Physical Review Letters

Random Exchange Heisenberg Chain for Classical and Quantum Spins

Akira Furusaki, Manfred Sigrist, Patrick A. Lee, Katsuhiro Tanaka, Naoto Nagaosa

Published Nov 7, 1994• 61 citations

Abstract

The Heisenberg chain with random $\ifmmode\pm\else\textpm\fi{}J$ bonds is studied for the quantum spin $s=\frac{1}{2}$ and in the classical limit. The former is treated by high-temperature expansion and transfer matrix calculation while the latter can be analyzed exactly. The disorder leads to a $\frac{1}{T}$ behavior of the low-temperature susceptibility in the classical system. For $s=\frac{1}{2}$ our analysis reveals a significant residual entropy at low temperature. From this we conclude that for quantum spins the susceptibility exhibits three different regimes in temperature and that the specific heat has a peak in the very low-temperature regime.

DOI: 10.1103/physrevlett.73.2622

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Physical Review Letters

Random Exchange Heisenberg Chain for Classical and Quantum Spins

Akira Furusaki, Manfred Sigrist, Patrick A. Lee, Katsuhiro Tanaka, Naoto Nagaosa

Published Nov 7, 1994• 61 citations

Publisher

Abstract

The Heisenberg chain with random

\ifmmode\pm\else\textpm\fi{}J

bonds is studied for the quantum spin

s=\frac{1}{2}

and in the classical limit. The former is treated by high-temperature expansion and transfer matrix calculation while the latter can be analyzed exactly. The disorder leads to a

\frac{1}{T}

behavior of the low-temperature susceptibility in the classical system. For

s=\frac{1}{2}

our analysis reveals a significant residual entropy at low temperature. From this we conclude that for quantum spins the susceptibility exhibits three different regimes in temperature and that the specific heat has a peak in the very low-temperature regime.

DOI: 10.1103/physrevlett.73.2622

Related papers

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The one-dimensional quantum Heisenberg model with random \ifmmode\pm\else\textpm\fi{}J bonds is studied for S=1/2 and S=1. The specific heat and the zero-field susceptibility are calculated by using high-temperature series expansions and the quantum transfer matrix method. The susceptibility shows a Curie-like temperature dependence at low temperatures as well as at high temperatures. The numerical results for the specific heat suggest that there are anomalously many low-lying excitations. The qualitative nature of these excitations is discussed based on the exact diagonalization of finite-size systems.The one-dimensional quantum Heisenberg model with random \ifmmode\pm\else\textpm\fi{}J bonds is studied for S=1/2 and S=1. The specific heat and the zero-field susceptibility are calculated by using high-temperature series expansions and the quantum transfer matrix method. The susceptibility shows a Curie-like temperature dependence at low temperatures as well as at high temperatures. The numerical results for the specific heat suggest that there are anomalously many low-lying excitations. The qualitative nature of these excitations is discussed based on the exact diagonalization of finite-size systems.

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Oleg A. Starykh, Anders W. Sandvik, Rajiv R. P. Singh

Combining high-temperature expansions with the recursion method and quantum Monte Carlo simulations with the maximum entropy method, we study the dynamics of the spin-1/2 Heisenberg chain at temperatures above and below the coupling J. By comparing the two sets of calculations, their relative strengths are assessed. At high temperatures, we find that there is a low-frequency peak in the momentum-integrated dynamic structure factor, due to diffusive long-wavelength modes. This peak is rapidly suppressed as the temperature is lowered below J. Calculation of the complete dynamic structure factor S(k,\ensuremath{\omega}) shows how the spectral features associated with the two-spinon continuum develop at low temperatures. We extract the nuclear spin-lattice relaxation rate 1/ ${\mathrm{T}}_{1}$ from the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, and compare with recent experimental results for ${\mathrm{Sr}}_{2}$ ${\mathrm{CuO}}_{3}$ and ${\mathrm{CuGeO}}_{3}$ . We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility \ensuremath{\chi}(k). We confirm the asymptotic low-temperature forms S(\ensuremath{\pi})\ensuremath{\sim}[ln (T) ${]}^{3\mathrm{/}2}$ and \ensuremath{\chi}(\ensuremath{\pi})\ensuremath{\sim}${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$[ln (T) ${]}^{1\mathrm{/}2}$ , expected from previous theoretical studies.Combining high-temperature expansions with the recursion method and quantum Monte Carlo simulations with the maximum entropy method, we study the dynamics of the spin-1/2 Heisenberg chain at temperatures above and below the coupling J. By comparing the two sets of calculations, their relative strengths are assessed. At high temperatures, we find that there is a low-frequency peak in the momentum-integrated dynamic structure factor, due to diffusive long-wavelength modes. This peak is rapidly suppressed as the temperature is lowered below J. Calculation of the complete dynamic structure factor S(k,\ensuremath{\omega}) shows how the spectral features associated with the two-spinon continuum develop at low temperatures. We extract the nuclear spin-lattice relaxation rate 1/${\mathrm{T}}_{1}$ from the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, and compare with recent experimental results for ${\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3}$ and ${\mathrm{CuGeO}}_{3}$. We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility \ensuremath{\chi}(k). We confirm the asymptotic low-temperature forms S(\ensuremath{\pi})\ensuremath{\sim}[ln (T)${]}^{3\mathrm{/}2}$ and \ensuremath{\chi}(\ensuremath{\pi})\ensuremath{\sim}${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$[ln (T)${]}^{1\mathrm{/}2}$, expected from previous theoretical studies.

