  We analyze the bootstrap approach (a dual optimization method to the
variational approach) to one-dimensional spin chains, leveraging semidefinite
programming to extract numerical results. We study how correlation functions in
the ground state converge to their true values at and away from criticality and
at relaxed optimality. We consider the transverse Ising model, the three state
Potts model, and other non-integrable spin chains and investigate to what
extent semidefinite methods can reliably extract numerical emergent physical
data, including conformal central charges, correlation lengths and scaling
dimensions. We demonstrate procedures to extract these data and show
preliminary results in the various models considered. We compare to exact
analytical results and to exact diagonalization when the system volume is small
enough. When we attempt to go to the thermodynamic limit, the semidefinite
numerical method with translation invariance imposed as a constraint finds the
solution with periodic boundary conditions even if these have not been
specified. This implies that the determination of all conformal data in
correlators has to be handled at finite volume. Our investigation reveals that
the approach has practical challenges. In particular, the correlation functions
extracted from the optimal solution, which function as slack variables in the
optimization, have convergence issues that suggest an underlying exponential
complexity in the system size.
