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Multi-point Correlation Functions in the Boundary XXZ Chain at Finite Temperature

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Abstract

We consider multi-point correlation functions in the open XXZ chain with longitudinal boundary fields and in a uniform external magnetic field. We show that, at finite temperature, these correlation functions can be written in the quantum transfer matrix framework as sums over thermal form factors. More precisely, and quite remarkably, each term of the sum is given by a simple product of usual matrix elements of the quantum transfer matrix multiplied by a unique factor containing the whole information about the boundary fields. As an example, we provide a detailed expression for the longitudinal spin one-point functions at distance m from the boundary. This work thus solves the long-standing problem of setting up form factor expansions in integrable models subject to open boundary conditions.

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Notes

  1. In particular, in case singular roots are present, one is in a situation where the functions \(1+\mathfrak {a}_{{\mathbb {Y}}}\) have additional poles inside of the contour located at the singular roots. This gives rise to new contributions upon applying the product/sum \(\hookrightarrow \) integral vs \( \mathfrak {a}_{{\mathbb {Y}}}^{\prime } / \big ( 1 + \mathfrak {a}_{{\mathbb {Y}}} \big ) \) technique described later in the paper. Hence, this would change the fine details of the expression for the infinite Trotter number limit of the bulk or boundary form factors appearing in (3.15). We, however, do not expect that this would impact on the overall properties of the site-m magnetisation and, in particular, on its low-T limit where one expects the appearance of a universal behaviour in the large-distance regime. Moreover, as demonstrated rigorously in [37], singular roots do not exist in the low-T limit for \(0\le \Delta <1\) and there are very good indications that the statement also holds true for \(-1<\Delta <0\).

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Acknowledgements

K.K.K. and V.T. acknowledge support from CNRS and are indebted to F. Göhmann for stimulating discussions.

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Authors and Affiliations

  1. Laboratoire de Physique, Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, 69342, Lyon, France

    Karol K. Kozlowski

  2. Université Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France

    Véronique Terras

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  1. Karol K. Kozlowski
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  2. Véronique Terras
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Correspondence to Karol K. Kozlowski.

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Kozlowski, K.K., Terras, V. Multi-point Correlation Functions in the Boundary XXZ Chain at Finite Temperature. Ann. Henri Poincaré 25, 1007–1046 (2024). https://doi.org/10.1007/s00023-023-01310-4

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