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arXiv:1711.06001 · Physical review. B./Physical review. B

Hund and anti-Hund rules in circular molecules

Marian Niţă, Mugurel Tolea, D. C. Marinescu, Andrei R. Manolescu

Published Dec 1, 2017• 3 citations
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Abstract

We study the validity of Hund's first rule for the spin multiplicity in circular molecules---made of real or artificial atoms such as quantum dots---by considering a perturbative approach in the Coulomb interaction in the extended Hubbard model with both on-site and long-range interactions. In this approximation, we show that an anti-Hund rule always defines the ground state in a molecule with $4N$ atoms at half-filling. In all other cases (i.e., number of atoms not a multiple of four, or a $4N$ molecule away from half-filling) both the singlet and the triplet outcomes are possible, as determined primarily by the total number of electrons in the system. In some instances, the Hund rule is always obeyed and the triplet ground state is realized mathematically for any values of the on-site and long-range interactions, while for other filling situations the singlet is also possible but only if the long-range interactions exceed a certain threshold, relatively to the on-site interaction.

DOI
10.1103/physrevb.96.235101

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Physical review. B, Condensed matter•Apr 15, 1980•5 citations
We study the validity of Hund's first rule for the spin multiplicity in circular molecules---made of real or artificial atoms such as quantum dots---by considering a perturbative approach in the Coulomb interaction in the extended Hubbard model with both on-site and long-range interactions. In this approximation, we show that an anti-Hund rule always defines the ground state in a molecule with 4N4N4N atoms at half-filling. In all other cases (i.e., number of atoms not a multiple of four, or a 4N4N4N molecule away from half-filling) both the singlet and the triplet outcomes are possible, as determined primarily by the total number of electrons in the system. In some instances, the Hund rule is always obeyed and the triplet ground state is realized mathematically for any values of the on-site and long-range interactions, while for other filling situations the singlet is also possible but only if the long-range interactions exceed a certain threshold, relatively to the on-site interaction.
Highly-symmetric molecules often exhibit degenerate tight-binding states at the Fermi edge. This typically results in a magnetic ground state if small interactions are introduced in accordance with Hund’s rule. In some cases, Hund’s rule may be broken, which signals pair binding and goes hand-in-hand with an attractive pair-binding energy. We investigate pair binding and Hund’s rule breaking for the Hubbard model on high-symmetry fullerenes C _{20} 20 , C _{20} 20 , C _{40} 40 , and C _{60} 60 by using large-scale density-matrix renormalization group calculations. We exploit the SU(2) spin symmetry, the U(1) charge symmetry, and optionally the \mathbb{Z}_N ℤ N spatial rotation symmetry of the problem. For C _{20} 20 , our results agree well with available exact-diagonalization data, but our approach is numerically much cheaper. We find a Mott transition at U_c\sim2.2t U c ∼ 2.2 t , which is much smaller than the previously reported value of U_c\sim4.1t U c ∼ 4.1 t that was extrapolated from a few datapoints. We compute the pair-binding energy for arbitrary values of U U and observe that it remains overall repulsive. For larger fullerenes, we are not able to evaluate the pair binding energy with sufficient precision, but we can still investigate Hund’s rule breaking. For C _{28} 28 , we find that Hund’s rule is fulfilled with a magnetic spin-2 ground state that transitions to a spin-1 state at U_{c,1}\sim 5.4t U c , 1 ∼ 5.4 t before the eventual Mott transition to a spin singlet takes place at U_{c,2}\sim 11.6t U c , 2 ∼ 11.6 t . For C _{40} 40 , Hund’s rule is broken in the singlet ground state at half filling, but is restored if the system is doped with one electron. Hund’s ru
We give a detailed analysis of the applicability of Hund's first rule for harmonic planar two-electron quantum dots by means of entanglement witnesses. We find that for purely harmonic confinement there is only one pair of singlet and triplet states for which it can be applied. We also discuss the origin and validity for this case and extend the discussion to a quartic confining potential, a hard-wall potential, and a combination of harmonic confinement with quartic perturbation. A generalized rule, the alternating rule, is found to be applicable and valid for vanishing angular momentum states in all these cases. Furthermore, we are able to clarify the role of entanglement in general harmonic two-electron models for vanishing interaction strength. This behavior can be attributed to the special separability properties of these models.
