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Journal of Statistical Mechanics Theory and Experiment

Frustration—no frustration crossover and phase transitions in 2D spin models with zig-zag structures

J. Szṅajd

Published May 1, 2024• 3 citations
Publisher

Abstract

Abstract Three 2D spin models made of frustrated zig-zag chains with competing interactions which, by exact summation with respect to some degrees of freedom, can be replaced by an effective temperature-dependent interaction, were considered. The first model, exactly solvable Ising chains coupled by only four-spin interactions, does not exhibit any finite temperature phase transition; nevertheless, temperature can trigger a frustration–no frustration crossover accompanied by gigantic specific heat. A similar effect was observed in several two-leg ladder models (Weiguo 2020 arXiv: 2006.08921v2 ; 2020 2006.15087v1 ). The anisotropic Ising chains coupled by a direct interchain interaction and, competing with it, indirect interaction via spins located between chains, are analyzed using the exact Onsager’s equation and linear perturbation renormalization group (LPRG). Depending on the parameter set, such a model exhibits one antiferromagnetic (AF) or ferromagnetic (FM) phase transition or three phase transitions with a re-entrant disordered phase between AF and FM ones. The LPRG method was also used to study coupled uniaxial XXZ chains which, for example, can be a minimal model to describe the magnetic properties of compounds in which uranium and rare earth atoms form zig-zag chains. As with the Ising model, for a certain set of parameters, the model can undergo three phase transitions. However, both intrachain and interchain plain interactions s i , j x s k , l x + s i , j y s k , l y can eliminate the re-entrant disordered phase, and then only one transition takes place. Additionally, the XXZ model can undergo temperature-induced metamagnetic transition.

