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Register a new userIsing Problem on Simple Cubic Lattice
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Authors
Dhananjay P. Mehendale
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Simple cubic lattice (SC lattice) can be viewed as plane triangular lattice (PT lattice) by viewing it along its principle diagonal lines. By viewing thus we establish the exact one-to-one correspondence between the closed graphs on SC lattice and the corresponding closed graphs on PT lattice. We thus see that the propagator for PT lattice (with suitable modifications) can be used to solve, at least in principle, the 3D Ising problem for SC lattice in the absence of external magnetic field. A new method is then proposed to generate high temperature expansion for the partition function. This method is applicable to 2D as well as 3D lattices. This method does not require explicit counting of closed graphs and this counting is achieved in an indirect way and thus exact series expansion can be achieved up to any sufficiently large order.
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Recent results for the Ising model with first ($J_1$) and second ($J_2$) neighbour interactions on the body-centered cubic (bcc) lattice suggest that this model can host signatures of strong frustration, including Schottky anomalies and residual entropy, as well as, a spin-liquid-like phase [E. Jurv{c}iv{s}inova and M. Jurv{c}iv{s}in, Phys. Rev. B, 101 214443 (2020)]. Motivated by these findings, we investigate phase transitions and thermodynamics of this model using a cluster mean-field approach. In this lattice, tuning $g=J_2/J_1$ leads to a ground-state transition between antiferromagnetic (AF) and superantiferromagnetic (SAF) phases at the frustration maximum $g=2/3$. Although the ordering temperature is reduced as $g to 2/3$, our findings suggest the absence of any Schottky anomaly and residual entropy, in good agreement with Monte Carlo simulations. We also find a direct transition between AF and SAF phases, ruling out the presence of the spin-liquid-like state. Furthermore, the cluster mean-field outcomes support a scenario with only continuous phase transitions between the paramagnetic state and the low-temperature long-range orders. Therefore, our results indicate the absence of strong frustration effects in the thermodynamics and in the nature of phase transitions, which can be ascribed to the higher dimensionality of the bcc lattice.
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