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Physical review. B, Condensed matter

Antiferromagnetic chains coupled by weak ferromagnetic interchain interactions

J. Szṅajd

Published Nov 18, 2003• 6 citations
Publisher

Abstract

The linear-perturbation renormalization-group transformation is used to study $s=\frac{1}{2}$ spin chains with antiferromagnetic intrachain interactions $J<0$ 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 $(XY,$ anisotropic Heisenberg, and Ising) has been found. The specific heat of the one-dimensional models ${(J}^{\ensuremath{'}}=0)$ exhibits a maximum at ${H=H}_{\mathrm{max}}$ which decreases and shifts to higher fields with increasing temperature, and a minimum which disappears for the temperature higher than some characteristic temperature ${T}_{g}.$ For the antiferromagnetic $\mathrm{XY}$ 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.

DOI
10.1103/physrevb.68.174420

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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{J}^{\ensuremath{'}}>0J\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){(J}^{\ensuremath{'}}=0)(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){(J}^{\ensuremath{'}}>0)(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 thermodynamic properties of the weakly interacting spin-1/2 and spin-1 chains with the intrachain coupling J and interchain coupling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$ are studied by using the linear renormalization-group method. The magnetic susceptibility of the weakly interacting s=1/2s=1/2s=1/2 quantum anisotropic Heisenberg chains in the longitudinal field (h) is investigated. The scaling shifts of the parallel susceptibility maxima in the external magnetic field are considered. The same method is used to study the specific heat and susceptibility of the weakly interacting S=1S=1S=1 Ising models. In the special case (Blume-Capel model) the phase diagram with critical and first-order transition lines is found. The location of the tricritical point for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}/J=0.2$ is estimated.
The linear real space renormalization group is applied to study alternating spin (1/2,1) chains. The temperature dependences of the specific heat and spin–spin correlation functions for the quantum easy-axis mixed-spin chain are found. The role of single-ion anisotropy is discussed. The phase diagram of Ising spin chains with antiferromagnetic intrachain interactions and weak ferromagnetic interchain interactions is presented. As one expects, the weak interchain coupling leads to antiferrimagnetic long-range order in several chains. The threshold value of single-ion anisotropy below which the system does not exhibit a phase transition is found.
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 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.
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The anisotropic XXZXXZXXZ spin-12\frac{1}{2}21​ Heisenberg chain is studied using renormalization-group theory. The specific heats and nearest-neighbor spin-spin correlations are calculated throughout the entire temperature and anisotropy ranges in both ferromagnetic and antiferromagnetic regions, obtaining a global description and quantitative results. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Ising-like ferromagnetic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of characteristics of purely quantum nature are found: The in-plane interaction sixsjx+siysjy{s}_{i}^{x}{s}_{j}^{x}+{s}_{i}^{y}{s}_{j}^{y}six​sjx​+siy​sjy​ induces an antiferromagnetic correlation in the out-of-plane siz{s}_{i}^{z}siz​ component, at higher temperatures in the antiferromagnetic XXZXXZXXZ chain, dominantly at low temperatures in the ferromagnetic XXZXXZXXZ chain, and, in-between, at all temperatures in the XYXYXY chain. We find that the converse effect also occurs in the antiferromagnetic XXZXXZXXZ chain: an antiferromagnetic sizsjz{s}_{i}^{z}{s}_{j}^{z}siz​sjz​ interaction induces a correlation in the sixy{s}_{i}^{xy}sixy​ component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of either type (Ising or XYXYXY) of anisotropy; and (ii) in complete contrast, in the ferromagnet, the value and temperature of the specific heat peak increase with either type of anisotropy.