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arXiv:cond-mat/0405699 · Physical Review B

Low-energy properties of two-leg spin-1 antiferromagnetic ladders with commensurate external fields and their extensions

Masahiro Sato

Published Jan 6, 2005• 11 citations
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Abstract

This study addresses low-energy properties of two-leg spin-1 ladders with antiferromagnetic (AF) intrachain coupling under a uniform or staggered external field $H$, and a few of their modifications. The generalization to spin-$S$ ladders is also discussed. In the strong AF rung(interchain)-coupling ${J}_{\ensuremath{\perp}}$ region, degenerate perturbation theory applied to spin-$S$ ladders predicts $2S$ critical curves in the parameter space $({J}_{\ensuremath{\perp}},H)$ for the staggered-field case, in contrast to $2S$ finite critical regions for the uniform-field case. All critical areas belong to a universality with central charge $c=1$. On the other hand, we employ Abelian and non-Abelian bosonization techniques in the weak rung-coupling region. They show that in the spin-1 ladder, a sufficiently strong uniform field engenders a $c=1$ critical state regardless of the sign of ${J}_{\ensuremath{\perp}}$, whereas the staggered field is expected not to yield any singular phenomena. From the bosonization techniques, new field-theoretical expressions of string order parameters in the spin-1 systems are also proposed.

DOI
10.1103/physrevb.71.024402

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This study addresses low-energy properties of two-leg spin-1 ladders with antiferromagnetic (AF) intrachain coupling under a uniform or staggered external field HHH, and a few of their modifications. The generalization to spin-SSS ladders is also discussed. In the strong AF rung(interchain)-coupling J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​ region, degenerate perturbation theory applied to spin-SSS ladders predicts 2S2S2S critical curves in the parameter space (J\ensuremath⊥,H)({J}_{\ensuremath{\perp}},H)(J\ensuremath⊥​,H) for the staggered-field case, in contrast to 2S2S2S finite critical regions for the uniform-field case. All critical areas belong to a universality with central charge c=1c=1c=1. On the other hand, we employ Abelian and non-Abelian bosonization techniques in the weak rung-coupling region. They show that in the spin-1 ladder, a sufficiently strong uniform field engenders a c=1c=1c=1 critical state regardless of the sign of J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​, whereas the staggered field is expected not to yield any singular phenomena. From the bosonization techniques, new field-theoretical expressions of string order parameters in the spin-1 systems are also proposed.
We study a two-leg antiferromagnetic spin-1/2 ladder in the presence of a staggered magnetic field. We consider two parameter regimes: strong (weak) coupling along the legs and weak (strong) coupling along the rungs. In both cases, the staggered field drives the Haldane spin-liquid phase of the ladder towards a Gaussian quantum criticality. In a generalized spin ladder with a non-Haldane, spontaneously dimerized phase, the staggered magnetic field induces an Ising quantum critical regime. In the vicinity of the critical lines, we derive low-energy effective field theories and use these descriptions to determine the dynamical response functions, the staggered spin susceptibility, and the string order parameter.
We have studied the ground state phase diagram of a spin ladder with alternating rung exchange J⊥n=J⊥[1+(−1)nδ]J^{n}_{\perp} = J_{\perp}[1 + (-1)^{n} δ]J⊥n​=J⊥​[1+(−1)nδ] in a magnetic filed, in the limit where the rung coupling is dominant. In this limit the model is mapped onto an XXZXXZXXZ Heisenberg chain in a uniform and staggered longitudinal magnetic fields, where the amplitude of the staggered field is ∼δ\sim δ∼δ. We have shown that the magnetization curve of the system exhibits a plateau at magnetization equal to the half of the saturation value. The width of a plateau scales as δνδ^νδν, where ν=4/5ν=4/5ν=4/5 in the case of ladder with isotropic antiferromagnetic legs and ν=2ν=2ν=2 in the case of ladder with isotropic ferromagnetic legs. We have calculated four critical fields (Hc1±H^{\pm}_{c1}Hc1±​ and Hc2±H^{\pm}_{c2}Hc2±​) corresponding to transitions between different magnetic phases of the system. We have shown that these transitions belong to the universality class of the commensurate-incommensurate transition.
We consider asymmetric spin-12\frac{1}{2}21​ two-leg ladders with nonequal antiferromagnetic (AF) couplings ${J}_{\ensuremath{\parallel}}$ and $\ensuremath{\kappa}{J}_{\ensuremath{\parallel}}$ along legs $(\ensuremath{\kappa}\ensuremath{\le}1)$ and ferromagnetic rung coupling, ${J}_{\ensuremath{\perp}}$. This model is characterized by a gap $\ensuremath{\Delta}$ in the spectrum of spin excitations. We show that in the large ${J}_{\ensuremath{\perp}}$ limit this gap is equivalent to the Haldane gap for the AF spin-1 chain, irrespective of the asymmetry of the ladder. The behavior of the gap at small rung coupling falls in two different universality classes. The first class, which is best understood from the case of the conventional symmetric ladder at $\ensuremath{\kappa}=1$, admits a linear scaling for the spin gap $\ensuremath{\Delta}\ensuremath{\sim}{J}_{\ensuremath{\perp}}$. The second class appears for a strong asymmetry of the coupling along legs, $\ensuremath{\kappa}{J}_{\ensuremath{\parallel}}⪡{J}_{\ensuremath{\perp}}⪡{J}_{\ensuremath{\parallel}}$ and is characterized by two energy scales: the exponentially small spin gap $\ensuremath{\Delta}\ensuremath{\sim}{J}_{\ensuremath{\perp}}\text{ }\text{exp}(\ensuremath{-}{J}_{\ensuremath{\parallel}}/{J}_{\ensuremath{\perp}})$, and the bandwidth of the low-lying excitations induced by a Suhl-Nakamura indirect exchange $\ensuremath{\sim}{J}_{\ensuremath{\perp}}^{2}/{J}_{\ensuremath{\parallel}}$. We report numerical results obtained by exact diagonalization, density-matrix renormalization group and quantum Monte Carlo simulations for the spin gap and various spin correlation functions. Our data indicate that the behavior of the string order parameter, characterizing the hidden AF order in Haldane phase, is different in the limiting cases of weak and strong asymmetries. On the basis of the numerical data, we propose a low-energy theory of effective spin-1 variables, pertaining to large blocks on a decimated lattice.
The experimental realization of various spin ladder systems has prompted their detailed theoretical investigations. Here we study the evolution of ground-state magnetization with an external magnetic field for two different antiferromagnetic systems: a three-legged spin-1/21/21/2 ladder, and a two-legged spin-1/21/21/2 ladder with an additional diagonal interaction. The finite system density-matrix renormalization-group method is employed for numerical studies of the three-chain system, and an effective low-energy Hamiltonian is used in the limit of strong interchain coupling to study the two- and three-chain systems. The three-chain system has a magnetization plateau at one-third of the saturation magnetization. The two-chain system has a plateau at zero magnetization due to a gap above the singlet ground state. It also has a plateau at half of the saturation magnetization for a certain range of values of the couplings. We study the regions of transitions between plateaus numerically and analytically, and find that they are described, at first order in a strong-coupling expansion, by an XXZ\mathrm{XXZ}XXZ spin-1/21/21/2 chain in a magnetic field; the second-order terms give corrections to the XXZ\mathrm{XXZ}XXZ model. We also study numerically some low-temperature properties of the three-chain system, such as the magnetization, magnetic susceptibility and specific heat.
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