Abstract
The interplay between interaction and inhomogeneity for electrons in solids generates many interesting phenomena, including insulating and metallic behaviour, magnetism, superconductivity, quantum criticality and more exotic phases^{1}. Many of the same phenomena appear in ultracold fermionic atoms in optical lattices^{2}, which provide clean, controlled and tunable ‘quantum simulators’ to explore the intriguing physics of fermionic systems. Although density functional theory^{3,4,5} (DFT) is widely used to calculate material properties^{6}, it has not yet been applied to cold atomic gases in optical lattices. Here we present a new density functional for shortrange interactions (as opposed to Coulomb interactions of electrons), which renders DFT suitable for atomic Fermi gases. This grants us access to an extensive toolset, previously developed for materials simulations, to calculate the static and dynamic properties of atomic Fermi gases in optical lattices and external potentials. Ultracold atom quantum simulators can in turn be used to explore limitations of DFT functionals, and to further improve hybrid functionals, thus forming a bridge between materials simulations and atomic physics.
Main
Ultracold atomic gases have several advantages over materials for systematically exploring fermionic quantum systems. In materials, strong interactions go hand in hand with increasing localization of electrons as one moves from weakly correlated materials with s and p electrons to transition metal compounds with d electrons, and lanthanides and actinides with even more localized f orbitals. In contrast, all relevant parameters can be almost arbitrarily changed for fermionic gases in optical lattices: increasing the intensity of the optical lattice tunes continuously from shallow lattices with extended wave functions to deep lattices with almost localized Wannier functions, and the band structure can be further modified by changing the laser wave form. The strength of the interatomic interactions can independently be varied by tuning an external magnetic field across Feshbach resonances^{7}. This facilitates a controlled realization of intriguing quantum phenomena in fermionic systems, and the systematic exploration of interaction and localization effects. The simplicity of atomic gases, the absence of core electrons and the ability to realize strongly interacting systems already in the lowest band makes optical lattice systems also much easier to treat from a theoretical point of view, avoiding the need for expensive allelectron calculations including the core electrons or the use of less reliable pseudopotentials incorporating their effects.
So far, the experimental and theoretical focus has been on deep optical lattices, which are described well by singleband Hubbard models and where exciting progress has been achieved. In experiments, a Mott insulating phase has been realized^{8,9}, and the detection of a Néel state seems feasible in the not too distant future. Progress in simulation methods^{10,11} now allows one to accurately simulate the Hubbard model in the same temperature regime^{12}, thus enabling direct comparison between experiments and simulation for the fermionic Hubbard model similar to what has previously been done for bosons^{13}.
Even more intriguing physics than in the Hubbard model is expected in shallow optical lattices, where—in analogy to materials—a rich variety of phases is expected as the bandgap closes and multiband and orbital effects become important. Although shallow lattices are easy to achieve experimentally by reducing the laser amplitude, they pose a challenge to theoretical treatments. As the bandgap is reduced a singleband description fails and there are fundamental problems for deriving reliable multiband Hubbard type models^{14}. We thus propose to investigate this largely unexplored regime using DFT with a new exchange correlation functional for atomic quantum gases. As a first application we will focus on the competition between paramagnetic, ferromagnetic and antiferromagnetic (AF) phases. Specifically, we will show that the ferromagnetic phase of repulsively interacting fermions, which was investigated in experiment^{15} and in simulations^{16}, is substantially enlarged in an optical lattice and a nontrivial phase diagram is obtained owing to band structure effects.
