Entanglement entropy in long-range harmonic oscillators

Abstract

-We study the Von Neumann and Rényi entanglement entropy of long-range harmonic oscillators (LRHO) by both theoretical and numerical means. We show that the entanglement entropy in massless harmonic oscillators increases logarithmically with the sub-system size as S = c ef f 3 log l. Although the entanglement entropy of LRHO's shares some similarities with the entanglement entropy at conformal critical points we show that the Rényi entanglement entropy presents some deviations from the expected conformal behaviour. In the massive case we demonstrate that the behaviour of the entanglement entropy with respect to the correlation length is also logarithmic as the short range case.Introduction. -While the entanglement entropy of short-range interacting quantum systems in one dimension is well studied with many different techniques [1], such as hamiltonian techniques [2][3][4][5], euclidean methods [6,7] and conformal field theory [8,9] there are few results concerning the long-range interacting systems. The main difficulty is the lack of exact solution for these systems which makes the problem much harder than the short-range counterparts. Because of the presence of non-trivial dynamical exponents the common euclidean techniques are usually useless in calculating the entanglement entropy in these systems. It is worth mentioning that although having a nontrivial dynamical exponents is not restricted to just longrange interacting systems, see for example [10,11], they usually provide a very natural knob to change it arbitrarily. Because of the huge finite size effects in these systems even the numerical calculations are not easy as their shortrange counterparts. Nevertheless, recently much progress has been made in different directions: In [12] the entanglement entropy in Lipkin-Meshkov-Glick (LMG) model is studied. This model has hamiltonian similar to the XY model but while in the latter model the interaction only takes place between nearest neighbors, in the LMG model, all spins interact among themselves. In [13], the interactions are restricted to the Ising-type without external magnetic field which allows to study both the static and the dynamical entanglement properties of the system. Eisert et. all [14] found a logarithmically divergent geometric en-

Quantum entanglement entropy and classical mutual information in long-range harmonic oscillators

“…(18) and (21) or equivalently Eqs. (26) and (27), which was first studied in 24 . In this respect, we follow the method explained in the last section.…”

“…Then we study the entanglement entropy numerically both at the purely discrete level and also at the level of discretization of the eigenvalue problem. This part of the paper is the extension of the work done in 24 . Then we study the finite size effects in different kind of situations such as, periodic boundary conditions and Dirichlet boundary conditions.…”

“…Recently using the methods of 7,9 we studied the entanglement entropy of block of long-range coupled harmonic oscillators 24 . We showed that the entanglement of the gapless system is logarithmically dependent to the system size and we calculated the prefactor of the logarithm in different situations.…”

Kaleidoscope of quantum phases in a long-range interacting spin-1 chain

“…One reason is that many analytically solvable lattice models become intractable when interactions are no longer short-ranged, a well-known example being the spin-1/2 XXZ model. Thus exact analytical studies are either restricted to non-interacting bosonic and fermionic systems with long-range hopping and pairing 33,[35][36][37] , or to certain contrived long-range interacting spin models which are difficult to realize in real systems [38][39][40][41] . In addition, to properly incorporate long-range interactions in low-energy effective theories, existing field theoretic treatments need to be modified and usually become more complicated 42,43 .…”

Discrete scale invariance and stochastic Loewner evolution

“…This comparison shows that the contour lines of 2D Weierstrass function with H = 0 could be related to the Loewner equation with the WM function as a drift. Here it is worth mentioning that we expect the same results for the 2D WM functions generated by the other periodic functions insteed of cos(x), we will discuss properties of the contour lines of 2D WM function for generic H for the different periodic functions in the forthcoming paper [30].…”