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arXiv:0904.2886 · Physical Review B

Spatial reflection and associated string order in quantum spin chains

Li-Xiang Cen

Published Oct 20, 2009• 7 citations
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Abstract

We investigate spatial reflection and associated nonlocal order in spin-chain quantum systems. The proposed string order parameters, e.g., reflected via operations of the spatial reflection or combinations of it with spin reflection, are able to characterize a variety of physical systems and allow us to gain insights to the statistical mechanism underlying phenomena such as the Haldane gap and quantum phase transitions. Besides revealing further the potential application of the generalized parity symmetry in numerical algorithm, we build an explicit scheme to determine the symmetry and the related string order for matrix-product states so that one can construct ansatz models with presumed properties.

DOI
10.1103/physrevb.80.132405

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We investigate spatial reflection and associated nonlocal order in spin-chain quantum systems. The proposed string order parameters, e.g., reflected via operations of the spatial reflection or combinations of it with spin reflection, are able to characterize a variety of physical systems and allow us to gain insights to the statistical mechanism underlying phenomena such as the Haldane gap and quantum phase transitions. Besides revealing further the potential application of the generalized parity symmetry in numerical algorithm, we build an explicit scheme to determine the symmetry and the related string order for matrix-product states so that one can construct ansatz models with presumed properties.
One-dimensional gapped spin chains with symmetry PSU(N)=SU(N)/ZNPSU(N)=SU(N)/{\mathbb{Z}}_{N}PSU(N)=SU(N)/ZN​ are known to possess NNN different topological phases. In this paper, we introduce a nonlocal string order parameter which characterizes each of these NNN phases unambiguously. Numerics confirm that our order parameter allows one to extract a quantized topological invariant from a given nondegenerate gapped ground state wave function. Discontinuous jumps in the discrete topological order that arise when varying physical couplings in the Hamiltonian may be used to detect quantum phase transitions between different topological phases.
One-dimensional gapped systems are often characterized by a ``hidden'' nonlocal order parameter, the so-called string order. Due to the gap, thermodynamic properties are robust against a weak higher-dimensional coupling between such chains or ladders. On the contrary, we find that the string order is not stable and decays for arbitrary weak interchain or interladder coupling. We investigate the vanishing of the order for three different systems: spin-1 Haldane chains, band insulators, and the transverse-field Ising model. Using perturbation theory and bosonization, we show that the fragility of the string order arises from nonlocal commutation relations between the nonlocal order parameter and the perturbation.
Quantum entanglement and non-local string order are investigated in the one-dimensional spin-1 bond-alternating Heisenberg model for quantum phase transitions. Employing the infinite matrix product state representation, ground state wavefunctions are numerically obtained by using the infinite time evolving block decimation approach in the infinite lattice system. From a bipartite entanglement measure, i.e., von Neumann entropy, a quantum phase transition point corresponding to the singular point of the von Neumann entropy between the Haldane phase and the dimerised phase is found to be δc=0.26. In addition, based on the infinite matrix product state algorithm, the non-local string order is calculated. The result shows that the non-local string order is able to characterize the topological phase transitions.
We investigate the ground-state properties of a newly discovered phase of one-dimensional lattice bosons with extended interactions [E. G. Dalla Torre et al., Phys. Rev. Lett. 97, 260401 (2006)]. The new phase, termed the Haldane insulator in analogy with the gapped phase of spin-1 chains, is characterized by a nonlocal order parameter, which can only be written as an infinite string in terms of the bosonic densities. We show that the string order can nevertheless be probed with physical fields that couple locally, via the effect those fields have on the quantum phase transitions separating the exotic phase from the conventional Mott and density wave phases. Using a field theoretical analysis, we show that a perturbation that breaks lattice inversion symmetry gaps the critical point separating the Mott and Haldane phases and eliminates the sharp distinction between them. This is remarkable given that neither of these phases involves broken inversion symmetry. We also investigate the evolution of the phase diagram with the tunable coupling between parallel chains in an optical lattice setup. We find that interchain tunneling destroys the direct phase transition between the Mott and Haldane insulators by establishing an intermediate superfluid phase. On the other hand, coupling the chains only by weak repulsive interactions does not modify the structure of the phase diagram. The theoretical predictions are confirmed with numerical calculations using the density matrix renormalization group.
Abstract The ground-state phase diagram and nonlocal order parameters of an infinite spin tetrahedron chain with inhomogeneous exchange couplings are investigated. It is shown that the phase boundaries of the three phases in the model can be determined precisely, in line with the precision of its ground-state energy. Numerical calculations using the regularized time-evolving block decimation (rTEBD) algorithm yield the locations of the two quantum critical points with an accuracy about 10 digits. Moreover, we explain how to calculate the parity-associated string order for the output wave function obtained through the rTEBD procedure, which not only reveals the presence of long-range correlations but also identifies the symmetry-protected topological order within the intermediate phase of the model.
We study the robustness of nonlocal string order in two paradigmatic disordered spin-chain models, a spin-1/2 cluster-Ising and a spin-1 XXZ Heisenberg chain. In the clean case, they both display a transition from antiferromagnetic to string order. Applying a disorder, which preserves the Hamiltonian symmetries, we find that the transition persists in both models. In the disordered cluster-Ising model, we can study the transition analytically---by applying the strongest coupling renormalization group ---and numerically---by exploiting integrability to study the antiferromagnetic and string order parameters. We map the model into a quadratic fermion chain, where the transition appears as a change in the number of zero-energy edge modes. We also explore its zero-temperature-singularity behavior and find a transition from a nonsingular to a singular region, at a point that is different from the one separating nonlocal and local ordering. The disordered Heisenberg chain can be treated only numerically: by means of MPS-based simulations, we are able to locate the existence of a transition between antiferromagnetic and string-ordered phase, through the study of order parameters. Finally, we discuss possible connections of our findings with many-body localization.