Quantum field theories in finite volume: Excited state energies

Abstract

We develop a method of computing the excited state energies in Integrable Quantum Field Theories (IQFT) in finite geometry, with spatial coordinate compactified on a circle of circumference R. The IQFT "commuting transfer-matrices" introduced in [1] for Conformal Field Theories (CFT) are generalized to non-conformal IQFT obtained by perturbing CFT with the operator Φ 1,3 . We study the models in which the fusion relations for these "transfer-matrices" truncate and provide closed integral equations which generalize the equations of Thermodynamic Bethe Ansatz to excited states. The explicit calculations are done for the first excited state in the "Scaling Lee-Yang Model".We have also shown that the "fusion relations" for the operators T j (λ)1 These operators appear as the continuous QFT versions of Baxter's commuting transfermatrices [7], [8] and therefore we maintain using this term although the original meaning of the term "transfer-matrix" here is apparently lost.

T T ¯ $$ \mathrm{T}\overline{\mathrm{T}} $$ -deformed 2D quantum field theories

“…Equations describing excited states have the same form but in general with deformed contours encircling a number of singularities {θ ± i } with 1 + e f ± (θ ± i ) = 0, see [37][38][39][40]. It is not difficult to check that E(R) and P (R) have exactly the form predicted by conformal invariance.…”

The full Quantum Spectral Curve for AdS4/CFT3

“…These equations were solved numerically for a particular operator in [39]. However, solving excited states TBA equations with high precision is a challenging task already for very simple models [40][41][42]. Besides, the form of the TBA equations depends on the state and possibly also on the range of the coupling considered, so that they can be studied only on a case-by-case basis.…”

Quench echo and work statistics in integrable quantum field theories

“…Since the TBA is an integral equation representation, we a priori expect that the analytical continuation would give the correct Loschmidt echo. Indeed, a similar approach has been successfully used before for the calculation of excitation energies in IQFT [109][110][111][112]. We perform this calculation in the sinh-Gordon model, arguably the simplest relativistic IQFT with non-trivial S-matrix.…”