We introduce a machine-learning density-functional-theory formalism for the spinless Hubbard model in one dimension at both zero and finite temperature. In the zero-temperature case this establishes a one-to-one relation between the site occupation and the total energy, which is then minimized at the ground-state occupation. In contrast, at finite temperature the same relation is defined between the Helmholtz free energy and the equilibrium site occupation. Most importantly, both functionals are semilocal, so that they are independent from the size of the system under investigation and can be constructed over exact data for small systems. These ``exact'' functionals are numerically defined by neural networks. We also define additional neural networks for finite-temperature thermodynamical quantities, such as the entropy and heat capacity. These can be either a functional of the ground-state site occupation or of the finite-temperature equilibrium site occupation. In the first case their equilibrium value does not correspond to an extremal point of the functional, while it does in the second case. Our work gives us access to finite-temperature properties of many-body systems in the thermodynamic limit.
We introduce a machine-learning density-functional-theory formalism for the spinless Hubbard model in one dimension at both zero and finite temperature. In the zero-temperature case this establishes a one-to-one relation between the site occupation and the total energy, which is then minimized at the ground-state occupation. In contrast, at finite temperature the same relation is defined between the Helmholtz free energy and the equilibrium site occupation. Most importantly, both functionals are semilocal, so that they are independent from the size of the system under investigation and can be constructed over exact data for small systems. These ``exact'' functionals are numerically defined by neural networks. We also define additional neural networks for finite-temperature thermodynamical quantities, such as the entropy and heat capacity. These can be either a functional of the ground-state site occupation or of the finite-temperature equilibrium site occupation. In the first case their equilibrium value does not correspond to an extremal point of the functional, while it does in the second case. Our work gives us access to finite-temperature properties of many-body systems in the thermodynamic limit.
The solution of complex many-body lattice models can often be found by defining an energy functional of the relevant density of the problem. For instance, in the case of the Hubbard model the spin-resolved site occupation is enough to describe the system's total energy. Similarly to standard density functional theory, however, the exact functional is unknown, and suitable approximations need to be formulated. By using a deep-learning neural network trained on exact-diagonalization results, we demonstrate that one can construct an exact functional for the Hubbard model. In particular, we show that the neural network returns a ground-state energy numerically indistinguishable from that obtained by exact diagonalization and, most importantly, that the functional satisfies the two Hohenberg-Kohn theorems: for a given ground-state density it yields the external potential, and it is fully variational in the site occupation.
Density functional theory underlies the most successful and widely used numerical methods for electronic structure prediction of solids. However, it has the fundamental shortcoming that the universal density functional is unknown. In addition, the computational result—energy and charge density distribution of the ground state—is useful for electronic properties of solids mostly when reduced to a band structure interpretation based on the Kohn-Sham approach. Here, we demonstrate how machine learning algorithms can help to free density functional theory from these limitations. We study a theory of spinless fermions on a one-dimensional lattice. The density functional is implicitly represented by a neural network, which predicts, besides the ground-state energy and density distribution, density-density correlation functions. At no point do we require a band structure interpretation. The training data, obtained via exact diagonalization, feeds into a learning scheme inspired by active learning, which minimizes the computational costs for data generation. We show that the network results are of high quantitative accuracy and, despite learning on random potentials, capture both symmetry-breaking and topological phase transitions correctly.
Density functional theory underlies the most successful and widely used numerical methods for electronic structure prediction of solids. However, it has the fundamental shortcoming that the universal density functional is unknown. In addition, the computational result---energy and charge density distribution of the ground state---is useful for electronic properties of solids mostly when reduced to a band structure interpretation based on the Kohn-Sham approach. Here, we demonstrate how machine learning algorithms can help to free density functional theory from these limitations. We study a theory of spinless fermions on a one-dimensional lattice. The density functional is implicitly represented by a neural network, which predicts, besides the ground-state energy and density distribution, density-density correlation functions. At no point do we require a band structure interpretation. The training data, obtained via exact diagonalization, feeds into an active learning scheme which minimizes the computational costs for data generation. We show that the network results are of high quantitative accuracy and, despite learning on random potentials, capture both symmetry-breaking and topological phase transitions correctly.
We revisit the machine-learning (ML) approach to the universal density functional F[n] of the one-dimensional Hubbard model with a site-dependent random potential v={vi}. We generate exact ground-state data via exact diagonalization for a periodic chain with L=8 in the paramagnetic sector (N↑,N↓)=(2,2), with site electron densities ni=ni↑=ni↓. The resulting density-potential dataset is analyzed. Using principal component analysis of the joint feature space (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]. 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 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.
The thermodynamics of the inhomogeneous one-dimensional repulsive fermionic Hubbard model with parabolic confinement is studied by a density-functional-theory approach, based on Mermin's generalization to finite temperatures. A local-density approximation (LDA), based on exact results for the homogeneous model, is used to approximate the correlation part in the Helmholtz free-energy, comprising the thermodynamic Bethe ansatz LDA (TBALDA). The general presentation of the method is given and some properties of the homogeneous model that are relevant to the DFT approach are analyzed. Extensive comparison between TBALDA and numerical exact diagonalization results for thermodynamic properties of small inhomogeneous chains is discussed. In the remaining, a classical thermodynamic treatment of the confined system is developed with the focus on global properties of large systems. It is observed that, depending on the temperature and specific volume, the system can have its temperature increased under isentropic expansion. Such unusual behavior becomes more pronounced with the increase of the intrasite interaction U and can be understood considering the peculiar behavior of the correlation entropy per site of the Hubbard model around half-filling. A related nonmonotonic variation of the temperature under isentropic change of U is also studied. At the end, the dependence of the double occupancy on temperature and volume per particle is discussed.
The ground-state properties of the half-filled Hubbard model are investigated in the framework of lattice density functional theory. The single-particle density matrix ${\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}$ is regarded as the central variable of the many-body problem, where i and j refer to the lattice sites and $\ensuremath{\sigma}$ to the spin. The interaction-energy functional $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ is calculated exactly for representative finite periodic systems by performing exact Lanczos diagonalizations. The relationship between $W[{\ensuremath{\gamma}}_{ij\ensuremath{\sigma}}]$ and the entropy $S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}]$ of independent fermions with natural-orbital occupations ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$ is analyzed. A simple approximation to the interaction energy of the half-filled Hubbard model is proposed, which takes the form $W=W(S[{\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}])$. Using this functional we derive the ground-state energy, kinetic energy, average number of double occupations, charge distribution, magnetic susceptibility, and field-induced spin polarization in one-, two-, and three-dimensional periodic lattices. The limit of infinite dimensions is also explored. The accuracy of the method is assessed by comparison with available exact numerical or analytical results. Goals, limitations, and possible extensions of the domain of applicability of the functional are discussed.