Generalizing the classical Thomson problem to the quantum regime provides an ideal model to explore the underlying physics regarding electron correlations. In this work, we systematically investigate the combined effects of the geometry of the substrate and the symmetry of the wave function on correlations of geometrically confined electrons. By the numerical configuration interaction method in combination with analytical theory, we construct symmetrized ground-state wave functions; analyze the energetics, correlations, and collective vibration modes of the electrons; and illustrate the routine for the strongly correlated, highly localized electron states with the expansion of the sphere. This work furthers our understanding about electron correlations on confined geometries and shows the promising potential of exploiting confinement geometry to control electron states.
The accuracy of large-scale ab initio calculations is always desired, but they are unfortunately not usually practical in many fields, such as studies of heterogeneous catalysis, photosynthesis, and functional mechanisms of biomolecules. DFT serves as a prevailing method for large calculations due to its compromise between cost and accuracy, but it has a number of failure cases, like bond breaking. On the other hand, the simple theoretical nature of force-field simulations, which often work well, reminds us that large systems are not necessarily exponentially complicated. A recurring theme in the electronic structure of large systems is therefore the use of fragment-based approaches. Existing fragment approaches typically treat the most powerful electron correlations perturbatively, on the same footing with dispersion forces (if at all). In our approach, electrons are not treated individually in the global calculation, but subunits will be chosen. (e.g., amino acid residues or individual water molecules). The most important local states are selected to account for inter-subunit interactions. Most familiar wavefunction methods (SCF, PT, coupled-cluster) can be applied directly to this internally correlated Hamiltonian, and the Hamiltonian itself is systematically improvable to exactitude. Furthermore, different parts of a system can be treated at different accuracy. Our initial investigations focus on the correlation structure of groups of harmonic oscillators, where the coupling inside of groups is stronger than between groups. This mimics discrete molecules while providing us with exact analytical solutions for comparison. Correlations between grouped oscillators will be characterized by both numerical and analytical methods, in order to shed light on building an algorithm for real molecules and systems.
It is common wisdom that a better understanding of how to quantify electron correlation should lead to more efficient treatments of the correlation problem. Here, we not only substantiate this view but turn it into a concrete, systematic framework for ground states. Since prominent correlation measures such as the nonfreeness depend only on the one-particle reduced density matrix, modeling the correlation energy as a function of such a measure naturally places the framework within reduced density matrix functional theory, while keeping the cost at mean-field level. Strikingly, the variable conjugate to the measure emerges as a ``correlation temperature'', thereby connecting correlation measures, functional theory, and effective temperatures. The framework rationalizes methods such as i-DMFT, and we illustrate its key concepts analytically for the Hubbard dimer and numerically for small molecular systems. This brings a fundamental question within reach: how much of the correlation problem can be solved at mean-field cost?
We introduce a systematic expansion tailored to systems with strong local interactions and capable of computing response functions, including finite DC transport, analytically. The expansion is controlled by a small parameter
s2 that measures the area of the momentum space region where kinematics of the theory is concentrated. In real space, this corresponds to single-particle or correlated hopping terms with amplitudes that decay over a length scale
1/s and scale in magnitude as
s2 in two dimensions. In the limit
s2≪1, long, self-avoiding tunneling paths dominate over paths revisiting the same site. This enables systematic controlled calculations of various physical quantities. We illustrate the method with three applications. (i) A Hubbard model with concentrated dispersion: we analytically obtain spectral broadening which scales as
s2 and identify a high-temperature bad metal with
T-linear resistivity coexisting with parametrically long-lived quasiparticles, as well as an intermediate-temperature "thermal FL*" with a small hole pocket that coexists with thermally disordered fluctuating local moments, all within a single controlled framework. (ii) A correlated-hopping model with interesting electron-trion dynamics. (iii) A model of Chern bands with concentrated Berry curvature, motivated by twisted bilayer graphene, which realizes a Mott semimetal where we compute the broadening for the electron and trion spectral functions. At the end, we discuss how our approach paves the way to addressing various challenging questions in strongly correlated systems and outline its various generalizations.