In the repulsive case, given a proper set of quantum numbers ,
the solution of the Bethe equations for the set of rapidities
exists and is unique 1969.Yang.JMP.10 due to the convexity of the
Yang-Yang action.
Furthermore all these solutions have real rapidities .
For the attractive case, the situation is completely different.
We will define as the interaction parameter; let us rewrite our Bethe equations as
Consider now a complex rapidity . The Bethe equation
for this rapidity is
We consider finite and . If , we have
on the left-hand side. Looking at the finite product on the right-hand side,
we conclude that there must thus be a rapidity such that .
On the other hand, if , we have on the left-hand side,
and there must thus be a rapidity such that .
The rapidities thus like to arrange themselves into clusters, the elements of each cluster being evenly-spaced by in the imaginary direction. Such clusters represent bound states of particles, and we will call them strings.
For a given number of atoms , we can construct eigenstates with fixed string content by partitioning
into strings of length , denoting the total number of strings as
. We clearly have
Specifically, we will parametrize the string rapidities as
with exponentially small deviations provided .
In our string notation, the index labels rapidities
within the string, and labels strings
of a given length.
We stress that perfect strings (i.e. with all the ) are
exact eigenstates in the limit with for arbitrary .
It is then natural to consider the limit at fixed .
This is different from what done in the repulsive case where the limit
at fixed density is performed. Here, the particles
remain strongly correlated and bound to one another even when .
These bound states should be viewed as individual particles of mass , with momentum and
energy of the string centered on given by
Such strings are known but not commonly discussed in the literature on the Bose gas,
since they do not appear in the repulsive case. However, their direct
equivalents exist in integrable spin chains,
where they have been extensively studied. The technology to treat them, at least on the level of
eigenstates, is thus completely standard.
Author: Jean-Sébastien Caux
Created: 2024-01-18 Thu 14:24