Quantum spin chains with composite spins have been used to approximate conventional chains with higher spins. For instance, a spin 1 (or [Formula: see text]) chain was sometimes approximated by a chain with two (or three) spin [Formula: see text]'s per site. However, little examination has been given as to whether this approximation, effectively assuming the first Hund rule per site, is valid and why. In this paper, the validity of this approximation is investigated numerically. We diagonalize the Hamiltonians of spin chains with a spin 1 and [Formula: see text] per site and with two and three spin [Formula: see text]'s per site. The low energy excitation spectrum for the spin chain with M spin [Formula: see text]'s per site is found to coincide with that of the corresponding conventional chain with one spin [Formula: see text] per site. In particular, we find that as the system size increases, an increasingly larger block of consecutive lowest energy states with maximal spin per site is observed, robustly supporting the first Hund rule even though the exclusion principle does not apply and the system does not possess Coulomb repulsion. As for why this approximation works, we show that this effective Hund rule emerges as a plausible consequence when applying to composite spin systems the Lieb-Mattis theorem, which is originally for the ground state of ferrimagnetic and antiferromagnetic spin systems.
Quantum spin chains with composite spins have been used to approximate conventional chains with higher spins. For instance, a spin 1 (or [Formula: see text]) chain was sometimes approximated by a chain with two (or three) spin [Formula: see text]'s per site. However, little examination has been given as to whether this approximation, effectively assuming the first Hund rule per site, is valid and why. In this paper, the validity of this approximation is investigated numerically. We diagonalize the Hamiltonians of spin chains with a spin 1 and [Formula: see text] per site and with two and three spin [Formula: see text]'s per site. The low energy excitation spectrum for the spin chain with M spin [Formula: see text]'s per site is found to coincide with that of the corresponding conventional chain with one spin [Formula: see text] per site. In particular, we find that as the system size increases, an increasingly larger block of consecutive lowest energy states with maximal spin per site is observed, robustly supporting the first Hund rule even though the exclusion principle does not apply and the system does not possess Coulomb repulsion. As for why this approximation works, we show that this effective Hund rule emerges as a plausible consequence when applying to composite spin systems the Lieb-Mattis theorem, which is originally for the ground state of ferrimagnetic and antiferromagnetic spin systems.
Excitonic phases described by the quantum condensation of electron-hole pairs (or excitons) are studied in the two-orbital Hubbard model with the Hund's rule coupling J . Using the variational cluster approximation, we calculate the ground-state energy and order parameters, and show that the Hund's rule coupling always stabilizes the excitonic spin-density- wave state and destabilizes the excitonic charge-density-wave state. The pair coherence lengths calculated in the spin-singlet and spin-triplet states indicate that only the spin-triplet excitons are paired more tightly with increasing J . The single-particle spectrum and density of states are also calculated to clarify the characters of these excitonic density-wave states.
The electronic and magnetic correlations in diatomic molecules are studied rigorously on the basis of a Hubbard Hamiltonian. The problem is treated in three phases separately, firstly for electrons with nondegenerate orbitals, then for electrons with degenerate orbitals, and finally for a mixture of both. In this last case special emphasis is put on the interaction between the electrons in the nondegenerate orbital levels with those in the degenerate orbital levels. In this context some qualitative features of how electronic correlations influence the cohesive forces are sketched. We highlight the extent to which various factors such as the total number of electrons in the system, the Pauli principle, Hund's rule, the level splitting, and the relative size of the transfer integrals are influential in determining the ground-state configurations. Some simple physical guidelines are provided for accurately guessing the various possible ground-state configurations.