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Physical review. B, Condensed matter

Low-energy behavior of a random classical Heisenberg chain

Ibrahim Avgin, David L. Huber

Published Apr 1, 1998• 6 citations
Publisher

Abstract

Low-energy magnons in a classical Heisenberg chain with random nearest-neighbor interactions are investigated using the negative eigenvalue counting technique. A detailed study is made for a range of concentrations of ``wrong sign'' bonds (i.e., $c<~0.1$ and $c>~0.9)$. It is found that low-energy magnons have an anomalous power-law (2/3) behavior in the spectrum for random sign-fixed magnitude exchange interactions. Combining these and previous results, a phenomenological fit is established describing the concentration dependence of the spectrum for a wide range of values of $c$. A discussion based on scaling arguments is presented to account for the anomalous power law for arbitrary concentrations. The general case where the magnitudes and the signs of the exchange interactions and the magnitudes of the spins are all random quantities is further investigated to check whether the anomalous power-law behavior holds. For uniform (flat), non-singular distributions of the random quantities, the 2/3 power in the spectrum is obtained but for singular distributions and mixed distributions (e.g., the magnitude of the spin has a singular distribution while the magnitude of the exchange interaction has a regular distribution) significant deviations from the 2/3 power law occur.

DOI
10.1103/physrevb.57.8269

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Low-energy magnons in a classical Heisenberg chain with random nearest-neighbor interactions are investigated using the negative eigenvalue counting technique. A detailed study is made for a range of concentrations of ``wrong sign'' bonds (i.e., c< 0.1c<~0.1c< 0.1 and c> 0.9)c>~0.9)c> 0.9). It is found that low-energy magnons have an anomalous power-law (2/3) behavior in the spectrum for random sign-fixed magnitude exchange interactions. Combining these and previous results, a phenomenological fit is established describing the concentration dependence of the spectrum for a wide range of values of ccc. A discussion based on scaling arguments is presented to account for the anomalous power law for arbitrary concentrations. The general case where the magnitudes and the signs of the exchange interactions and the magnitudes of the spins are all random quantities is further investigated to check whether the anomalous power-law behavior holds. For uniform (flat), non-singular distributions of the random quantities, the 2/3 power in the spectrum is obtained but for singular distributions and mixed distributions (e.g., the magnitude of the spin has a singular distribution while the magnitude of the exchange interaction has a regular distribution) significant deviations from the 2/3 power law occur.
The nearest-neighbor random-exchange Heisenberg antiferromagnetic Heisenberg chain (s=12s=\frac{1}{2}s=21​) is studied at low temperatures via an approximate renormalization-group method. This entails renormalization of the random-exchange coupling constant JJJ and of the probability law for JJJ. P0(J){P}_{0}(J)P0​(J). After nnn iterations we find that the renormalized probability function Pn(J(n)){P}_{n}({J}^{(n)})Pn​(J(n)), a function of the renormalized coupling J(n){J}^{(n)}J(n), develops singular behavior for small J(n){J}^{(n)}J(n), independent of the initial form of P0(J){P}_{0}(J)P0​(J). This happens both for P0(J){P}_{0}(J)P0​(J) that diverge or go to zero as $J\ensuremath{\rightarrow}0$. The singular form of Pn(J(n)){P}_{n}({J}^{(n)})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 (TTT) 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 nnn-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){P}_{0}(J)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.
The authors present a calculation of the Lyapunov exponent for transverse magnon modes in a Heisenberg spin-glass chain with positive and negative exchange interactions, both analytically as a microcanonical average and also numerically in the usual canonical statistical ensemble. In the positive- omega regime they find that the localization length obeys xi ( omega ) varies as omega -1 at omega to 0 if p>q, where p, q=1-p are the +1 and -1 spin concentrations, respectively. It is shown that the previously reported xi ( omega ) varies as omega -2/3 singularity, related to the anomalous dispersion law omega varies as k 3/2 , holds only in the limit of equal concentrations p=q=0.5. The corresponding densities of states diverge as rho ( omega ) varies as omega -1/2 and varies as omega -1/3 for p>q and p=q, respectively.
We study one-magnon excitations in a random ferromagnetic Heisenberg chain with long-range correlations in the coupling constant distribution. By employing an exact diagonalization procedure, we compute the localization length of all one-magnon states within the band of allowed energies E. The random distribution of coupling constants was assumed to have a power spectrum decaying as $S(k)\ensuremath{\propto}{1/k}^{\ensuremath{\alpha}}.$ We found that for $\ensuremath{\alpha}<1,$ one-magnon excitations remain exponentially localized with the localization length $\ensuremath{\xi}$ diverging as 1/E.1/E.1/E. For $\ensuremath{\alpha}=1$ a faster divergence of $\ensuremath{\xi}$ is obtained. For any $\ensuremath{\alpha}>1,$ a phase of delocalized magnons emerges at the bottom of the band. We characterize the scaling behavior of the localization length on all regimes and relate it with the scaling properties of the long-range correlated exchange coupling distribution.
The high-field and zero-field behavior of the linearized magnetic excitations (harmonic magons) in a one-dimensional \ifmmode\pm\else\textpm\fi{}J Heisenberg spin glass are studied. In the high-field limit---a field strong enough to ensure complete alignment of the ground state---the density of states (DOS), the inverse localization length (ILL), and the dynamic structure factor are calculated over the interval -4JE-H4J (H is applied field) by employing the coherent-exchange approximation (CEA), negative-eigenvalue counting, and matrix diagonalization. In the low-energy regime (0.001E-H0.1J), the CEA closely approximates the exact results reproducing, in particular, the anomalous power-law behavior of the DOS \ensuremath{\rho}(E)\ensuremath{\sim}(E-H)1/3{)}^{1/3})1/3 and the ILL 1/L(E)\ensuremath{\sim}(E-H)2/3{)}^{2/3})2/3 for the symmetric distribution of the exchange interactions (concentration c=0.5). In zero field, the DOS and the ILL are calculated using negative-eigenvalue counting for 0E4J and 0.0001E0.01J. For c=0.5, a connection is established between the zero-field and the high-field limits, and for other concentrations, a phenomenological approach is developed in the low-energy regime where it is found that \ensuremath{\rho}(E)=f(c)${\mathit{E}}^{\mathrm{\ensuremath{-}}1/3}$ and 1/L(E)=g(c)E2/3{\mathit{E}}^{2/3}E2/3.
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Low temperature properties of random antiferromagnetic Heisenberg spin-1/2 systems with a wide distribution of exchange couplings are examined using a scaling technique. The method iteratively discards the irrelevant high energy excitations of strongly coupled clusters, while retaining the low lying states which are reproduced by an equivalent smaller cluster. The method is applied to n-doped semiconductors, a spatially random three dimensional system of spin-1/2 donors with Heisenberg exchange varying exponentially with separation. It shows how the system does not order down to temperatures well below the median nearest neighbor exchange. Comparison is made with the corresponding system in two dimensions and the one-dimensional random antiferromagnetic chain (applicable to certain organic charge transfer salts), as well as with the case of Ising spins in three dimensions.