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arXiv:cond-mat/0201118 · Physical review. B, Condensed matter

Griffiths effects in random Heisenberg antiferromagnetic S=1 chains

Kedar Damle

Published Sep 27, 2002• 25 citations
arXivHTMLPDF

Abstract

I consider the effects of enforced dimerization on random Heisenberg antiferromagnetic $S=1$ chains. I argue for the existence of novel Griffiths phases characterized by two independent dynamical exponents that vary continuously in these phases; one of the exponents controls the density of spin-1/2 degrees of freedom in the low-energy effective Hamiltonian, while the other controls the corresponding density of spin-1 degrees of freedom. Moreover, in one of these Griffiths phases, the system has very different low temperature behavior in two different parts of the phase which are separated from each other by a sharply defined crossover line; on one side of this crossover line, the system ``looks'' like a $S=1$ chain at low energies, while on the other side, it is best thought of as a $S=1/2$ chain. A strong-disorder renormalization group analysis makes it possible to analytically obtain detailed information about the low temperature behavior of physical observables such as the susceptibility and the specific heat, as well as identify an experimentally accessible signature of this crossover.

DOI
10.1103/physrevb.66.104425

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I consider the effects of enforced dimerization on random Heisenberg antiferromagnetic S=1S=1S=1 chains. I argue for the existence of novel Griffiths phases characterized by two independent dynamical exponents that vary continuously in these phases; one of the exponents controls the density of spin-1/2 degrees of freedom in the low-energy effective Hamiltonian, while the other controls the corresponding density of spin-1 degrees of freedom. Moreover, in one of these Griffiths phases, the system has very different low temperature behavior in two different parts of the phase which are separated from each other by a sharply defined crossover line; on one side of this crossover line, the system ``looks'' like a S=1S=1S=1 chain at low energies, while on the other side, it is best thought of as a S=1/2S=1/2S=1/2 chain. A strong-disorder renormalization group analysis makes it possible to analytically obtain detailed information about the low temperature behavior of physical observables such as the susceptibility and the specific heat, as well as identify an experimentally accessible signature of this crossover.
We present a new perturbative real space renormalization group (RG) to study random quantum spin chains and other one-dimensional disordered quantum systems. The method overcomes problems of the original approach which fails for quantum random chains with spins larger than S=1/2. Since it works even for weak disorder, we are able to obtain the zero temperature phase diagram of the random antiferromagnetic Heisenberg spin-1 chain as a function of disorder. We find a random singlet phase for strong disorder. As the disorder decreases, the system shows a crossover from a Griffiths to a disordered Haldane phase.
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.
The low energy properties of the spin-1/2 random Heisenberg chain with ferromagnetic and antiferromagnetic interactions are studied by means of the density matrix renormalization group (DMRG) and real space renormalization group (RSRG) method for finite chains. The results of the two methods are consistent with each other. The deviation of the gap distribution from that of the random singlet phase and the formation of the large-spin state is observed even for relatively small systems. For a small fraction of the ferromagnetic bond, the effect of the crossover to the random singlet phase on the low temperature susceptibility and specific heat is discussed. The crossover concentration of the ferromagnetic bond is estimated from the numerical data.
The effect of quenched disorder on the low-energy properties of various antiferromagnetic spin-ladder models is studied by a numerical strong disorder renormalization-group method and by density-matrix renormalization. For strong enough disorder the originally gapped phases with finite topological or dimer order become gapless. In these quantum Griffiths phases the scaling of the energy, as well as the singularities in the dynamical quantities, are characterized by a finite dynamical exponent, z, which varies with the strength of disorder. At the phase boundaries, separating topologically distinct Griffiths phases the singular behavior of the disordered ladders is generally controlled by an infinite randomness fixed point.
Using a numerical implementation of a strong-disorder renormalization group, we study the low-energy, long-distance properties of layers and bilayers of S=1∕2S=1∕2S=1∕2 Heisenberg antiferromagnets with different types of disorder: bond randomness, site dilution, and dimer dilution. Generally the systems exhibit an ordered and a disordered phase separated by a phase boundary on which the static critical exponents appear to be independent of bond randomness in the strong-disorder regime, while the dynamical exponent is a continuous function of the bond disorder strength. The low-energy fixed points of the off-critical phases are affected by the actual form of the disorder, and the disorder-induced dynamical exponent depends on the disorder strength. As the strength of the bond disorder is increased, there is a set of crossovers in the properties of the low-energy singularities. For weak disorder quantum fluctuations play the dominant role. For intermediate disorder nonlocalized disorder fluctuations are relevant, which become localized for even stronger bond disorder. We also present some quantum Monte Carlo simulation results to support the strong-disorder renormalization approach.
The effect of quenched disorder on the low-energy and low-temperature properties of various two- and three-dimensional Heisenberg models is studied by a numerical strong disorder renormalization-group method. For strong enough disorder we have identified two relevant fixed points, in which the gap exponent, $\ensuremath{\omega},$ describing the low-energy tail of the gap distribution $P(\ensuremath{\Delta})\ensuremath{\sim}{\ensuremath{\Delta}}^{\ensuremath{\omega}}$ is independent of disorder, the strength of couplings, and the value of the spin. The dynamical behavior of nonfrustrated random antiferromagnetic models is controlled by a singletlike fixed point, whereas for frustrated models the fixed point corresponds to a large spin formation and the gap exponent is given by $\ensuremath{\omega}\ensuremath{\approx}0.$ Another type of universality class is observed at quantum critical points and in dimerized phases but no infinite randomness behavior is found, in contrast to that of one-dimensional models.