3,689 research outputs found
On condensation properties of Bethe roots associated with the XXZ chain
I prove that the Bethe roots describing either the ground state or a certain
class of "particle-hole" excited states of the XXZ spin- chain in any
sector with magnetisation exist and form, in the
infinite volume limit, a dense distribution on a subinterval of .
The results holds for any value of the anisotropy . In fact, I
establish an even stronger result, namely the existence of an all order
asymptotic expansion of the counting function associated with such roots. As a
corollary, these results allow one to prove the existence and form of the
infinite volume limit of various observables attached to the model -the
excitation energy, momentum, the zero temperature correlation functions, so as
to name a few- that were argued earlier in the literature.Comment: 54 pages, 2 figures. Some details in proof adde
On lacunary Toeplitz determinants
By using Riemann--Hilbert problem based techniques, we obtain the asymptotic
expansion of lacunary Toeplitz determinants generated by holomorhpic symbols, where (resp. )
except for a finite subset of indices (resp. ). In addition to the usual Szeg\"{o} asymptotics, our answer involves a
determinant of size .Comment: 11 page
Fine structure of the asymptotic expansion of cyclic integrals
The asymptotic expansion of -dimensional cyclic integrals was expressed as
a series of functionals acting on the symmetric function involved in the cyclic
integral. In this article, we give an explicit formula for the action of these
functionals on a specific class of symmetric functions. These results are
necessary for the computation of the O(1) part in the long-distance asymptotic
behavior of correlation functions in integrable models.Comment: 13 page
Surface free energy of the open XXZ spin-1/2 chain
We study the boundary free energy of the XXZ spin-\tf{1}{2} chain subject
to diagonal boundary fields. We first show that the representation for its
finite Trotter number approximant obtained by Bortz, Frahm and G\"{o}hmann is
related to the partition function of the six-vertex model with reflecting ends.
Building on the Tsuchiya determinant representation for the latter quantity we
are able to take the infinite Trotter number limit. This yields a
representation for the surface free energy which involves the solution of the
non-linear integral equation that governs the thermodynamics of the XXZ
spin-1/2 chain subject to periodic boundary conditions. We show that this
integral representation allows one to extract the low- asymptotic behavior
of the boundary magnetization at finite external magnetic field on the one hand
and numerically plot this function on the other hand.Comment: 35 pages, 11 figures, V3: some new plots adde
Asymptotic behaviour of two-point functions in multi-species models
We extract the long-distance asymptotic behaviour of two-point correlation
functions in massless quantum integrable models containing multi-species
excitations. For such a purpose, we extend to these models the method of a
large-distance regime re-summation of the form factor expansion of correlation
functions. The key feature of our analysis is a technical hypothesis on the
large-volume behaviour of the form factors of local operators in such models.
We check the validity of this hypothesis on the example of the
-invariant XXX magnet by means of the determinant representations for
the form factors of local operators in this model. Our approach confirms the
structure of the critical exponents obtained previously for numerous models
solvable by the nested Bethe Ansatz.Comment: 45 pages, 1 figur
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