Simulations of the low-dimensional magnetic systems by the quantum transfer-matrix technique

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Abstract

A numerical quantum transfer-matrix approach based on the Suzuki–Trotter formula and the checker-board decomposition is presented in the framework of quantum statistical mechanics. It is applied to the supramolecular cluster Mn6 (i.e. [Mn(hfac)2NITPh]6) and to a number of the macroscopic quasi-one-dimensional magnets. The latter include: (i) the macroscopic Haldane-gap spin S=1 chains and molecular-based magnetic spin S=1 chains (with uniform and alternating interaction couplings); (ii) the spin-Peierls CuGeO3 and Pb[Cu(SO4)(OH)2] compounds subject to frustration; (iii) Yb4As3 which is a new semimetallic material being the first example of the antiferromagnetic S=1/2 spin chain with the induced staggered field appearing as a result of the antisymmetric Dzyaloshinsky–Moriya interaction. The compounds in question can be characterized within the spin Heisenberg models and their thermodynamic properties at finite temperatures are calculated using our own codes. Our simulation results are compared with the available experimental results and a quantitative agreement has been established. The large-scale numerical simulations were carried out on the Cray and Silicon Graphics supercomputers, using the parallelized and vectorized codes, exploiting the Parallel Virtual Machine (PVM) and the Message Passing Interface (MPI) system libraries.

Introduction

The magnetic features of many magnetic materials may be quite accurately represented by Heisenberg model. In the presence of the external magnetic field and with the nearest neighbor and the next-nearest neighbor interactions, a one-dimensional system can be described by the HamiltonianH=−J∑i=1N(Si·Si+1Si·Si+2)−D∑i=1N(Siz)2−gνμBB∑i=1NSiν,where Si is interpreted as the spin located at the ith site of a one-dimensional lattice of N equally spaced sites. J denotes the nearest neighbor exchange integral (negative for the antiferromagnetic coupling), α is the ratio of the next-nearest neighbor exchange integral to the nearest neighbor one and D stands for the anisotropy parameter. B is the external magnetic field which can be applied along the chain (ν=z) or in the perpendicular direction (ν=x;y), gν is the corresponding gyromagnetic ratio and N is the size of a given one-dimensional system (the chain or the ring). The spin values Si may be uniform or non-uniform.

In this article we present results of numerical simulations based on the quantum transfer-matrix (QTM) technique for both the macroscopic quasi-one-dimensional magnets and the single-molecule magnets and we compare them with known experimental results. We also describe the methodology of our simulations. The domain of applications of the QTM method is very wide and includes simulations of the thermodynamic properties of the low-dimensional quantum systems.

First we consider the compound [Mn(hfac)2NITPh]6 (hfac, hexafluoroacetyl acetonate; NITPh, 2-phenyl-4,4,5,5-tetramethyl-4,5-dihydro-1H-imidazolyl-1-oxy -3-oxide) which belongs to a class of nanocompounds actively investigated for their magnetic properties [1], [2]. Synthesis of polynuclear metal complexes with oxygen atom bridges has resulted in a series of new molecules with unusual geometric symmetries and patterns [3]. Their magnetic properties, associated to a large number of interacting paramagnetic centers in a single aggregate, have significantly stimulated the research effort with the prospect of technological applications [1], [4], [5], [6]. The interest in spin assemblies stems from the fact that they set the low-size limit for magnetic nanoparticles. They can display magnetic quantum tunneling [3] and quantum-size effects in the thermodynamical properties [7]. Spin aggregates are embedded as the cores in macromolecules with well-defined size, shape and stoichiometry. They are separated from one another by shells of ligand molecules and can be synthesized in macroscopically large samples of regular structures [1]. Polynuclear clusters are magnetic materials of a size intermediate between that of isolated dimers or trimers and that of bulk magnetic materials. They occur in the mesoscopic scale in which the quantum effects may coexist with classical behaviour and various new properties. Also, large metal ion clusters are present in biological systems (e.g. ferritin [1]) and modelling of their properties is under way [8].

