Entanglement properties and momentum distributions of hard-core anyons on a ring

Abstract

We study the one-particle von Neumann entropy of a system of N hard-core anyons on a ring. The entropy is found to have a clear dependence on the anyonic parameter which characterizes the generalized fractional statistics described by the anyons. This confirms the entanglement is a valuable quantity to investigate topological properties of quantum states. We derive the generalization to anyonic statistics of the Lenard formula for the one-particle density matrix of N hard-core bosons in the large N limit and extend our results by a numerical analysis of the entanglement entropy, providing additional insight into the problem under consideration.In recent years an intense research activity has been devoted to the study of entanglement in many-body states. Initially, this effort has been mostly motivated by the fact that quantum correlated many-body states, which appear in various solid-state models, can be valuable resources for information processing and quantum computation [1,2]. The theory of entanglement is now attracting even more attention because of its fundamental implication for the development of new efficient numerical methods for quantum systems [3,4,5] and for the characterization of quantum critical phases [6,7,8].Generally speaking, entanglement measures nonlocal properties of composite quantum systems and it can provide additional information to that obtained by investigating local observables or traditional correlation functions. In this respect entanglement might be a sensitive probe into the topological properties of quantum states. A particularly significant quantity is the entanglement entropy S A , which is defined in a bipartite system A ∪ B and quantified as the von Neumann entropy S A = −Trρ A ln ρ A associated to the reduced density matrix ρ A of a subsystem A. In two-dimensional systems a firm connection between topological order and entanglement entropy has been established in [9,10], where the entanglement entropy was defined by spatial partitioning. Recent studies on Laughlin states [11,12] have considered the entanglement entropy associated to particle partitioning [11,12]. Also in this case, the entanglement entropy turns out to reveal important aspects of the topological order in Laughlin States.The two-dimensional case is of particular interest due to the existence of models whose elementary excitations exhibit generalized fractional statistics. Anyons, the particles obeying such statistics, play a fundamental role in the description of the fractional quantum Hall effect [13]. Although this concept is essentially twodimensional, anyons can also occur in one-dimensional (1D) systems [14,15,16,17,18,19,20], where statistics and interactions are inextricable, leading to strong shortrange correlations. The 1D anyonic models have proven useful to study persistent charge and magnetic currents in 1D mesoscopic rings [15]. This possibility and their own pure theoretical interest lead us to investigate the effects of the anyonic statistics on the entanglement entropy in the present Letter....

Junctions of anyonic Luttinger wires

“…Thus, when κ = 1 the model is the well-known fermionic TL model, while the bosonic limit κ → 0 is not well defined in this formalism as will be clearer in the following. We stress that this anyonic model is different from the gases discussed elsewhere 42,43,46,49,56 , that also have a Luttinger liquid description. As in the fermionic case, the model is naturally solved exactly through bosonization 27 .…”

“…For n = 3, as we said in the introduction the literature is enormous. The boundary conditions we used are equivalent to those of the "auxiliary model" of Nayak et al 1 for g = 1 [in fact, expanding the exponential defining the auxiliary model 1 and keeping only up to the quadratic terms, neglecting irrelevant higher orders, we arrive to the Hamiltonian (46) where the symmetry of the boundary terms is just the Kirchhoff's rule]. We predict two possible stable fixed points: Neumann and mixed.…”

“…The more natural one, as done elsewhere 1,9,23 , relies on calculating the scaling dimension of the perturbing operator at a given fixed point. Since our problem can be thought as n independent half-lines with n − p Neumann boundary conditions and p Dirichlet ones, the problem is just equivalent to understand the stability of Neumann or Dirichlet against a Robin term as in the Hamiltonian (46). This is a standard problem.…”

“…The reason for this generalization is twofold. On one hand the study of 1D anyonic model is attracting a renewed interest [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59] , mainly motivated by possible experiments with cold atoms 60 . On the other hand, the transport of wires joined with a quantum Hall island is driven by anyonic excitations 12 .…”

Dynamical properties of hard-core anyons in one-dimensional optical lattices

“…Kundu proved that a 1D Bose gas interacting through δ-function potential combined with double δ-function potential and derivative δ-function potential is equivalent to the anyon gas interacting via δ-function potential [21]. This stimulated many research interests on δ-anyon gas [22][23][24][25][26][27][28][29][30][31][32]. It turns out that the ground state density distribution of δ-anyon gas displays similar behavior as that of Bose gas with the increasing interaction.…”