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arXiv:cond-mat/0112261 · Physical review. B, Condensed matter

Strongly disordered spin ladders

R. Mélin, Yu‐Cheng Lin, Péter Lajkó, Heiko Rieger, Ferenc Iglói

Published Feb 13, 2002• 49 citations
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Abstract

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.

DOI
10.1103/physrevb.65.104415

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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.
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.
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.
We study by the strong disorder renormalization group (RG) method the low-energy properties of the one-dimensional Hubbard model with random-hopping matrix-elements tmin<t<tmax{t}_{min}<t<{t}_{max}tmin​<t<tmax​, and with random onsite Coulomb repulsion terms $0\ensuremath{\leqslant}{U}_{min}<U<{U}_{max}$. There are two critical phases, corresponding to an infinite randomness spin random singlet for strong interactions (Umin>tmax)({U}_{min}>{t}_{max})(Umin​>tmax​) and to an orbital infinite randomness fixed point for vanishing interactions $({U}_{max}∕{t}_{max}\ensuremath{\rightarrow}0)$. To each critical infinite randomness fixed point is connected a Griffiths phase, the correlation length, and dynamical exponent of which have well defined asymptotic dependences on the corresponding quantum control parameter. The theoretical predictions for the scaling in the vicinity of the critical points compare well to numerical RG simulations.
Critical properties of quantum spin chains with varying degrees of disorder are studied at zero temperature by analytical and extensive density matrix renormalization methods. Generally the phase diagram is found to contain three phases. The weak disorder regime, where the critical behavior is controlled by the fixed points of the pure system, and the strong disorder regime, which is attracted by an infinite randomness fixed point, are separated by an intermediate disorder regime, where dynamical scaling is anisotropic and the static and dynamical exponents are disorder dependent.
We introduce a strong-disorder renormalization group (RG) approach suitable for investigating the quasiparticle excitations of disordered superconductors in which the quasiparticle spin is not conserved. We analyze one-dimensional models with this RG and with elementary transfer matrix methods. We find that such models with broken spin rotation invariance generically lie in one of two topologically distinct localized phases. Close enough to the critical point separating the two phases, the system has a power-law divergent low-energy density of states (with a nonuniversal continuously varying power law) in either phase, due to quantum Griffiths singularities. This critical point belongs to the same infinite-disorder universality class as the one-dimensional particle-hole symmetric Anderson localization problem, while the Griffiths phases in the vicinity of the transition are controlled by lines of strong (but not infinite) disorder fixed points terminating in the critical point.
We have developed a very efficient numerical algorithm of the strong disorder renormalization group method to study the critical behaviour of the random transverse field Ising model, which is a prototype of random quantum magnets. With this algorithm we can renormalize an N-site cluster within a time NlogN, independently of the topology of the graph, and we went up to N ∼ 4 × 10(6). We have studied regular lattices with dimension D ≤ 4 as well as Erdős-Rényi random graphs, which are infinite dimensional objects. In all cases the quantum critical behaviour is found to be controlled by an infinite disorder fixed point, in which disorder plays a dominant role over quantum fluctuations. As a consequence the renormalization procedure as well as the obtained critical properties are asymptotically exact for large systems. We have also studied Griffiths singularities in the paramagnetic and ferromagnetic phases and generalized the numerical algorithm for other random quantum systems.