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We employ the Stabilizer Renyi Entropy (SRE) to characterize a quantum phase transition that has so far eluded any standard description and can thus now be explained in terms of the interplay between its non-stabilizer properties and entanglement. The transition under consideration separates a region with a unique ground state from one with a degenerate ground state manifold spanned by states with finite and opposite (intensive) momenta. We show that SRE has a jump at the crossing points, while the entanglement entropy remains continuous. Moreover, by leveraging on a Clifford circuit mapping, we connect the observed jump in SRE to that occurring between standard and generalized $W$-states with finite momenta. This mapping allows us to quantify the SRE discontinuity analytically., Comment: 8 pages including supplementary material, 3 figures

For systems consisting of distinguishable particles, there exists an agreed upon notion of entanglement which is fundamentally based on the possibility of addressing individually each one of the constituent parties. Instead, the indistinguishability of identical particles hinders their individual addressability and has prompted diverse, sometimes discordant definitions of entanglement. In the present review, we provide a comparative analysis of the relevant existing approaches, which is based on the characterization of bipartite entanglement in terms of the behaviour of correlation functions. Such a a point of view provides a fairly general setting where to discuss the presence of non-local effects; it is performed in the light of the following general consistency criteria: i) entanglement corresponds to non-local correlations and cannot be generated by local operations; ii) when, by "freezing" suitable degrees of freedom, identical particles can be effectively distinguished, their entanglement must reduce to the one that holds for distinguishable particles; iii) in absence of other quantum resources, only entanglement can outperform classical information protocols. These three requests provide a setting that allows to evaluate strengths and weaknesses of the existing approaches to indistinguishable particle entanglement and to contribute to the current understanding of such a crucial issue. Indeed, they can be classified into five different classes: four hinging on the notion of particle and one based on that of physical modes. We show that only the latter approach is consistent with all three criteria, each of the others indeed violating at least one of them.

We consider the reduced density matrix of a large block of consecutive spins in the ground states of the XY spin chain on an infinite lattice. We derive the spectrum of the density matrix using the expression of the Renyi entropy in terms of modular functions. The eigenvalues \lambda_n form an exact geometric sequence. For example, for strong magnetic field \lambda_n = C \exp{(-\pi \tau_0 n)}, here \tau_0>0 and C > 0 depend on the anisotropy and the magnetic field. Different eigenvalues are degenerated differently. The largest eigenvalue is unique, but the degeneracy g_n increases sub-exponentially as eigenvalues diminish: g_n \sim \exp{(\pi \sqrt{n/3})}. For weak magnetic field expressions are similar., Comment: 13 pages, 5 figures

We calculate the Emptiness Formation Probability (EFP) in the spin-Calogero Model (sCM) and Haldane-Shastry Model (HSM) using their hydrodynamic description. The EFP is the probability that a region of space is completely void of particles in the ground state of a quantum many body system. We calculate this probability in an instanton approach, by considering the more general problem of an arbitrary depletion of particles (DFP). In the limit of large size of depletion region the probability is dominated by a classical configuration in imaginary time that satisfies a set of boundary conditions and the action calculated on such solution gives the EFP/DFP with exponential accuracy. We show that the calculation for sCM can be elegantly performed by representing the gradientless hydrodynamics of spin particles as a sum of two spin-less Calogero collective field theories in auxiliary variables. Interestingly, the result we find for the EFP can be casted in a form reminiscing of spin-charge separation, which should be violated for a non-linear effect such as this. We also highlight the connections between sCM, HSM and $\lambda=2$ spin-less Calogero model from a EFP/DFP perspective., Comment: 26 pages, no figures

The fully nonlinear dynamics of spin and charge in spin-Calogero model is studied. The latter is an integrable one-dimensional model of quantum spin-1/2 particles interacting through inverse-square interaction and exchange. Classical hydrodynamic equations of motion are written for this model in the regime where gradient corrections to the exact hydrodynamic formulation of the theory may be neglected. In this approximation variables separate in terms of dressed Fermi momenta of the model. Hydrodynamic equations reduce to a set of decoupled Riemann-Hopf (or inviscid Burgers') equations for the dressed Fermi momenta. We study the dynamics of some non-equilibrium spin-charge configurations for times smaller than the time-scale of the gradient catastrophe. We find an interesting interplay between spin and charge degrees of freedom. In the limit of large coupling constant the hydrodynamics reduces to the spin hydrodynamics of the Haldane-Shastry model., Comment: 26 pages, 14 figures, 2 tables

We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power $\alpha$ of the density matrix of the block. We calculate the asymptotic for $L \to \infty$ analytically in terms of Klein's elliptic $\lambda$ - function. We study the limiting entropy as a function of its parameter $\alpha$. We show that up to the trivial addition terms and multiplicative factors, and after a proper re-scaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map $\alpha \to \alpha^{-1}$ and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when $\alpha$ is a integer power of 2, this includes the purity $tr \rho^2$. We also analyze the behavior of the entropy as $\alpha \to 0$ and $\infty$ and at the critical magnetic field and in the isotropic limit [XX model]., Comment: 28 Pages, 1 Figure

Entanglement in the ground state of the XY model on the infinite chain can be measured by the von Neumann entropy of a block of neighboring spins. We study a double scaling limit: the size of the block is much larger then 1 but much smaller then the length of the whole chain. The entropy of the block has an asymptotic limit. We study this limiting entropy as a function of the anisotropy and of the magnetic field. We identify its minima at product states and its divergencies at the quantum phase transitions. We find that the curves of constant entropy are ellipses and hyperbolas and that they all meet at one point (essential critical point). Depending on the approach to the essential critical point the entropy can take any value between 0 and infinity. In the vicinity of this point small changes in the parameters cause large change of the entropy., Comment: Revised Version, 20 pages, 8 figures, 1 table

Entanglement in the ground state of the XY model on the infinite chain can be measured by the von Neumann entropy of a block of neighboring spins. We study a double scaling limit: the size of the block is much larger then 1 but much smaller then the length of the whole chain. In this limit, the entropy of the block approaches a constant. The limiting entropy is a function of the anisotropy and of the magnetic field. The entropy reaches minima at product states and increases boundlessly at phase transitions., Comment: 6 pages, 1 figure