Essential singularity in the Renyi entanglement entropy of the one-dimensionalXYZspin-12chain

Abstract

We study the Renyi entropy of the one-dimensional XY Z spin-1/2 chain in the entirety of its phase diagram. The model has several quantum critical lines corresponding to rotated XXZ chains in their paramagnetic phase, and four tricritical points where these phases join. Two of these points are described by a conformal field theory and close to them the entropy scales as the logarithm of its mass gap. The other two points are not conformal and the entropy has a peculiar singular behavior in their neighbors, characteristic of an essential singularity. At these nonconformal points the model undergoes a discontinuous transition, with a level crossing in the ground state and a quadratic excitation spectrum. We propose the entropy as an efficient tool to determine the discontinuous or continuous nature of a phase transition also in more complicated models. In the past several years there has been a constantly increasing interest in quantifying and studying the entanglement of virtually every physical system [1][2][3]. This interest is not surprising, as entanglement provides valuable insights from many different perspectives.As it is often measured as entanglement entropy, that is, the Von Neumann entropy of the reduced density matrix of a system S = −Trρ lnρ, its origin lies in the context of quantum information theory [4,5]. Essentially, it quantifies the "quantumness" of a state and therefore it provides a measure of its suitability for efficient quantum algorithms, in the ongoing quest for quantum computing.In the physics of strongly interacting systems, entanglement has been welcomed as a new interesting correlation function, different in nature compared to the traditional ones, due to its nonlocal structure. In particular, it has provided a new challenge for the integrable model community on one side [6][7][8] and has led to new, more efficient approaches for numerical simulations (tensor network states) on the other [9][10][11]. For statistical physics, the entanglement entropy has also been proposed as a numerically efficient way to characterize a system, due to its diverging behavior across a phase transition [12,13].If we concentrate on the bipartite entanglement, that is, the entanglement entropy between two complementary regions, a lot is understood about its qualitative behavior. In general, it satisfies the so-called area law [2], that is, it is proportional to the area of the boundary dividing the two regions. This is naively understood considering that the entanglement is carried by correlations, that, in general, decay exponentially with the distance with a rate given by the correlation length ξ. If the volume of the two regions is much bigger than ξ, entanglement is localized at the boundary. Hence the area law. This behavior is modified when correlations decay more slowly, like close to phase transitions, where ξ → ∞.The one-dimensional (1D) case is most interesting.As the boundary between regions consists of individual points, in a massive phase the entropy saturates to a finite value for sufficien...

Nontransverse factorizing fields and entanglement in finite spin systems

“…where lead to similar side-limits for the concurrence C ij between any two spins i = j (see (35) below), i.e.,…”

Dynamics of entanglement entropy and entanglement spectrum crossing a quantum phase transition

“…It is well known that the final efficiency of such methods is related to the amount of entanglement of the considered state 9 , a quantity which is expected to diverge when getting closer to a phase transition. However, at least in the static case, the behaviour of entanglement (and more specifically of entanglement entropy) has an universal character so that it can be used as an estimator of quantum correlations 10 and to detect as well as to classify quantum phase transitions also in fully interacting models [11][12][13][14][15][16] . Thus, it is natural to ask whether the dynamical behaviour of a closed quantum system, especially when crossing a phase transition, can be described by looking at the dynamics of entanglement entropy and entanglement spectrum, a topic on which there are only a few general results [17][18][19][20] .…”

Exact phase boundaries and topological phase transitions of the XYZ spin chain

“…The above gapped phases must be separated by a gapless line in the plane of (∆, λ) 18 . Indeed Ercolessi and coworkers using the exact solution of Baxter [1][2][3] calculated the Renyi entropy and identify lines of essential singularity 23 ending at tricritical points where the lines join. They find that two of the tricritical points are conformally invariant 23,24 .…”