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arXiv:cond-mat/0702064 · Physical Review B

Fragility of string orders

Fabrizio Anfuso, Achim Rosch

Published Aug 22, 2007• 50 citations
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Abstract

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.

DOI
10.1103/physrevb.76.085124

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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.
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.
The ground state of spin-1 Haldane chains is characterized by the so-called string order. We show that the same hidden order is also present in ordinary one-dimensional band insulators. We construct a family of Hamiltonians which connect adiabatically band insulators to two topologically nonequivalent spin models, the Haldane chain and the antiferromagnetic spin-1∕21∕21∕2 ladder. We observe that the localized spin-1∕21∕21∕2 edge-state characteristic of spin-1 chains is smoothly connected to a surface-bound state of band insulators and its emergence is not related to any bulk phase transition. Furthermore, we show that the string order is absent in any dimensions higher than 1.
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.
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.
We analyzed quantum phase transitions in the antiferromagnetic dimerized spin-1/2 XY chain and two-leg ladders. Their gapped phases possess distinct nonlocal string order parameters (SOPs) and the hidden symmetry breaking in each phase is identified. After mapping of the original spin Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and the mean-field approximation (for ladders), the essential physics of nonlocal orders can be grasped analytically. By utilizing duality transformations the calculation of SOPs is mapped onto the local order problem in some dual representation and done without further approximations. The phases with hidden orders can also be distinguished by topological winding numbers. The latter provide a complimentary description. Our main result is the framework to treat nonlocal orders and hidden symmetries which unifies the key elements of the Landau paradigm with the new concept of topological order. This unified framework can be straightforwardly applied for spin chain and ladders, topological insulators and superconductors. Applications of this theory to other systems are under way.
The author studies the S=1 antiferromagnetic Heisenberg chain with uniaxial anisotropy, which exhibits the Haldane gap phenomenon. He argues that, when a magnetic field in a direction transverse to the anisotropy axis is applied, a sharp phase transition takes place at a critical value of the field. The transition is characterized by vanishing of the hidden antiferromagnetic order which can be measured by the string order parameter of den Nijs and Rommelse. The existence of such a transition distinguishes a Haldane gap system from other disordered quantum spin systems.