DFT is the workhorse for electronic structure simulations and has been widely and highly successfully used for many decades^{6}. Although there are open challenges in calculating spectral properties of strongly correlated systems, DFT works extremely well in the weakly correlated regime and gives accurate ground state energies and densities also in the presence of strong correlations. Following Hohenberg and Kohn^{3} (HK), the exact groundstate energy E and ground state density ρ can be determined by minimizing an energy functional of the density:
where the first term is the potential energy due to the external potential V (r) at location r. In the context of materials V (r) is the electrostatic potential (pseudopotential) of nuclei (and core electrons), whereas in ultracold atomic gases V (r) is the optical lattice potential (see Methods). The second term is an unknown but universal functional F which includes the interaction and kinetic energies, but does not explicitly depend on V (r). Generalizing to potentially spinpolarized systems one introduces separate densities of the two spin components ρ_{↑} and ρ_{↓}:
An explicit (but rather inaccurate) expression for F[ρ_{↑},ρ_{↓}] is given by the HKlocal spin density approximation^{3} (LSDA)
where ε(ρ_{↑},ρ_{↓}) is the groundstate energy density of a homogeneous Fermi gas with the given spin densities. The Thomas–Fermi approximation is recovered as a meanfield approximation for ε(ρ_{↑},ρ_{↓}). A better approximation is to use the results of a quantum Monte Carlo (QMC) simulation with the given density (seeMethods). Although in the quantum gases community this is referred to as the ‘local density approximation’ (LDA), we will explicitly call it the HKLSDA, to avoid confusion with the common use of ‘LDA’ in electronic structure calculations.
The kinetic part is usually highly nonlocal and cannot be treated well under the local approximation. Therefore, Kohn and Sham^{4} (KS) proposed a more accurate functional by explicitly including the exact kinetic energy T_{0} of noninteracting fermions. What is left is the interaction energy E_{HXC}, combining the usual Hartree (meanfield term) E_{H} and the exchange–correlation correction E_{XC}:
A simple yet often reliable treatment of E_{HXC} is the LSDA
where the functional is replaced by an integral over the interaction energy of a uniform system with the same local density. For contact interactions, and unlike the Coulomb case, the Hartree term depends only on the local densities, and we can thus combine it with the exchange–correlation terms into one interaction energy term ε_{HXC}. We will refer to this functional explicitly as the KSLSDA functional.
The LSDA functionals have been obtained using fixednode diffusion Monte Carlo simulations, similar to previous calculations for the 3D electron gas^{17}. In the Supplementary Information we report an explicit parametrization of this functional which accurately fits the QMC results for the energy density and reproduces the known behaviour in the limiting cases of small coupling, and small and large polarization. By imposing stationarity of the functional equation (1) with respect to variations of the densities ρ_{↑} and ρ_{↓} one obtains a set of coupled Schrödingertype equations—the KS equations (see Supplementary Information)—of an effective noninteracting system. From the eigenstates and eigenvalues of the KS equations one can compute the density profiles, the groundstate energy and the band structure.
As a first application we consider a repulsive Fermi gas in shallow and moderately deep optical lattices. Repulsive interactions are realized in experiments when the gas is prepared in the excited branch of the Feshbach resonance. We parametrize the interactions by the ratio a/d of the scattering length a and the lattice spacing d, and the density ρ in units of the band filling n=ρ d^{3}. In Fig. 1 we compare KSLSDA results to direct QMC simulations in an optical lattice. The excellent agreement demonstrates that DFT calculations with a LSDA functional are reliable in weak and moderate optical lattices. To compare HKLSDA and KSLSDA, we first show, in Fig. 2, the density of states at halffilling (n=1) in a moderately deep optical lattice with intensity V_{0}=3E_{R}. Here, interactions lead to a partial polarization of the system, seen as a shift between the density of states of the two components: there are more spinup than spindown atoms if we fill the system to the Fermi level. The cruder HKLSDA approximation (green line), on the other hand, predicts zero polarization, and misses the existence of a bandgap that is clearly visible in both the free fermion and KSLSDA results.
