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arXiv:2104.03208 · Physical Review Research

Machine learning universal bosonic functionals

Jonathan Schmidt, Matteo Fadel, Carlos L. Benavides-Riveros

Published Sep 13, 2021• 22 citations
arXivHTMLPDF

Abstract

The one-body reduced density matrix $γ$ plays a fundamental role in describing and predicting quantum features of bosonic systems, such as Bose-Einstein condensation. The recently proposed reduced density matrix functional theory for bosonic ground states establishes the existence of a universal functional $\mathcal{F}[γ]$ that recovers quantum correlations exactly. Based on a novel decomposition of $γ$, we have developed a method to design reliable approximations for such universal functionals: our results suggest that for translational invariant systems the constrained search approach of functional theories can be transformed into an unconstrained problem through a parametrization of an Euclidian space. This simplification of the search approach allows us to use standard machine-learning methods to perform a quite efficient computation of both $\mathcal{F}[γ]$ and its functional derivative. For the Bose-Hubbard model, we present a comparison between our approach and Quantum Monte Carlo.

DOI
10.1103/physrevresearch.3.l032063

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The one-body reduced density matrix γγγ plays a fundamental role in describing and predicting quantum features of bosonic systems, such as Bose-Einstein condensation. The recently proposed reduced density matrix functional theory for bosonic ground states establishes the existence of a universal functional F[γ]\mathcal{F}[γ]F[γ] that recovers quantum correlations exactly. Based on a novel decomposition of γγγ, we have developed a method to design reliable approximations for such universal functionals: our results suggest that for translational invariant systems the constrained search approach of functional theories can be transformed into an unconstrained problem through a parametrization of an Euclidian space. This simplification of the search approach allows us to use standard machine-learning methods to perform a quite efficient computation of both F[γ]\mathcal{F}[γ]F[γ] and its functional derivative. For the Bose-Hubbard model, we present a comparison between our approach and Quantum Monte Carlo.
Based on a generalization of Hohenberg-Kohn's theorem, we propose a ground state theory for bosonic quantum systems. Since it involves the one-particle reduced density matrix γ as a variable but still recovers quantum correlations in an exact way it is particularly well suited for the accurate description of Bose-Einstein condensates. As a proof of principle we study the building block of optical lattices. The solution of the underlying v-representability problem is found and its peculiar form identifies the constrained search formalism as the ideal starting point for constructing accurate functional approximations: The exact functionals F[γ] for this N-boson Hubbard dimer and general Bogoliubov-approximated systems are determined. For Bose-Einstein condensates with N_{BEC}≈N condensed bosons, the respective gradient forces are found to diverge, ∇_{γ}F∝1/sqrt[1-N_{BEC}/N], providing a comprehensive explanation for the absence of complete condensation in nature.
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Density functional theory constitutes the workhorse of modern electronic structure calculations due to its favourable computational cost despite the fact that it usually fails to describe strongly correlated systems. A particularly promising approach to overcome those difficulties is reduced density matrix functional theory (RDMFT): It abandons the complexity of the NNN-particle wave function and at the same time explicitly allows for fractional occupation numbers. It is the goal of this thesis to initiate and establish a bosonic RDMFT for both ground state and excited state energy calculations. Motivated by the Onsager and Penrose criterion which identifies RDMFT as a particularly suitable approach to describe Bose-Einstein condensates (BECs), we derive the universal functional for a homogeneous BEC in the Bogoliubov regime. Remarkably, the gradient of the universal functional is found to diverge repulsively in the regime of complete condensation. This introduces the new concept of a BEC force, which provides a universal explanation for quantum depletion since it is merely based on the geometry of quantum states. In the second part of the thesis, we propose and work out an ensemble RDMFT targeting excitations in bosonic quantum systems. This endeavour further highlights the potential of convex analysis for the development of functional theories in the future. Indeed by resorting to several concepts from convex analysis, we succeeded to provide a comprehensive foundation of w\boldsymbol{w}w-ensemble RDMFT for bosons which is further based on a generalization of the Ritz variational principle and a constrained search formalism. In particular, we solve the emerging NNN-representability problem leading to a generalization of Pauli's famous exclusion principle to bosonic mixed states.
