For pt.II see ibid., vol.15, p.1693 (1982). The methods developed by Williams (1981) to describe the ground-state properties of a random-bond infinite Ising chain are extended to apply to a system of finite-length random-bond Ising chains. It is shown how to evaluate the magnetisation, ground-state energy, and entropy as a function of magnetic field. It is shown that the steps in the magnetisation and entropy when B=2J/r behave differently for r even and r odd. The magnetisation and entropy remain finite as B becomes very small.
The transfer matrix method is used to describe the properties of the random bond of a simple periodic frustrated Ising chain in a uniform magnetic field. Doman and Williams(1982) are found expressions for the thermodynamic functions of the system at low temperature. Their methods is extended to apply to a system of five-antiferromagnetic bond Ising chain terminated by a ferromagnetic bond at low temperatures. New expressions for the magnetization, energy, and entropy as a functions of magnetic field is found. All results are in agreement with the previous results.
The methods of Williams,Doman and Williams,and Doman and Habeeb are used to describe the ground state properties of infinite ferromagnetic Ising chains in a random field.Analytic results for the ground state energy,the magnetisation per spin in the direction of the local field and the entropy as a function of magnetic field are obtained. When the probability factor of the zero local field is zero,our results are in agreement with thut obtained previously by Williams. terminators.A terminator could be defined as the simplest group of spins whose direction is indepen- dent of that of its neighbours,but depends on the value of the magnetic field.The behaviour of the system is now dominated by these subchains and so we take into account all possible configurations for such subchains.The idea of superspins and superb- onds introduced by Williams(1981)and applied later to examine the properties of many Ising
The low-temperature behavior ($T\ensuremath{\ll}h, J$) of a random-bond Ising chain in a magnetic field is considered [$H=\ensuremath{-}(\frac{1}{2})J\ensuremath{\Sigma}{T}_{i}{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{i+1}\ensuremath{-}h\ensuremath{\Sigma}{\ensuremath{\sigma}}_{i}, {\ensuremath{\sigma}}_{i}=\ifmmode\pm\else\textpm\fi{}1$,
Ti is a fixed random sequence of numbers +1 and -1 with concentrations
c1 and ${c}_{2}=1\ensuremath{-}{c}_{1}$, respectively]. The ground-state energy
E0, magnetization ${\ensuremath{\mu}}_{0}$ and zero-point entropy
S0 are calculated exactly. It is shown that ${\ensuremath{\mu}}_{0}$ and
S0 are discontinuous functions of magnetic field having jumps at
h=nJ,
n=1,2,….
For the spin-glass chain in an external field
h, a nonzero weight at the origin of the bond distribution $\ensuremath{\rho}(J)$ is known to induce a nonanalytical magnetization at zero temperature; for $\ensuremath{\rho}(J)\ensuremath{\sim}A{\ensuremath{\mid}J\ensuremath{\mid}}^{\ensuremath{\mu}\ensuremath{-}1}$ near $J\ensuremath{\rightarrow}0$, the magnetization follows the Chen-Ma scaling $M\ensuremath{\sim}{h}^{\ensuremath{\mu}∕(2+\ensuremath{\mu})}$. In this paper, we numerically revisit this model to obtain detailed statistical information on the ground-state configuration and on the low-energy two-level excitations that govern the low-temperature properties. The ground state consists of long unfrustrated intervals separated by weak frustrated bonds. We accordingly compute the strength distribution of these frustrated bonds, as well as the length and magnetization distributions of the unfrustrated intervals. We find that the low-energy excitations are of two types: (i) one frustrated bond of the ground state may have two positions that are nearly degenerate in energy and (ii) two neighboring frustrated bonds of the ground state may be annihilated or created with nearly zero energy cost. For each excitation type, we compute its probability density as a function of its length. Moreover, we show that the contributions of these excitations to various observables (specific heat, Edwards-Anderson order parameter, susceptibility) are in full agreement with the direct transfer matrix evaluations at low temperature. Finally, following the special bimodal case $\ifmmode\pm\else\textpm\fi{}J$, where a Ma-Dasgupta, RG procedure has been previously used to compute explicitly the above observables, we discuss the possibility of an extended RG procedure. We find that the ground state can be seen as the result of a hierarchical ``fragmentation'' procedure that we describe.
The magnetic and thermodynamic properties of spin-
1∕2 Heisenberg diamond chains are investigated in three different cases: (a)
J1,
J2,
J3>0 (frustrated), (b)
J1,
J3<0,
J2>0 (frustrated), and (c)
J1,
J2>0,
J3<0 (nonfrustrated), where the bond coupling
Ji (i=1,2,3)>0 stands for an antiferromagnetic (AF) interaction, and
<0 for a ferromagnetic (F) interaction. The density-matrix renormalization-group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer-matrix renormalization-group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin-correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor
S(q) shows peaks at wave vectors $q=a\ensuremath{\pi}∕3$
(a=0,1,2,3,4,5) for different couplings in a zero magnetic field, which, however, in the magnetic fields where the magnetization plateau with
m=1∕6 pertains, exhibits the peaks only at
q=0, $2\ensuremath{\pi}∕3$, and $4\ensuremath{\pi}∕3$, which are found to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility, and specific heat show various behaviors for different couplings. The double-peak structures of the susceptibility, and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the
XXZ anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat, and magnetization for the compound
Cu3(CO3)2(OH)2 are fairly compared with our TMRG results.