DOI
10.1088/1742-5468/ad3a5d

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Abstract Three 2D spin models made of frustrated zig-zag chains with competing interactions which, by exact summation with respect to some degrees of freedom, can be replaced by an effective temperature-dependent interaction, were considered. The first model, exactly solvable Ising chains coupled by only four-spin interactions, does not exhibit any finite temperature phase transition; nevertheless, temperature can trigger a frustration–no frustration crossover accompanied by gigantic specific heat. A similar effect was observed in several two-leg ladder models (Weiguo 2020 arXiv: 2006.08921v2 ; 2020 2006.15087v1 ). The anisotropic Ising chains coupled by a direct interchain interaction and, competing with it, indirect interaction via spins located between chains, are analyzed using the exact Onsager’s equation and linear perturbation renormalization group (LPRG). Depending on the parameter set, such a model exhibits one antiferromagnetic (AF) or ferromagnetic (FM) phase transition or three phase transitions with a re-entrant disordered phase between AF and FM ones. The LPRG method was also used to study coupled uniaxial XXZ chains which, for example, can be a minimal model to describe the magnetic properties of compounds in which uranium and rare earth atoms form zig-zag chains. As with the Ising model, for a certain set of parameters, the model can undergo three phase transitions. However, both intrachain and interchain plain interactions s i , j x s k , l x + s i , j y s k , l y can eliminate the re-entrant disordered phase, and then only one transition takes place. Additionally, the XXZ model can undergo temperature-induced metamagnetic transition.
Three 2D spin models made of frustrated zig-zag chains with competing interactions which by exact summation with respect to some degrees of freedom can be replaced by an effective temperature-dependent interaction were considered. The first model, exactly solvable Ising chains coupled by only four-spin interactions does not exhibit any finite temperature phase transition, nevertheless temperature can trigger a frustration - no frustration crossover accompanied by gigantic specific heat. A similar effect was observed in several two-leg ladder models [Yin Weiguo, arXiv:2006.08921v2 (2020), 2006.15087v1 (2020)]. The anisotropic Ising chains coupled by a direct interchain interaction and competing with it indirect interaction via spins located between chains are analyzed by using exact Onsager's equation and linear perturbation renormalization group (LPRG). Depending on the parameter set such a model exhibits one antiferromagnetic (AF) or ferromagnetic (FM) phase transition or three phase transitions with reentrant disordered phase between AF and FM ones. The LPRG method was also used to study coupled uniaxial XXZXXZXXZ chains which, for example, can be a minimal model to describe the magnetic properties of compounds in which uranium and rare earth atoms form zig-zag chains. As with the Ising model for a certain set of parameters the model can undergo three phase transitions. However, both inchain and interchain plain interactions si,jxsk,lx+si,jysk,lys_{i,j}^x s_{k,l}^x+s_{i,j}^y s_{k,l}^ysi,jx​sk,lx​+si,jy​sk,ly​ can eliminate the reentrant disordered phase and then only one transition takes place. Additionally, the XXZXXZXXZ model can undergo temperature-induced metamagnetic transition.
As is well known, one-dimensional spin systems with short-range interactions cannot undergo a phase transition at finite temperature. However, in many 1D Ising spin models with frustration, at a certain crossover point, the thermodynamic quantities show anomalies that resemble such a transition W. Yin [Phys. Rev. Res. 6, 013331 (2024)2643-156410.1103/PhysRevResearch.6.013331]. Recently, Yin [Phys. Rev. Res. 6, 013331 (2024)2643-156410.1103/PhysRevResearch.6.013331] has shown, using decorated ladders as an example, that a one-dimensional Ising model can be arbitrarily close to the genuine phase transition. Using the transfer matrix method and real space renormalization group technique, we find that crossover resembling a phase transition accompanied by gigantic specific heat can also occur in two-dimensional systems. We consider coupled Ising spin chains with a zig-zag structure and a four-spin j_{4} interaction, which allows us to control the temperature at which the spins become unbound in the model with short-range interactions. The crossover point is defined by two parameters: the crossover temperature t=t^{*} and the crossover interaction j_{4}=j_{4}^{*}, and occurs when the temperature at which the effective interaction between the spin subsystems zeros out coincides with the temperature at which the spins become unbound. In systems in which a genuine phase transition occurs at the critical temperature t=t_{c}, there are two possibilities: if t_{c}>t^{*} the system undergoes a sequence of three phase transitions with a reentrant transition to disordered phase, while for t_{c}<t^{*} the phase transition is preceded by the considered crossover. We study the key role in the appearance of anomalies in the thermodynamic quantities of the four-spin interactions which are suspected to have a significant impact on the properties of magnetic two-dimensional lattices, but possible also on the construction of social systems modeling the phenomenon of social validation or peer pressure.
The linear-perturbation real space renormalization transformation (LPRG) is presented and applied to the study of quantum spin chains coupled by interchain interaction (k1) weaker than intrachain one (k). The method is examined in two exact solvable cases: Ising chains on the square and triangular lattices and quan-tum XY chain. For the Ising model, in the second order in the cumulant epansion, the deviation of the critical temperature from the exact value is less than 1 % for 0:5k> k1> 0:15k, but even in the case of the standard Ising model (k1 = k) we found the value of Tc which differs by 2 % from the exact one. For the quantum XY chain the deviation of the free energy value found by using LPRG from the exact Katsura result is less than 1 % for T=J> 1, and for rather low temperature T=J = 0:08 is about 6%. The LPRG is used to study the effects of interchain frustration on the phase transition in 2D Heisenberg spin chains with easy axis along the z direction. It is shown that contrary to the pure Ising model in systems with in-plane interactions (XY), the interchain frustration does not destroy the nite-temperature transition. However, such a frustration changes the character of the phase transition from Ising-like to, probably, Kosterlitz-Thouless-like. We have also ap-plied the LPRG method to the calculation of the isothermal magnetocaloric coefcient (MT) for several spin models in disordered phases. Is is demonstrated that in the presence of antiferromagnetic uctuations, MT changes sign at some value of the magnetic eld. Generally, MT is negative if magnetic eld competes with a short-range order, and consequently it can be an indicator of the change in the short-range correlation. Key words: Heisenberg chains, renormalization group, interchain frustration, magnetocaloric effect
The finite-temperature phase transition in the Heisenberg spin chains with easy axis along the zzz direction coupled by interchain interactions weaker than intrachain interactions is studied by using linear-perturbation renormalization group transformation. Contrary to the pure Ising model in systems with in-plane interactions (XY)(XY)(XY), the interchain frustration does not destroy the finite-temperature phase transition. However, such a frustration changes the character of the phase transition from Ising-like to, probably, Kosterlitz--Thouless-like.
The linear-perturbation real space renormalization transformation (LPRG) is presented and applied to the study of quantum spin chains coupled by interchain interaction (k 1 ) weaker than intrachain one (k).The method is examined in two exact solvable cases: Ising chains on the square and triangular lattices and quantum XY chain.For the Ising model, in the second order in the cumulant epansion, the deviation of the critical temperature from the exact value is less than 1% for 0.5k > k 1 > 0.15k, but even in the case of the standard Ising model (k 1 = k) we found the value of Tc which differs by 2% from the exact one.For the quantum XY chain the deviation of the free energy value found by using LPRG from the exact Katsura result is less than 1% for T /J > 1, and for rather low temperature T /J = 0.08 is about 6%.The LPRG is used to study the effects of interchain frustration on the phase transition in 2D Heisenberg spin chains with easy axis along the z direction.It is shown that contrary to the pure Ising model in systems with in-plane interactions (XY), the interchain frustration does not destroy the finite-temperature transition.However, such a frustration changes the character of the phase transition from Ising-like to, probably, Kosterlitz-Thouless-like.We have also applied the LPRG method to the calculation of the isothermal magnetocaloric coefficient (M T ) for several spin models in disordered phases.Is is demonstrated that in the presence of antiferromagnetic fluctuations, M T changes sign at some value of the magnetic field.Generally, M T is negative if magnetic field competes with a short-range order, and consequently it can be an indicator of the change in the short-range correlation.
The linear perturbation renormalization group (LPRG) is used to study the phase transition of the weakly coupled Ising chains with intrachain (J) and interchain nearest-neighbor (J_{1}) and next-nearest-neighbor (J_{2}) interactions forming the triangular and rectangular lattices in a field. The phase diagrams with the frustration point at J_{2}=-J_{1}/2 for a rectangular lattice and J_{2}=-J_{1} for a triangular lattice have been found. The LPRG calculations support the idea that the phase transition is always continuous except for the frustration point and is accompanied by a divergence of the specific heat. For the antiferromagnetic chains, the external field does not change substantially the shape of the phase diagram. The critical temperature is suppressed to zero according to the power law when approaching the frustration point with an exponent dependent on the value of the field.