Toolkit for General 2d Scalar Potential in LCT

Abstract

We present efficient algorithms for obtaining the Hamiltonian in Lightcone Conformal Truncation (LCT) for a 2d scalar field with a generic potential. We apply this method to the sine-Gordon and sinh-Gordon models in $1+1d$, and find precise agreement with integrability results when the scaling dimension $\hat{I}”$ of the deforming cosine/cosinh potential is in the range $\hat{I}”\le 1$. The agreement provides additional evidence for a recent conjecture for how to compute the effective lightcone Hamiltonian in this class of models. In addition, to high precision, we provide the first direct confirmation for the conjectured self-duality of the sinh-Gordon model ($\hat{I}”<0)$, which relates $\hat{I}”\leftrightarrow 4/\hat{I}”$. As the dimension approaches the upper limit $\hat{I}”=1$ from below, we show analytically that the Hamiltonian matrix elements exactly reproduce those of a free Majorana fermion, demonstrating how bosonization is manifested in the LCT basis. We comment on the possible extension of the approach to $\hat{I}”>1$.

30 pages, 11 figures, v2: minor fixes, v3: minor revision