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Physical review. B, Condensed matter

Hidden order and symmetry breaking in the ground state of a spin-1/2 antiferromagnetic Heisenberg ladder

Hiroshi K. WATANABE

Published Nov 1, 1995• 18 citations
Publisher

Abstract

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-${\mathit{T}}_{\mathit{c}}$ superconductors. The inverse correlation length is further calculated for any ratio of the interchain and intrachain couplings under shifted boundary conditions.

DOI
10.1103/physrevb.52.12508

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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.
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.
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 properties of antiferromagnetic Heisenberg S= 1/2 ladders with two, three and four chains are expanded in the ratio of the intrachain and interchain coupling constants. A simple mapping procedure is introduced to relate the four- and two-chain ladders, which keeps to moderate values of the expansion parameters. A second-order calculation of the spin gap to the lowest triplet excitation in the two- and four-chain ladders is found to be quite accurate, even at the isotropic point where the couplings are equal. Similar expansions and mapping procedures are presented for the three-chain ladders, which are in the same universality class as single chains.
The ground-state properties of the spin-1/2 Heisenberg chain, which has two exchange couplings (J>J') alternatingly, are studied. This model interpolates the spin-1 antiferromagnetic Heisenberg chain (AFHC) (J'\ensuremath{\rightarrow}-\ensuremath{\infty}, J>0), the dimerized spin-1/2 AFHC (J'\ensuremath{\simeq}J>0), and the uniform spin-1/2 AFHC (J'=J>0) continuously. The string order introduced by den Nijs--Rommelse and Tasaki changes continuously through the Haldane phase and the dimer phase to vanish at J=J'. It is at a maximum for J'=0. This implies that the string order is related to the local singlet correlation in terms of the spin-1/2 model. The lowest excitation can be regarded as the renormalized triplet wave. The reversal of the anisotropy effect on the excitation spectrum of the spin-1 AFHC is also explained in a intuitive way based on this picture.
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 $\ensuremath{\beta}$.
The antiferromagnetic Heisenberg spin systems on the three-leg ladder are investigated. Periodic boundary condition is imposed in the rung direction. The system has an excitation gap for all antiferromagnetic inter-chain coupling (J⊥>0J_{\perp}>0J⊥​>0). The estimated gap for the strong coupling limit (J⊥/J1→∞J_{\perp}/J_1 \to \inftyJ⊥​/J1​→∞) is 0.28J1J_1J1​. Although the interaction is homogeneous and only nearest-neighbor, the ground states of the system are dimerized and break the translational symmetry in the thermodynamic limit. Introducing the next-nearest neighbor coupling (J2J_2J2​), we can see that the system is solved exactly. The ground state wave function is completely dimer-ordered. Using density matrix renomalization group algorithm, we show numerically that the original model (J2=0J_2=0J2​=0) has the same nature with the exactly solvable model. The ground state properties of the ladder with a higher odd number of legs are also discussed.