients in the following table:

Cdamp1 = 1.8 Cχ1 = 0.6366 Cm∗1 = 0.0807Cdamp2 = 0.5 Cχ2 = 0.2911 Cm∗2 = 1.59CE1 = 0.3536 Cχ3 = 0.56 CF = 0.419CE2 = 0.1855 CA1 = 0.4244CE3 = 0.307 CA2 = 0.2026CE4 = −0.115 CA3 = 0.105

III. QUANTUM MONTE CARLO ALGORITHM

FOR FERMIONS IN OPTICAL LATTICES

To test the accuracy of DFT in optical lattices,we extend our previous Diffusion Monte Carlo studyof the homogeneous system [8]. To simulate Fermigases is optical lattices, we employ the trial wavefunction (19) using the Bloch states (6) (obtained bysolving the equation (3) with a = 0 but finite V (r)) assingle-particle orbitals. The Bloch waves are expandedin a plane-wave basis, as in equation (10), using up to133 states. Differently from the Monte Carlo techniquesfor the single-band Fermi-Hubbard model, which is – inprinciple – reliable only for deep optical lattices, thiscontinuous-space Monte Carlo method allows to simulatealso moderate and shallow lattices. It is analogous toa recent bosonic Monte Carlo method based on theground-state Path-Integral Monte Carlo algorithm,which has been used to perform a continuous-spacesimulation of the superfluid-to-insulator transition ofhard-sphere bosons in optical lattices, going beyond thesingle-band approximation [17].

IV. QUANTUM MONTE CARLO DATA USED

TO EXTRACT THE FUNCTIONAL

Here we summarize the quantum Monte Carlo dataused to obtain the Hartree and exchange-correlation en-ergy function �HXC (ρ↑, ρ↓). Most of the data providedhere have been obtained as part of the work done inreference [8]. Table I contains the data for the polaron

kF↑a A [3/5EF↑] m∗/m0.25 0.208 (1) 1.018 (10)0.40 0.353 (3) 1.019 (6)0.50 0.468 (2) 1.025 (4)0.60 0.591 (2) 1.034 (6)0.70 0.733 (3) 1.047 (9)0.80 0.880 (2) 1.056 (14)0.90 1.038 (3) 1.075 (12)1.00 1.222 (4) 1.069 (10)1.10 1.415 (4) 1.112 (8)1.20 1.631 (4) 1.118 (55)1.25 1.731 (7) 1.083 (16)1.30 1.845 (6) 1.104 (36)1.40 2.073 (5) 1.131 (24)

TABLE I: (a) Polaron chemical potential at zero concentra-tion A, (b) Polaron effective mass m∗/m, vs. interactionstrength kF↑a.

kF a e [3/5ρEF ]0.125 1.0482 (1)0.25 1.1049 (2)0.4 1.1887 (2)0.5 1.2544 (3)0.6 1.3296 (4)0.65 1.3713 (8)0.7 1.4153 (5)0.75 1.4622 (7)0.8 1.5104 (6)0.85 1.5648 (4)0.9 1.6195 (6)1 1.7300 (7)1.1 1.8330 (8)

TABLE II: Energy per volume e vs. interaction strength kFaat P = 0, ie. unpolarized gas.

chemical potential and effective mass shown in figure 3,while table II reports the energy density of the unpolar-ized gas (see figure 2) and table III the energy density ofthe (partially) polarized gas (see figure 1).

[1] Astrakharchik, G. E., Boronat, J., Casulleras, J. &Giorgini, S. Equation of state of a Fermi gas in the BEC-BCS crossover: A quantum Monte Carlo study. Phys.

Rev. Lett. 93, 200404 (2004).[2] Chang, S.-Y., Pandharipande, V. R., Carlson, J. &

Schmidt, K. E. Quantum Monte Carlo studies of su-

perfluid Fermi gases. Phys. Rev. A 70, 043602 (2004).[3] Carlson, J. & Reddy, S. Asymmetric two-component

fermion systems in strong coupling. Phys. Rev. Lett. 95,060401 (2005).

[4] Lobo, C., Recati, A., Giorgini, S. & Stringari, S. Normalstate of a polarized Fermi gas at unitarity. Phys. Rev.

5

Lett. 97, 200403 (2006).[5] Pilati, S. & Giorgini, S. Phase separation in a polarized

Fermi gas at zero temperature. Phys. Rev. Lett. 100,030401 (2008).

[6] Navon, N., Nascimbene, S., Chevy, F. & Salomon, C.The Equation of State of a Low-Temperature Fermi Gaswith Tunable Interactions. Science 328, 729–732 (2010).

[7] Nascimbene, S. et al. Fermi-Liquid Behavior of the Nor-mal Phase of a Strongly Interacting Gas of Cold Atoms.Phys. Rev. Lett. 106, 215303 (2011).

[8] Pilati, S., Bertaina, G., Giorgini, S. & Troyer, M. Itin-erant Ferromagnetism of a Repulsive Atomic Fermi Gas:A Quantum Monte Carlo Study. Phys. Rev. Lett. 105,030405 (2010).

[9] Huang, K. & Yang, C. N. Quantum-mechanical many-body problem with hard-sphere interaction. Phys. Rev.

105, 767–775 (1957).[10] Lee, T. D. & Yang, C. N. Many-body problem in quan-

tum mechanics and quantum statistical mechanics. Phys.Rev. 105, 1119–1120 (1957).

[11] Duine, R. A. & MacDonald, A. H. Itinerant Ferromag-netism in an Ultracold Atom Fermi Gas. Phys. Rev. Lett.95, 230403 (2005).

