The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum. Here we explore in depth this degeneracy for the spin-1 AKLT and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the entanglement spectrum can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.
Angular-time evolution and edge-spin dynamics in the Haldane phase of the S=1 bilinear-biquadratic chain
Graduate School of Science and Technology, Niigata University（新潟大学科学技术研究科）；Department of Physics, Osaka Metropolitan University（大阪公立大学物理学系）；Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University（大阪公立大学南部阳一郎理论与实验物理研究所）
Here we investigate the phase diagram of the SO(n) bilinear-biquadratic quantum spin chain by studying the global quantum correlations of the ground state. We consider the cases of n=3,4 and 5 and focus on the geometric entanglement in the thermodynamic limit. Apart from capturing all the known phase transitions, our analysis shows a number of novel distinctive behaviors in the phase diagrams which we conjecture to be general and valid for arbitrary n. In particular, we provide an intuitive argument in favor of an infinite entanglement length in the system at a purely-biquadratic point. Our re…
Using a matrix product state algorithm with infinite boundary conditions, we compute high-resolution dynamic spin and quadrupolar structure factors in the thermodynamic limit to explore the low-energy excitations of isotropic bilinear-biquadratic spin-1 chains. Haldane mapped the spin-1 Heisenberg antiferromagnet to a continuum field theory, the non-linear sigma model (NL$σ$M). We find that the NL$σ$M fails to capture the influence of the biquadratic term and provides only an unsatisfactory description of the Haldane phase physics. But several features in the Haldane phase can be explained by …
We present a low-energy effective field theory to describe the SO(n) bilinear-biquadratic spin chain. We start with n=6 and construct the effective theory by using six Majorana fermions. After determining various correlation functions we characterize the phases and establish the relation between the effective theories for SO(6) and SO(5). Together with the known results for n=3 and 4, a reduction mechanism is proposed to understand the ground state for arbitrary SO(n). Also, we provide a generalization of the Lieb-Schultz-Mattis theorem for SO(n). The implications of our results for entangleme…
Response functions $\langle A_x(t) B_y(0)\rangle$ for one-dimensional strongly correlated quantum many-body systems can be computed with matrix product state (MPS) techniques. Especially, when one is interested in spectral functions or dynamic structure factors of translation-invariant systems, the response for some range $|x-y|<\ell$ is needed. We demonstrate how the number of required time-evolution runs can be reduced substantially: (a) If finite-system simulations are employed, the number of time-evolution runs can be reduced from $\ell$ to $2\sqrt{\ell}$. (b) To go beyond, one can employ …
We study the spin-1 bilinear-biquadratic model on the complete graph of N sites, i.e., when each spin is interacting with every other spin with the same strength. Because of its complete permutation invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir operators of su(3) and su(2). Using group representation theory, we explicitly diagonalize the Hamiltonian and map out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, is obtained analytically for any number of sites.
Using a generalized Jordan-Wigner transformation combined with the defining representation of the SO(N) spin, we map the SO(N) bilinear-biquadratic(BLBQ) spin chain into the N-color bosonic particle model. We find that, when the Jordan-Wigner transformation disentangles the symmetry-protected topological entanglement, this bosonic model becomes negative-sign free in the context of quantum Monte-Carlo simulation. For the SO(3) case, moreover, the Kennedy-Tasaki transformation for the S=1 BLBQ chain, which is also a topological disentangler, derives the same bosonic model through the dimer-R bas…
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 novel phases. Previous studies have analyzed the effect of such uniaxial anysotropy 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 anysotropy. To this aim, we employ the density matrix renormalization group (DMRG) together with a…
By using the concept of negativity, we study entanglement in spin-one Heisenberg chains. Both the bilinear chain and the bilinear-biquadratic chain are considered. Due to the SU(2) symmetry, the negativity can be determined by two correlators, which greatly facilitate the study of entanglement properties. Analytical results of negativity are obtained in the bilinear model up to four spins and the two-spin bilinear-biquadratic model, and numerical results of negativity are presented. We determine the threshold temperature before which the thermal state is doomed to be entangled.
We introduce an order parameter for symmetry-protected phases in one dimension which allows to directly identify those phases. The order parameter consists of string-like operators and swaps, but differs from conventional string order operators in that it only depends on the symmetry but not on the state. We verify our framework through numerical simulations for the SO(3) invariant spin-1 bilinear-biquadratic model which exhibits a dimerized and a Haldane phase, and find that the order parameter not only works very well for the dimerized and the Haldane phase, but it also returns a distinct si…
We consider a one-dimensional trapped gas of strongly interacting few spin-1 atoms which can be described by an effective spin chain Hamiltonian. Away from the SU(3) integrable point, where the energy spectrum is highly degenerate, the rules of ordering and crossing of the energy levels and the symmetry of the eigenstates in the regime of large but finite repulsion have been elucidated. We study the spin-mixing dynamics which is shown to be very sensitive to the ratio between the two channel interactions g0/g2 and the effective spin chain transfers the quantum states more perfectly than the He…
The incipient quantum phase transitions of relevance to nonzero fluctuations and entanglement in Heisenberg clusters are studied in this paper by exploiting negativity as a measure in bipartite and frustrated spin-1 anisotropic Heisenberg clusters with bilinear-biquadratic exchange, single-ion anisotropy and magnetic field. Using the exact diagonalization technique, it is shown that quantum critical points signaled by qualitative changes in behavior of magnetization and particle number are ultimately related to microscopic entanglement and collective excitations. The plateaus and peaks in spin…
We present an explicit construction of a family of steady state density matrices for an open integrable spin-1 chain with bilinear and biquadratic interactions, also known as the Lai-Sutherland model, driven far from equilibrium by means of two oppositely polarizing Markovian dissipation channels localized at the boundary. The steady state solution exhibits n+1 fold degeneracy, for a chain of length n, due to existence of (strong) Liouvillian U(1) symmetry. The latter can be exploited to introduce a chemical potential and define a grand canonical nonequilibrium steady state ensemble. The matri…
We explore quantum and classical correlations along with coherence in the ground states of spin-1 Heisenberg chains, namely the one-dimensional XXZ model and the one-dimensional bilinear biquadratic model, with the techniques of density matrix renormalization group theory. Exploiting the tools of quantum information theory, that is, by studying quantum discord, quantum mutual information and three recently introduced coherence measures in the reduced density matrix of two nearest neighbor spins in the bulk, we investigate the quantum phase transitions and special symmetry points in these model…
A spin-1 atomic gas in an optical lattice, in the unit-filling Mott Insulator (MI) phase and in the presence of disordered spin-dependent interaction, is considered. In this regime, at zero temperature, the system is well described by a disordered rotationally-invariant spin-1 bilinear-biquadratic model. We study, via the density matrix renormalization group algorithm, a bounded disorder model such that the spin interactions can be locally either ferromagnetic or antiferromagnetic. Random interactions induce the appearance of a disordered ferromagnetic phase characterized by a non-vanishing va…
