Abstract:
In this paper we calculate the block entanglement entropies of spin models whose ground states have perfect antiferromagnetic or ferromagnetic long-range order. In the latter case the definition of entanglement entropy is extended to properly take into account the ground state degeneracy. We find in both cases the entropy grows logarithmically with the block size. Implication of our results on states with general long-range order will be discussed.

Abstract:
The phase diagram of spins 1/2 embedded in a magnetic field mutually interacting antiferromagnetically is determined. Contrary to the ferromagnetic case where a second order quantum phase transition occurs, a first order transition is obtained at zero field. The spectrum is computed for a large number of spins and allows one to study the ground state entanglement properties which displays a jump of its concurrence at the critical point.

Abstract:
We consider a system of mutually interacting spin 1/2 embedded in a transverse magnetic field which undergo a second order quantum phase transition. We analyze the entanglement properties and the spin squeezing of the ground state and show that, contrarily to the one-dimensional case, a cusp-like singularity appears at the critical point $\lambda_c$, in the thermodynamic limit. We also show that there exists a value $\lambda_0 \geq \lambda_c$ above which the ground state is not spin squeezed despite a nonvanishing concurrence.

Abstract:
We use the entanglement negativity, a measure of entanglement for mixed states, to probe the structure of entanglement in the ground state of a topologically ordered system. Through analytical calculations of the negativity in the ground state(s) of the toric code model, we explicitly show that the entanglement of a region $A$ and its complement $B$ is the sum of two types of contributions. The first type of contributions consists of \textit{boundary entanglement}, which we see to be insensitive to tracing out the interior of $A$ and $B$. It therefore entangles only degrees of freedom in $A$ and $B$ that are close to their common boundary. As it is well-known, each boundary contribution is proportional to the size of the relevant boundary separating $A$ and $B$ and it includes an additive, universal correction. The second contribution appears only when $A$ and $B$ are non-contractible regions (e.g. on a torus) and it consists of long-range entanglement, which we see to be destroyed when tracing out a non-contractible region in the interior of $A$ or $B$. Only the long-range contribution to the entanglement may depend on the specific ground state under consideration.

Abstract:
The multi-scale entanglement renormalisation ansatz (MERA) is argued to provide a natural description for topological states of matter. The case of Kitaev's toric code is analyzed in detail and shown to possess a remarkably simple MERA description leading to distillation of the topological degrees of freedom at the top of the tensor network. Kitaev states on an infinite lattice are also shown to be a fixed point of the RG flow associated with entanglement renormalization. All these results generalize to arbitrary quantum double models.

Abstract:
We investigate the build-up of quasi-long-range order in the XX chain with transverse magnetic field at finite size. As the field is varied, the ground state of the system displays multiple level crossings producing a sequence of entanglement jumps. Using the partial fidelity and susceptibility, we study the transition to the thermodynamic limit and argue that the topological order can be described in terms of kink-antikink pairs and marked by edge spin entanglement.

Abstract:
We present a study of magnetic field induced quantum phase transitions in insulating systems. A generalized scaling theory is used to obtain the temperature dependence of several physical quantities along the quantum critical trajectory ($H=H_{C}$, $T\to0$) where $H$ is a longitudinal external magnetic field and $H_{C}$ the critical value at which the transition occurs. We consider transitions from a spin liquid at a critical field $H_{C1}$ and from a fully polarized paramagnet, at $H_{C2}$, into phases with long range order in the transverse components. The transitions at $H_{C1}$ and $H_{C2}$ can be viewed as Bose-Einstein condensations of magnons which however belong to different universality classes since they have different values of the dynamic critical exponent $z$. Finally, we use that the magnetic susceptibility is an entanglement witness to discuss how this type of correlation sets in as the system approaches the quantum critical point along the critical trajectory, $H=H_{C2}$, $T\to0$.

Abstract:
We show that the quantum order parameters (QOP) associated with the transitions between a normal conductor and a superconductor in the BCS and eta-pairing models and between a Mott-insulator and a superfluid in the Bose-Hubbard model are directly related to the amount of entanglement existent in the ground state of each system. This gives a physical meaningful interpretation to these QOP, which shows the intrinsically quantum nature of the phase transitions considered.

Abstract:
We investigate macroscopic entanglement in an infinite XX spin-1/2 chain with staggered magnetic field, B_l=B+e^{-i\pi l}b. Using single-site entropy and by constructing an entanglement witness, we search for the existence of entanglement when the system is at absolute zero, as well as in thermal equilibrium. Although the role of the alternating magnetic field b is, in general, to suppress entanglement as do B and T, we find that when T=0, introducing b allows the existence of entanglement even when the uniform magnetic field B is arbitrarily large. We find that the region and the amount of entanglement in the spin chain can be enhanced by a staggered magnetic field.

Abstract:
We study the effect of inhomogeneities in the magnetic field on the thermal entanglement of a two spin system. We show that in the ferromagnetic case a very small inhomogeneity is capable to produce large values of thermal entanglement. This shows that the absence of entanglement in the ferromagnetic Heisenberg system is highly unstable against inhomogeneoity of magnetic fields which is inevitably present in any solid state realization of qubits.