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Physical review. B, Condensed matter

Universal low-energy behavior of disordered quantum spin chains: Exact analytic results

Andrei A. Zvyagin

Published Sep 1, 2000• 11 citations
Publisher

Abstract

Low energy asymptotics of the behavior of static and dynamic susceptibility, specific heat, correlation length, Knight shift, etc. (which can be measured experimentally) of the integrable disordered quantum spin $\frac{1}{2}$ 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.

DOI
10.1103/physrevb.62.r6069

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Low energy asymptotics of the behavior of static and dynamic susceptibility, specific heat, correlation length, Knight shift, etc. (which can be measured experimentally) of the integrable disordered quantum spin 12\frac{1}{2}21​ 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.
The one-dimensional isotropic quantum Heisenberg spin systems with random couplings and random spin sizes are investigated using a real-space renormalization-group scheme. It is demonstrated that these systems belong to a universality class of disordered spin systems, characterized by weakly coupled large effective spins. In this large-spin phase the uniform magnetic susceptibility diverges as ${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$ with a nonuniversal Curie constant at low temperatures T, while the specific heat vanishes as ${\mathrm{T}}^{\mathrm{|}\mathrm{\ensuremath{\alpha}}\mathrm{|}}$|ln T| for T\ensuremath{\rightarrow}0. For a broad range of initial distributions of couplings and spin sizes the distribution functions approach a single fixed-point form, where \ensuremath{\alpha}\ensuremath{\approx}-0.44. For some singular initial distributions, however, the fixed-point form of distributions becomes nonuniversal, suggesting that there is a line of fixed points.
Quenched disorders can strongly influence the physical properties of quantum many-body systems. The real-space strong-disorder renormalization group (SDRG) analysis has shown that the spin-1/2 random Heisenberg chain is controlled by the infinite-randomness fixed point (IRFP) and forms a random singlet (RS) ground state. Motivated by recent thermal transport experiments on the quasi-one-dimensional antiferromagnet copper benzoate [B. Y. Pan et al., Phys. Rev. Lett. 129, 167201 (2022)], we adapt the SDRG to study the low-temperature properties of the random Heisenberg chain by assuming that its low-energy excited states are captured by the parent Hamiltonian of the RS ground state as well. We find that while the specific heat coefficient and the uniform magnetic susceptibility scale as $C/T\ensuremath{\sim}{T}^{\ensuremath{-}{\ensuremath{\alpha}}_{c}}$ and $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}{\ensuremath{\alpha}}_{s}}$ with $0<{\ensuremath{\alpha}}_{c,s}<1$, indicating a divergent low-energy density of states, the thermal and the spin conductivities scale as $\ensuremath{\kappa}/T\ensuremath{\sim}T$ and ${\ensuremath{\sigma}}_{s}\ensuremath{\sim}T$, which implies a vanishing density of extended states in the low-energy limit. We believe that such a disparity in the thermodynamic and transport properties is a common feature of random systems controlled by the IRFP.
In this work we consider several exactly solvable models of magnetic impurities in critical quantum antiferromagnetic spin chains and multichannel Kondo impurities. Their ground-state properties are studied, and the finite set of nonlinear integral equations which exactly describe the thermodynamics of the models is constructed. We obtain several analytical low-energy expressions for the temperature, magnetic field, and frequency dependences of important characteristics of exactly solvable disordered quantum spin models and disordered multichannel Kondo impurities with essential many-body interactions. We show that the only low-energy parameter that gets renormalized is the velocity of the low-lying excitations (or the effective crossover scale connected with each impurity); the others appear to be universal. In our study several kinds of strong disorder important for experiments were used. Some of them produce low divergences in certain characteristics of our strongly disordered critical systems (compared with finite values for the homogeneous case or a single impurity). For weak disorder, or for narrow distributions of the local Kondo temperatures, our exact results reveal the presence of Kondo screening of disordered ensembles of magnetic impurities by low-lying excitations of the host. We point out that our results qualitatively coincide with the data of experiments on real disordered quasi-one-dimensional antiferromagnetic systems and with the similar behavior of some heavy metallic alloys.
Quantum Heisenberg spin chains with random couplings and spin sizes are studied using a real-space renormalization group technique. These systems belong to a new universality class of disordered quantum spin systems realized in, e.g., ${\mathrm{Sr}}_{3}\mathrm{Cu}{\mathrm{Pt}}_{1\ensuremath{-}x}{\mathrm{Ir}}_{x}{\mathrm{O}}_{6}$. The low-energy fixed point is characterized by the formation of weakly coupled large effective spins. At low temperature TTT the entropy obeys a power law ${T}^{\ensuremath{\delta}}$ ($\ensuremath{\delta}\ensuremath{\approx}0.44$), and the susceptibility follows a Curie-like behavior.
In the framework of the exactly solvable model, using nonperturbative methods, the temperature and magnetic field behavior of the NMR relaxation rate (and low-energy dynamical structure factor) for the antiferromagnetic Heisenberg spin-1/2 chain with magnetic impurities is calculated. For a single magnetic impurity, totally screened at low temperatures by spinons of the chain, the critical exponents, which determine the low-temperature and magnetic field behavior, get renormalized. For the finite concentration of impurities with the random distribution of the impurity-chain coupling the weak disorder in the distribution does not produce essential changes to the dynamical structure factor. Contrary, the strong disorder yields drastically different behavior of the low-energy dynamical characteristics, comparing to the homogeneous chain. Our results are generic, and they qualitatively agree in limiting cases with the results of perturbative calculations for disordered spin chains and with the data of experiments on quasi-one-dimensional spin compounds.
The Heisenberg chain with random $\ifmmode\pm\else\textpm\fi{}J$ bonds is studied for the quantum spin s=12s=\frac{1}{2}s=21​ 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 1T\frac{1}{T}T1​ behavior of the low-temperature susceptibility in the classical system. For s=12s=\frac{1}{2}s=21​ 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.