Frank Göhmann, Karol K. Kozlowski, Jesko Sirker, Junji Suzuki
SciPost Phys. 12, 158 (2022) ·
published 12 May 2022

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We present a series representation for the dynamical twopoint function of
the local spin current for the XXZ chain in the antiferromagnetic massive
regime at zero temperature. From this series we can compute the correlation
function with very high accuracy up to very long times and large distances.
Each term in the series corresponds to the contribution of all scattering
states of an even number of excitations. These excitations can be interpreted
in terms of an equal number of particles and holes. The lowest term in the
series comprises all scattering states of one hole and one particle. This term
determines the longtime largedistance asymptotic behaviour which can be
obtained explicitly from a saddlepoint analysis. The spacetime Fourier
transform of the twopoint function of currents at zero momentum gives the
optical spin conductivity of the model. We obtain highly accurate numerical
estimates for this quantity by numerically Fourier transforming our data. For
the oneparticle, onehole contribution, equivalently interpreted as a
twospinon contribution, we obtain an exact and explicit expression in terms of
known special functions. For large enough anisotropy, the twospinon
contribution carries most of the spectral weight, as can be seen by calculating
the fsum rule.
SciPost Phys. Lect. Notes 16 (2020) ·
published 17 August 2020

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This script is based on the notes the author prepared to give a set of six
lectures at the Les Houches School "Integrability in Atomic and Condensed
Matter Physics" in the summer of 2018. The school had its focus on the
application of integrability based methods to problems in nonequilibrium
statistical mechanics. The lectures were meant to complement this subject with
background material on the equilibrium statistical mechanics of quantum spin
chains from a vertex model perspective. The author was asked to provide a
minimal introduction to quantum spin systems including notions like the reduced
density matrix and correlation functions of local observables. He was further
asked to explain the graphical language of vertex models and to introduce the
concepts of the Trotter decomposition and the quantum transfer matrix. This was
basically the contents of the first four lectures presented at the school. In
the remaining two lectures these notions were filled with life by deriving an
integral representation of the free energy per lattice site for the
HeisenbergIsing chain (alias XXZ model) using techniques based on nonlinear
integral equations.
Dr Göhmann: "Thank you for your positive an..."
in Submissions  report on Statistical mechanics of integrable quantum spin systems