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arXiv:1104.5234 · Physical Review B

Bilinear-biquadratic spin-1 chain undergoing quadratic Zeeman effect

Gabriele De Chiara, Maciej Lewenstein, Anna Sanpera

Published Aug 15, 2011• 58 citations
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Abstract

The Heisenberg model for spin-1 bosons in one dimension presents many different quantum phases, including the famous topological Haldane phase. Here we study the robustness of such phases in front of a SU(2) symmetry-breaking field as well as the emergence of unique phases. Previous studies have analyzed the effect of such uniaxial anisotropy in some restricted relevant points of the phase diagram. Here we extend those studies and present the complete phase diagram of the spin-1 chain with uniaxial anisotropy. To this aim, we employ the density-matrix renormalization group together with analytical approaches. The complete phase diagram can be realized using ultracold spinor gases in the Mott insulator regime under a quadratic Zeeman effect.

DOI
10.1103/physrevb.84.054451

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The Heisenberg model for spin-1 bosons in one dimension presents many different quantum phases, including the famous topological Haldane phase. Here we study the robustness of such phases in front of a SU(2) symmetry-breaking field as well as the emergence of unique phases. Previous studies have analyzed the effect of such uniaxial anisotropy in some restricted relevant points of the phase diagram. Here we extend those studies and present the complete phase diagram of the spin-1 chain with uniaxial anisotropy. To this aim, we employ the density-matrix renormalization group together with analytical approaches. The complete phase diagram can be realized using ultracold spinor gases in the Mott insulator regime under a quadratic Zeeman effect.
Two recent publications report different boundaries for the dimerized phase of the bilinear-biquadratic spin-1 Heisenberg model with quadratic Zeeman anisotropy (see Refs. [9] and [10]). We numerically address these discrepancies for the purely biquadratic model with quadratic Zeeman term using methods based on matrix product states (MPS). We extrapolate the results beyond the purely biquadratic model in order to determine the phase boundaries of the dimerized phase and propose a modified phase diagram. Our MPS implementation incorporates SU(2) symmetry explicitly and we discuss its implementation into the MPS representation in the presence of the Zeeman anisotropy.
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Here we study the emergence of different symmetry-protected topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})+T$ symmetry group, where Z2{\mathbb{Z}}_{2}Z2​ is some $\ensuremath{\pi}$ rotation in the spin internal space and TTT is time reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an SO(5)SO(5)SO(5)-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous and can be described by a c=2c=2c=2 conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using matrix product states via the infinite time-evolving block decimation method, as well as by effective field theory considerations. Our findings show that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.
We study the phase diagram of the SOq(3){\text{SO}}_{q}(3)SOq​(3) quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of q>1q>1q>1. Numerical computations suggest that the chain has at least three clearly distinguished phases: A chiral analog of the Haldane phase, a dimerized phase, and a ferromagnetic phase. In contrast, the counterpart of the extended critical region that is known to exist for q=1q=1q=1 remains elusive. Our results show that the Haldane phase fails to exhibit a twofold degeneracy in the entanglement spectrum but that the degeneracy is restored upon a suitable qqq deformation of the entanglement Hamiltonian, which can be interpreted as a Zeeman field. The structure of the phase diagram is confirmed through analytical calculations in the extreme anisotropic limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results suggest that symmetries of the form Uq[su(2)]{\mathcal{U}}_{q}\left[su(2)\right]Uq​[su(2)] for distinct choices of qqq should be interpreted as one single family instead of separate symmetries when defining SPT phases, leading naturally to the notion of a qqq-SPT phase.
Spinor Bose condensates loaded in optical lattices have a rich phase diagram characterized by different magnetic order. Here we apply the density matrix renormalization group to accurately determine the phase diagram for spin-1 bosons loaded on a one-dimensional lattice. The Mott lobes present an even or odd asymmetry associated to the boson filling. We show that for odd fillings the insulating phase is always in a dimerized state. The results obtained in this work are also relevant for the determination of the ground state phase diagram of the S = 1 Heisenberg model with biquadratic interaction.