Calculating the groundstate polarization for a range of lattice depths V_{0}, band fillings n, and interaction strengths a/d we obtain the phase diagrams of Fig. 3. In a shallow lattice with V_{0}=0.5E_{R} (Fig. 3a) we see three phases: a paramagnetic phase at weak interactions (white), partially polarized (shown as pink gradations), and fully polarized (ferromagnetic, shown in solid red). The phase boundaries in this shallow lattice are similar to those of the homogeneous system V_{0}=0 (ref. 16), indicated by the green and blue lines. In deeper optical lattices (V_{0}=2E_{R} in Fig. 3b and V_{0}=4E_{R} in Fig. 3c) polarization sets in at much weaker interactions, indicating that the optical lattice strongly favours ferromagnetism.
We can see two prominent features due to the presence of an optical lattice. The first is the much bigger extent of the polarized phases, which is due to the higher local density at the potential minima in the optical lattice and increases the local density beyond the critical value for polarization. Comparing Fig. 3c and d we see that including the accurate kinetic energy in the KSLSDA this effect is even stronger than in the simpler HKLSDA approximation. A second striking effect is the nonmonotonic behaviour of the phase boundary: there is a large fully polarized region at densities up to halffilling (n≤1), which rapidly shrinks at higher filling. This phenomenon is due to band structure effects and a gap between upspin and downspin subbands. It is thus completely absent in the HKLSDA approximation (Fig. 3d), which ignores band structure effects.
Thus we next calculate the detailed band structure of the interacting system, shown in the left panels of Fig. 4 for a weak optical lattice (V_{0}=2E_{R}) without a bandgap and on the right for a moderate optical lattice with a bandgap (V_{0}=4E_{R}). Weak interactions (a=0.04d) change the band structure only slightly. Increasing the interaction to a=0.08d (second row) we find a partially polarized state in the deeper lattice: the two spin subbands split and the band structure is substantially changed. At even stronger interaction a=0.16d (third row) the gas is partially polarized also in the shallower lattice, and becomes fully polarized in the deeper lattice. Note that here the fermions are fully polarized up to half band filling n=1, as only the upspin subband gets occupied. Notice also that in the fully polarized state the first band is fully occupied and the system is insulating owing to the gap between the first and second subbands. Filling the bands further puts fermions in the next band with opposite spin, resulting in a partially polarized state. This explains the sharp feature around n=1 in the phase diagram in Fig. 3c. To recover full polarization for n>1 one needs to increase either the interaction strength or lattice depth to push the energy of the lowest downspin subband above the second upspin subband.
To see antiferromagnetism competing with ferromagnetism at half band filling n=1 we need to consider a unit cell consisting of two lattice sites, and compare the energies of antiferromagnetic and uniform configurations. We find, as shown in Fig. 5, that antiferromagnetic ordering is preferred at intermediate interaction strengths and half band filling, matching with the single band Hubbard model physics, which becomes valid in the upper lefthand corner of the phase diagram shown.
There are many further applications of DFT for atomic gases. Already in this simple system we have seen striking effects, such as substantially enhanced ferromagnetism and strikingly nonmonotonic behaviour of the phase boundaries, which is not present in a simple HKLSDA approximation.
Hybrid functionals, which are successfully used in electronic structure calculations, can be adapted to atomic gases based on our LSDA functional. Examples are Hedin’s GW method^{18}, the LDA+U method^{19}, which combines LSDA with a Hartree–Fock approximation, and the LDA+DMFT (ref. 20), which combines an LSDA functional with a dynamical mean field theory^{21} (DMFT) treatment of the correlated orbitals.
Another extension is functionals for attractive interactions using the local pair density approximation^{22}, which has been developed to describe superconducting electronic systems, including not only the fermionic densities ρ_{↑} and ρ_{↓}but also a density for paired fermions ρ_{p}, or the superfluid LDA method of Bulgac^{23}. Potentially most useful are generalizations of DFT to finite temperatures^{24} (see also refs 25, 26 for a recent review) and using timedependent DFT (ref. 27) to study thermal and nonequilibrium properties of quantum gases.
