Abstract
We present a conjecture for the exact expression of finite volume expectation values in excited states in integrable quantum field theories, which is an extension of an earlier conjecture to the case of general diagonal factorized scattering with bound states and a nontrivial bootstrap structure. The conjectured expression is a spectral expansion which uses the exact form factors and the excited state thermodynamic Bethe Ansatz as building blocks. The conjecture is proven for the case of the trace of the energy-moment tensor. Concerning its validity for more general operators, we provide numerical evidence using the truncated conformal space approach. It is found that the expansion fails to be well-defined for small values of the volume in cases when the singularity structure of the TBA equations undergoes a non-trivial rearrangement under some critical value of the volume. Despite these shortcomings, the conjectured expression is expected to be valid for all volumes for most of the excited states, and as an expansion above the critical volume for the rest.
Highlights
Another method to obtain finite temperature correlators is based on finite temperature form factors [12, 13]; so far, this approach seems limited to free theories such as the Ising model
We present a conjecture for the exact expression of finite volume expectation values in excited states in integrable quantum field theories, which is an extension of an earlier conjecture to the case of general diagonal factorized scattering with bound states and a nontrivial bootstrap structure
The finite volume form factor formalism introduced in [19, 20] only included the corrections that decay with a power of the volume; exponential corrections were neglected
Summary
The Leclair-Mussardo series for the finite volume vacuum expectation value of a local operator in an integrable model with diagonal scattering and k species of massive particles takes the following form [1]:. Let us state the conjecture for the general form of the finite volume expectation values in excited states It contains two kind of quantities, the “dressed version” of the diagonal form factors and the densities of the active singularities. Excited TBA systems of the form (2.6), (2.7) are much more general with the imaginary parts of the singularity positions themselves being dynamically determined by the quantization conditions (2.7) This corresponds to finite volume corrections known as μ-terms, which were considered in the form factor context in [35]. The “dressed” diagonal form factors reduce to connected diagonal form factors, and for theories where particles are represented by a single active singularity, formula (2.15) reduces to the results obtained in [19, 20] for the finite-volume diagonal matrix elements, which are valid up to exponential corrections in the volume. First we explicitly evaluate the TBA prediction for Θ , recast it in a form which can be matched with the dependence of (2.15) on the densities, and prove that the rest of the formula matches the dressed form factors of Θ
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