The large-$m$ limit, and spin liquid correlations in kagome-like spin models

It is noted that the pair correlation matrix $hat{chi}$ of the nearest neighbor Ising model on periodic three-dimensional ($d=3$) kagome-like lattices of corner-sharing triangles can be calculated partially exactly. Specifically, a macroscopic number $1/3 , N+1$ out of $N$ eigenvalues of $hat{chi}$ are degenerate at all temperatures $T$, and correspond to an eigenspace $mathbb{L}_{-}$ of $hat{chi}$, independent of $T$. Degeneracy of the eigenvalues, and $mathbb{L}_{-}$ are an exact result for a complex $d=3$ statistical physical model. It is further noted that the eigenvalue degeneracy describing the same $mathbb{L}_{-}$ is exact at all $T$ in an infinite spin dimensionality $m$ limit of the isotropic $m$-vector approximation to the Ising models. A peculiar match of the opposite $m=1$ and $mrightarrow infty$ limits can be interpreted that the $mrightarrowinfty$ considerations are exact for $m=1$. It is not clear whether the match is coincidental. It is then speculated that the exact eigenvalues degeneracy in $mathbb{L}_{-}$ in the opposite limits of $m$ can imply their quasi-degeneracy for intermediate $1 leqslant m < infty$. For an anti-ferromagnetic nearest neighbor coupling, that renders kagome-like models highly geometrically frustrated, these are spin states largely from $mathbb{L}_{-}$ that for $mgeqslant 2$ contribute to $hat{chi}$ at low $T$. The $mrightarrowinfty$ formulae can be thus quantitatively correct in description of $hat{chi}$ and clarifying the role of perturbations in kagome-like systems deep in the collective paramagnetic regime. An exception may be an interval of $T$, where the order-by-disorder mechanisms select sub-manifolds of $mathbb{L}_{-}$.

Heisenberg antiferromagnet with anisotropic exchange on the Kagome lattice: Description of the magnetic properties of volborthite

We study the properties of the Heisenberg antiferromagnet with spatially anisotropic nearest-neighbour exchange couplings on the kagome net, i.e. with coupling J in one lattice direction and couplings J along the other two directions. For J/J > 1, this model is believed to describe the magnetic properties of the mineral volborthite. In the classical limit, it exhibits two kinds of ground states: a ferrimagnetic state for J/J < 1/2 and a large manifold of canted spin states for J/J > 1/2. To include quantum effects self-consistently, we investigate the Sp(N) symmetric generalisation of the original SU(2) symmetric model in the large-N limit. In addition to the dependence on the anisotropy, the Sp(N) symmetric model depends on a parameter kappa that measures the importance of quantum effects. Our numerical calculations reveal that in the kappa-J/J plane, the system shows a rich phase diagram containing a ferrimagnetic phase, an incommensurate phase, and a decoupled chain phase, the latter two with short- and long-range order. We corroborate these results by showing that the boundaries between the various phases and several other features of the Sp(N) phase diagram can be determined by analytical calculations. Finally, the application of a block-spin perturbation expansion to the trimerised version of the original spin-1/2 model leads us to suggest that in the limit of strong anisotropy, J/J >> 1, the ground state of the original model is a collinearly ordered antiferromagnet, which is separated from the incommensurate state by a quantum phase transition.

Spatially anisotropic Heisenberg Kagome antiferromagnet

In the search for spin-1/2 kagome antiferromagnets, the mineral volborthite has recently been the subject of experimental studies [Hiroi et al.,2001]. It has been suggested that the magnetic properties of this material are described by a spin-1/2 Heisenberg model on the kagome lattice with spatially anisotropic exchange couplings. We report on investigations of the Sp(N) symmetric generalisation of this model in the large N limit. We obtain a detailed description of the dependence of possible ground states on the anisotropy and on the spin length S. A fairly rich phase diagram with a ferrimagnetic phase, incommensurate phases with and without long range order and a decoupled chain phase emerges.

A Marginal Dimension of a Weakly Diluted Quenched m-Vector Model

We calculate a marginal order parameter dimension $m_c$ which in a weakly diluted quenched $m$-vector model controls the crossover from a universality class of a ``pure model ($m>m_c$) to a new universality class ($m<m_c$). Exploiting the Harris criterion and the field-theoretical renormalization group approach allows us to obtain $m_c$ as a five-loop $epsilon$-expansion as well as a six-loop pseudo-$epsilon$ expansion. In order to estimate the numerical value of $m_c$ we process the series by precisely adjusted Pade--Borel--Leroy resummation procedures. Our final result $m_c=1.912pm0.004<2$ stems from the longer and more reliable pseudo-$epsilon$ expansion, suggesting that a weak quenched disorder does not change the values of $xy$-model critical exponents as it follows from the experiments on critical properties of ${rm He}^4$ in porous media.

Critical exponents of a three dimensional weakly diluted quenched Ising model

We discuss universal and non-universal critical exponents of a three dimensional Ising system in the presence of weak quenched disorder. Both experimental, computational, and theoretical results are reviewed. Special attention is paid to the results obtained by the field theoretical renormalization group approach. Different renormalization schemes are considered putting emphasis on analysis of divergent series obtained.

Pseudo--epsilon expansion of six--loop renormalization group functions of an anisotropic cubic model

Six-loop massive scheme renormalization group functions of a d=3-dimensional cubic model (J.M. Carmona, A. Pelissetto, and E. Vicari, Phys. Rev. B vol. 61, 15136 (2000)) are reconsidered by means of the pseudo-epsilon expansion. The marginal order parameter components number N_c=2.862(5) as well as critical exponents of the cubic model are obtained. Our estimate N_c<3 leads in particular to the conclusion that all ferromagnetic cubic crystals with three easy axis should undergo a first order phase transition.

Effective and Asymptotic Critical Exponents of Weakly Diluted Quenched Ising Model: 3d Approach Versus $epsilon^{1/2}$-Expansion

We present a field-theoretical treatment of the critical behavior of three-dimensional weakly diluted quenched Ising model. To this end we analyse in a replica limit n=0 5-loop renormalization group functions of the $phi^4$-theory with O(n)-symmetric and cubic interactions (H.Kleinert and V.Schulte-Frohlinde, Phys.Lett. B342, 284 (1995)). The minimal subtraction scheme allows to develop either the $epsilon^{1/2}$-expansion series or to proceed in the 3d approach, performing expansions in terms of renormalized couplings. Doing so, we compare both perturbation approaches and discuss their convergence and possible Borel summability. To study the crossover effect we calculate the effective critical exponents providing a local measure for the degree of singularity of different physical quantities in the critical region. We report resummed numerical values for the effective and asymptotic critical exponents. Obtained within the 3d approach results agree pretty well with recent Monte Carlo simulations. $epsilon^{1/2}$-expansion does not allow reliable estimates for d=3.