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Analytical results on the Heisenberg spin chain in a magnetic field | Etienne Granet
; Jesper Lykke Jacobsen
; Hubert Saleur
; | Date: |
17 Jan 2019 | Abstract: | We obtain the ground state magnetization of the Heisenberg and XXZ spin
chains in a magnetic field $h$ as a series in $sqrt{h_c-h}$, where $h_c$ is
the smallest field for which the ground state is fully polarized. All the
coefficients of the series can be computed in closed form through a recurrence
formula that involves only algebraic manipulations. The radius of convergence
of the series in the full range $0<hleq h_c$ is studied numerically.
To that end we express the free energy at mean magnetization per site
$-1/2leq langle sigma^z_i
angleleq 1/2$ as a series in $1/2-langle
sigma^z_i
angle$ whose coefficients can be similarly recursively computed in
closed form. This series converges for all $0leq langle sigma^z_i
angleleq
1/2$. The recurrence is nothing but the Bethe equations when their roots are
written as a double series in their corresponding Bethe number and in
$1/2-langle sigma^z_i
angle$. It can also be used to derive the corrections
in finite size, that correspond to the spectrum of a free compactified boson
whose radius can be expanded as a similar series.
The method presumably applies to a large class of models: it also
successfully applies to a case where the Bethe roots lie on a curve in the
complex plane. | Source: | arXiv, 1901.5878 | Services: | Forum | Review | PDF | Favorites |
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