We initiate the recently proposed w w -ensemble one-particle reduced density matrix functional theory ( w w -RDMFT) by deriving the first functional approximations and illustrate how excitation energies can be calculated in practice. For this endeavour, we first study the symmetric Hubbard dimer, constituting the building block of the Hubbard model, for which we execute the Levy-Lieb constrained search. Second, due to the particular suitability of w w -RDMFT for describing Bose-Einstein condensates, we demonstrate three conceptually different approaches for deriving the universal functional in a homogeneous Bose gas for arbitrary pair interaction in the Bogoliubov regime. Remarkably, in both systems the gradient of the functional is found to diverge repulsively at the boundary of the functional’s domain, extending the recently discovered Bose-Einstein condensation force to excited states. Our findings highlight the physical relevance of the generalized exclusion principle for fermionic and bosonic mixed states and the curse of universality in functional theories.
From drug design to material science, computational chemistry has become a valuable tool for many fields of applications in the past decades. Quantum chemistry in particular, can provide accurate descriptions of molecules and solids by relying on principles of quantum mechanics with the Schrödinger equation at its core. Unfortunately this equation becomes prohibitively expensive to solve for systems made of more than a few electrons and one has to rely on other approaches. Among the most popular is Density Functional Theory, relatively inexpensive and exact in principle. One has, however, to introduce approximations in practice, deteriorating its accuracy for challenging, so-called strongly correlated, systems. On the contrary, other methods like CASSCF remain reliable for these systems, but although cheaper than the solution of the Schrödinger equation, remain too expensive for extended systems. There is therefore room for methods in between, that could handle large correlated systems with reasonable accuracy. One such method is the One Body Reduced Density Matrix (1-RDM) Functional Theory (RDMFT), based on the fact that an electronic system can, in principle exactly, be described by its 1-RDM only. Although its theory dates back to the 1970s, computational hindrances have prevented this method from shining so far. Indeed, computations in RDMFT rely on a constrained optimization of the eigenvalues and eigenvectors of the 1-RDM, that can be reformulated as an unconstrained optimization by parameterizing the latter but in both cases suffers from convergence problems. In this thesis, the role of this parametrization is analyzed and motivates a now second order optimization algorithm, which improves the convergence speed by two orders of magnitude w.r.t. the state of the art algorithm. This understanding of the problem also inspires an alternative approach with the 1-RDM as the optimization variable. This however requires the development of new approximations and a first investigation in that direction, based on Machine Learning is performed. Lastly, a generalization of the method to excited states is attempted, highlighting some key computational challenges.
We revisit the machine-learning (ML) approach to the universal density functional F[n]F[\mathbf{n}]F[n] of the one-dimensional Hubbard model with a site-dependent random potential v={vi}\mathbf{v}=\{v_{i}\}v={vi​}. We generate exact ground-state data via exact diagonalization for a periodic chain with L=8L=8L=8 in the paramagnetic sector (N↑,N↓)=(2,2)(N_\uparrow,N_\downarrow)=(2,2)(N↑​,N↓​)=(2,2), with site electron densities ni=ni↑=ni↓n_{i} = n_{i\uparrow}=n_{i\downarrow}ni​=ni↑​=ni↓​. The resulting density-potential dataset is analyzed. Using principal component analysis of the joint feature space (n,v)(\mathbf n,\mathbf v)(n,v), we identify the intrinsic low-dimensional structure of the data. Then, we restricted the study of the dataset with an energy-based filtering criterion to concentrate the data around weakly perturbed energy values with zero potential. A compact one-dimensional convolutional neural network is trained to learn the universal functional considering the lattice periodicity through unilateral wrapping and enforce the lattice symmetries by data augmentation (translations and mirror reflections), achieving near-exact predictions of F[n]F[\mathbf n]F[n]. Finally, we address the fact that accurate functional values do not necessarily imply accurate functional derivatives. By augmenting training with a variational consistency term that constrains the Euler-Lagrange relation between ∂F/∂ni\partial F/\partial n_i∂F/∂ni​ and the gauge-fixed potential we reconstruct the external potentials from automatic differentiation. These results clarify the roles of dataset geometry, symmetry, gauge fixing, and derivative-based constraints in learning physically consistent density functionals.