Ising antiferromagnet with mobile, pinned and quenched defects

Motivated by recent experiments on (Sr,Ca,La)_14 Cu_24 O_41, a two-dimensional Ising antiferromagnet with mobile, locally pinned and quenched defects is introduced and analysed using mainly Monte Carlo techniques. The interplay between the arrangement of the defects and the magnetic ordering as well as the effect of an external field are studied.

In particular, a simple Ising model on a square lattice with mobile defects has been introduced [ 5], with the chain direction corresponding to one of the axes of the lattice. The spins, S (i,j) = ±1 at lattice site (i, j), correspond to the magnetic Cu 2+ ions, and the defects, S (i,j) = 0, to those Cu ions which are believed to be spinless due to holes (Zhang-Rice singlets). In the model neighbouring spins are supposed to be coupled ferromagnetically in each chain and antiferromagnetically in adjacent chains. In addition, each pair of next-nearest neighbour spins in the same chain separated by a defect is presumed to interact strongly antiferromagnetically. The (mobile) defects are allowed to hop to neighbouring sites in the chains, with the energy barriers of these moves given merely by the magnetic couplings.
Without defects, the model describes a two-dimensional Ising antiferro-or metamagnet, with ferromagnetic ordering in the chains and antiferromagnetic ordering between the chains in the low-temperature phase [ 9]. The defects form, at low temperatures, nearly straight stripes, perpendicular to the CuO 2 chains, separating antiferromagnetic domains. The coherency of the stripes gets lost at a phase transition of first order [ 5,6].
To mimic the possible pinning of the defects, due to the La ions, variants of the model may be considered with local pinning positions at periodic [ 6,7] or random sites. For indefinitely large pinning strength, the defects will be quenched at fixed sites. Obviously, pinning or even quenching may affect the stability of the defect stripes, the magnetic ordering, and related phase transitions. Note that the model, albeit being motivated experimentally, is thought to be of genuine theoretical interest as well, describing the interplay of magnetic structures and defect arrangements. In that respect, the analysis of the model belongs to the intriguing studies dealing with various aspects on randomness in magnets, ranging, say, from site diluted ferromagnets to spin glasses [ 10,11,12,13,14,15].
Concerning the interpretation of experiments on (Sr,La,Ca) 14 Cu 24 O 41 , attention may be drawn also to current theoretical analyses based on two-dimensional anisotropic Heisenberg models [ 3,16,17].
The article is organized as follows: In the next section, the Ising model with defects is introduced. Results are discussed in Section 3, reviewing as well as illustrating previous, recent findings and presenting new results on the full model with quenched defects. A brief summary concludes the paper.

