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Physical review. B, Condensed matter

Renormalization-group approach to weakly interacting spin chains

J. Szṅajd

Published Sep 19, 2002• 8 citations
Publisher

Abstract

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 $s=1/2$ 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 $S=1$ 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.

DOI
10.1103/physrevb.66.104420

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The thermodynamic properties of the weakly interacting spin-1/2 and spin-1 chains with the intrachain coupling J and interchain coupling J\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}J\ifmmode~\else˜\fi​ 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 J\ifmmode~\else˜\fi/J=0.2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}/J=0.2J\ifmmode~\else˜\fi​/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 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.
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 global phase diagram of the S=1S=1S=1 Ising model in a transverse field is studied using a mean-field approximation. The critical behavior has also been considered by real-space renormalization-group techniques. This model is used to analyze the phase diagram of weakly coupled ($\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}<<1$) classical Blume-Emery-Griffith chains. It is found that such systems display a first-order transition only in a very limited range of their parameters.
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 frustrated ferromagnetic spin-1∕21∕21∕2 Heisenberg chain is studied by means of a low-energy field theory as well as the density-matrix renormalization group and exact diagonalization methods. First, we study the ground-state phase diagram in a magnetic field and find an ``even-odd'' (EO) phase characterized by bound pairs of magnons in the region of two weakly coupled antiferromagnetic chains. A jump in the magnetization curves signals a first-order transition at the boundary of the EO phase, but otherwise the curves are smooth. Second, we discuss thermodynamic properties at zero field, where we confirm a double-peak structure in the specific heat for moderate frustrating next-nearest-neighbor interactions.