Scale invariance in disordered systems: the example of the random-field Ising model

We show by numerical simulations that the correlation function of the random-field Ising model (RFIM) in the critical region in three dimensions has very strong fluctuations and that in a finite volume the correlation length is not self-averaging. This is due to the formation of a bound state in the underlying field theory. We argue that this nonperturbative phenomenon is not particular to the RFIM in 3D. It is generic for disordered systems in two dimensions and may also happen in other three-dimensional disordered systems.     

A Note on Exponential Decay in the Random Field Ising Model

For the two-dimensional random field Ising model (RFIM) with bimodal (i.e., two-valued) external field, we prove exponential decay of correlations either (i) when the temperature is larger than the critical temperature of the Ising model without external field and the magnetic field strength is small or (ii) at any temperature when the magnetic field strength is sufficiently large. Unlike previous work on exponential decay, our approach is not based on cluster expansions but rather on arguably simpler methods; these combine an analysis of the Kertész line and a coupling of Ising measures (and also their random cluster representations) with different boundary conditions. We also show similar but weaker results for the RFIM with a general field distribution and in any dimension.     

Thermal fluctuations in some random field models

Exact identities are derived for a family of models including (a) a domain wall in a random field Ising model (RFIM), and (b) the random anisotropyXY model in the no-vortex approximation. In particular, the second moment of thermal fluctuations is not affected by frozen randomness. It is checked in a one-dimensional model that higher moments are on the contrary strongly enhanced. Thus, thermal fluctuations are strongly non-Gaussian. This reflects excursions between remote potential wells in the phase space. It is shown exactly that the Imry-Ma argument yields a correct evaluation of the field-induced fluctuations for the one-dimensional model.     

The effects of random fields on the critical and bicritical behavior of Mn x Zn1−x F2

We have carried out a comprehensive study of the static and dynamic spin-spin correlations of Mn _x Zn_1–x F₂ in a magnetic field. Samples withx=0.75 andx=0.5 have been studied. This system exhibits behavior closely related, if not identical, to that of the Random Field Ising Model (RFIM). An additional feature of Mn _x Zn_1–x F₂ is that it exhibits an easily accessible bicritical point; thus one can study the changeover from the RFIM to the uniformXY model with a transverse random field. Quite generally, the instantaneous spin-spin correlations in a field are described by a combination of Lorentzian, Lorentzian-squared and delta function terms the latter corresponds to the long range order (LRO) component. In the Ising phase one finds history dependent behavior as discussed previously. In theXY phase, except very near the spin-flop boundary, one finds ergodic behavior withXY LRO and Lorentzian squared Ising fluctuations. Rather complicated instability effects are found all along the spin-flop boundary. Further, when one establishes LRO in theXY phase and lowers the field through the spin-flop value, one obtains a LRO Ising state in thex=0.75 sample whereas one obtains the field-cooled domain state in thex=0.50 sample. This dramatic difference in behavior is not understood. Our results on the RFIM aspects of the problem are consistent with our previous studies. The transition is dominated by the metastability effects with an underlying equilibrium transition which is either first order or weakly second order (0). The underlying transition manifests itself directly in measurements of the dynamic response nearT _N (H). From the data above the metastability boundary we deduce for the static correlation length exponentv=1.4±0.3 in good agreement with theory. We find for the RFIM crossover exponent _RF=1.5±0.2 where the errors represent the spread in values obtained from different techniques. Finally, we have determined in detail the field-temperature phase diagram of thex=0.5 sample including the critical behavior along the spin-flop line; the latter transition appears to be second order for an extended region.     

Generalized off-equilibrium fluctuation-dissipation relations in random Ising systems

We show that the numerical method based on the off-equilibrium fluctuation-dissipation relation does work and is very useful and powerful in the study of disordered systems which show a very slow dynamics. We have verified that it gives the right information in the known cases (diluted ferromagnets and random field Ising model far from the critical point) and we used it to obtain more convincing results on the frozen phase of four-dimensional spin glasses. Moreover we used it to study the Griffiths phase of the diluted and the random field Ising models. Received 1 December 1998 and Received in final form 17 February 1999     

The hierarchical random field Ising model

We introduce and solve explicitly a hierarchical approximation to the random field Ising model. This approximation is defined in terms of Peierls' contours. It exhibits a spontaneous magnetization ind>2 and illustrates some of the ideas used in the proof of that result for the real RFIM. Ind=2, there is no spontaneous magnetization, but a very slow decay of correlations. However, we argue that this latter property is an artifact of the approximation. For the real RFIM, we expect exponential decay of correlations for any value of the disorder.     

