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A boundary-field gap for the spin-one Heisenberg chain

OpenAI

Published Oct 6, 2026• 0 citations
Publisher

Abstract

We prove a uniform spectral gap for odd open spin-one antiferromagnetic Heisenberg chains with the same fixed magnetic field at both endpoints. The field h=3/5 gives a gap greater than \log(10)/392 on every chain of 2L+1 sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.

DOI
10.83295/x86g-78g4

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We prove a uniform spectral gap for odd open spin-one antiferromagnetic Heisenberg chains with the same fixed magnetic field at both endpoints. The field h=3/5 gives a gap greater than \log(10)/392 on every chain of 2L+1 sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.
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On the uniqueness of translation-invariant ground states of the S=1S=1S=1 Heisenberg and JJJ--DDD antiferromagnetic chains
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Linear Heisenberg antiferromagnets (HAFs) are chains of spin-SSS sites with isotropic exchange JJJ between neighbors. Open and periodic boundary conditions return the same ground-state energy per site in the thermodynamic limit, but not the same spin SG{S}_{G}SG​ when $S\ensuremath{\ge}1$. The ground state of open chains of NNN spins has SG=0{S}_{G}=0SG​=0 or SSS, respectively, for even or odd NNN. Density-matrix renormalization-group calculations with different algorithms for even and odd NNN are presented up to N=500N=500N=500 for the energy and spin densities $\ensuremath{\rho}(r,N)$ of edge states in HAFs with S=1S=1S=1, 3/2, and 2. The edge states are boundary-induced spin density waves (BI-SDWs) with $\ensuremath{\rho}(r,N)\ensuremath{\propto}{(\ensuremath{-}1)}^{r\ensuremath{-}1}$ for r=1,2,...,Nr=1,2,...,Nr=1,2,...,N. The SDWs are in phase when NNN is odd, are out of phase when NNN is even, and have finite excitation energy $\mathrm{\ensuremath{\Gamma}}(N)$ that decreases exponentially with NNN for integer SSS and faster than 1/N1/N1/N for half integer SSS. The spin densities and excitation energy are quantitatively modeled for integer SSS chains longer than $5\ensuremath{\xi}$ spins by two parameters, the correlation length $\ensuremath{\xi}$ and the SDW amplitude, with $\ensuremath{\xi}=6.048$ for S=1S=1S=1 and 49.0 for S=2S=2S=2. The BI-SDWs of S=3/2S=3/2S=3/2 chains are not localized and are qualitatively different for even and odd NNN. Exchange between the ends for odd NNN is mediated by a delocalized effective spin in the middle that increases $|\mathrm{\ensuremath{\Gamma}}(N)|$ and weakens the size dependence. The nonlinear sigma model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$) has been applied to the HAFs, primarily to S=1S=1S=1 with even NNN, to discuss spin densities and exchange between localized states at the ends as $\mathrm{\ensuremath{\Gamma}}(N)\ensuremath{\propto}{(\ensuremath{-}1)}^{N}exp(\ensuremath{-}N/\ensuremath{\xi})$. S=1S=1S=1 chains with odd NNN are fully consistent with the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$; S=2S=2S=2 chains have two gaps $\mathrm{\ensuremath{\Gamma}}(N)$ with the same $\ensuremath{\xi}$ as predicted whose ratio is 3.45 rather than 3; the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ is more approximate for S=3/2S=3/2S=3/2 chains with even NNN and is modified for exchange between ends for odd NNN.