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arXiv:cond-mat/0106073 · Physical review. B, Condensed matter

Plaquette-singlet solid state and topological hidden order in a spin-1 antiferromagnetic Heisenberg ladder

Synge Todo, Munehisa Matsumoto, Chitoshi Yasuda, Hajime Takayama

Published Nov 20, 2001• 59 citations
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Abstract

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.

DOI
10.1103/physrevb.64.224412

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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.
The ground state of S=1 antiferromagnetic Heisenberg (AFH) two-leg ladder has been shown to be gapful, in a whole range from the weak rung coupling limit to the strong rung coupling limit. The well-known valence-bond solid (VBS) picture for single AFH chains does not hold for this disordered ground state. We generalize the VBS picture to the plaquette-singlet solid (PSS) one to obtain a good description of the S=1 ladder. The validity of the PSS picture for this system is also supported by the finiteness of a newly-introduced hidden order parameter analogous to the string order parameter. We observe that there is also no critical point in the ground state of ladders with in-phase leg-bond alternation.
Dimerized antiferromagnetic spin-12\frac{1}{2}21​ ladders are known to exhibit a quantum critical phase transition in the ground state, the existence or absence of which is dependent on the dimerization pattern of the ladder. The gapped phases cannot be distinguished by the conventional Landau long-range order parameter. However, they possess a nonlocal (hidden) string-order parameter, which is nonzero in one phase and vanishes in the other. We use an exact diagonalization technique to calculate ground-state energies, energy gaps, and string-order parameters of dimerized two- and three-leg Heisenberg ladders, as well as a critical scaling analysis to yield estimates of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$.
Thermodynamic properties of spin=1/2 two-leg ladders with alternating chains including ferromagnetic and antiferromagnetic interactions are discussed on the basis of the numerical results obtained by quantum Monte Carlo simulations. As well as the role of exchange interaction of rungs, the essential role of the bond alternation in the chain on ladder systems is investigated in connection with the appearance of characteristic properties in magnetic susceptibilities and heat capacities. Furthermore, quantum effects on the ground-state energy are obtained. On the basis of calculated results, the relationship between singlet dimer phases and Haldane states is discussed.
We investigate an S =(1/2) ladder model with a ferromagnetic interchain coupling (-λ) and an antiferromagnetic intrachain coupling based on the bosonization method, which reduces to the S =1 antiferromagnetic Heisenberg model in the limit λ→∞. We argue with the aid of Wilson's renormalization group method that the ground state is the Haldane state with an excitation gap for λ>0, that the gap opens at λ=0 exponentially and that the critical index of the coherence length is unity (ν=1) on the critical line between the antiferromagnetic and the Haldane phases. Furthermore, it is found that the string order parameter proposed by den Nijs and Rommelse is expressed by a local order parameter in the bosonization method.
Dimerized antiferromagnetic spin-1/2 chains and ladders demonstrate quantum critical phase transition, the existence or absence of which is dependent on the dimerization and the dimerization pattern of the chain and the ladder, respectively. The gapped phases can not be distinguished by the conventional Landau long-range order parameters. However, they possess non-local topological string order parameters which can be used to classify different phases. We utilize the self-consistent free fermionic approximation and some standard results for exactly solved models to analytically calculate the string order parameters of dimerized spin chains. As a complement parameter the gapped phases possess the topological number, called the winding number and they are characterized by different integer values of the winding number. In order to calculate the string order parameters and winding numbers in dimerized spin chains and two-leg ladders we use analytical methods such as the Jordan-Wigner transformation, mean-field approximation, duality transformations, and some standard results available for the exactly 1D solve models. It is shown that the winding number provides the complementary framework to the string order parameter to characterize the topological gapped phases.
It is shown that the ground state of a spin-1/2 antiferromagnetic ladder model is characterized by a string order parameter which is similar to but differs from that introduced by den Nijs and Rommelse. This reveals the presence of a hidden antiferromagnetic order and symmetry breaking even in the ground state which relates to high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors. The inverse correlation length is further calculated for any ratio of the interchain and intrachain couplings under shifted boundary conditions.