Induced random fields in the LiHoxY1-xF4 quantum Ising magnet in a transverse magnetic field

The LiHoxY1-xF4 magnetic material in a transverse magnetic field Bx x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in a random disordered system. We show that the Bx-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline symmetries, generates, via the predominant dipolar interactions between Ho3+ ions, random fields along the Ising z direction. This identifies LiHoxY1-xF4 in Bx as a new random field Ising system. The random fields explain the rapid decrease of the critical temperature in the diluted ferromagnetic regime and the smearing of the nonlinear susceptibility at the spin-glass transition with increasing Bx and render the Bx-induced quantum criticality in LiHoxY1-xF4 likely inaccessible.     

Fluctuations in the Curie-Weiss version of the random field Ising model

The fluctuations of the order parameter in the Curie-Weiss version of the Ising model with random magnetic field are computed. Away from criticality or at first-order critical points they have a Gaussian distribution with random (i. e.,sample-dependent) mean, thermal fluctuations contributing in same order as the fluctuations of the field; at second- or higher-order critical points, non-Gaussian sample-dependent distributions appear, and the fluctuations of the fields are enhanced, dominating over the thermal ones.     

Phase diagram of the 3D bimodal random-field Ising model

The one-parametric Wang-Landau (WL) method is implemented together with an extrapolation scheme to yield approximations of the two-dimensional (exchange-energy, field-energy) density of states (DOS) of the 3D bimodal random-field Ising model (RFIM). The present approach generalizes our earlier WL implementations, by handling the final stage of the WL process as an entropic sampling scheme, appropriate for the recording of the required two-parametric histograms. We test the accuracy of the proposed extrapolation scheme and then apply it to study the size-shift behavior of the phase diagram of the 3D bimodal RFIM. We present a finite-size converging approach and a well-behaved sequence of estimates for the critical disorder strength. Their asymptotic shift-behavior yields the critical disorder strength and the associated correlation length's exponent, in agreement with previous estimates from ground-state studies of the model.     

Ordering dynamics in Disordered systems

We review analytical and numerical studies of phase ordering dynamics or domain growth in systems with quenched disorder. These studies are usually based on kinetic versions of the random-exchange Ising model (REIM) or random-field Ising model (RFIM). We also present detailed numerical results which clarify the nature of domain growth in random magnets. These results demonstrate that domain walls are trapped by disorder barriers with a logarithmic dependence on the domain size.     

Magnetic-field induced quantum critical point in YbRh(2)Si(2)

We report low-temperature calorimetric, magnetic, and resistivity measurements on the antiferromagnetic (AF) heavy-fermion metal YbRh(2)Si(2) ( T(N)=70 mK) as a function of magnetic field B. While for fields exceeding the critical value B(c0) at which T(N)-->0 the low-temperature resistivity shows an AT2 dependence, a 1/(B-B(c0)) divergence of A(B) upon reducing B to B(c0) suggests singular scattering at the whole Fermi surface and a divergence of the heavy quasiparticle mass. The observations are interpreted in terms of a new type of quantum critical point separating a weakly AF ordered from a weakly polarized heavy Landau-Fermi liquid state.     

Odd-integer quantum Hall effect in graphene: interaction and disorder effects

We study the competition between the long-range Coulomb interaction, disorder scattering, and lattice effects in the integer quantum Hall effect (IQHE) in graphene. By direct transport calculations, both nu=1 and nu=3 IQHE states are revealed in the lowest two Dirac Landau levels. However, the critical disorder strength above which the nu=3 IQHE is destroyed is much smaller than that for the nu=1 IQHE, which may explain the absence of a nu=3 plateau in recent experiments. While the excitation spectrum in the IQHE phase is gapless within numerical finite-size analysis, we do find and determine a mobility gap, which characterizes the energy scale of the stability of the IQHE. Furthermore, we demonstrate that the nu=1 IQHE state is a Dirac valley and sublattice polarized Ising pseudospin ferromagnet, while the nu=3 state is an xy plane polarized pseudospin ferromagnet.     

