Orbital-Free Quasi-Density Functional Theory

Wigner functions are broadly used to probe non-classical effects in the macroscopic world. Here we develop an orbital-free functional framework to compute the 1-body Wigner quasi-probability for both fermionic and bosonic systems. Since the key variable is a quasi-density, this theory is particularly well suited to circumvent the problem of finding the Pauli potential or approximating the kinetic energy in orbital-free density functional theory. As proof of principle, we find that the universal functional for the building block of optical lattices results from a translation, a contraction, and a rotation of the corresponding functional of the 1-body reduced density matrix, indicating a strong connection between these functional theories. Furthermore, we relate the concepts of Wigner negativity and $v$-representability, and find a manifold of ground states with negative Wigner functions.

Introduction.-Detecting and understanding quantum features at the macroscopic level is one of the main theoretical and technological challenges of modern quantum sciences.Nowadays, state-of-the-art experiments can directly observe non-classical behavior (as quantum superposition) in systems with a truly macroscopic number of particles, with as many as 10 16 atoms [1][2][3][4].A powerful theoretical and computational strategy to detect that quantumness is by directly measuring the system's corresponding Wigner function.Although normalized to unity, Wigner functions are quasi-probability distributions that can take negative values, a phenomenon that has no classical counterpart.Hence, negativity in the Wigner functions has been linked to non-classical features of quantum states and is considered a distinctive signature of quantum entanglement [5][6][7], contextuality [8][9][10], quantum computation [11], quantum steering [12,13], or even quantum gravity [14].
Due to the exponentially large Hilbert spaces of quantum many-body systems, finding the corresponding Wigner function is, in general, a computationally prohibitive task.Yet for identical particles it is possible to circumvent the Hilbert space's exponential growth by means of a universal functional of certain reduced, more manageable, quantities, like, e.g., the density.Based on the important observation that electronic systems are fully determined by the ground-state density [15], density functional theory (DFT) is a prominent methodology in electronic structure calculations, with applications ranging from quantum chemistry and material science [16,17] to self-driving labs [18].Quite remarkably, orbital-free DFT achieves a computational linear scaling with the system size [19].But, unfortunately, from the density alone it is not possible to reconstruct the Wigner function, and therefore standard DFT is, in general, not suitable for describing non-classical features of quantum many-body systems.
A recent phase-space formulation of DFT employs, as the central variable, the 1-particle Wigner quasi-density, which is in a one-to-one correspondence with the respective ground state for interacting many-fermion/boson systems [20].Its main feature is that the 1-body Wigner function can be accessed directly, without pre-computing the full wave function.This Wigner quasi-density functional theory (quasi-DFT) is a promising theoretical tool to model many-body problems while accounting for nonclassical features, strong interactions, and quantum correlations, with the same computational cost as standard DFT.As we will show below, the theory has also the potential of bypassing well-known problems of orbital-free DFT.To date, however, there are neither orbital-free nor orbital-dependent functionals for quasi-DFT.
Here, we will obtain equations for the fermionic/bosonic 1-particle quasi-density.This is, we will argue, the initial step to developing a full orbital-free framework for Wigner quasi-DFT.As one of our main results, we will show that ω(r, p), the 1-particle Wigner quasi-density, satisfies the following, effective, eigen-equation: where h eff = 1 2 p 2 +v ext (r)+v eff (r, p), v ext (r) is the external potential, v eff (r, p) is certain effective potential that we introduce below, and the symbol ⋆ is the so-called star product of phase-space quantum mechanics.
The letter is organized as follows: First, we review both the orbital-free functional theories and the Wigner formulation of DFT.Second, we derive an Euler-Lagrange equation for the 1-body Wigner quasi-density.Next, we derive an equation using the Moyal product.We then employ the Hubbard model to present for the first time a functional realization of quasi-DFT.We conclude with a summary and discuss some implications of our results.In the Appendixes, we provide additional technical details.
Functional theories.-Theenormous success of DFT in electronic structure calculations is mainly due to the existence of a set of self-consistent 1-particle equations that allow for the computation of the density from 1particle orbitals [21].Although it is much cheaper than wave-function methods, this Kohn-Sham DFT still has an unfavorably computational scaling with the cube of the number of electrons [22].In turn, orbital-free DFT allows a much more favorable, linear scaling with the system size [17,19], but this computational advantage is counterbalanced by the fact that the quantum mechanical kinetic energy functional is unknown, and it is written as a classical, approximate function of the electron density.A parallel intellectual effort is the 1-particle reduced density matrix functional theory (1-RDMFT) that exploits the full 1-particle picture of the many-body problem by seeking a universal functional of the 1-body reduced density matrix (1-RDM), [23][24][25], for fermionic [26][27][28], bosonic [29][30][31][32], or relativistic [33] interacting particles.Similar to DFT, 1-RDMFT is based on a one-to-one correspondence between the ground state and its corresponding 1-RDM.Although this theory is in a better position than DFT to tackle strong correlations [34], its broad use has been hampered by the absence of Kohn-Sham-like equations for the natural orbitals (i.e., the eigenvectors of the 1-RDM) [25,35].It, therefore, comes as no surprise that fermionic 1-RDMFT is computationally much more demanding than DFT [36].Unfortunately, there are quite a few orbital-free formulations of 1-RDMFT (the most notable being the exchange part of the Hartree-Fock functional).The development of an orbital-free perspective of 1-RDMFT could boost its broad applicability.
