Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory. Part II. Multi-particle states

Abstract

We study the excited state Rényi entropy and subsystem Schatten distance in the two-dimensional free massless non-compact bosonic field theory, which is a conformal field theory. The discretization of the free non-compact bosonic theory gives the harmonic chain with local couplings. We consider the field theory excited states that correspond to the harmonic chain states with excitations of more than one quasiparticle, which we call multi-particle states. This extends the previous work by the same authors to more general excited states. In the field theory we obtain the exact Rényi entropy and subsystem Schatten distance for several low-lying states. We obtain short interval expansion of the Rényi entropy and subsystem Schatten distance for general excited states, which display different universal scaling behaviors in the gapless and extremely gapped limits of the non-compact bosonic theory. In the locally coupled harmonic chain we calculate numerically the excited state Rényi entropy and subsystem Schatten distance using the wave function method. We find excellent matches of the analytical results in the field theory and numerical results in the gapless limit of the harmonic chain. We also make some preliminary investigations of the Rényi entropy and the subsystem Schatten distance in the extremely gapped limit of the harmonic chain.

Entanglement of magnon excitations in spin chains

“…Using exact calculations on free fermions and bosons, i.e. Hamiltonian being equal to the number operator, these results were later confirmed and generalized in [50,52,54,55]. In particular it was shown that for the excited states with more than one mode excited in the most generic circumstances apart from the universal terms there are extra terms.…”

“…We could also calculate the Schatten distance from the wave function method [41,42]. One could also see [54,55]. With the number operator (3.2) as the Hamiltonian, there sets up a permanent formula for the Rényi entropy [54]…”

Entanglement of magnon excitations in spin chains

“…Hamiltonian being equal to the number operator, these results were later confirmed and generalized in [50,52,54,55]. In particular it was shown that for the excited states with more than one mode excited in the most generic circumstances apart from the universal terms there are extra terms.…”

Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory -- I. Single-particle states

“…To calculate the same quantities efficiently for multi-particle excited states of the harmonic chains one needs to use the full-fledged wave function method as in [59,60], and in the corresponding CFT one needs to consider higher level descendant states. We hope to come back to this problem in the future [93].…”