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Register a new userMachine learning density functional theory for the Hubbard model
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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 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.
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According to the Hohenberg-Kohn theorem of density-functional theory (DFT), all observable quantities of systems of interacting electrons can be expressed as functionals of the ground-state density. This includes, in principle, the spin polarization (magnetization) of open-shell systems; the explicit form of the magnetization as a functional of the total density is, however, unknown. In practice, open-shell systems are always treated with spin-DFT, where the basic variables are the spin densities. Here, the relation between DFT and spin-DFT for open-shell systems is illustrated and the exact magnetization density functional is obtained for the half-filled Hubbard trimer. Errors arising from spin-restricted and -unrestricted exact-exchange Kohn-Sham calculations are analyzed and partially cured via the exact magnetization functional.
We present in full detail a newly developed formalism enabling density functional perturbation theory (DFPT) calculations from a DFT+$U$ ground state. The implementation includes ultrasoft pseudopotentials and is valid for both insulating and metallic systems. It aims at fully exploiting the versatility of DFPT combined with the low-cost DFT+$U$ functional. This allows to avoid computationally intensive frozen-phonon calculations when DFT+$U$ is used to eliminate the residual electronic self-interaction from approximate functionals and to capture the localization of valence electrons e.g. on $d$ or $f$ states. In this way, the effects of electronic localization (possibly due to correlations) are consistently taken into account in the calculation of specific phonon modes, Born effective charges, dielectric tensors and in quantities requiring well converged sums over many phonon frequencies, as phonon density of states and free energies. The new computational tool is applied to two representative systems, namely CoO, a prototypical transition metal monoxide and LiCoO$_2$, a material employed for the cathode of Li-ion batteries. The results show the effectiveness of our formalism to capture in a quantitatively reliable way the vibrational properties of systems with localized valence electrons.
The Bethe-Ansatz local density approximation (LDA) to lattice density functional theory (LDFT) for the one-dimensional repulsive Hubbard model is extended to current-LDFT (CLDFT). The transport properties of mesoscopic Hubbard rings threaded by a magnetic flux are then systematically investigated by this scheme. In particular we present calculations of ground state energies, persistent currents and Drude weights for both a repulsive homogeneous and a single impurity Hubbard model. Our results for the ground state energies in the metallic phase compares favorably well with those obtained with numerically accurate many-body techniques. Also the dependence of the persistent currents on the Coulomb and the impurity interaction strength, and on the ring size are all well captured by LDA-CLDFT. Our study demonstrates that CLDFT is a powerful tool for studying one-dimensional correlated electron systems with high accuracy and low computational costs.
Lattice Monte Carlo calculations of interacting systems on non-bipartite lattices exhibit an oscillatory imaginary phase known as the phase or sign problem, even at zero chemical potential. One method to alleviate the sign problem is to analytically continue the integration region of the state variables into the complex plane via holomorphic flow equations. For asymptotically large flow times the state variables approach manifolds of constant imaginary phase known as Lefschetz thimbles. However, flowing such variables and calculating the ensuing Jacobian is a computationally demanding procedure. In this paper we demonstrate that neural networks can be trained to parameterize suitable manifolds for this class of sign problem and drastically reduce the computational cost. We apply our method to the Hubbard model on the triangle and tetrahedron, both of which are non-bipartite. At strong interaction strengths and modest temperatures the tetrahedron suffers from a severe sign problem that cannot be overcome with standard reweighting techniques, while it quickly yields to our method. We benchmark our results with exact calculations and comment on future directions of this work.
Machine learning is a powerful tool to design accurate, highly non-local, exchange-correlation functionals for density functional theory. So far, most of those machine learned functionals are trained for systems with an integer number of particles. As such, they are unable to reproduce some crucial and fundamental aspects, such as the explicit dependency of the functionals on the particle number or the infamous derivative discontinuity at integer particle numbers. Here we propose a solution to these problems by training a neural network as the universal functional of density-functional theory that (i) depends explicitly on the number of particles with a piece-wise linearity between the integer numbers and (ii) reproduces the derivative discontinuity of the exchange-correlation energy. This is achieved by using an ensemble formalism, a training set containing fractional densities, and an explicitly discontinuous formulation.
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