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arXiv:2603.00802

Revisiting the machine-learning density functional for the one-dimensional Hubbard model with random external potential

Octavio D. Rodrı́guez Salmon, Minos A. Neto, J. Roberto Viana, Griffith Mendonça

Published Feb 28, 2026• 0 citations
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Abstract

We revisit the machine-learning (ML) approach to the universal density functional $F[\mathbf{n}]$ of the one-dimensional Hubbard model with a site-dependent random potential $\mathbf{v}=\{v_{i}\}$. We generate exact ground-state data via exact diagonalization for a periodic chain with $L=8$ in the paramagnetic sector $(N_\uparrow,N_\downarrow)=(2,2)$, with site electron densities $n_{i} = n_{i\uparrow}=n_{i\downarrow}$. The resulting density-potential dataset is analyzed. Using principal component analysis of the joint feature space $(\mathbf n,\mathbf v)$, 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 $F[\mathbf n]$. 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 $\partial F/\partial n_i$ 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.

DOI
10.48550/arxiv.2603.00802

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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 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.
A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites Na,{N}_{a},Na​, band filling Ne,{N}_{e},Ne​, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on Na,{N}_{a},Na​, Ne,{N}_{e},Ne​, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive (U>0)(U>0)(U>0) and attractive (U<0)(U<0)(U<0) interactions are contrasted.
We present a deep neural network (DNN) -based model (HubbardNet) to variationally find the ground-state and excited-state wave functions of the one-dimensional and two-dimensional Bose-Hubbard model. Using this model for a square lattice with MMM sites, we obtain the energy spectrum as an analytical function of the on-site Coulomb repulsion, UUU, and the total number of particles, NNN, from a single training. This approach bypasses the need to solve a new Hamiltonian for each different set of values (U,N)(U,N)(U,N) and generalizes well even for out-of-distribution (U,N)(U,N)(U,N). Using HubbardNet, we identify the two ground-state phases of the Bose-Hubbard model (Mott insulator and superfluid). We show that the DNN-parametrized solutions are in excellent agreement with results from the exact diagonalization of the Hamiltonian, and it outperforms exact diagonalization in terms of computational scaling. These advantages suggest that our model is promising for efficient and accurate computation of exact phase diagrams of many-body lattice Hamiltonians.