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arXiv:cond-mat/0205408 · Journal of Physics A Mathematical and General

The possibility of gapless excitations in antiferromagnetic spin chains with long-range interactions

Tigran Hakobyan

Published Nov 11, 2003• 2 citations
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Abstract

The Lieb–Shultz–Mattis theorem is extended to Heisenberg chains with long-range interactions. We prove that the half-integer spin chain has no gap, if it possesses unique ground state and the exchange decays faster than the inverse-square of distance between spins. The results can be extended to a wide class of one-dimensional models.

DOI
10.1088/0305-4470/36/47/l01

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The Lieb–Shultz–Mattis theorem is extended to Heisenberg chains with long-range interactions. We prove that the half-integer spin chain has no gap, if it possesses unique ground state and the exchange decays faster than the inverse-square of distance between spins. The results can be extended to a wide class of one-dimensional models.
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The Lieb-Schultz-Mattis theorem states that a finite-range Hamiltonian in translationally and spin-rotationally invariant spin-1/2 systems cannot have a unique gapped ground state. Here, the author extends the theorem to include Hamiltonians with terms that act on a finite number of sites, but which may be far apart, provided that the interaction strength decays with distance faster than a certain power law. As the spatial dimension increases, the power law required to guarantee the result demands faster decay.
We study antiferromagnetic spin chains with unfrustrated long-range interactions that decay as power laws with exponent $\ensuremath{\beta},$ using the spin-wave approximation. We find for sufficiently large spin S that the Neel order is stable at T=0T=0T=0 for $\ensuremath{\beta}<3,$ and survives up to a finite Neel temperature for $\ensuremath{\beta}<2,$ validating the spin-wave approach in these regimes. We estimate the critical values of S and T for the Neel order to be stable. The spin-wave spectra are found to be gapless but have nonlinear momentum dependence at long wavelength, which is responsible for the suppression of quantum and thermal fluctuations and stabilizing the Neel state. We also show that for $\ensuremath{\beta}<~1$ and for a large but finite-size system size L, the excitation gap of the system approaches zero slower than ${L}^{\ensuremath{-}1},$ a behavior that is in contrast to the Lieb-Schultz-Mattis theorem.
We study quantum many-body systems in the presence of an exotic antiunitary translation or inversion symmetry involving time reversal. Based on a symmetry-twisting method and spectrum robustness, we propose that a half-integer spin chain that respects any of these two antiunitary crystalline symmetries in addition to the discrete Z_{2}×Z_{2} global spin-rotation symmetry must either be gapless or possess degenerate ground states. This explains the gaplessness of a class of chiral spin models not indicated by the Lieb-Schultz-Mattis theorem and its known extensions. Moreover, we present symmetry classes with minimal sets of generators that give nontrivial Lieb-Schultz-Mattis-type constraints, argued by the bulk-boundary correspondence in 2D symmetry-protected topological phases as well as lattice homotopy. Our results for detecting the ingappability of 1D quantum magnets from the interplay between spin-rotation symmetries and magnetic space groups are applicable to systems with a broader class of spin interactions, including Dzyaloshinskii-Moriya and triple-product interactions.