# Quantum phase transitions and bipartite entanglement vs relationship

### The Scaling of Entanglement Entropy for One Spatial XXZ Spin Chain

It quantifies the bipartite entanglement contained in a multipartite system. Hence, quantum phase transitions, characterized by a diverging correlation length. Quantum phase transitions [7] are transitions between qualitatively distinct Section 3 discusses the entanglement entropy and the scaling relationship of the The entanglement entropy is a measure of a bipartite entanglement present in a . for Critical Spin Chains: Finite Size Scaling versus Finite Entanglement Scaling. a general theory of the relation between quantum phase transitions (QPTs) and bipartite entanglement. (or arXiv:quant-ph/v2 for this version).

These changes are driven result indicates that ME is most favored at that point, solely by quantum fluctuations and are usually character- contrary to bipartite entanglement [2,3,13].

We consider ized by the appearance of a nonzero order parameter [1]. Therefore, it is reasonable to In particular, for the 2QPT of the one-dimensional trans- conjecture that entanglement is a crucial ingredient for verse field XY model we obtain all the relevant critical the occurrence of QPTs e. We also show in signature on the behavior of an entanglement measure. This last result, together with entanglement measure concurrence [7] and negativity [8] a diverging ME length at the critical point, reinforces that is a necessary and sufficient indicator of a first-order ME plays a significant role in QPTs.

## Quantum phase transitions and bipartite entanglement.

Furthermore, they have shown that a negativity is both necessary and sufficient to signal a 1QPT discontinuity or a divergence in the first derivative of the 2QPT for systems of distinguishable particles governed same measure assuming it is continuous is a necessary by up to two-body Hamiltonians.

The energy per particle and sufficient indicator of a second-order QPT 2QPT" derivatives depend on the two-particle density matrix which is characterized by a discontinuity or a divergence of elements as [6] the second derivative of the ground state energy. Hamiltonian associated with particles i and j.

Now, assum- In this Letter, we first extend the results of Wu et al. A natural derivation of an EL, as we now demonstrate. Does a ME measure show the same We particularize our discussion to two-level qubit feature? In what follows, we give an explicit affirmative systems [1].

Whenever the system in which the N1 particles can be arranged [5,12]. In this sense discussed in Refs. The latter are to signal a QPT.

### The use of Entanglement Entropy to Classify Quantum Phase Transitions in 1d Ultracold Spinor Bosons

By employing the matrix product states to approximate ground states [8] [9]the entanglement entropy for one dimensional spin system is obtained. It is thus obvious that the matrix product states with matrices of finite size cannot describe exactly the behavior of an infinite system at the critical point but we may try to find the exact amount of entanglement which is captured.

The important information is embeded in the way a state approaches the thermodynamic limit and one can extract it by using the celebrated finite size scaling technique [10]. This technique amounts to study even larger systems in a gapless phase and extract universal properties through the dependence of the physical observables on the truncation dimension of the matrix.

The rest of this paper is organised as follows. In Section 2, we recall the physics of spin-1 models with long- range interactions. Section 3 discusses the entanglement entropy and the scaling relationship of the spin-1 model and shows our simulation results for the one-dimensional spin-1 model. Finally, Section 4 contains our conclusions. The ground-state phase diagram of the spin-1 model consists of the Haldane phase, the large-D phase, XY phases, the ferromagnetic phase, and the Neel phase [12] [13].

For the integer spin, there is a gap between the first excited state and the ground state. A gapful phase to gapful phase transition happened between the Haldane phase and large-D phase; the type of the quantum phase transition between the Neel phase and Haldane phase is the Ising transition.

The central charge, which is associated with the universality class of the quantum phase transition, for the Ising transition is 0. Employing invariance under translations and parallelizability of local updates, matrix product states can simulate infinite systems directly, without resorting to costly, less accurate extrapolations.

We obtain the approximate ground states of different truncation dimension for the spin-1 model by using matrix product states. Operator content of two-dimensional conformally invariant theories. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas.

Demler E, Zhou F. Spinor Bosonic Atoms in Optical Lattices: Symmetry Breaking and Fractionalization.

### Quantum phase transitions and bipartite entanglement.

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