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arXiv:1908.09704

Finite volume corrections of non-diagonal form factors

István Vona

Published Aug 26, 2019• 0 citations
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Abstract

This thesis presents Lüscher's $μ$- and $F$-term corrections to volume dependence of non-diagonal finite volume form factors in the scaling Lee-Yang model. An explicit calculation proves the suspected relation that the $μ$-terms known previously from bound state quantization can be obtained from the $F$-term integrals by modifying the contour of integration such that it picks up residues of appropriate poles in the integrand. The fact that these two different approaches for getting the $μ$-terms give the same result underpins the formal derivation of the $F$-term in arXiv:1904.00492 which was not known until recently. In the meantime, the notions of integrable quantum field theories and those related to their treatment in finite volume are introduced to help understand the topic for readers not familiar with it.

DOI
10.48550/arxiv.1908.09704

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Abstract We derive the leading exponential finite volume corrections in two dimensional integrable models for non-diagonal form factors in diagonally scattering theories. These formulas are expressed in terms of the infinite volume form factors and scattering matrices. If the particles are bound states then the leading exponential finite-size corrections (μ-terms) are related to virtual processes in which the particles disintegrate into their constituents. For non-bound state particles the leading exponential finite-size corrections (F-terms) come from virtual particles traveling around the finite world. In these F-terms a specifically regulated infinite volume form factor is integrated for the momenta of the virtual particles. The F-term is also present for bound states and the μ-term can be obtained by taking an appropriate residue of the F-term integral. We check our results numerically in the Lee-Yang and sinh-Gordon models based on newly developed Hamiltonian truncations.
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In this thesis we investigate finite size effects in 1+1 dimensional integrable QFT. In particular we consider matrix elements of local operators (finite volume form factors) and vacuum expectation values and correlation functions at finite temperature. In the first part of the thesis we give a complete description of the finite volume form factors in terms of the infinite volume form factors (solutions of the bootstrap program) and the S-matrix of the theory. The calculations are correct to all orders in the inverse of the volume, only exponentially decaying (residual) finite size effects are neglected. We also consider matrix elements with disconnected pieces and determine the general rule for evaluating such contributions in a finite volume. The analytic results are tested against numerical data obtained by the truncated conformal space approach in the Lee-Yang model and the Ising model in a magnetic field. In a separate section we also evaluate the leading exponential correction (the μμμ-term) associated to multi-particle energies and matrix elements. In the second part of the thesis we show that finite volume factors can be used to derive a systematic low-temperature expansion for correlation functions at finite temperature. In the case of vacuum expectation values the series is worked out up to the third non-trivial order and a complete agreement with the LeClair-Mussardo formula is observed. A preliminary treatment of the two-point function is also given by considering the first nontrivial contributions.
In this paper we present sets of linear integral equations which make possible to compute the finite volume expectation values of the trace of the stress energy tensor (Θ) and the U ( 1 ) current ( J μ ) in any eigenstate of the Hamiltonian of the sine-Gordon model. The solution of these equations in the large volume limit allows one to get exact analytical formulas for the expectation values in the Bethe-Yang limit. These analytical formulas are used to test an earlier conjecture for the Bethe-Yang limit of expectation values in non-diagonally scattering theories. The analytical tests have been carried out upto three particle states and gave agreement with the conjectured formula, provided the definition of polarized symmetric diagonal form-factors is modified appropriately. Nevertheless, we point out that our results provide only a partial confirmation of the conjecture, and further investigations are necessary to confirm all of its subtleties. The most important missing piece in the confirmation is the mathematical proof of the finiteness of the symmetric diagonal limit of form-factors in a non-diagonally scattering theory.