The phase diagram of the next-neighbour Ising model of the face-centred cubic lattice

We use Monte Carlo simulation to determine the stable structures in the second-neighbour Ising model on the face-centred cubic lattice. Those structures are for strongly antiferromagnetic second-neighbour interactions and for ferromagnetic and weakly antiferromagnetic second neighbours. We find a third stable “intermediate” antiferromagnetic phase with symmetry, and calculate the paramagnetic transition temperature for each. The transition temperature depends strongly on second-neighbour interactions which are not frustrated. We determine a sublattice structure suitable for solving this problem with mean-field theory.

where stands for summation over NNs, and for NNNs. Ising spins S i are taken as ±1. H is the magnetic field which we consider only in the ground-state analysis; simulations are at zero field (H = 0). In the case of the fcc lattice, the sets of NN and NNN bonds have the same F m3m as the lattice. The Hamiltonian in the above eq. (1) can be analysed as a function of two dimensionless (a) E-mail: gjackland@ed.ac.uk (corresponding author) quantities: the ratio of the interactions relative to each other, and to the temperature: Without loss of generality, we choose units such that |J 1 | = 1.
Calculation of phase stability in the antiferromagnetic Ising model is challenging because of the existence of many possible antiferromagnetic arrangements. Furthermore, the face-centred cubic lattice can be viewed as ABC stacking of triangular lattices, leading to frustration: when two spins on the triangle are different, the third cannot be simultaneously different from both. Furthermore, there exists an ordering without translational symmetry for the AFM triangular and fcc lattice which has lower energy than any periodic one ( fig. 1), which inhibits nucleation and growth of periodic structures in a Monte Carlo simulation. Although the Hamiltonian H has full F m3m symmetry, the antiferromagnetic arrangement of spins will normally have lower symmetry. The two main approaches to the problem are Monte Carlo simulation and mean-field theories [1][2][3][4][5][6][7]. Monte Carlo correctly includes all correlation effects, within the 6912 independent sites, but being a numerical method, it cannot determine the phase boundary analytically [8,9]. By contrast, effective mean-field approaches [10] are typically built on cluster approaches which limits the spatial range of correlations. Crucial to this is the choice of sublattice structure, which 41003-p1 Fig. 1: Aperiodic ordering on the triangular lattice with lower energy than any periodic order for the nearest-neighbour AFM Ising model. Central site has six unlike neighbours: when extended in a bullseye pattern all other sites have four unlike neighbours. The lowest-energy AFM periodic structures have four unlike neighbours at each site. The generalisation to fcc is straightforward -each subsequent layer is coloured to be different from the majority of sites below. restricts the possible antiferromagnetic symmetry breakings. The sublattice structure must therefore be chosen with reference to possible solutions for H.
Many previous authors have looked at the nearneighbour only case [11][12][13][14][15][16][17]. In our previous work [10], we analysed the case where α is positive, i.e., secondneighbour interactions are ferromagnetic. We also considered non-zero field, creating a three-dimensional α, T, H phase diagram. In that system the possible phases are L1 0 , L1 2 and paramagnetic. Those phases were examined in mean-field theory using a conventional (4-atom) fcc cell in which the four sites are treated as independent sublattices. A superdegenerate point exists at H = 4, T = 0 where L1 0 , and L1 2 are degenerate, as are a range of point and extended defects.
A recent paper by Jurčišinová and Jurčišin (JJ) entitled "Prediction of the existence of an intermediate phase in the antiferromagnetic J 1 -J 2 system on the face-centered cubic lattice" [18] tackled the harder problem of α < 0, where second-neighbour interactions are also antiferromagnetic, simplifying matters by setting H = 0. Despite the title, they actually considered a Hamiltonian which has P m3m symmetry with two inequivalent sites (L1 2 in Strukturbericht designation). To investigate symmetry breaking due to antiferromagnetism, they used a threesite sublattice structure with one sublattice comprising the face-centres, and two sublattices on cube corners ( fig. 2). They reported that the phase diagram has two "antiferromagnetic" phases (named AFM1 and AFM2) and a third "well-defined" intermediate phase. Here we investigate whether any intermediate state of the type found in the P m3m Ising model is also present in the more familiar fcc lattice.
Ground-state structures. -First we consider only the T = 0 case, attempting to identifying the possible stable structures. According to the third law of thermodynamics, an ordered state must be the most stable. Identifying these candidate states is a necessary precursor to  [18]. Discussing with the authors after the current paper was completed, it transpires that the lattice they considered has only the interactions shown in the figure, i.e., no interactions between atoms on the face-centred sublattice C. Moreover, the corner sites were doubly weighted.
making a sensible definition of order parameters or sublattice structures. At T = 0, these can be generated by hand, looking at colourings of sites on the appropriate lattice which maximise unlike first and/or second neighbours. Some orderings are long established from the nearneighbour problem and taken from previous work (here refs. [10,18] were used). Other structures were constructed by colouring-in drawings of the fcc lattice with a crayon, maximising the number of unlike second neighbours either absolutely (L1 1 ) or subject to maximised near neighbours (I4 1 /amd). The relevant phases are shown in fig. 3 with details given in table 1 and figs. 2 and 4.
If we consider the reported states of the JJ structures, we see that AF1 has m A = m B = −m C . This is the L1 2 structure, which can be obtained in the four-sublattice model with m 1 = m 2 = m 3 = −m 4 . In fcc, the L1 2 structure has a ground-state energy which can be written in the three-sublattice decomposition as or in the four-site decomposition as For antiferromagnetic J 2 this is less stable than randomly oriented spins, and therefore L1 2 (AF1) should not appear in this region of the phase diagram, since it 41003-p2   Table 1: Perfect crystal energies at T = 0 from eq. (1). Candidate phases from [10] AFM1 and AFM2 are from ref. [18]. "Stability" indicates the region of the phase diagram where the phase is expected. The horizontal line separates phases observed in this work from others reported elsewhere.

