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arXiv:0708.4181 · Physical Review B

Spin-Peierls instability in the spin- 1∕2 Heisenberg three-leg ladder

Mohamed Moussadek Azzouz, Khalada Shahin, Gennady Y. Chitov

Published Oct 26, 2007• 20 citations
arXivHTMLPDF

Abstract

Because the three-leg ladder behaves like a renormalized single Heisenberg chain, we argue that a spin-Peierls instability must occur in this system when it is coupled to three-dimensional phonons. Using the bond-mean-field theory, we show that this is indeed the case. The dimerized state below the spin-Peierls transition temperature forms into the columnar dimerized phase instead of the staggered one. This contrasts with the argument based on antiferromagnetism. A physical argument based rather on spin bonding into singlets could explain why the columnar configuration is favored. No quantum criticality (gaplessness) can occur in the columnar arrangement of the dimerized chains.

DOI
10.1103/physrevb.76.132410

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Because the three-leg ladder behaves like a renormalized single Heisenberg chain, we argue that a spin-Peierls instability must occur in this system when it is coupled to three-dimensional phonons. Using the bond-mean-field theory, we show that this is indeed the case. The dimerized state below the spin-Peierls transition temperature forms into the columnar dimerized phase instead of the staggered one. This contrasts with the argument based on antiferromagnetism. A physical argument based rather on spin bonding into singlets could explain why the columnar configuration is favored. No quantum criticality (gaplessness) can occur in the columnar arrangement of the dimerized chains.
We analyze the possibility of quantum criticality (gaplessness) in dimerized antiferromagnetic two- and three-leg spin-12\frac{1}{2}21​ ladders. Contrary to earlier studies of these models, we examine different dimerization patterns in the ladder. We find that ladders with the columnar dimerization order have lower zero-temperature energies, and they are always gapped. For the staggered dimerization order, we find the quantum critical lines, in agreement with earlier analyses. The bond mean-field theory we apply demonstrates its quantitative accuracy and agrees with available numerical results. We conclude that unless some mechanism for locking dimerization into the energetically less favorable staggered configuration is provided, the dimerized ladders do not order into the phase where the quantum criticality occurs.
A two-leg S = 1/2 spin ladder with ferromagnetic rung coupling is investigated to reveal the phase transition between the Haldane and columnar dimer phase. The elastic lattice with the elastic force K is introduced into the system, which induces unstable spin chains towards the spontaneous dimerization. When the rung coupling is strong enough, the dimerization along the legs is suppressed and the spin ladder undergoes a phase transition. The dimerization amplitude is calculated self-consistently by the density-matrix renormalization group method. To determine the phase transition boundary, the spin gap, the columnar dimer order parameter and the block-block entanglement entropy are calculated. Our results show that the phase boundary between the columnar dimer phase and Haldane phase follows the power law J t ∼ K −α .
We study the effect of dimerization (due to, e.g., spin-Peierls instability) on the phase diagram of a frustrated antiferromagnetic spin-1/2 ladder, with weak transverse and diagonal rung coupling. Our analysis focuses on a one-dimensional version of the model (i.e., a single two-leg ladder) where we consider two forms of dimerization on the legs: columnar dimers (CDs) and staggered dimers (SDs). We examine in particular the regime of parameters (corresponding to an intermediate XXZXXZXXZ anisotropy) in which the leg dimerization and the rung coupling terms are equally relevant. In both the CD and SD cases, we find that the effective field theory describing the system is a self-dual sine-Gordon model, which favors ordering and the opening of a gap to excitations. The order parameter, which reflects the interplay between the leg and rung dimerization interactions, represents a crystal of 4-spin plaquettes on which longitudinal and transverse dimers are in a coherent superposition. Depending on the leg dimerization mode, these plaquettes are closed or open, however both types spontaneously break reflection symmetry across the ladder. The closed plaquettes are stable, while the open plaquette order is relatively fragile and the corresponding gap may be tuned to zero under extreme conditions. We further find that a first-order transition occurs from the plaquette order to a valence bond crystal (VBC) of dimers on the legs. This suggests that in a higher-dimensional version of this system, this variety of distinct VBC states with comparable energies leads to the formation of domains. Effectively one-dimensional gapless spinon modes on domain boundaries may account for the experimental observation of spin-liquid behavior in a physical realization of the model.
We investigate the spin-Peierls instability of some periodic 1D Heisenberg spin systems having a gapless energy spectrum at different values of coupling J between the unit cells. Using the density-matrix renormalization group method we numerically study the dependence of critical exponents p of spin-Peierls transition of above spin systems on the value of J. In contrast to chain systems, we find significantly non-monotonous dependence p (J) for three-legs ladder system. In the limit of weak coupling J we derive effective spin s chain Hamiltonians describing the low-energy states of the system considered by means of perturbation theory. The value of site spin s coincides with the value of the ground-state spin of the isolated unit cell of the system considered. This means that at small J values all the systems with the singlet ground state and the same half-integer value of s should have a similar critical behavior which is in agreement with our numerical study. The presence of gapped excitations inside the unit cells at small values of J should give, for our spin systems, at least one intermediate plateau in field dependence of magnetization at low temperatures. The stability of this plateau against the increase of the values of J and temperature is studied using the quantum Monte-Carlo method.
Dimerized spin-1/2 ladders exhibit a variety of phase structures, which depend on the intra-chain and inter-chain spin exchange energies as well as on the dimerization pattern of the ladder. Using the density matrix renormalization group (DMRG) algorithm, we study critical properties of the bond-alternating two-leg Heisenberg spin ladder with diagonal interaction J ×. Two types of spin systems, staggered dimerized antiferromagnetic ladder and columnar dimerized ferro-antiferromagnetic couplings ladder, are investigated. To clarify the phase transition behaviors, we simultaneously analyze the string order parameter (SOP), the twisted order parameter (TOP), as well as a measurement of the quantum information analysis. Based on measuring this different observables, we establish the phase diagram accurately and give the fitting functions of the phase boundaries. In addition, the phase transition of cross-coupled spin ladder (in the absence of intrinsic dimerization) is also discussed.
The antiferromagnetic Heisenberg model on the two-leg ladder with exchange interactions along the chains, rungs, and diagonals is studied using the Jordan-Wigner transformation and bond-mean-field theory. The inclusion of all three couplings introduces frustration to the system and depending on their relative strengths the ladder can adopt one of three possible magnetically disordered gapped states. The phase diagram found in this mean-field approach is in very good agreement with the one calculated by Weihong et al. [Phys. Rev. B 57, 11439 (1998)] using the Lanczos exact diagonalization method. By analyzing the ground-state energy, we study quantum criticality when the coupling parameters are varied at zero temperature. We study the effect of temperature on the phase boundaries and find that the system shows thermally induced criticality for some values of the rung and diagonal coupling constants. All the phase transitions encountered in this system occur between disordered phases and are all caused by frustration.