We study the phase diagram for spin-2 bosons loaded in a one-dimensional optical lattice. By using the nonAbelian density matrix renormalization group (DMRG) method we identify three possible phases: ferromagnetic, dimerized, and trimerized phases. We sketch the phase boundaries based on DMRG. We illustrate two methods for identifying the phases. The first method is based on the spin-spin correlation function, while in the second method one observes the excitation gap as a dimerization, or a trimerization superlattice is imposed. The advantage of the second method is that it also can be easily implemented in experiments. By using the scattering lengths in the literature we estimate that 83 Cold atomic gases have been actively studied in recent years because they offer different possibilities for studying quantum many-body systems [1]. In the very early experiments of a dilute Bose gas in a trap, Bose-Einstein condensation was observed directly [2]. We have also witnessed the realization of the Bose-Hubbard model and the observation of the superfluidMott transition [3], a phenomenon theoretically predicted long ago but only observed recently. In this case, the presence of a lattice and the interatomic interaction actually destroy the superfluid, resulting in an "insulating" state. In Ref. [3] and also in the many following experiments, the bosons are spin polarized and so they are effectively spinless. However, Bose-Einstein condensation of bosons with spin degree of freedom has also been realized [4]. Hence it is natural to ask what would be the spin ordering of such bosons in a lattice in the Mott-insulating state, where, even though one is confined to an integral number of particles per site, the spins of the neighboring sites can still interact via virtual tunneling. Indeed, this question is of much theoretical interest, as it can be easily shown that the effective spin Hamiltonians one can realize for spinor bosons loaded in an optical lattice are very different from the Heisenberg-like Hamiltonians that have been much studied in electronic systems [5]. Similarly, a two-component Bose system allows us to realize the XXZ spin-1/2 model [6], which has been discussed considerably in the theoretical literature.In this Rapid Communication, we consider spin-2 bosons in a one-dimensional lattice. Spin-2 systems are already available and have been experimentally studied [7][8][9][10]. The theoretical phase diagram of spin-2 condensates is a function of the scattering lengths a S in the spin S = 0, 2, 4 channels [11,12]. It is divided into three regions, which are named ferromagnetic (F), polar (P), and cyclic (C) in Ref. [11]. For spin-2 bosons with one particle per site in a higher dimensional lattice in the insulating phase, it can be easily shown that again the phase diagram is divided into three regions in the mean-field limit, in direct analogy to the Bose-condensed case [13][14][15][16] (see also Ref. [17]). In one dimension (1D), however, strong quantum fluctuations are expected to substantially modify the ...

Abstract. We study the entanglement entropy scaling of the XXZ chain. While in the critical XY phase of the XXZ chain the entanglement entropy scales logarithmically with a coefficient that is determined by the associated conformal field theory, at the ferromagnetic point, however, the system is not conformally invariant yet the entanglement entropy still scales logarithmically albeit with a different coefficient. We investigate how such an nontrivial scaling at the ferromagnetic point influences the estimation of the central charge c in the critical XY phase. In particular we use the entanglement scaling of the finite or infinite system, as well as the finite-size scaling of the ground state energy to estimate the value of c. In addition, the spin-wave velocity and the scaling dimension are also estimated. We show that in all methods the evaluations are influenced by the nearby ferromagnetic point and result in crossover behavior. Finally we discuss how to determine whether the central charge estimation is strongly influenced by the crossover behavior and how to properly evaluate the central charge.Entanglement entropy scaling of the XXZ chain 2

We study the quantum critical phase of an SU(2) symmetric spin-2 chain obtained from spin-2 bosons in a one-dimensional lattice. We obtain the scaling of the finite-size energies and entanglement entropy by exact diagonalization and density-matrix renormalization group methods. From the numerical results of the energy spectra, central charge, and scaling dimension we identify the conformal field theory describing the whole critical phase to be the SUð3Þ 1 Wess-Zumino-Witten model. We find that, while the Hamiltonian is only SU(2) invariant, in this critical phase there is an emergent SU(3) symmetry in the thermodynamic limit. DOI: 10.1103/PhysRevLett.114.145301 PACS numbers: 67.85.-d, 03.65.Ud, 11.25.Hf, 65.40.gd Cold atomic gases in optical lattices have become an ideal framework for studying quantum many-body systems in recent years [1]. In particular, various schemes have been proposed to study quantum magnetism [2]. For spin 1=2 systems, simulation of the Ising model has been realized using bosons in a tilted optical lattice [3]. It has also been proposed that the spin 1=2 XYZ Heisenberg model can be realized using p-orbital bosons [4]. This rapid progress in cold atomic physics results in a considerable renewal of interest to study models with higher spins or higher symmetries, especially for models that are potentially realizable by cold atomic systems. A natural direction is to study spinor bosons and their novel phases. For example, it has been proposed that the spin-1 bilinear biquadractic (BB) model can be engineered using spin-1 cold bosons in an optical lattice [5,6]. Furthermore, the phase diagram of spin-1 bosons in a one-dimensional (1D) lattice has been studied numerically and compared to the spin-1 BB model [7]. Since spin-2 bosons are available and have been experimentally studied [8][9][10][11], it is of great interest to explore the phases realizable by spin-2 bosons. On the other hand, it has also been pointed out that fermions with hyperfine spin F ¼ 3=2 can be used to realize models with SO(5) symmetry [12], or to realize an SU(3) spin chain by effectively suppressing the occupation of one of the middle states [13]. Possibilities to realize higher SU(N) symmetry have also been proposed [14,15]. Along this line, spin dynamics and correlation have been studied experimentally using cold fermions with effective spin ranging from 1=2 to 9=2 [16,17]. Another interesting problem is to explore symmetries that emerge in the low energy limit of the models. Indeed, different aspects of emergent symmetries have been discussed widely in the recent literature. Examples include SO(5) and SO (8) symmetries in high temperature superconductors and two-leg ladders [18,19] Recently we studied the phase diagram of spin-2 bosons in a 1D optical lattice with one particle per site and identified three possible phases for a finite system: ferromagnetic, dimerized, and trimerized phases [25]. Within the trimerized phase, if the system size is a multiple of 3, the ground state is a spin singlet with a finite-siz...