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 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.
The linear-perturbation renormalization-group transformation is used to study
s=21 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=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. For the antiferromagnetic
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 thermodynamics of coupled frustrated ferromagnetic chains is studied within a spin-rotation-invariant Green's-function approach. We consider an isotropic Heisenberg spin-half system with a ferromagnetic in-chain coupling
J1<0 between nearest neighbors and a frustrating antiferromagnetic next-nearest-neighbor in-chain coupling
J2>0. We focus on the moderate strength of frustration
J2<∣J1∣/4 such that the in-chain spin-spin correlations are predominantly ferromagnetic. We consider two interchain couplings (ICs) ${J}_{\ensuremath{\perp},y}$ and ${J}_{\ensuremath{\perp},z}$, corresponding to the two axes perpendicular to the chain, where ferromagnetic as well as antiferromagnetic ICs are taken into account. We discuss the influence of frustration on the ground-state properties for antiferromagnetic ICs, where the ground state is of a quantum nature. The major part of our study is devoted to the finite-temperature properties. We calculate the critical temperature
Tc as a function of the competing exchange couplings ${J}_{2},{J}_{\ensuremath{\perp},y},{J}_{\ensuremath{\perp},z}$. We find that for fixed ICs,
Tc decreases monotonically with increasing frustration
J2, where as ${J}_{2}\ensuremath{\rightarrow}|{J}_{1}|/4$ the
Tc(J2) curve drops down rapidly. To characterize the magnetic ordering below and above
Tc, we calculate the spin-spin correlation functions $\ensuremath{\langle}{\mathbf{S}}_{0}{\mathbf{S}}_{\mathbf{R}}\ensuremath{\rangle}$, the magnetic order parameter
M, the uniform static susceptibility ${\ensuremath{\chi}}_{0}$, as well as the correlation length $\ensuremath{\xi}$. Moreover, we discuss the specific heat
CV and the temperature dependence of the excitation spectrum ${\ensuremath{\omega}}_{\mathbf{q}}$. As ${J}_{2}\ensuremath{\rightarrow}|{J}_{1}|/4$, some unusual frustration-induced features were found, such as an increase of the in-chain spin stiffness (in the case of ferromagnetic ICs) or of the in-chain spin-wave velocity (in the case of antiferromagnetic ICs) with growing temperature.
We have numerically studied the thermodynamic properties of the spin
21 XXZ chain in the presence of a transverse (noncommuting) magnetic field. The thermal, field dependence of specific heat and correlation functions for chains up to 20 sites has been calculated. The area where the specific heat decays exponentially is considered as a measure of the energy gap. We have also obtained the exchange interaction between chains in a bulk material using the random phase approximation and derived the phase diagram of the three-dimensional material with this approximation. The behavior of the structure factor at different momenta verifies the antiferromagnetic long-range order in the
y direction for the three-dimensional case. Moreover, we have concluded that the low temperature Lanczos results [M. Aichhorn et al., Phys. Rev. B 67, 161103(R) (2003)] are more accurate for low temperatures and closer to the full diagonalization ones than the results of finite temperature Lanczos method [J. Jaklic and P. Prelovsek, Phys. Rev. B 49, 5065 (1994)].