DFT can thus form an important bridge between atomic physics and materials simulation, much stronger than the Hubbard model, which is at the centre of attention now but relevant only to a small class of materials. Innovations in DFT for materials science provide valuable tools for the investigation of the intriguing physics of quantum gases. In return, atomic gases provide an ideal test bed to address the challenges faced in the simulation of correlated fermionic systems and will help to further improve functionals for strongly correlated systems—thus realizing the promise of optical lattice quantum simulators to be a useful tool for materials science.
Methods
Fermions in an optical lattice.
In the presence of an optical field created by three pairs of counterpropagating laser beams, the neutral fermions experience a potential due to dipolefield interaction, that is , where λ is the laser wavelength and V_{0} the laser intensity, conveniently expressed in units of the recoil energy E_{R}=(ℏ^{2}/2m)(2π/λ)^{2} (ℏis the reduced Planck constant and m is the atomic mass). Here, fermions with like spins follow the Pauli exclusion principle and those with unlike spins scatter with the scattering length a, in units of the lattice spacing d=λ/2.
Solving the KS equations.
Owing to cubic translational symmetry, the coupled spinup and spindown KS Hamiltonians are diagonalized selfconsistently via the Bloch ansatz in a 11^{3} plane wave basis, with 20^{3}kpoints in the first Brillouin zone of the simple cubic lattice. To observe antiferromagnetism, the unit cell has to be doubled, and the doubled unit cells form a facecentred cubic lattice. Details of the KS equations in this basis are presented in the Supplementary Information.
Calculation of the LSDA functional.
To determine the LSDA functional we have used the the fixednode diffusion Monte Carlo method^{28} to calculate the groundstate energy of a homogeneous Fermi gas. We simulate a twocomponent gas with repulsive interspecies interactions (the swave scattering between identical particles is inhibited by the Pauli exclusion principle), modelling the interaction by the hardsphere potential , if r<a, and 0otherwise. The diameter of the sphere a is equal to the swave scattering length. Although this model neglects the effect of the lowenergy bound states of the true interatomic potential, it represents an improvement with respect to the conventional description of atomic gases in optical lattices, which is based on the Born approximation^{14}. The role of nonuniversal corrections, which depend on details of the potential beyond a, have been analysed in ref. 16 by employing different models with the same scattering length. In the physically relevant regime (k_{F}a≲1, with a Fermi wave vector k_{F}=(3π^{2}ρ)^{1/3}) these nonuniversal corrections are very small (<3%). As in previous work^{16,29}, we fix the nodal surface of the groundstate using a Jastrow–Slater trial wave function, using the Jastrow factor of the exact solution of the twobody problem. The two Slater determinants (one for the spinup and one for the spindown particles) are filled with the singleparticle eigenstates, namely the plane waves or the Bloch functions (up to the Fermi energy) for simulations in free space or in the optical lattice, respectively. We use up to 81 particles per spin species and up to 6^{3} lattice sites, and we find that the finitesize effects are negligible if one follows Fermi liquid theory^{30} and includes the finitesize correction terms of a noninteracting Fermi gas.
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Acknowledgements
We acknowledge useful discussions with J. Hutter, L. Pollet, T. Schulthess andJ. van de Vondele. This work was supported by the Swiss National Science Foundation, the National Competence Centers in Research (NCCRs) ‘Materials with Novel Electronic Properties’ (MaNEP) and ‘Quantum Science and Technology’ (QSIT) and by a grant from the Army Research Office through the DARPA Optical Lattice Emulator (OLE) program.
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M.T. and X.D. developed the DFT for ultracold quantum gases. S.P. performed the quantum Monte Carlo simulations. S.P. and M.T. extracted the LSDA functional.P.N.M. performed the DFT calculations. All authors contributed to writing the manuscript and the results presented.
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Ma, P., Pilati, S., Troyer, M. et al. Density functional theory for atomic Fermi gases. Nature Phys 8, 601–605 (2012). https://doi.org/10.1038/nphys2348
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DOI: https://doi.org/10.1038/nphys2348
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