Model
The Ising model with mobile, pinned or quenched defects is defined on a square lattice with one axis corresponding to the chain direction, say, the horizontal, i-axis, as shown in Fig. 1. Each lattice site, (i, j), is occupied either by a spin, S (i,j) = ±1, or by a defect, S (i,j) = 0. The concentration of defects is fixed to be ten percent of the lattice sites, as it seems to be the case in La 5 Ca 9 Cu 24 O 41 . For simplicity, we shall assume the same concentration of defects in each chain. The interactions, as sketched in Fig. 1, are as follows. Neighbouring spins are coupled ferromagnetically, J > 0, along the chains, and antiferromagnetically, J a < 0, perpendicular to them. In addition, next-nearest neighbouring spins in the same chain separated by a defect interact antiferromagnetically, J 0 < 0, as suggested by experiments. A local pinning potential, E p (i p , j p ) at fixed sites (i p , j p ) may act upon defects, reducing the energy by the pinning strength. Here we shall assume that the concentration of pinning sites is identical to the concentration of defects, with the same number of pinning sites and defects in each chain. Furthermore, the pinning strength E p is taken to be the same at each pinning site. If E p = 0, the defects are called mobile, at finite values of E p > 0, they are pinned, and at E p −→ ∞, the defects are quenched. For finite pinning strength, defects are allowed to diffuse along the chains, keeping a minimal distance of two lattice spacings.
When J and |J 0 | are large compared to |J a |, as suggested by experiments, one may arrive at the 'minimal version' of the model, where spins along a chain are assumed to have the same sign between two defects, reversing sign at a defect. The only relevant energy parameter is then J a [ 5].
In the following, we shall present properties of the minimal model with mobile defects and defects pinned at periodic pinning lines perpendicular to the chains, extending recent work [ 5,6,7,8,18].
The full model, where all couplings are finite, will be discussed without pinning as well as with defects quenched at periodic pinning lines and at random sites. In the following, we take J a = −0.3J and J 0 = −6.25J, choices based on experimental input [ 5].
Analyses are based on ground state considerations, the free-fermion method to describe thermal properties at low temperatures, the transfer matrix approach, and Monte Carlo simulations.
Simulations are rather demanding, because of, usually, slow fluctuations due to the defects. Typically, runs over at least 10 6 Monte Carlo steps per site were performed, averaging then over several realizations, simulating lattices with linear dimension L in the i direction, and M in the j direction, see Fig. 1. In the following we take L = M. In the case of pinning at random sites, an ensemble average has to be taken. Finally, different system sizes have to be simulated especially to determine accurately the phase transition temperatures.
Physical quantities of interest [ 5,6] include the specific heat, C, the susceptibility, χ, and spin correlation functions parallel to the chains, and perpendicular to the chains, Without pinning as well as in the case of defects quenched at random sites, the defect positions are expected to be uncorrelated, so that there is translational invariance with the spin correlations not depending on i. Note that in the thermodynamic limit (L, M −→ ∞) for infinitely large distance r the correlations determine the (sublattice) magnetization.  We also calculated less common microscopic quantities which describe the stability of the defect stripes and the ordering of the defects in the chains [ 5,6], including the average minimal distance d m between each defect in chain j, at position (i d , j), and those in the next chain, at (i ′ d , j + 1), and the cluster distribution n d (l) denoting the probability of a cluster with l consecutive spins of equal sign in a chain (as considered, e.g., in percolation theory [ 19]). Our main emphasis will be on pairs of defects with l = 1. Finally, it turned out to be quite useful to visualize the microscopic spin and defect configurations as encountered during the simulation.

