Abstract
We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.
Highlights
Bethe root which corresponds to a boundary mode and that we call boundary root. The latter is localized, up to exponentially small corrections in L, at the zero of one of the two boundary factors appearing in the Bethe equations, and represents a collective magnonic excitation pinned at one of the two edges of the chain, whose wave function has exponentially decreasing tails away from the boundary [18, 19]. We show that this boundary mode is responsible for the ground state degeneracy, which in particular has two main physical consequences: 1. The boundary magnetization in the ground state for even size L depends on the value of both boundary fields, even in the infinite chain limit L → ∞
In this paper we have shown that the physics of the open XXZ chain at zero temperature and in the antiferromagnetic regime ∆ > 1 is strongly influenced by the presence of its boundary modes
We have shown that for chains of even size there exists a regime, |h±| < h(c1r), in which such a boundary root is present both in the Bethe solutions for the ground state and for the first excited state
Summary
The study of condensed matter theory involves understanding many-body systems starting from their elementary constituents. The latter is localized, up to exponentially small corrections in L, at the zero of one of the two boundary factors appearing in the Bethe equations, and represents a collective magnonic excitation pinned at one of the two edges of the chain, whose wave function has exponentially decreasing tails away from the boundary [18, 19] We show that this boundary mode is responsible for the ground state degeneracy, which in particular has two main physical consequences: 1.
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