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.
We investigate the quasiparticle excited state R\'enyi entropy of single, double and more intervals in the fermionic chain, bosonic chain and XY chain. When the gap of the theory is large or all the momenta of the quasiparticles are large, the R\'enyi Entropy takes a super universal form, which is independent of, among others, the model, the quasiparticle momenta, and the subsystem connectedness. We calculate analytically the R\'enyi Entropy in the extremely gapped limit by writing the excited states in terms of the local excitations and find different additional contributions to the super universal R\'enyi entropy in various models. The additional contributions to the super universal R\'enyi entropy cannot be neglected when the momentum differences of the excited quasiparticles are small, however, they are negligible when all the momentum differences are large. The R\'enyi entropy with additional terms derived in the extremely gapped limit is universal in the sense that it is valid in the slightly gapped and critical models as long as all the momenta of the excited quasiparticles are large. In the case of double interval in the XY chain we find new exact super universal as well as the universal results. In the case of the bosonic chain in the extremely massive limit we find analytically a novel formula, written as a permanent of a certain matrix, for the R\'enyi entropy. We support all of our analytical results with numerical calculations.
The Rényi entanglement entropies of quasiparticle excitations in the many-body gapped systems show a remarkable universal picture which is independent of the model, the quasiparticle momenta, and the connectedness of the subsystem. In this letter we calculate exactly the single-interval and double-interval Rényi entanglement entropies of quasiparticle excitations in the many-body gapped fermions, bosons, and XY chains and find additional contributions to the universal Rényi entanglement entropy in the excited states with quasiparticles of different momenta. The additional terms are different in different models, depend on the momentum differences of the quasiparticles, and are different for the single interval and the double interval. We derive the analytical Rényi entanglement entropy in the extremely gapped limit, matching perfectly the numerical results as long as either the intrinsic correlation length of the model or all the de Broglie wavelengths of the quasiparticles are small. When the momentum difference of any pair of distinct quasiparticles is small, the additional terms are non-negligible. We argue that the derived formulas can be applied to a wider range of models than those discussed here.
We report on a systematic replica approach to calculate the subsystem trace distance for a generic quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero, P. Calabrese, Phys. Rev. Lett. 122, 141602 (2019)], of which this work is a completion. The trace distance between two reduced density matrices
ρA and
σA is obtained from the moments
tr(ρA−σA)n and taking the limit
n→1 of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length
ℓ embedded in the low lying eigenstates of a one-dimensional critical system of length
L, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a {\it scale invariant universal function} of
ℓ/L. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten
n-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of
ℓ/L. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.
Abstract We investigate the subsystem Schatten distance, trace distance and fidelity between the quasiparticle excited states of the free and the nearest-neighbor coupled fermionic and bosonic chains and the ferromagnetic phase of the spin-1/2 XXX chain. The results support the scenario that in the scaling limit when one excited quasiparticle has a large energy it decouples from the ground state and when two excited quasiparticles have a large momentum difference they decouple from each other. From the quasiparticle picture, we get the universal subsystem distances that are valid when both the large energy condition and the large momentum difference condition are satisfied, by which we mean each of the excited quasiparticles has a large energy and the momentum difference of each pair of the excited quasiparticles is large. In the free fermionic and bosonic chains, we use the subsystem mode method and get efficiently the subsystem distances, which are also valid in the coupled fermionic and bosonic chains if the large energy condition is satisfied. Moreover, under certain limit the subsystem distances from the subsystem mode method are even valid in the XXX chain. We expect that the results can be also generalized for other integrable models.
Abstract We study the Rényi entropy and subsystem distances on one interval for the finite size and thermal states in the critical XY chains, focusing on the critical Ising chain and XX chain with zero transverse field. We construct numerically the reduced density matrices and calculate the von Neumann entropy, Rényi entropy, subsystem trace distance, Schatten two-distance, and relative entropy. As the continuum limit of the critical Ising chain and XX chain with zero field are, respectively, the two-dimensional free massless Majorana and Dirac fermion theories, which are conformal field theories, we compare the spin chain numerical results with the analytical results in conformal field theories and find perfect matches in the continuum limit.