Results
We first present briefly some main results on the minimal model [ 5,6,7,18], illustrating crucial features by showing Monte Carlo data for typical equilibrium configurations, see Fig. 2, and for correlation functions.
The model with mobile defects, E p = 0, is known to form, at T = 0, straight defect stripes perpendicular to the chains with arbitrary separation between the stripes. Correlations along the chains, G 1 , oscillate, and the amplitude decays exponentially due to the large degeneracy of the ground state (compare Fig. 3). On the other hand, perpendicular to the chains the spins are fully correlated, G 2 (i, r) = 0.9(−1) r , reflecting the antiferromagnetic ordering. As temperature T is increased the stripes will meander, tending to keep, caused by entropic repulsion, on average their largest possible distance. The amplitude of the correlations decays to zero with distance r algebraically, i.e. there is no long-range antiferromagnetic order. At a phase transi-  tion of first order, the stripes will break up, accompanied by a pairing of defects. The pairing of the defects, as monitored in the temperature dependence of the probability of next-nearest defect pairs n d (1), results from an attractive effective interaction between neighbouring defects in a chain. This interaction, mediated by the magnetic coupling J a , occurs for strongly fluctuating stripes [ 5].
Let us now introduce, in the minimal model, a local pinning, E p > 0, of the defects at the sites of straight equidistant lines perpendicular to the chains. Then, at low temperatures, the stripes stay close to these pinning lines and long-range antiferromagnetic order is observed. The phase transition remains to be of first order, driven, again, by the enhanced pairing of defects [ 6]. Certainly, the transition shifts to higher temperatures as E p is increased.
The distinction between the algebraic and long-range order at low temperatures without and with pinning is illustrated in Fig. 3. There correlations along the chains, G 1 , are depicted at temperatures below, close to, and above the transition temperature. At low temperatures for E p = 0, the height of the maxima in the correlations falls off even at large distance r, Fig. 3a, while it reaches quickly a non-zero constant value in the pinned case E p > 0, see Fig. 3b.
The impact of an external field on the arrangement of the defect stripes and on the phase transition, both for mobile and periodically pinned defects, has been discussed before [ 7,18]. In particular, at low temperatures, the defect stripes are straight at low fields, acquire a zig-zag structure at larger fields and break up into defect pairs when further increasing the field. The influence of random pinning sites on thermal properties of the minimal model has not been studied in detail yet. Of course, the limiting case of defects quenched at random sites is rather trivial, where the amplitude of the correlations decays exponentially along and perpendicular to the chain, independent of temperature. Similarly, periodic quenching leads to full antiferromagnetic order at any temperature in the minimal model.
We now turn to the full model with J a = −0.3J and J 0 = −6.25J. The two spins next to a defect still tend to have different sign, because J 0 is assumed to be rather large. However, spins between two defects may flip quite easily, with the flip energy being determined by the intrachain coupling J. Indeed, those quasi onedimensional spin excitations may mask the phase transition in some thermodynamic quantities, as may be seen, for instance, in the specific heat [ 5,18]. However, by analysing microscopic quantities describing the stability of the defect stripes, like the minimal distance d m and the probability of encountering defect pairs separated by merely one spin n d (1), in the full model with mobile defects, one observes again the phase transition driven by the stripe instability due to defect pairing at a similar temperature as in the minimal model [ 5,18].
Introducing quenched defects in the full model, quite interesting thermal properties are observed due to the spin excitations, in contrast to the trivial situation in the minimal model. In the case of quenching the defects at periodically placed lines perpendicular to the chains, we determine, using standard Monte Carlo techniques, the phase diagram in the (temperature, field)-plane, as shown in Fig. 4. In the ground state, when applying and increasing a field, H, the antiferromagnetic configuration eventually transforms into a (predominantly) ferromagnetic structure. For sufficiently strong antiferrogmagnetic coupling at the defects, J 0 , as it is the case here, one of the two spins next to a defect will still point against the direction of the field (so that the antiferromagnetic ordering next to a defect is still preserved), while all other spins are aligned parallel to the field. Of course, further increase of the field, at T = 0, will lead to full ferromagnetic order even for strong couplings J 0 [ 18].
While at zero field, H = 0, the phase transition temperature may be determined exactly by analytical means [ 20], Monte Carlo simulations are useful to map the entire phase diagram. Here, we obtained the phase transition line by finitesize analyses on data for the specific heat as well as the sublattice magnetization. Note that one encounters, of course, long-range antiferromagnetic ordering in the low-temperature and low-field phase, as observed, e.g., in the correlation function perpendicular to the chains, see Fig. 5a.
When quenching defects at random sites, the antiferromagnetic order seems to be destroyed even at zero temperature. Ground state analyses for finite systems in- dicate that the amplitude of the correlations, Eqs. (1) and (2), falls off exponentially with large distance r. However, the behaviour deviates quantitatively from that of the minimal model with randomly quenched defects. Indeed, at T = 0, to minimize the energy of the full model, the antiferromagnetic arrangement of the spins in adjacent chains, suppressed locally in the minimal model, may be partly restored by turning over spins between two consecutive defects. The amplitudes of the simulated spin correlations are observed to decay rapidly with r even at very low temperatures, as illustrated in Fig. 5b. In fact, there is no evidence for a transition from an algebraically or even long-range ordered antiferromagnetic low-temperature phase to a disordered phase. Instead, the disordered phase seems to extend down to T = 0. The absence of a phase transition is also reflected in the temperature dependence of the specific heat C. As depicted in Fig. 6, C displays a Schottky-type maximum at k B T /J ≈ 1, whose height depends very weakly on the system size. Around that temperature, short-range spin correlations get reduced significantly, due to the thermally enhanced number of spin flips.

Summary
In this article, we studied a two-dimensional Ising model with ferromagnetic interactions between nearest neighbouring spins along one axis, the chain direction, and somewhat weaker antiferromagnetic couplings between spins in adjacent chains. The model has an antiferro-or metamagnetic low-temperature phase, with a con-tinuous transition to the disordered phase.
Introducing mobile, pinned or quenched defects with a strong antiferromagnetic interaction between next-nearest neighbour spins in a chain separated by a defect leads to a variety of interesting features, both in the minimal and the full models.
In both models, mobile defects tend to form stripes at low temperatures. At a phase transition of first order the stripes become unstable, losing their coherency. The antiferromagnetic ordering in the low-temperature phase is reduced, characterized by algebraically decaying spin correlations.
Long-range order for antiferromagnetic domains may be restored by pinning the defects at (almost) straight lines perpendicular to the chains.
Obviously, properties of the minimal model with quenched defects do not depend on temperature. By quenching the defects in the full model at random sites, the magnetic ordering at low temperatures is destroyed, and we find no evidence for a phase transition.