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arXiv:cond-mat/0609051 · Physical Review B

String order and adiabatic continuity of Haldane chains and band insulators

Fabrizio Anfuso, Achim Rosch

Published Apr 19, 2007• 73 citations
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Abstract

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-$1∕2$ ladder. We observe that the localized spin-$1∕2$ 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.

DOI
10.1103/physrevb.75.144420

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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.
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.
The antiferromagnetic Heisenberg spin chain with integer spin has a short-range magnetic order and an excitation energy gap above the ground state. This so-called Haldane gap is proportional to the exchange coupling constant J of the Heisenberg chain. In this study, we discuss recent results about the spin dependence of the Haldane gap and conjecture an analytical formula valid asymptotically for large spin values. We then discuss the robustness of the edge states of the spin-1 chain by studying a spin-1 ladder using the density matrix renormalization group (DMRG) algorithm. We show that the peculiar hidden topological order of the spin-1 chain disappears smoothly by increasing the ladder rung coupling without any intervening phase transition. This is evidence for the fragile character of the topological order of the spin-1 chain.
Ground-state properties of the spin-1 two-leg antiferromagnetic ladder are investigated precisely by means of the quantum Monte Carlo method. It is found that the correlation length along the chains and the spin gap both remain finite regardless of the strength of interchain coupling, i.e., the Haldane state and the spin-1 dimer state are connected smoothly without any quantum phase transitions between them. We propose a plaquette-singlet solid state, which qualitatively describes the ground state of the spin-1 ladder quite well, and also a corresponding topological hidden order parameter. It is shown numerically that the proposed hidden order parameter remains finite up to the dimer limit, though the conventional string order defined on each chain vanishes immediately when infinitesimal interchain coupling is introduced.
Motzkin spin chains are frustration-free models whose ground state is a combination of Motzkin paths. The weight of such path contributions can be controlled by a deformation parameter ttt. As a function of the latter, these models, besides the formation of domain wall structures, exhibit gapped Haldane topological orders with a constant decay of string order parameters for t<1t<1t<1. A behavior compatible with a Berezinskii-Kosterlitz-Thouless phase transition at t=1t=1t=1 is also presented. By means of numerical calculations we show that the topological properties of the Haldane phases depend on the spin value. This allows one to classify different kinds of hidden antiferromagnetic Haldane gapped regimes associated with nontrivial features such as symmetry-protected topological order. On one hand, our results allow one to clarify the physical properties of Motzkin frustration-free chains, and on the other hand, suggest them as an interesting and paradigmatic class of local spin Hamiltonians.
The Haldane phase for antiferromagnetic spin-1 chains is a celebrated topological state of matter, featuring gapped excitations and fractional spin-1/2 edge states. Here, we provide numerical evidence that this phase can be realized with a Hubbard model at half filling, where each s=1s=1s=1 spin is stored in a four-site fermionic structure. We find that the noninteracting limit of our proposed model describes a one-dimensional topological insulator, and we conjecture it to be adiabatically connected to the Haldane phase. Our work suggests a route to build spin-1 networks using Hubbard model quantum simulators.
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.