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Construction of average lattices for quasi-periodic structures by the section method.
Periodic surfaces in the description of quasi-crystals. Journal de physique 51 23, Suppl. Bounded interpolations between lattices. Frustrated Ising models on quasiperiodic lattices in two dimensions. Coincidence lattices and associated shear transformations. Acta cryst. Pavages et quasicristaux , chap. Comprendre, — Encyclopaedia Universalis Ground-states of frustrated ising quasi-crystals. Crystallographic aspects of the bonnet transformation for periodic minimal-surfaces and crystals of films. Symmetry reduction in homopolymeric dna - implications for dna fine-structure. Atomic migrations induced by phasons in quasicrystals.
In Janot, C and Mosseri, R, ed. ICQ5 ,. Deformations of periodic minimal surfaces. Kluwer Euclidean invariance of classic states of equilibrium. I Thermodynamics of a frustrated spin system the renormalized triangular lattice. The K-dependent function G K is intro- duced in eq. Furthermore, the invariance of the partition functions under the renormalization leads to 3 where the sum runs over all spin configurations.
The size-dependent coefficients A, B, denote the degeneracy of a configuration with energy M, - 4m K; e. The three smallest clusters used in this paper.
The renormalization procedure we use in this work consists of dividing the lattice into elementary clusters of size L as those in fig. In fig. Furthermore, in analogy to the invariance requirement for the partition functions of the lattice as a whole eq. Moreover we should notice that the sites spins of the new lattice will not in general coincide with some of the sites in the original lattice, as may be suggested by inspection of fig.
In fact, we have the freedom e. In fact, in analogy to eq.
Holdings: Statistical Mechanics of Periodic Frustrated Ising Systems
Lacerdu et al. I Thermodynamics of a frustrated spin system Iterating eq. We should notice that the renormalization group transformation, eq. Actually they represent desirable features of the transformation: eq. By replacing 18 into 16 and using the flow equation 12 we can obtain the free energy f, K for general values of K.
Furthermore, by evaluating the first and second derivatives of eq. We also plot the corresponding exact results dashed lines on the triangular lattice for comparison. The dashed line is the exact result . The specific heat per spin as function of K and for values of L from 2 to 6 full lines. The dashed line is the exact result. The L-dependent quantities A,,and B, are the coefficients of the leading terms of eqs.
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Results for S,,, and E,,,Lfor several sizes of clusters are displayed in table I. Also we include our extrapolated L- 00 values to be compared with the exact results. Finite-size effect: scaling In this section we will propose a simple form of finite size scaling [14, to relate the thermodynamic functions calculated in section 2 for finite triangular Table I Characteristic temperature T, see text and values related to the ground-state energy E,,, and the ground-state entropy S,,, obtained from clusters with different sizes.
Statistical Mechanics of Periodic Frustrated Ising Systems
S:TI refers to the renormalization group estimates eq. For this purpose we first notice that the above approximate theory yields a few relations which either do depend only on the parameter L or is a constant.
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For the frustrated triangular Ising system we are studying, it seems quite natural to choose such characteristic temperatures as those for which the specific heat has a maximum. The extrapolated closed symbols and exact crosses values are indicated. Data collapsing plot for the thermal behaviour of the free energy per spin according to the scaling hypothesis 33 and We conclude our presentation of results by showing in fig. The entropy per spin in the scaled space defined by 33 and In spite of our data full lines not displaying the same degree of agreement with the exact result dashed line as for the free energy in fig.
Difference between the free energy calculated within the present approximation scheme and the exact result. Discussion and conclusions We have employed a renormalization group approximation scheme to calcu- late the thermodynamic properties of the nearest-neighbour antiferromagnetic Ising model on a two-dimensional triangular lattice. The calculations presented in section 2 for a given family of triangular clusters recover the main features of such a fully frustrated spin system.
For example, the flow diagram in the case of antiferromagnetic coupling exhibits no order-disorder transition and we only observe those broad maxima in the curves fig. On the other hand the finite values of the ground-state entropy see table I we obtain are indicative of at least for the family of clusters of fig. We should also notice that our estimates for the thermal behaviour of the thermodymacic functions show a general qualitative agreement with exact results. In fact the rate of convergence may be measured by eq. In section 3 we have proposed a simple form of finite-size scaling [14, which enables us to proceed with a finite size analysis of our renormalization group data, and hence to extract the thermodynamic behaviour from results obtained for finite clusters.
Such a scaling hypothesis may be easiest under- stood if we rewrite eq.
Domain walls in spin glass
It should be emphasized that the pre-factor occurring in 37 , which follows from the observed appropriate analytic L dependence of the ground-state energy eq. In what concerns the plots of fig. For the free energy the agreement with the exact curve full line is also very good, where the errors fig. I Thermodynamics of a frustrated spin system errors are larger but we still have a satisfactory data collapsing notice the relative difference in scales of figs.
Now we would like to discuss the dependence, shown in fig. In our view, the super-estimate extrapolated values for 7ext and SiTL, might be mainly attributed to approximations inherent to the renormalization group approach used in estimating the characteristic temperature FL rather than the small departures from linear behaviour that exist for the small values of L used to perform the extrapolation shown in fig. As a final remark we express our belief that the present work indicates that an appropriate combination of the real-space renormalization group technique and scaling theory may be a useful approach to study the thermal behaviour of a spin system with frustration, for temperatures not only in the regime where critical behaviour is expected to occur.
Therefore generalization of the present formalism to another frustrated spin system would be possible. FGBM is grateful to R. Stinchcombe for useful suggestions and helpful conversations in the early stages of this work. References Th. Niemeijer and J. Domb and M. Green, eds. Academic Press. New York, Domb and J. Lebowitz, eds.