Robust Hund rule without Coulomb repulsion and exclusion principle in quantum antiferromagnetic chains of composite half spins

Quantum spin chains with composite spins have been used to approximate conventional chains with higher spins. For instance, a spin 1 (or 32) chain was sometimes approximated by a chain with two (or three) spin 12’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 32 per site and with two and three spin 12’s per site. The low energy excitation spectrum for the spin chain with M spin 12’s per site is found to coincide with that of the corresponding conventional chain with one spin M2 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.


Introduction
Quantum spin chain models are suitable for studying interacting quantum many-body systems with strong electron correlations, broken translational symmetry, and quantum fluctuations. One physical problem that is best studied in such models is magnetism [1,2]; another example of growing interest in using spin chain models is the study of quantum information transfer and quantum information processing, the core parts of quantum information theory [3].
Typically a conventional Heisenberg spin chain is described by the Hamiltonian (1) where S i is the spin-s operator at the i th site, positive (negative) J ij denotes antiferromagnetic (ferromagnetic) coupling between spins at sites i and j, and the coupling strength |J ij | generally depends on the separation distance |i -j|. In this study, we assume J ij = J > 0 if i and j are nearest neighbors and 0 otherwise.
For the antiferromagnetic nearest neighbor coupling that we assume, the infinite length limit of the spin 1 2 chain is exactly solvable via the Bethe ansatz [4] or by using the quantum inverse method [5]. However, for chains of higher spins the exact solution is unattainable in the infinite size limit. Very extensive studies have been done both theoretically and experimentally since Haldane [6][7][8] pointed out the drastic differences between integer and half-integer spin chains. In the infinite length limit, the energy difference (gap) between the first excited state and the ground state remains finite for an integer spin chain but vanishes for a half-integer spin chain.
For finite size spin-chains, a fairly large number of studies have also been done to investigate different aspects of magnetic materials. These include the fundamental origins of magnetism in metallic ions [9], impurity effects on chains [10], boundary effects [11], and the effect of even-odd number of sites on the eigenpairs of the chain [12].
In a somewhat different context, there have been earlier studies to approximate a spin 1 (or 3 2 ) chain by a spin-chain with two (or three) spin 1 2 's per site to study different aspects of finite size antiferromagnetic clusters [13][14][15][16]. However, little examination has been given as to whether this approximation, effectively assuming the first Hund rule per site, is valid and why should it work except that replacing one spin M 2 by M spin 1 2 's maps the Hamiltonian into a solvable portion (made of spin 1 2 chains) and interaction portions that some authors [15,16] treated perturbatively. Evidently, if the multiple spin 1 2 's on the same site interact strongly in a ferromagnetic manner, the spins on the same site would tend to line up and the first Hund rule emerges effectively. What is fascinating, however, is that the first Hund rule persists without any on-site ferromagnetic interaction, not to mention that the spin chains possess neither exclusion principle nor Coulomb repulsion, which are the foundation for the first Hund rule in atoms with multiple electrons. In this paper we investigate numerically the robustness of the first Hund rule in spin chains using chains of even number of sites with both periodic and open boundary condition. In addition to showing numerically that the first Hund rule is more pronounced in the periodic chain than in the open chain, we also provide an explanation for the first Hund rule based on the Lieb-Mattis theorem [17].

The model
The one-dimensional model we study here originates from [15] where I 2 is the identity matrix of rank 2 and σ a=x,y,z are the two dimensional Pauli matrices.
The spin operator at the ith site acts on the full Hilbert space but non-trivially only on site i. In short we write the ith spin as where I 2 M is the identity matrix (in 2 M dimension) and S i (M) is the spin operator obtained by taking the tensor product of M spin 1 2 operators, as shown in equations (2) and (3) means that only the high spin state at each site contributes to the lower energy eigenstates. This is the first Hund rule that is observed in atomic and molecular systems except that we do not have Coulomb repulsion or exclusion principle. Table 1 shows the low energy spectra of periodic chains with two spin In a similar fashion, if we have γ ⩾ 2 impurities in the system, the spin chain can be effectively opened up at up to γ number of sites. Cells containing energies corresponding to such states are colored in yellow.
In table 1, one may also enumerate n s , the number of states lie below the lowest energy impurity state. It is clear that n s increases as the chain size increases. These numbers are shown in table 2. The fact that n s increases with size indicates that in the low energy sector of a long chain, the two spin  table 3. In this case, since the spins at site 1 and site N can no longer contribute a negative energy due to the absence of coupling, the ground state energy will appear higher in comparison with the corresponding periodic chain. This can be easily seen by comparing the ground state energies in table 3 with those in table 1. One important thing to note is that when there is one impurity, one expects that having the impurity on one of the edge sites is energetically more favorable. This is because an impurity site in the middle of the chain interrupts two couplings while an impurity on the edge only interrupts one. Another interesting point to note is that when the single impurity is placed on the edge of the chain, it effectively reduces the open chain size by one. The energies corresponding to states of one impurity site on the edge are placed in cells colored in blue. These energies are also found in table 1  identified states corresponding to one impurity sitting at an interior site; the corresponding energies are displayed in cells colored in cyan. For states corresponding to two or more impurities, we again display their energies in cells colored in yellow.
In table 3, one may again enumerate n s , the number of states lie below the lowest energy impurity state. One still observes that n s increases as the chain size increases. These n s numbers are shown in table 4. It has not escaped our attention that for open chains n s does not grow with the system size as fast as for periodic chains. This can be understood as follows: for the same chain length the ground state energy of an open chain is higher than the periodic chain while the lowest energy impurity states have the same energies for both periodic and open chains. That is, the energy difference between the lowest energy state and the lowest energy impurity state is larger for the periodic chains, hence able to accommodate more impurity-free states before encountering the lowest energy impurity state. Nevertheless, because n s increases with size still, in the low energy sector of a long chain, the two spin We have also obtained a similar result when we put three spin Lieb and Mattis [17] considered a generic Heisenberg spin system, similar to (1), within which two sublattices A and B can be identified as follows: the spin coupling among spins within the same sublattice is smaller than g 2 (J i(A),j(A) ⩽ g 2 and J i(B),j(B) ⩽ g 2 ) while the coupling between spins in different sublattices is larger than g 2 (J i(A),j(B) ⩾ g 2 ). They worked Duki and Yu Page 6 out the case for g 2 = 0 and then extend it to general g 2 values. For our purpose, however, we only need to consider and review the g 2 = 0 case. Let us also note that for open boundary condition the even numbered sites and the odd numbered sites constitute the two sublattices needed.
Even though an open chain is allowed to have an odd number of total sites, for a periodic chain, the total number of sites must be even in order to have two clearly defined sublattices. It is for this particular reason that all tables, shown in the previous section, are for even number of sites.
One of the key steps involves identifying two operators S tot it is a linear combination of all basis states yielding S tot z = ℳ with positive coefficients.
However, we can look at H′ from a different perspective. Given that H′ can be written as , we know its lowest energy state has maximum S A and S B but with