Structure
Free energy Magnetization is not stable at T = 0, and has lower entropy than the disordered paramagnetic state. DO 22 is always more stable than L1 2 , but it may only be stabilised by an external field [10]. We can contrast this with the L1 0 phase which comprises alternating (001) planes of different spins; using our sublattice structure it is m 1 = m 2 = −m 3 = −m 4 , but L1 0 cannot be represented within the three-sublattice assumption. In L1 0 all sites have equal energy E = −4J 1 + 6J 2 . This is the unique stable state at zero field for ferromagnetic J 2 , and extends some way into the antiferromagnetic J 2 region (fig. 5). Clearly, for 6J 2 > 4J 1 this L1 0 structure is higher than zero, so some other ordered phase must exist which favours unlike second neighbours.
It seemed unlikely that L1 0 , which has all NNN aligned, could persist when J 2 is antiferromagnetic. For 41003-p3 near-neighbour only interactions L1 0 has zero-energy stacking faults [10], and by considering an array of stacking faults we found an intermediate phase with I4 1 /amd symmetry which does not appear in the Strukturbericht designation. This is degenerate with L1 1 at J 2 = J 1 /2 and L1 0 J 2 = 0, and more stable between those values. We note that in the limit J 1 → 0 the fcc structure breaks into four unconnected simple cubic lattices, which can be made independently antiferromagnetic in the B1 (NaCl) structure without frustration. L1 1 can be viewed as four interpenetrating NaCl lattices.