Phase-space quantum mechanics.-Inthe phase-space formulation of quantum mechanics, observables are represented by symbols, i.e., functions of position r and momentum p coordinates.Out of many choices, Wigner functions host the most natural representation of quantum mechanics [37].In the classical limit, it turns out to be the phase-space distributions of statistical mechanics [38].In this formulation, quantum operators correspond uniquely to phase-space classical functions via the Weyl correspondence, while operator products correspond to ⋆products.This noncommutative star (twisted or Moyal) product is commonly defined by the phase-space pseudodifferential operator [39]: ; the arrows denote that a given derivative acts only on a function standing on the left/right.This product is defined by 1-body Wigner quasi-density.-Bydefinition, the 1body Wigner quasi-densities are given in terms of the 1-RDM γ(r, σ; r ′ , σ ′ ), by the relation: where σ ∈ {↑, ↓} are the spin variables.Notice that the marginal σ ω σσ (r, p) d 3 p gives exactly the density n(r) (the central object of DFT) [42].
Wigner quasi-DFT.-Ageneralization of the Hohenberg-Kohn [15] and Gilbert [23] theorems to Hamiltonians of the form H = h + V , with a fixed two-particle interaction V , proves the existence of a universal Wigner functional F V [ω] of the 1-body quasi-density ω [20].Indeed, for any choice of the 1-particle phase-space Hamiltonian h(r, p) = 1 2 p 2 + v ext (r, p) the energy functional: is bounded from below by the exact ground-state energy.The equality in Eq. ( 3) holds exactly when E[ω] is evaluated using the ground-state 1-body quasi-density ω gs .As in standard DFT, the functional F V [ω] is completely independent of any external (phase-space) potential v(r, p).As in 1-RDMFT, it is also completely independent of the kinetic energy and depends only on the fixed two-particle interaction V .The (universal) functional F V [ω] obeys a constrained-search formulation, by considering only many-body wave-functions that integrate to the same ω: While the functional is unknown, it is known that it has some better scaling properties than the functionals in DFT.For instance, by defining, [20], a fact that lies in the exact knowledge of the kinetic energy functional.
Representability condition of ω.-In a quite natural way, 1-body quasi-densities inherit the representability conditions of the 1-RDM, γ.Due to unitary invariance, those can be expressed as conditions on the eigenvalues of γ [43].Therefore, it is convenient to use the spectral representation of ω (i.e., ω = i n i f i ) to find its representability conditions.In general, n i ≥ 0. In addition, in the case of fermions: which is just a consequence of the Pauli exclusion principle [44].
Equation for the 1-body quasi-density.-Wenow exhibit an exact equation for the phase-space 1-body quasidensity.Let E[ω] be the energy functional of the Wigner function (3), subject to the constraint dΩ ω(r, p) = N .The N -particle phase-space density which minimizes such a functional is found by applying a functional derivative of the Lagrangian E[ω]−µN with respect to ω, yielding the Euler-Lagrange equation of Wigner quasi-DFT: There is an important consequence of this result.As is well known, one of the central problems in orbital-free DFT is approximating the kinetic energy functional in terms of the density [45,46] or, alternatively, the Pauli potential [47,48].It is, indeed, particularly crucial that the Pauli principle be captured precisely in the kinetic energy.As we can see in Eq. ( 6), this important problem is completely absent in the phase-space formalism.First, the kinetic energy and the external potential are exact, rather simple phase-space functions, and no approximation is needed.Second, the representability condition of the Wigner function (5) guarantees that the Pauli principle is fulfilled.As a consequence, our orbital-free quasi-DFT needs only to approximate the universal functional ⋆-eigenequation for ω.-Inspired by the work of Levy, Perdew and Sahni [49] we exhibit now an exact ⋆-eigenequation for the 1-particle quasi-density.As explained in Appendix C, by computing the directional functional derivative of E[ω] at the point ω in the direction of ω one can show that ω gs fulfills the following equation: where The simplicity of this formula can be compared with the one from orbital-free DFT for n(r) [50].Noticeably, the formula ( 7) allows for a Wigner-Moyal expansion of the equation for the quasi-density: -To the best of our knowledge, there are no explicit functionals of ω.Although 1-RDMFT functionals could be Wigner transformed, almost all of them are written in terms of natural orbitals [51][52][53][54][55][56][57][58][59][60], so they are not suited for our purposes.Let us, therefore, illustrate the potential of orbital-free quasi-DFT by discussing the generalized Bose-Hubbard dimer, whose standard version has been broadly used to unveil aspects of functional theories [29,[61][62][63][64][65].The interacting Hamiltonian, containing all particle-conserving quartic terms, can be written with 3 parameters in the following way: where b † j , b j and nj are the creation, anhilitation and particle-number operators on the left/right sites j ∈ {l, r}.Normalizing to 1 and assuming real-valued matrix elements, the 1-RDM can be represented in the lattice-site basis |l⟩, |r⟩ as where ⃗ γ = (γ lr , 0, γ ll − 1 2 ), ⃗ σ = (σ x , σ y , σ z ) is the vector of Pauli matrices, and γ ij = ⟨i|γ|j⟩.
