SciPost Phys. 10, 006 (2021) ·
published 12 January 2021

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We explain how to compute correlation functions at zero temperature within
the framework of the quantum version of the Separation of Variables (SoV) in
the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted
(quasiperiodic) boundary conditions. We first detail all steps of our method
in the case of antiperiodic boundary conditions. The model can be solved in
the SoV framework by introducing inhomogeneity parameters. The action of local
operators on the eigenstates are then naturally expressed in terms of multiple
sums over these inhomogeneity parameters. We explain how to transform these
sums over inhomogeneity parameters into multiple contour integrals. Evaluating
these multiple integrals by the residues of the poles outside the integration
contours, we rewrite this action as a sum involving the roots of the Baxter
polynomial plus a contribution of the poles at infinity. We show that the
contribution of the poles at infinity vanishes in the thermodynamic limit, and
that we recover in this limit for the zerotemperature correlation functions
the multiple integral representation that had been previously obtained through
the study of the periodic case by Bethe Ansatz or through the study of the
infinite volume model by the qvertex operator approach. We finally show that
the method can easily be generalized to the case of a more general nondiagonal
twist: the corresponding weights of the different terms for the correlation
functions in finite volume are then modified, but we recover in the
thermodynamic limit the same multiple integral representation than in the
periodic or antiperiodic case, hence proving the independence of the
thermodynamic limit of the correlation functions with respect to the particular
form of the boundary twist.
Sebastian Grijalva, Jacopo De Nardis, Veronique Terras
SciPost Phys. 7, 023 (2019) ·
published 20 August 2019

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We study the open XXZ spin chain in the antiferromagnetic 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
halfinfinite 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 quasidegenerate 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.