Numerical simulations. -We ran Metropolis Monte
Carlo [19] simulations on a 12× 12 × 12 × 4 atom supercell. The model parameters are J 2 and T and there are two cases: ferromagnetic J 1 = 1 and antiferromagnetic J 1 = −1. No external field was applied (H = 0). Updates were single-site flips, of randomly chosen sites. At each temperature we equilibrate for 10 6 attempted flips and collect data for 10 9 .
In fig. 5 we show the phase diagram found by monitoring the temperature variation of fluctuations in the energy, and detecting peaks therein. To detect transitions between ordered phases we monitor fluctuations in the NNN contribution to the energy only.
The simulations revealed just four distinct ordered phases, all of which were as anticipated from the analytic ground-state calculations: The AFM1 and AFM2 structures found by JJ on their P m3m lattice are not observed in fcc, F m3m with antiferromagnetic second neighbours. Our intermediate I4 1 /amd structure is also different from the JJ intermediate structure.
Peak detection is not completely straightforward, because a high variation of H can occur if there is a domain structure which rearranges itself during a simulation. Such an event produces a bimodal distribution and consequent high value for ( H ∈ − H 2 ) at a single temperature, whereas a thermodynamic phase transition produces a characteristic lambda transition across a range of temperatures. For this reason, c(T ) cannot always be associated with a specific heat capacity. To address this, we plot in fig. 5 the temperatures corresponding to the two highest values of c(T ) as points on a graph of J 2 vs. T . This traces out the phase boundaries with a sharp line, and also shows 41003-p4 a diffuse region corresponding to the "annealing temperature", at which point the single-flip algorithm is able to anneal out a domain structure. We also plot examples of c(T ) and H(T ) from single runs which show the domain formation events as single peaks. In all the AFM phases, the sites are equivalent except for the sign of the spin, so the sublattice magnetisation is simply the square root of fraction of the T = 0 binding energy (i.e., energy with negative sign).
The phase lines are rather straight, with the PM transition temperature lowest at the "maximally frustrated" value of J 2 where two ordered structures are degenerate.
Sublattice structures. -A mean-field treatment of the antiferromagnetic second-neighbour Ising model will require a sublattice decomposition which permits all possible ground states: alternating (001) layers and alternating (111) layers, and the I4 1 /amd. Each have two independent sublattices, so a supercell which can describe them all requires at least eight sublattices. One such structure is shown in fig. 3. Compared to the conventional fcc cell it has a = (1, 1, 0) b = (1, −1, 0) c = ( 1 2 , 1 2 , 1). To include L1 2 and DO 22 structures a still larger set of sublattices is needed, based on a 16-atom cell a = (1, 1, 0) The changing domain structure of the Monte Carlo simulation precludes assignment of sites to sublattices, but one can obtain a mean-field estimate of sublattice magnetisation m from inverting eq. (1) using the T = 0 energies in table 1, i.e., m = H(T )/H(0). This is valid only in the ordered phase, and follows the typical Ising-model behaviour.
Discussion and conclusions. -We find four different ordered phases in the second-neighbour (J 1 , J 2 ) Ising model on the fcc lattice: ferromagnetic fcc, and ordered AFM phases I4 1 /amd, L1 1 , and L1 0 . All of these are stable at zero temperature, and with increased temperature, all transform directly to a paramagnetic state.
Numerical simulations show that the stable structures with antiferromagnetic J 1 interactions all have zero magnetisation (assuming H = 0). Spontaneous magnetisation is observed only for ferromagnetic J 1 .
The Monte Carlo simulations also reveal a reasonably well-defined temperature at which specific defects, such as stacking faults and microdomains, start to be generated or annealed out. While interesting, it is likely that this temperature is sensitive to the single-flip algorithm, and its exact position is both ill-defined and sensitive to finitesize effects [9].
A recent mean-field calculation, which also reported two AFM states and an intermediate structure in the "facecentred cubic lattice" was, in fact, considering a different lattice, i.e., L1 2 with no interactions between face-centred sites. There is no discrepancy between these results, but we note that the 3-sublattice decomposition assumed in that work does not permit the L1 0 , I4 1 /amd and L1 1 ground states of the antiferromagnetic fcc lattice, and cannot sensibly be applied to the Hamiltonian considered here. Similarly, the 4-sublattice decomposition which was used previously [9] in the ferromagnetic J 2 case would also be inappropriate for the antiferromagnetic J 2 case. We demonstrate that an effective mean-field theory treatment covering all possibilities for the second-neighbour fcc Ising model would require eight sublattices.
The paramagnetic transition temperature is strongly dependent on J 2 , even if J 1 is held fixed. It takes its lowest value at the point where two competing ordered structures have identical ground-state enthalpy. This is true regardless of whether T is measured in units of |J 1 | or an average interaction weighted by number of neighbours, i.e., |J 1 |+|J 2 |/2. The disproportionate effect of J 2 on the transition temperature follows from the absence of frustration in NNN interactions. * * * Funding for this work was provided by ERC grant Hecate. The author thanks Hossein Ehteshami for bringing this problem to his attention. The author thanks M. Jurčišin for useful discussions around the model in ref. [18] and fig. 2. For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.

Data availability statement:
The data that support the findings of this study are available upon reasonable request from the authors.