Exact finite volume expectation values of local operators in excited states

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.

Exact finite volume expectation values of $$ \overline{\varPsi}\varPsi $$ in the massive Thirring model from light-cone lattice correlators

“…The reason for this is that formally the NLIE equations (2.34) are very similar to the Thermodynamic Bethe Ansatz (TBA) equations of a purely diagonally scattering theory of two types of particles. This analogy, the actual form of the large volume series of the U(1) current of the theory [17] together with the all order large volume series conjectures for the diagonal form factors of purely elastic scattering theories [10,11,[14][15][16], led to the following large volume series conjecture for the diagonal multi-fermion (soliton) expectation values of local operators in the MT (SG) models [17]:…”

“…The process of the evaluation of (5.1) and (5.2) in the large volume limit is very similar to the method used for computing the diagonal matrix elements of the trace of the stressenergy tensor in purely elastic scattering theories [11]. The reason for this is that formally the NLIE equations (2.34) are very similar to the Thermodynamic Bethe Ansatz (TBA) equations of a purely diagonally scattering theory of two types of particles.…”

“…Though the structure of exponentially small in volume corrections for the non-diagonal matrix elements is still unknown, inspired by [14] for the diagonal matrix elements, a nice series representation was proposed in [10,11]. However the proposal is valid only to purely elastic scattering theories and its extension to non-diagonally scattering theories is still unknown in general.…”

“…the sum in (5.4) runs for all bipartite partitions of the rapidities of the sandwiching state: .4) is called the dressed form-factor [11] and it is given by an infinite sum in terms of the connected diagonal form-factors of the theory: (H 1 , H 2 , . .…”

“…In the past decade a remarkable progress has been made in the computation of finite volume form factors in integrable quantum field theories [4][5][6][7][8][9][10][11]. Most of the methods use the infinite volume form factors [12] as a starting point and the finite volume form factors are to be determined in the form of a systematic large volume series.…”