[12] Recati, A. & Stringari, S. Spin fluctuations, susceptibil-ity, and the dipole oscillation of a nearly ferromagneticfermi gas. Phys. Rev. Lett. 106, 080402 (2011).

[13] Kanno, S. Criterion for the ferromagnetism of hardsphere fermi liquid. ii. Progress of theoretical physics 44,813–815 (1970).

[14] Mora, C. & Chevy, F. Normal Phase of an ImbalancedFermi Gas. Phys. Rev. Lett. 104, 230402 (2010).

[15] Cui, X. & Zhai, H. Stability of a fully magnetized ferro-magnetic state in repulsively interacting ultracold Fermigases. Phys. Rev. A 81, 041602 (2010).

[16] Recati, A. Private communication (2010).[17] Pilati, S. & Troyer, M. Bosonic superfluid-insulator tran-

sition in continuous space. Phys. Rev. Lett. 108, 155301(2012).

P e [3/5ρEF ]kF a=0.25 kF a=0.50 kFa=0.60 kF a=0.65 kF a=0.70 kF a=0.75 kF a=0.80 kF a=0.82 kF a=0.85 kFa=0.90 kF a=1.00

0.000 1.1049 (2) 1.2544 (3) 1.3296 (4) 1.3713 (8) 1.4153 (5) 1.4622 (7) 1.5105 (6) 1.5324 (5) 1.5648 (4) 1.6195 (6) 1.7300 (7)0.061 1.1069 (3) 1.2544 (5) 1.3288 (5) 1.3704 (8) 1.4151 (3) 1.4622 (9) 1.5105 (4) 1.5310 (6) 1.5620 (6) 1.6158 (7) 1.7269 (7)0.121 1.1118 (3) 1.2579 (4) 1.3315 (3) 1.3734 (8) 1.4167 (5) 1.4634 (4) 1.5107 (3) 1.5299 (5) 1.5619 (5) 1.6152 (6) 1.7257 (7)0.182 1.1199 (3) 1.2637 (4) 1.3351 (4) 1.3767 (9) 1.4188 (5) 1.4659 (6) 1.5113 (5) 1.5315 (4) 1.5610 (5) 1.6152 (6) 1.7248 (6)0.242 1.1318 (3) 1.2722 (3) 1.3417 (4) 1.3805 (6) 1.4224 (6) 1.4662 (9) 1.5121 (4) 1.5318 (6) 1.5603 (7) 1.6110 (3) 1.7191 (6)0.303 1.1467 (1) 1.2811 (4) 1.3486 (4) 1.3868 (7) 1.4253 (6) 1.4678 (9) 1.5117 (4) 1.5299 (4) 1.5586 (7) 1.6065 (6) 1.7074 (9)0.364 1.1653 (3) 1.2934 (4) 1.3580 (4) 1.3925 (7) 1.4297 (5) 1.4723 (6) 1.5125 (4) 1.5305 (4) 1.5562 (9) 1.6026 (7) 1.6963 (9)0.424 1.1867 (3) 1.3069 (10) 1.3690 (3) 1.4028 (7) 1.4370 (4) 1.4747 (6) 1.5145 (3) 1.5307 (5) 1.5557 (7) 1.5986 (5) 1.6827 (9)0.485 1.2118 (3) 1.3248 (3) 1.3810 (3) 1.4125 (4) 1.4446 (4) 1.4796 (7) 1.5143 (3) 1.5292 (6) 1.5516 (5) 1.5888 (9) 1.6678 (10)0.545 1.2412 (2) 1.3446 (3) 1.3955 (6) 1.4234 (8) 1.4528 (4) 1.4845 (7) 1.5160 (4) 1.5298 (4) 1.5504 (8) 1.5861 (4) 1.6507 (10)0.606 1.2734 (1) 1.3662 (3) 1.4122 (4) 1.4379 (4) 1.4642 (4) 1.4926 (7) 1.5215 (3) 1.5341 (6) 1.5525 (5) 1.5823 (3) 1.6475 (10)0.667 1.3099 (3) 1.3916 (3) 1.4321 (4) 1.4538 (5) 1.4766 (4) 1.5029 (4) 1.5276 (2) 1.5387 (4) 1.5554 (5) 1.5825 (5) 1.6379 (8)0.727 1.3502 (2) 1.4192 (3) 1.4536 (3) 1.4713 (7) 1.4924 (3) 1.5114 (4) 1.5349 (3) 1.5432 (3) 1.5573 (4) 1.5817 (4) 1.6303 (7)0.788 1.3947 (1) 1.4499 (2) 1.4771 (3) 1.4921 (5) 1.5076 (3) 1.5243 (4) 1.5426 (2) 1.5493 (4) 1.5608 (4) 1.5808 (4) 1.6208 (5)0.848 1.4427 (2) 1.4839 (2) 1.5040 (3) 1.5143 (6) 1.5263 (4) 1.5384 (5) 1.5516 (3) 1.5572 (3) 1.5660 (3) 1.5792 (2) 1.6089 (4)0.909 1.4962 (1) 1.5217 (2) 1.5340 (2) 1.5409 (4) 1.5482 (3) 1.5560 (2) 1.5634 (1) 1.5664 (3) 1.5718 (3) 1.5804 (3) 1.5986 (3)0.970 1.5552 (1) 1.5641 (1) 1.5683 (1) 1.5706 (2) 1.5730 (1) 1.5756 (1) 1.5784 (1) 1.5794 (1) 1.5810 (1) 1.5838 (1) 1.5905 (2)1.000 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0) 1.5874 (0)

TABLE III: Energy per volume e vs. polarization P at various interaction strengths kF a.