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We investigate the R\'enyi entropy of the excited states produced by thecurrent and its derivatives in the two-dimensional free massless non-compactbosonic theory, which is a two-dimensional conformal field theory. We alsostudy the subsystem Schatten distance between these states. The two-dimensionalfree massless non-compact bosonic theory is the continuum limit of the finiteperiodic gapless harmonic chains with the local interactions. We identify theexcited states produced by current and its derivatives in the massless bosonictheory as the single-particle excited states in the gapless harmonic chain. Wecalculate analytically the second R\'enyi entropy and the second Schattendistance in the massless bosonic theory. We then use the wave functions of theexcited states and calculate the second R\'enyi entropy and the second Schattendistance in the gapless limit of the harmonic chain, which match perfectly withthe analytical results in the massless bosonic theory. We verify that in thelarge momentum limit the single-particle state R\'enyi entropy takes auniversal form. We also show that in the limit of large momenta and largemomentum difference the subsystem Schatten distance takes a universal form butit is replaced by a corrected form when the momentum difference is small.Finally we also comment on the mutual R\'enyi entropy of two disjoint intervalsin the excited states of the two-dimensional free non-compact bosonic theory.

Excited state R\'enyi entropy and subsystem distance in two-dimensional non-compact bosonic theory -- I. Single-particle states