152 research outputs found
Semi-classical analysis of the inner product of Bethe states
We study the inner product of two Bethe states, one of which is taken
on-shell, in an inhomogeneous XXX chain in the Sutherland limit, where the
number of magnons is comparable with the length L of the chain and the magnon
rapidities arrange in a small number of macroscopically large Bethe strings.
The leading order in the large L limit is known to be expressed through a
contour integral of a dilogarithm. Here we derive the subleading term. Our
analysis is based on a new contour-integral representation of the inner product
in terms of a Fredholm determinant. We give two derivations of the sub-leading
term. Besides a direct derivation by solving a Riemann-Hilbert problem, we give
a less rigorous, but more intuitive derivation by field-theoretical methods.
For that we represent the Fredholm determinant as an expectation value in a
Fock space of chiral fermions and then bosonize. We construct a collective
field for the bosonized theory, the short wave-length part of which may be
evaluated exactly, while the long wave-length part is amenable to a
expansion. Our treatment thus results in a systematic 1/L expansion of
structure factors within the Sutherland limit.Comment: 22 pages, 0 figure
CFT and topological recursion
We study the quasiclassical expansion associated with a complex curve. In a
more specific context this is the 1/N expansion in U(N)-invariant matrix
integrals. We compare two approaches, the CFT approach and the topological
recursion, and show their equivalence. The CFT approach reformulates the
problem in terms of a conformal field theory on a Riemann surface, while the
topological recursion is based on a recurrence equation for the observables
representing symplectic invariants on the complex curve. The two approaches
lead to two different graph expansions, one of which can be obtained as a
partial resummation of the other.Comment: Minor correction
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