Abstract
We investigate the Rényi entropy of the excited states produced by the current and its derivatives in the two-dimensional free massless non-compact bosonic theory, which is a two-dimensional conformal field theory. We also study the subsystem Schatten distance between these states. The two-dimensional free massless non-compact bosonic theory is the continuum limit of the finite periodic gapless harmonic chains with the local interactions. We identify the excited states produced by current and its derivatives in the massless bosonic theory as the single-particle excited states in the gapless harmonic chain. We calculate analytically the second Rényi entropy and the second Schatten distance in the massless bosonic theory. We then use the wave functions of the excited states and calculate the second Rényi entropy and the second Schatten distance in the gapless limit of the harmonic chain, which match perfectly with the analytical results in the massless bosonic theory. We verify that in the large momentum limit the single-particle state Rényi entropy takes a universal form. We also show that in the limit of large momenta and large momentum difference the subsystem Schatten distance takes a universal form but it is replaced by a new corrected form when the momentum difference is small. Finally we also comment on the mutual Rényi entropy of two disjoint intervals in the excited states of the two-dimensional free non-compact bosonic theory.
Highlights
Harmonic chain basics: ground and single-particle statesWe review the textbook properties of the discrete version of the 2D free massive bosonic theory, i.e. the harmonic chains with the local couplings, which will help us to fix the notation
Rényi entropy of a length interval on a one-dimensional infinity gapless system in the ground state takes the logarithmic formula [10, 15, 18, 19, 21]
We investigate the Rényi entropy of the excited states produced by the current and its derivatives in the two-dimensional free massless non-compact bosonic theory, which is a two-dimensional conformal field theory
Summary
We review the textbook properties of the discrete version of the 2D free massive bosonic theory, i.e. the harmonic chains with the local couplings, which will help us to fix the notation. We consider the 2D free non-compact bosonic theory with the Lagrangian density. With the metric ημν = diag(−1, 1), derivatives ∂μ = (∂t, ∂u), real temporal coordinate t, spatial coordinate u, and the mass (or equivalently gap) m. The energy eigenstates can be obtained by applying the raising operators on the ground state. In this paper we only consider the states with the excitation of only one quasiparticle. The wave function of the single-particle state |k is. The density matrix of the total system for the single-particle state is. Where Q|ρG|Q is the ground state density matrix (2.16) and Vk = vkvk† is an L × L hermitian matrix
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