A numerical method is developed to study the scaling of distribution of couplings of highly random antiferromagnetic Ising and quantum Heisenberg spin-\textonehalf{} systems. The method shows how freezing into inert local singlets prevents ordering down to temperatures well below the median nearest-neighbor coupling or bare exchange percolation threshold in positionally disordered systems with Heisenberg exchange varying exponentially with distance (e.g., doped semiconductors, quasi one-dimensional salts). This is contrasted with the Ising system.
The low-temperature thermodynamic properties of a random antiferromagnetic Heisenberg chain in zero magnetic field are studied approximately by a renormalisation group method. The couplings J obey a probability distribution function P(J) approximately 1/J alpha , with alpha a constant. It is found that the susceptibility chi and the specific heat C behave like chi approximately T - alpha chi and C approximately T 1- alpha c at low temperatures alpha chi and alpha c are weakly varying functions of temperature and slowly approach unity as T goes to zero. Comparison of these results with other theoretical analysis as well as experimental results on organic charge transfer compounds is also discussed.
The nearest-neighbor random-exchange Heisenberg antiferromagnetic Heisenberg chain (
s=21) is studied at low temperatures via an approximate renormalization-group method. This entails renormalization of the random-exchange coupling constant
J and of the probability law for
J.
P0(J). After
n iterations we find that the renormalized probability function
Pn(J(n)), a function of the renormalized coupling
J(n), develops singular behavior for small
J(n), independent of the initial form of
P0(J). This happens both for
P0(J) that diverge or go to zero as $J\ensuremath{\rightarrow}0$. The singular form of
Pn(J(n)) is such that it leads to a specific heat and susceptibility that behave like $C\ensuremath{\sim}{T}^{1\ensuremath{-}{\ensuremath{\alpha}}_{C}(T)}$ and $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}{\ensuremath{\alpha}}_{\ensuremath{\chi}}(T)}$. ${\ensuremath{\alpha}}_{C}(T)$ and ${\ensuremath{\alpha}}_{\ensuremath{\chi}}(T)$ are exponents weakly dependent on temperature (
T) that go to one as $T\ensuremath{\rightarrow}0$ and satisfy ${\ensuremath{\alpha}}_{\ensuremath{\chi}}(T)>{\ensuremath{\alpha}}_{C}(T)$, for arbitrary initial probability laws. The classical
n-vector models, and in particular the classical Heisenberg model are also studied using a renormalization-group approach and it is found that their behavior is different than that of the quantum model: singular behavior in the thermodynamic properties is only obtained if the starting
P0(J) is singular. An explanation of these results in terms of the scaling theory of localization is suggested. The relevance of our results to the experimental findings on the magnetic properties of tetracyanoquinodimethanide complexes is also discussed.
A random antiferromagnetic spin-1 chain with both Heisenberg and biquadratic nearest neighbor couplings is found to exhibit a novel ferromagnetic-like instability at low energies, if there is competition between the Heisenberg and biquadratic couplings, even though each individual nearest neighbor coupling is such as to produce a singlet ground state for the isolated pair. This instability leads to the generation of ferromagnetic couplings, and thus to the formation of weakly coupled large moments at low temperatures. A perturbative calculation (up to 4th order) is performed to investigate whether such a situation is physically realizable in spin-1 systems.
We give a full description of a recently developed efficient Monte Carlo Approach to low-lying excitations of one-dimensional quantum spin systems. The idea is in a word expressed as extracting the lower edge of the excitation spectrum from imaginary-time quantum Monte Carlo data at a sufficiently low temperature. First, the method is applied to the antiferromagnetic Heisenberg chains of S=1/2, 1, 3/2, and 2. In the cases of S=1/2 and S=1, comparing the present results with the previous findings, we discuss the reliability of the method. The spectra for S=3/2 and S=2 turn out to be massless and massive, respectively. In order to demonstrate that our method is very good at treating long chains, we calculate the S=2 chain with length up to 512 spins and give a precise estimate of the Haldane gap. Second, we show its fruitful use in studying quantum critical phenomena of bond-alternating spin chains. Using the conformal invariance of the system as well, we calculate the central charge of the critical S=1 chain, which results in the Gaussian universality class. Third, we study an alternating-spin system composed of two kinds of spins S=1 and 1/2, which shows the ferrimagnetic behavior. We find a quadratic dispersion relation in the small-momentum region. The numerical findings are qualitatively explained well in terms of the spin-wave theory. Finally, we argue a possibility of applying the method to the higher excitations, where we again deal with the S=1 Heisenberg antiferromagnet and inquire further into its unique low-energy structure. All the applications demonstrate the wide applicability of the method and its own advantages.
The quantum renormalization-group technique recently developed by Chakravarty and the present author is used to study weak random couplings in quantum-mechanical and classical nearest-neighbor Heisenberg ferromagnets in D=2+\ensuremath{\epsilon} dimensions and at low temperatures T. Momentum-shell recursion relations are obtained by considering the renormalization-group transformations of the cumulants of the probability distribution of the random exchange couplings. From a one-loop calculation it is found that, if the second cumulants ${\ensuremath{\sigma}}^{2}$ of the random couplings are smaller than a critical value ${\ensuremath{\sigma}}_{c}^{2}$, the only modification due to randomness in the low-temperature behavior of the order-parameter correlation length \ensuremath{\xi} in two dimensions is a renormalization of the spin stiffness ${\ensuremath{\rho}}_{s}$ according to ${\ensuremath{\rho}}_{s}$(X)=(1-X${)}^{\ensuremath{\varphi}}$${\ensuremath{\rho}}_{s}$(0), where X=${\ensuremath{\sigma}}^{2}$/${\ensuremath{\sigma}}_{c}^{2}$ and \ensuremath{\varphi}=(1/3 within the one-loop approximation. At zero temperature \ensuremath{\xi} diverges for with ${\ensuremath{\nu}}_{x}$=1/D. Precisely at X=1 the critical behavior of the system is characterized by new exponents; the dynamical critical exponent z is given by z=2+\ensuremath{\varphi}/${\ensuremath{\nu}}_{x}$. For D(1-\ensuremath{\varphi})>2 a new phase appears in the T-X phase diagram, and it is argued that this phase is characterized by spin-glass order.