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arXiv:cond-mat/0510552 · Journal of Statistical Mechanics Theory and Experiment

Néel and disordered phases of coupled Heisenberg chains withS= ½ toS= 4

S. Moukouri

Published Feb 6, 2006• 11 citations
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Abstract

We use the two-step density-matrix renormalization group method to study the effects of frustration in Heisenberg models for S = ½ to 4 in a two-dimensional anisotropic lattice. We find that as for S = ½ studied previously, the system is made up of nearly disconnected chains at the maximally frustrated point, J d / J ⊥ = 0.5, i.e., the transverse spin–spin correlations decay exponentially. This leads to the following consequences: (i) all half-integer spins systems are gapless, behaving like a sliding Luttinger liquid as for S = ½; (ii) for integer spins, there is an intermediate disordered phase with a spin gap, with the width of the disordered state roughly proportional to the 1D Haldane gap.

DOI
10.1088/1742-5468/2006/02/p02002

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We use the two-step density-matrix renormalization group method to study the effects of frustration in Heisenberg models for S = ½ to 4 in a two-dimensional anisotropic lattice. We find that as for S = ½ studied previously, the system is made up of nearly disconnected chains at the maximally frustrated point, J d / J ⊥ = 0.5, i.e., the transverse spin–spin correlations decay exponentially. This leads to the following consequences: (i) all half-integer spins systems are gapless, behaving like a sliding Luttinger liquid as for S = ½; (ii) for integer spins, there is an intermediate disordered phase with a spin gap, with the width of the disordered state roughly proportional to the 1D Haldane gap.
We use a recently proposed perturbative numerical renormalization group algorithm to investigate ground-state properties of a frustrated three dimensional Heisenberg model on an anisotropic lattice. We analyze the ground state energy, the finite size spin gap and the static magnetic structure factor. We find in two dimensions a frustration-induced gapless spin liquid state which separates two magnetically ordered phases. In the spin liquid state, the magnetic structure factor shows evidence that this state is made of nearly disconnected chains. This spin liquid state is unstable against unfrustrated interplane couplings.
Using a recently proposed perturbative numerical renormalization-group algorithm, we explore the connection between quantum criticality and the emergence of Luttinger liquid physics in t−Jt-Jt−J chains coupled by frustrated interactions. This study is built on an earlier finding that at the maximally frustrated point, the ground state of weakly-coupled Heisenberg chains is disordered, the transverse exchanges being irrelevant. This result is extended here to transverse couplings up to J⊥=0.6J_{\perp}=0.6J⊥​=0.6, and we argue that it may also be valid at the isotropic point. A finite size analysis of coupled Heisenberg chains in the vicinity of the maximally frustrated point confirms that the transverse spin-spin correlations decay exponentially while the longitudinal ones revert to those of decoupled chains. We find that this behavior persists upon moderate hole doping $x \alt 0.75$. For larger doping, the frustration becomes inactive and the quantum critical point is suppressed.
I use an improved version of the two-step density-matrix renormalization group method to study ground-state properties of the two-dimensional (2D) Heisenberg model on the checkerboard lattice. In this version, the Hamiltonian is projected on a tensor product of two-leg ladders instead of chains. This allows investigations of 2D isotropic models. I show that this method can describe both the magnetically disordered and ordered phases. The ground-state phases of the checkerboard model as J2{J}_{2}J2​ increases are (i) N\'eel with $Q=(\ensuremath{\pi},\ensuremath{\pi})$, (ii) a valence-bond crystal (VBC) of plaquettes, (iii) N\'eel with $Q=(\ensuremath{\pi}∕2,\ensuremath{\pi})$, and (iv) a VBC of crossed dimers. In agreement with previous results, I find that at the isotropic point J2=J1{J}_{2}={J}_{1}J2​=J1​, the ground state is made of weakly interacting plaquettes with a large gap $\ensuremath{\Delta}\ensuremath{\approx}0.67{J}_{1}$ to triplet excitations. The same approach is also applied to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model. There is no evidence of a columnar dimer phase in the highly frustrated regime.
Using a recently proposed perturbative numerical renormalization-group algorithm, we explore the connection between quantum criticality and the emergence of Luttinger liquid physics in t – J chains coupled by frustrated interactions. This study is built on an earlier finding that at the maximally frustrated point the ground state of weakly coupled Heisenberg chains is disordered, with the transverse exchanges being irrelevant. This result is extended here to transverse couplings up to , and we argue that it may also be valid at the isotropic point. A finite size analysis of coupled Heisenberg chains in the vicinity of the maximally frustrated point confirms that the transverse spin–spin correlations decay exponentially while the longitudinal ones revert to those of decoupled chains. We find that this behaviour persists upon moderate hole doping . For larger doping, the frustration becomes inactive and the quantum critical point is suppressed.
The ground-state phase transitions of a frustrated S=1 Heisenberg chain with the uniaxial single-ion-type anisotropy and the frustrating next-nearest-neighbor coupling are studied. For the system, it has been shown that there are gapless and gapped chiral phases in which the chirality κ_l = S^x_l S^y_{l+1} - S^y_l S^x_{l+1} exhibits a finite long-range order (LRO) and the spin correlation decays either algebraically or exponentially. In this study, the transitions between the Haldane and chiral phase and between the large-D (LD) and chiral phase are investigated using the infinite-system density-matrix renormalization group method. It is found that there exist two types of gapped chiral phases, "chiral Haldane" and "chiral LD" phases, in which the string LRO coexists with the chiral LRO and the string correlation decays exponentially, respectively.
The ground-state phase transitions of a frustrated S = 1 Heisenberg chain with the uniaxial single-ion-type anisotropy and the frustrating next-nearest-neighbor coupling are studied. For the system, it has been shown that there are gapless and gapped chiral phases in which the chirality κl = Slx Syl+1 – Sly Sxl+1 exhibits a finite long-range order (LRO) and the spin correlation decays either algebraically or exponentially. In this study, the transitions between the Haldane and chiral phases and between the large-D (LD) and chiral phases are investigated using the infinite-system density-matrix renormalization group method. It is found that there exist two types of gapped chiral phases, "chiral Haldane" and "chiral LD" phases, in which the string LRO coexists with the chiral LRO and the string correlation decays exponentially, respectively. PACS No.: 75.30