We also consider the spin S=1 antiferromagnetic Heisenberg chains. The ground state of integer spin chains was predicted disordered with a gap in the excitation spectrum and the spin-correlation function decaying exponentially. This Haldane conjecture shows the difference between the ground state of integer and half-integer spins. Development of molecular magnetism has determined a dramatic increase in the number and in the types of one-dimensional magnetic materials that can be described by Heisenberg model. An interesting aspect of these compounds is that the intrinsic low symmetry of the building blocks easily affords not only uniform chains, i.e., those in which the nearest neighbor pairs are all identically coupled to each other, but also non-uniform chains, i.e., systems in which a given building block has different coupling to its left and right nearest neighbors.

Another class compounds is that of one-dimensional systems with S=1/2 which have attracted the interest of chemists and physicists for more than three decades. The theory of the ideal uniform S=1/2 antiferromagnetic Heisenberg chain in the magnetic field is well established and usually well describes the observed properties in real systems. The majority of these systems are organic and inorganic compounds with chains of 3d and 4f ions. Recently, a new class of rare-earth pnictide compounds like Yb4As3 have become the focus of attention.

In the present paper, neglecting the interchain coupling, we calculate the field-dependent specific heat of Yb4As3 in a wide range of temperatures (above 4 K, where uncertainties of our extrapolations are below 3%), using the Heisenberg model with the Dzyaloshinsky–Moriya interaction [9], [10]. We would like to check if the experimental specific heat data obtained for a polydomain sample of the compound studies can be accounted for within this model. An alternative model with intrachain dipolar interactions [11] or the sine-Gordon model [12] should also be considered.

We also study another one-dimensional system with frustrated spin S=1/2 chains. In this case we consider the Heisenberg model with the next-nearest neighbor interactions, applicable to CuGeO3 and Pb[Cu(SO4)(OH)2] compounds. We obtained a classical system with spin σ=3/2 and effective interaction between nearest neighbors only, using the Suzuki–Trotter formula.

Our paper is organized as follows. In the next section we describe details of the QTM method. In particular, we show a way of calculation of the partition function for various classes of the low-dimensional magnetic systems. In Section 3 we present results of the speed-up and efficiency of the parallel calculations performed using our parallelized algorithm on CRAY T3E-900 supercomputer. Our main results of simulations are presented in Section 4 and compared to the experimental data for real compounds. The paper is concluded with a discussion of the results and indications for further applications of our technique.

Section snippets

Methodology and description of simulations

The advantages of the QTM simulation method have been demonstrated for the macroscopic Haldane-gap [13] and molecular-based [14] magnetic chains. The results are not subject to any statistical nor systematic errors and the macroscopic limit can be directly evaluated from the largest eigenvalue of the transfer matrix. For the finite rings, however, all terms in the definition of the partition function bring some contribution, so that the computational complexity of the QTM method increases

Parallel implementation of the transfer-matrix simulations

The main numerical problem of our simulations is the calculation of the partition function Zm which is given as a sum of the corresponding diagonal elements in Eq. (7). The algorithm was optimised so that the global transfer operators W1 and W2 (8) act only on the vectors different from the point of view of symmetry. In this way the summation in Eq. (7) is reduced N/2 times (N/2 is the number of pairs in the ring). The global transfer operators W1 and W2 can be expressed as product sparse

Physical applications and simulation results

First, the QTM approach is applied to calculate the susceptibility of the ring with alternating spins SA=1/2, SB=5/2. A representative physical realization of this model is the [Mn(hfac)2NITPh]6 compound. The complex is referred to as the supramolecular cluster Mn6. This molecule contains 12 paramagnetic centers [6], namely six manganese (II) ions, with S=5/2, connected with six organic radicals NITPh with non-paired S=1/2 electron. The two types of spins are strongly antiferromagnetically

Conclusions

We have worked out QTM approach to characterize the finite temperature magnetic properties of the high nuclearity cyclic spin clusters with large and alternating spins and a number of the macroscopic quasi-one-dimensional magnets. For Mn6 cluster the QTM technique provides the numerically exact results. The dimensionality of the multispin space allows us to perform the trace operation according to the definition of the partition function so that a quantitative interpretation could be made. For

Acknowledgments

This work was partially supported by the Committee for Scientific Research via the grant 4 T11F 014 24. Numerical simulations were performed in the Poznań Supercomputing and Networking Center.

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