Summary and conclusion
In summary, we considered an antiferromagnetic chain of M spin been implicitly assumed/used for decades, no reference was made to the Lieb-Mattis theorem.

Acknowledgments
This research was supported by the Intramural Research Program of the National Institutes of Health, National Library of Medicine. We thank Tim Savannah formerly in IRB support at the NCBI. We also thank the administrative group of the National Institutes of Health Biowulf Clusters, where most of the computational tasks were carried out using the Trilinos software package [21,22].

Appendix A. Hilbert space decomposition
In this appendix we show how the Hilbert space of M spin 1 2 per site gets decomposed into a direct sum of irreducible representations. Suppose that V is the Hilbert for a single spin 1 2 .
When considering two spin where M 2 represents the largest integer that is smaller than or equal to The Hilbert space of the system of N sites is naturally written as  . Evidently S l 2 commutes with S i ⋅ S j provided that l ≠ i and l ≠ j. If l equals i, due to the antisymmetric nature of the ϵ tensor. Other terms in H have the form S l ⋅ S l′ . We where the indices l and l′ of ∑ l ≠ l′ ′ do not overlap with i or j. Evidently, the first three terms of (B.3) commute with S i ⋅ S j . However, S l ⋅ ( S i + S j ) and ( S i + S j ) ⋅ S l′ both commute with S i ⋅ S j . As an example, one may work out With the transformed Hamiltonian and the defined basis set, if one defines then K βα < 0, i.e. K αβ = −|K αβ |. To see this, we note that in order for K αβ = 0, a term in H 1 must be able to bring the state |ϕ α ⟩ into |ϕ β ⟩. If |ϕ α ⟩ has at site i and j respectively the spin components m i + 1 and m j − 1 along the z direction while |ϕ β ⟩ has at site i and j respectively .) The realness of K βα along with the fact that K βα = K αβ * implies that K αβ = K βα .
Denote the lowest energy state in the ℳ subspace by ψ with energy E ℳ , we can expand ψ in terms of the complete set ϕ α with amplitude f α Multiplying both sides by ⟨ϕ α | we obtain The variational energy of any trial function exceeds E ℳ , unless it is also the lowest energy eigenfunction. LM mentioned that is a trial wave function with energy E ℳ as well. We shall explicitly illustrate this point before carrying on with the analysis. Let and we know E ℳ ′ ⩾ E ℳ by definition. (B.12) can be rewritten as If we multiply both sides of (B.10) by f α * and sum over α, we obtain This implies E ℳ ′ = E ℳ and the Schrödinger equation (B.10) satisfied by the eigenstate |ψ′⟩ This is possible only when f β ⩾ 0 for all β (or f β ⩽ 0 for all β). We may take the former without loss of generality. In fact, for the type of Hamiltonian we consider, we have f β > 0 for all β. For if one f α = 0, then from (B.16) we have which is only possible if all |ϕ β ⟩ that can be reached by applying H 1 to |ϕ α ⟩ have f β = 0 (zero amplitude). Then the states can be reached by applying H 1 to those |ϕ β ⟩ having f β = 0 must all have zero amplitude as well. Since for our case, successive application of H 1 eventually covers every basis state in the ℳ subspace, this implies that one must have f β > 0 for all β. Therefore, all amplitudes are positive and nonvanishing, hence E ℳ is nondegenerate in the ℳ subspace. This is because one cannot construct a state orthogonal to |ψ⟩ without some changes of signs. That is, any states orthogonal to |ψ⟩ must have amplitudes of opposite signs, hence not qualified as the lowest energy state (which we known must have positive and nonvanishing amplitude only).
We therefore arrive at the important conclusion of LM.  The energy eigenvalues of periodic chains with two spin  The number of impurity free states (n s ) with energies lower than that of any impurity state. Periodic spin chains with two spin