Modular Hamiltonian of the massive scalar field on the half line: A numerical approach

TL;DR

This work numerically analyzes the modular Hamiltonian of an interval for the ground state of a massive free scalar on the half-line with Robin boundary conditions, focusing on two configurations: intervals adjacent to the boundary and intervals separated from it. Building on the continuum one-particle framework and the two-block modular Hamiltonian structure, the authors implement a robust discretization of the operator $D^{-1/4}$ and related kernels, enabling arcoth-based construction of the modular Hamiltonian blocks and their smear against log-Gaussian test functions. The results show that, in the massless case, the modular Hamiltonian is local for Dirichlet bc but non-local for Neumann and Robin bc near the boundary, while in the massive regime the adjacent-interval Hamiltonian remains non-local across bc, tending toward a local linear form at large mass. For intervals separated from the boundary, non-locality is pervasive in both massless and massive regimes, with boundary effects diminishing as the interval distance grows. Overall, the paper extends a continuum-based numerical method to include boundaries, providing quantitative insights into locality properties that align with BCFT expectations in special limits and reveal rich non-local structure in general massive settings.

Abstract

We study the modular Hamiltonian of an interval for the ground state of a massive free scalar field on the half line with Robin boundary conditions, by employing a numerical method. When the interval is adjacent to the boundary, we find numerical evidence that the modular Hamiltonian is non-local, except for the limiting cases of the massless scalar satisfying either Dirichlet or Neumann boundary conditions. When the interval is separated from the boundary, the numerical analysis indicates that the modular Hamiltonian is non-local for all these boundary conditions and any value of the mass.

Figures (21)

• Figure 2.1: Two sets of functions on the half line occurring in the numerical method discussed in Fig. \ref{['sec:HalfLine.NumericalMethod']}, given in \ref{['eq:DiscretizationBasis']} (red lines) and \ref{['eq:LogGaussian']} (blue lines).
• Figure 2.1: Contours and poles in the complex momentum plane occurring in the Fourier-type integral of $D^{-1}$ (inverse of the modified Helmholtz operator) for the massive regime.
• Figure 2.2: Scheme for the construction of the kernels through the analytic approach of QFT (top) and the numerical approach employed in this work (bottom).
• Figure 3.1: Interval adjacent to the boundary, massless regime for either Dirichlet b.c. (left) or Neumann b.c. (right). Output of the numeric algorithm obtained with discretization parameters $n = 256$ and $\varLambda = 16 \ell$. The top (bottom) panels report the data before (after) the smearing procedure (see \ref{['eq:HalfLine.ModularHamiltonian.SmearedBlock.CompactNotation']}) through the log-Gaussian test functions \ref{['eq:LogGaussian.Discretization']}.
• Figure 3.1: Contours and poles for $D^{-\frac{1}{2}}_{\textrm{\tiny bdy}}$ and $D^{-\frac{1}{4}}_{\textrm{\tiny bdy}}$ in the complex plane of the rescaled momentum, in the left and right panel respectively.