arXiv:cond-mat/9701052Physical review. B, Condensed matterJun 1, 199766 citations

The Heisenberg chain with random

\ifmmode\pm\else\textpm\fi{}J

bonds is studied for the quantum spin

s=\frac{1}{2}

and in the classical limit. The former is treated by high-temperature expansion and transfer matrix calculation while the latter can be analyzed exactly. The disorder leads to a

\frac{1}{T}

behavior of the low-temperature susceptibility in the classical system. For

s=\frac{1}{2}

{T}^{2/3}

with a coefficient that depends on the relative fractions of positive and negative bonds. The coefficient is calculated exactly for the equal-probability distribution. There is also a brief analysis of the destabilizing effects of an applied magnetic field.

One-dimensional Heisenberg spin 1/2 chains with random ferro- and antiferromagnetic bonds are realized in systems such as

Sr_3 CuPt_{1-x} Ir_x O_6

. We have investigated numerically the thermodynamic properties of a generic random bond model and of a realistic model of

Sr_3 CuPt_{1-x} Ir_x O_6

by the quantum Monte Carlo loop algorithm. For the first time we demonstrate the separation into three different temperature regimes for the original Hamiltonian based on an exact treatment, especially we show that the intermediate temperature regime is well-defined and observable in both the specific heat and the magnetic susceptibility. The crossover between the regimes is indicated by peaks in the specific heat. The uniform magnetic susceptibility shows Curie-like behavior in the high-, intermediate- and low-temperature regime, with different values of the Curie constant in each regime. We show that these regimes are overlapping in the realistic model and give numerical data for the analysis of experimental tests.

\frac{1}{2}

Heisenberg chain are calculated analytically. For some experimentally essential distributions of the strong disorder we show how the latter renormalizes the universal low-energy characteristics of the system and causes the low temperature divergencies.

{S}^{z}

{S}^{z}

susceptibility \ensuremath{\chi} which was previously predicted by real-space renormalization-group (RSRG) treatments. Over the finite temperature range studied, these results are consistent with a 1/(T

{\mathrm{ln}}^{2}

T) behavior of \ensuremath{\chi}, and hence in qualitative agreement with the RSRG results. As in the XY-Heisenberg regime, we also find a disorder-induced enhancement of the low-T susceptibility for a system with Ising-like exchange coupling which, over the finite temperature range studied, is again consistent with RSRG results. However, there are inconsistencies between the RSRG predictions in the Ising-like regime at very low temperatures, and the exact results for the random-exchange Ising chain and the low-temperature behavior of \ensuremath{\chi} in the Ising-like regime may in fact be more complicated than predicted by RSRG. Finally, we also present results for the antiferromagnetic susceptibility and structure factor. For both Heisenberg and Ising-like systems, we find that disorder suppresses the long-range antiferromagnetic correlations at low T.

{\mathrm{T}}_{1}

from the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, and compare with recent experimental results for

{\mathrm{Sr}}_{2}

{\mathrm{CuO}}_{3}

and

{\mathrm{CuGeO}}_{3}

. We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility \ensuremath{\chi}(k). We confirm the asymptotic low-temperature forms S(\ensuremath{\pi})\ensuremath{\sim}[ln (T)

{]}^{3\mathrm{/}2}

and \ensuremath{\chi}(\ensuremath{\pi})\ensuremath{\sim}${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$[ln (T)

{]}^{1\mathrm{/}2}

, expected from previous theoretical studies.