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arXiv:cond-mat/0607472 · physica status solidi (b)

Weak universality, bicritical points and reentrant transitions in the critical behaviour of a mixed spin‐1/2 and spin‐3/2 Ising model on the Union Jack (centered square) lattice

Jozef Strečka

Published Jan 5, 2006• 17 citations
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Abstract

Abstract The mixed spin‐1/2 and spin‐3/2 Ising model on the Union Jack lattice is solved by establishing a mapping correspondence with the eight‐vertex model. It is shown that the model under investigation becomes exactly soluble as a free‐fermion eight‐vertex model when the parameter of uniaxial single‐ion anisotropy tends to infinity. Under this restriction, the critical points are characterized by critical exponents from the standard Ising universality class. In a certain subspace of interaction parameters, which corresponds to a coexistence surface between two ordered phases, the model becomes exactly soluble as a symmetric zero‐field eight‐vertex model. This surface is bounded by a line of bicritical points having interaction‐dependent critical exponents that satisfy a weak universality hypothesis. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssb.200541318

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Abstract The mixed spin‐1/2 and spin‐3/2 Ising model on the Union Jack lattice is solved by establishing a mapping correspondence with the eight‐vertex model. It is shown that the model under investigation becomes exactly soluble as a free‐fermion eight‐vertex model when the parameter of uniaxial single‐ion anisotropy tends to infinity. Under this restriction, the critical points are characterized by critical exponents from the standard Ising universality class. In a certain subspace of interaction parameters, which corresponds to a coexistence surface between two ordered phases, the model becomes exactly soluble as a symmetric zero‐field eight‐vertex model. This surface is bounded by a line of bicritical points having interaction‐dependent critical exponents that satisfy a weak universality hypothesis. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract The mixed spin‐(1/2, S ) Ising model on the Union Jack (centered square) lattice is investigated by establishing the mapping relationship with its corresponding eight‐vertex model. An interplay between the nearest‐neighbour interaction, the competing next‐nearest‐neighbour interaction and the single‐ion anisotropy gives rise to a rather complex critical behaviour displayed in the reentrant phase transitions, the weak universal critical behaviour, as well as, a presence of first‐ and second‐order phase transitions. The most interesting finding to emerge from the present study relates to a variation of the weak‐universal critical exponents along the line of bicritical points, which is being twice as large for the mixed spin‐(1/2, S ) systems with the integer spin‐S atoms as for the ones with the half‐odd‐integer spin‐S atoms. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The mixed spin-1/2 and spin-3/2 Ising model on the extended Kagomé lattice is solved by establishing a mapping correspondence with the eight-vertex model. Letting the parameter of uniaxial single-ion anisotropy tend to infinity, the model becomes exactly soluble as a free-fermion eight-vertex model. Under this restriction, the critical points are characterized by critical exponents from the standard Ising universality class. In a certain subspace of interaction parameters that corresponds to a coexistence surface between two ordered phases, the model becomes exactly soluble as a symmetric zero-field eight-vertex model. This surface is bounded by a line of bicritical points that have non-universal interaction-dependent critical exponents.
The mixed spin-1/2 and spin-S Ising model on the Union Jack (centered square) lattice with four different three-spin (triplet) interactions and the uniaxial single-ion anisotropy is exactly solved by establishing a rigorous mapping equivalence with the corresponding zero-field (symmetric) eight-vertex model on a dual square lattice. A rigorous proof of the aforementioned exact mapping equivalence is provided by two independent approaches exploiting either a graph-theoretical or spin representation of the zero-field eight-vertex model. An influence of the interaction anisotropy as well as the uniaxial single-ion anisotropy on phase transitions and critical phenomena is examined in particular. It is shown that the considered model exhibits a strong-universal critical behaviour with constant critical exponents when considering the isotropic model with four equal triplet interactions or the anisotropic model with one triplet interaction differing from the other three. The anisotropic models with two different triplet interactions, which are pairwise equal to each other, contrarily exhibit a weak-universal critical behaviour with critical exponents continuously varying with a relative strength of the triplet interactions as well as the uniaxial single-ion anisotropy. It is evidenced that the variations of critical exponents of the mixed-spin Ising models with the integer-valued spins S differ basically from their counterparts with the half-odd-integer spins S.
Spin‐1/2 Ising‐Heisenberg model withXYZ Heisenberg pair interaction and two different Ising quartic interactions is exactly solved with the help of the generalized star‐square transformation, which establishes a precise mapping equivalence with the corresponding eight‐vertex model on a square lattice generally satisfying Baxter’s zero‐field (symmetric) condition. The investigated model exhibits a remarkable weak‐universal critical behavior with two marked wings of critical lines along which critical exponents vary continuously with the interaction parameters. Both wings of critical lines merge together at a very special quantum critical point of the infinite order, which can be characterized through diverging critical exponents. The possibility of observing reentrant phase transitions in a close vicinity of the quantum critical point is related to a relative strength of the exchange anisotropy in the XYZ Heisenberg pair interaction.
The mixed spin-1/2 and spin-3/2 Ising model on the extended Kagom é lattice is solved by establishing a mapping correspondence with the eight-vertex model.When the parameter of uniaxial single-ion anisotropy tends to infinity, the model system becomes exactly solvable as the staggered eight-vertex model satisfying the free-fermion condition.The critical points within this manifold can be characterized by critical exponents from the standard Ising universality class.The critical points within another subspace of interaction parameters, which corresponds to a coexistence surface between two ordered phases, can be approximated by corresponding results of the uniform eight-vertex model satisfying the zero-field condition.This coexistence surface is bounded by a line of bicritical points that have non-universal continuously varying critical indices.