To write the corresponding (discrete) Wigner transformation we follow Refs.[66][67][68][69] where the Wigner function is represented on a grid of twice the dimension of the underlying Hilbert space {j, n}.For the momentum basis, we choose the one in which the hopping term of the Hubbard Hamiltonian is diagonal: The 1-body Wigner quasi-density can now be computed: As it should be, the marginal densities are recovered by the partial sums: n ω j,n = γ jj and j ω j,n = γnn , where γnn is the momentum density.Since ω r,1 = 1 2 −ω l,0 and ω r,0 = 1 2 − ω l,1 , we take ω l,0 and ω l,1 as our two degrees of freedom.It is straightforward to check that the representability condition reads: Since this is the equation of a disk of radius 1/ √ 8 centered in ( 14 , 1 4 ), one can parameterize the discrete Wigner function with a radius and an angle, namely, ω l,0 (R, ϕ) In Fig. 1 are presented two different realizations of the Hamiltonian (8) for both 1-RDMFT and quasi-DFT.It can be seen that the functional of quasi-DFT results from the respective 1-RDMFT functional after a translation, a contraction, and a rotation of 45°.This result seems to be general for lattice systems, as indicated in Appendix A. After applying Eq. ( 6) (or a discrete version of ( 7)) one can find ω gs for specific values of t (the strength of the hopping term) and v l − v r (the external potential).
Negativity & v-representability.-Quiteremarkably, the quasi-DFT presented here can relate two important concepts in Wigner and functional theories: Wigner negativity and v-representability: Which Wigner-negative 1body quasi-densities come from ground states?We answer explicitly this question for 2 and 3 bosons for the standard Bose-Hubbard dimer in Fig. 2: There are 4 disconnected ground-state regions of Wigner negativity!Relating these two important concepts seems to be new in the literature.
Conclusion.-Unveiling the role of quantum effects at the classical level is a crucial problem for developing quantum technologies.The Wigner quasi-probability is usually employed as a probe of such quantumness.This letter showed that ω(r, p), the (fermionic or bosonic) 1body Wigner quasi-density, can be inserted into a functional-theoretical framework in an orbital-free manner.
By providing an Euler-Lagrange equation and a Wigner-Moyal eigen-equation, we showed that ω(r, p) can be computed as a stationary point without referring to orbital equations, circumventing some known problems of orbital-free DFT (e.g., approximating the kinetic energy functional or finding the Pauli potential).In this framework, the concepts of Wigner negativity and vrepresentability can be related.We would like to emphasize that one of the most important aspects of our results is that the ⋆-product gives a very rich structure for extracting the corresponding 1-particle Wigner function.In that sense, quasi-DFT is a functional theory that can connect directly with DFT and with semiclassical expansions of the many-body problem.There are several potential research directions from our results: First, one could develop machine learning quasi-DFT functionals which is now current practice for standard DFT [70][71][72][73].Second, since Wigner negativities carry important quantum information it will be interesting to see what information they can unveil for electronic correlations [74,75] or fermionic entanglement [76].Finally, it could be quite promising to tackle -within this orbital-free framework-quantum excitations in the same spirit of state-average calculations [77] or the recently formulated w-1-RDMFT [78].

ACKNOWLEDGMENTS
I gratefully thank Luis Colmenarez, Julia Liebert, Eli Kraisler, and Jeff Maki for insightful discussions and for providing important comments on the manuscript, and acknowledge the European Union's Horizon Europe Research and Innovation program under the Marie Sk lodowska-Curie grant agreement n°101065295.I also thank Ana Maria Rey and the warm atmosphere of her group at JILA where this paper took its final shape.Here we apply the discrete Wigner formalism to the Hubbard model of L sites.This is defined in the L-dimensional Hilbert space H L whose position basis is S = {|1⟩, ..., |L⟩}.Another orthonormal basis for the same Hilbert space is {|ϕ 0 ⟩, ..., |ϕ L−1 ⟩}, defined by the Fourier transform: with ϕ m = 2π L m.The set of pairs {n, ϕ m } n,m constitutes a L × L grid.This is the phase space Γ L associated with the Hilbert space H L [67].
The map between f (n, ϕ m ), a real function in Γ L , and f , an operator in H L , is realized by means of the following relation: Eq. (A5) can now be used to find the Wigner quasi-distribution.Since the average value of the observable represented by the operator f in a state defined by the density operator γ reads a natural definition for the Wigner quasi-probability (for the kernel arises: ω(n, ϕ m ) = Tr γ Ωκ (n, ϕ m ) .From this definition, one can write: where Since γ ll + γ rr = 1 (by normalization) and γ lr = γ rl the 1RDM is fully determined by two free parameters.We represent the 1-RDM as: The only two degrees of freedom of this matrix can be represented in a vector form: |γ⟩⟩ = γ ll γ lr .