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Physical review. B./Physical review. B

Magnetic-field control of electric polarization in coupled spin chains with three-site interactions

J. Szṅajd

Published Jun 11, 2018• 12 citations
Publisher

Abstract

The linear perturbation renormalization group (LPRG) is used to study coupled $XY$ chains with Dzyaloshinskii-Moriya (DM) and three-spin interactions in a magnetic field. Starting with a minimal model exhibiting the magnetoelectric effect, a spin-$\frac{1}{2} XY$ chain with nearest, next-nearest $({J}_{2}^{x})$, and DM $({D}_{1}^{y})$ interactions in a magnetic field, the recursion relations for all effective interactions generated by the LPRG transformation are found. The evaluation of these relations allows us to analyze, among others, the influence of ${J}_{2}^{x},\phantom{\rule{0.16em}{0ex}}{D}_{1}^{y}$, three-spin (${S}_{i}^{x}{S}_{i+1}^{y}{S}_{i+2}^{z}\ensuremath{-}{S}_{i}^{y}{S}_{i+1}^{x}{S}_{i+2}^{z}$), and interchain interactions on the thermodynamic properties. The field and temperature dependences of the polarization, specific heat, and correlation functions are found. It is shown that an interchain coupling triggers a phase transition indicated by the divergence of the renormalized coupling parameters.

DOI
10.1103/physrevb.97.214410

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The linear perturbation renormalization group (LPRG) is used to study coupled XYXYXY chains with Dzyaloshinskii-Moriya (DM) and three-spin interactions in a magnetic field. Starting with a minimal model exhibiting the magnetoelectric effect, a spin-12XY\frac{1}{2} XY21​XY chain with nearest, next-nearest (J2x)({J}_{2}^{x})(J2x​), and DM (D1y)({D}_{1}^{y})(D1y​) interactions in a magnetic field, the recursion relations for all effective interactions generated by the LPRG transformation are found. The evaluation of these relations allows us to analyze, among others, the influence of J2x,D1y{J}_{2}^{x},\phantom{\rule{0.16em}{0ex}}{D}_{1}^{y}J2x​,D1y​, three-spin (SixSi+1ySi+2z\ensuremath−SiySi+1xSi+2z{S}_{i}^{x}{S}_{i+1}^{y}{S}_{i+2}^{z}\ensuremath{-}{S}_{i}^{y}{S}_{i+1}^{x}{S}_{i+2}^{z}Six​Si+1y​Si+2z​\ensuremath−Siy​Si+1x​Si+2z​), and interchain interactions on the thermodynamic properties. The field and temperature dependences of the polarization, specific heat, and correlation functions are found. It is shown that an interchain coupling triggers a phase transition indicated by the divergence of the renormalized coupling parameters.
We consider a simple integrable model of a spin chain exhibiting the magnetoelectric effect (MEE). Starting from the periodic S=1/2S=1/2S=1/2 XXZ chain with Dzyaloshinskii-Moriya terms, which we consider as a local electric polarization in the spirit of the Katsura-Nagaosa-Baladsky (KNB) mechanism, we perform the mapping onto the conventional XXZ chain with twisted boundary conditions. Using the techniques of quantum transfer matrix and nonlinear integral equations we obtain the magnetization, electric polarization, and magnetoelectric tensor as functions of magnetic and electric field for arbitrary temperatures. We investigate these dependencies as well as the thermal behavior of the above-mentioned physical quantities, especially in the low-temperature regime. We found several regimes of polarization. Adjusting the magnetic field one can switch the system from one regime to another. The features of the critical properties connected with the MEE are also illustrated.
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.
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 transformation is used to study s=12s=\frac{1}{2}s=21​ spin chains with antiferromagnetic intrachain interactions J<0J<0J<0 coupled by weak ferromagnetic interchain interactions ${J}^{\ensuremath{'}}>0$ in the longitudinal external magnetic field H. The field dependence of the specific heat for several models (XY,(XY,(XY, anisotropic Heisenberg, and Ising) has been found. The specific heat of the one-dimensional models ${(J}^{\ensuremath{'}}=0)$ exhibits a maximum at H=Hmax{H=H}_{\mathrm{max}}H=Hmax​ which decreases and shifts to higher fields with increasing temperature, and a minimum which disappears for the temperature higher than some characteristic temperature Tg.{T}_{g}.Tg​. For the antiferromagnetic XY\mathrm{XY}XY and Ising models the specific-heat maximum is shifted linearly, whereas for the Heisenberg model the shift cannot be fitted to a single power law. For an arbitrary small ferromagnetic interchain interaction ${(J}^{\ensuremath{'}}>0)$ the specific heat as a function of temperature has a singularity characterizing the antiferromagnetic phase transition. From the flow diagram analysis the dependence of the transition temperature on the interchain interaction and the external field has been found.
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
A spin-1/2 XYXYXY chain model of magnetoelectrics on a zigzag chain is considered rigorously. The magnetoelectric coupling is described within the Katsura-Nagaosa-Balatsky mechanism. In the zigzag geometry it leads to the staggered Dzyaloshinskii-Moriya interaction. Using nonuniform spin rotations, the model is reduced to a dimerized XYXYXY chain and solved exactly using the Jordan-Wigner transformation. We analyze the ground-state phase diagram of the model and the zero- and finite-temperature magnetoelectric effects and obtain the magnetization and polarization curves versus magnetic and electric fields, as well as the parameters of anisotropic dielectric and magnetoelectric response. It is also shown that the electric field may enhance the magnetocaloric effect in the model.