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Physics of Atomic Nuclei

Energy-level ordering for frustrated spin ladder models

Tigran Hakobyan

Published Feb 1, 2010• 2 citations
Publisher

Abstract

The Lieb—Mattis theorem about ordering of energy levels is generalized to frustrated antiferromagnetic spin-1/2 ladder model with diagonal and four-spin interaction.
DOI
10.1134/s1063778810020213

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The Lieb-Mattis theorem is generalized to an antiferromagnetic spin-ladder model with four-spin cyclic exchange interaction. We prove that for J>2KJ>2\phantom{\rule{0.3em}{0ex}}\text{K}J>2K, the antiferromagnetic ordering of energy levels takes place separately in two sectors, which remain symmetric and antisymmetric under the reflection with respect to the longitudinal axis of the ladder. We prove also that at the self-dual point J=2KJ=2\phantom{\rule{0.3em}{0ex}}\text{K}J=2K, the Lieb-Mattis rule holds in the sectors with fixed number of rung singlets. In both cases, it agrees with the similar rule for Haldane chain with appropriate spin number.
Simple formulation and a straightforward proof of the Lieb-Mattis theorem (LMT) do not indicate how powerful a tool it is.For more than fifty years, this theorem has been mainly discussed in its 'strong' form and applied mainly to many type of infinite spin lattices.It can be easily proved that in such cases, geometrically frustrated systems have to be excluded.However, it has been recently shown that the so-called 'weak'or general form of the LMT can be exploited considering some small, geometrically frustrated quantum spin systems.Moreover, many systems, which do not satisfy assumptions of the LMT, show identical features, including the level order characteristic for bipartite spin systems.It yields a question about possible generalizations or modifications of this theorem to involve a larger class of problems.To begin, algebraic aspects have to be understood with the invaluable role of the Perron-Frobenius theorem.The latter theorem is investigated, discussed, modified etc. in immense number of works.Many of them are important in physical applications and ways of reasoning they present can be exploited in different approaches to the LMT.
We prove ferromagnetic ordering of energy levels for XXX Heisenberg chains with spins of arbitrary magnitude, thus extending our previous result for the spin-1/2 chain. Ferromagnetic ordering means that the minimum energies in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin. This result provides a ferromagnetic analogue of the well-known theorem by Lieb and Mattis about ordering of energy levels in antiferromagnetic and ferrimagnetic systems on bipartite graphs.
The Lieb-Mattis theorem about antiferromagnetic ordering of energy levels on bipartite lattices is generalized to finite-size two-leg spin-1∕21∕21∕2 ladder model frustrated by diagonal interactions. For reflection-symmetric model with site-dependent interactions, we prove exactly that the lowest energies in sectors with fixed total spin and reflection quantum numbers are monotone increasing functions of total spin. The nondegeneracy of most levels is also proven. We also establish the uniqueness and obtain the spin value of the lowest level multiplet in the whole sector formed by reflection-symmetric (antisymmetric) states. For a wide range of coupling constants, we prove that the ground state is a unique spin singlet. For other values of couplings, it may be also a unique spin triplet or may consist of both multiplets. Similar results have been obtained for the ladder with arbitrary boundary impurity spin. Some partial results have also been obtained in the case of periodical boundary conditions.
A theorem about the spin properties of the ground state and the ordering of the energy levels for different spins of an interacting-electron model including an arbitrary diagonal interacting potential, an antiferromagnetic exchange, and an electron pair-hopping term for particles on a one-dimensional lattice is stated and proved. We show that when the number of electrons N=4n+2 (n an integer) with periodic boundary conditions or N=4n with antiperiodic boundary conditions, the ground state is a nondegenerate singlet. This theorem generalizes a similar theorem Lieb and Mattis proved for the one-dimensional interacting electron system.