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Condensed Matter Physics

Thermodynamics, geometrical frustration and quantum fluctuations in coupled spin chains

J. Szṅajd

Published Jan 1, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.5488/cmp.12.4.697

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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 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 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.
Motivated by the recently developed [T. Plackowski et al., Rev. Sci. Instrum. 73, 2755 (2002)] technique for measuring isothermal magnetocaloric coefficient (MT)({M}_{T})(MT​), we revisit the thermodynamic properties of coupled spin chains in a field. The coupled Ising antiferromagnetic chains in two and three dimensions, quantum XYXYXY chain in a transverse field, and Heisenberg chain are considered. For the Ising model the shift of the critical temperature under magnetic field and dependence of MT{M}_{T}MT​ on interchain coupling are found. The field dependence of MT{M}_{T}MT​ for several models is presented. It is demonstrated that in the disordered phase in the presence of antiferromagnetic fluctuations, MT{M}_{T}MT​ changes sign at some value of the magnetic field. Generally, MT{M}_{T}MT​ 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 renormalization-group transformation is proposed to study critical temperatures and temperature dependence of the thermodynamic values (free energy, specific heat) of the weakly interacting spin chains. The method is examined in two classical systems: the standard Ising model on a rectangular lattice and the Ising spin chains coupled by pair-pair interactions. In the latter case the transformation does not exhibit any nontrivial fixed point. This result agrees with the exactly determined free energy of the system under consideration. The method is also applied to the weakly interacting quantum XY spin chains. The critical temperature as a function of the interchain interactions and the temperature dependence of the specific heat are found.
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.