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arXiv:1507.05362 · Physical review. B./Physical review. B

Extension of Lieb-Schupp theorem to Heisenberg models with higher-order interactions

Kengo Tanaka

Published Oct 3, 2016• 0 citations
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Abstract

We extend the Lieb-Schupp theorem to Heisenberg models with higher-order interactions on nonfrustrated or frustrated finite lattices. These lattices are constructed by even-numbered rings with or without crossing bonds and have reflection symmetry. The results show that the ground state has total spin zero in wide interaction parameter regions which are not covered by the results of Marshall-Lieb-Mattis-type arguments.

DOI
10.1103/physrevb.94.144401

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We extend the Lieb-Schupp theorem to Heisenberg models with higher-order interactions on nonfrustrated or frustrated finite lattices. These lattices are constructed by even-numbered rings with or without crossing bonds and have reflection symmetry. The results show that the ground state has total spin zero in wide interaction parameter regions which are not covered by the results of Marshall-Lieb-Mattis-type arguments.
We suggest a general rule for the shift quantum numbers k of the relative ground states of antiferromagnetic Heisenberg spin rings. This rule generalizes the well-known results of Marshall, Peierls, Lieb, Schultz, and Mattis for even rings. Our rule is confirmed by numerical investigations and rigorous proofs for special cases, including systems with a Haldane gap for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ Implications for the total spin quantum number S of relative ground states are discussed as well as generalizations to the XXZ model.
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Lieb, Schultz and Mattis (LSM) (1961 Ann. Phys., NY 16 407 ) studied the S = 1/2 XXZ spin chain. The theorems of LSM's paper can be applied to broader models. In the original LSM theorem the nonfrustrating system was assumed. However, reconsidering the LSM theorem, we can extend the LSM theorem for frustrating systems. Next, several researchers tried to extend the LSM theorem for excited states. In the cases , the lowest energy eigenvalues are continuous for wave number q . But we found that their proofs were insufficient, and improve upon them. In addition, we can prove the LSM theory without the assumption of the discrete symmetry, which means that LSM-type theorems are applicable for Dzyaloshinskii–Moriya type interactions or other nonsymmetric models.
The celebrated Lieb-Schultz-Mattis theorem excludes a nondegenerate gapped ground state of a spin system with an odd number of s=12s=\frac{1}{2}s=21​ spins per unit cell. Here, the author presents a new family of Lieb-Schultz-Mattis type theorems for fermionic systems that have no analog in spin systems. An example is a particle-hole symmetric system of spinless fermions on a lattice with an odd number of fermionic modes per unit cell. The author shows that a symmetric gapped ground state must be topologically ordered, and the lattice translation symmetries must permute anyon types, resulting in non-Abelian zero modes bound to lattice dislocations. These results can be interpreted as quantum anomalies on the surface of three-dimensional fermionic symmetry-protected phases.
We will study properties of the ferrimagnetic state depending on types of interaction. In non-frustrated case, the Lieb–Mattis type of ferrimagnetic order appears, where a gap exists between the ground state and the first excited state. On the other hand, in some frustrated lattices, so called non-collinear type of ferrimagnetic order appears, where no gap exists. We will study a model where the type of interaction changes from the former to the latter when one of parameters is varied, and study the dependence of the ground state magnetization on the parameter. We will also study other types of spontaneous magnetization in the ground state in the diamond lattice.