Off-perturbative states in disordered systems

The systematic approach for the off-perturbative calculations in disordered systems is developed. The proposed scheme is applied for the random temperature and the random field ferromagnetic Ising models. It is shown that away from the critical point, in the paramagnetic phase of the random temperature model, and in the ferromagnetic phase of the random field one, the free energy contains non-analytic contributions which have the form of essential singularities. It is demonstrated that these contributions appear due to localized in space instanton-like excitations.     

Effect of local field fluctuations on orientational ordering in random-site dipole systems

Some peculiarities of dipole ordering in systems with uniaxial or cubic anisotropy with an arbitrary degree of dilution are analyzed in terms of random local field theory. The approach takes into account the effect of thermal and spatial fluctuations of the local fields acting on each particle from its neighbors with an accuracy corresponding to that of the Bethe-Paierls pair clusters approach. We show that ferromagnetic (ferroelectric) structure for uniaxial Ising dipoles distributed on a simple cubic lattice is intrinsically unstable against the fluctuations of the local fields for any concentration of the dipoles. This result is quite different from the prediction of the mean-field theory which implies the possibility of ferromagnetic ordering as a metastable state in field-cooled experiments. The local field fluctuations do not exclude, however, antiferromagnetic ordering above a certain critical concentration. Ferromagnetic ordering is possible for other types of lattice geometries and for an amorphous-like dipole distribution above a certain critical concentration. A simple physical explanation of such behavior is given based on the specific angular dependence of the dipole-dipole interaction that results in a relatively high value of the local field second moment for simple cubic lattice.     

Renormalization group study of random quantum magnets

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 Erd?s-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.     

On evaluation of transverse spin fluctuations in quantum magnets

A numerical method is described for evaluating transverse spin correlations in the random phase approximation. Quantum spin-fluctuation corrections to sublattice magnetization are evaluated for the antiferromagnetic ground state of the half-filled Hubbard model in two and three dimensions in the whole U/t range. Extension to the case of defects in the AF is also discussed for spin vacancies and low-U impurities. In the limit, the vacancy-induced enhancement in the spin fluctuation correction is obtained for the spin-vacancy problem in two dimensions, for vacancy concentration up to the percolation threshold. For low-U impurities, the overall spin fluctuation correction is found to be strongly suppressed, although surprisingly spin fluctuations are locally enhanced at the low-U sites. Received 27 April 1998 and Received in final form 13 August 1998     

Spin wave propagation in the domain state of a random field magnet

Inelastic neutron scattering with high wave-vector resolution has characterized the propagation of transverse spin wave modes near the antiferromagnetic zone center in the metastable domain state of a random field Ising magnet. A well-defined, long wavelength excitation is observed despite the absence of long-range magnetic order. Direct comparisons with the spin wave dispersion in the long-range ordered antiferromagnetic state reveal no measurable effects from the domain structure. This result recalls analogous behavior in thermally disordered anisotropic spin chains but contrasts sharply with that of the phonon modes in relaxor ferroelectrics. Received 2 November 2002 / Received in final form 4 February 2003 Published online 11 April 2003 RID="a" ID="a"leheny@pha.jhu.edu     

Dynamics of irregular copolymers

Reptation dynamics of AB copolymers with irregular chemical structure are considered theoretically. It is shown that interactions between A and B monomers could result in a significant slowdown of copolymer dynamics in the disordered (macroscopically homogeneous) state. The dynamical copolymer length N* showing the crossover to the strongly retarded dynamics is calculated. It is shown that contour-length fluctuations (internal reptation modes) give rise to a strong reduction of the slowdown effect and to a strong increase of N* which becomes unrealistically high in the case of a genuinely random chemical structure. The following scaling dependence of N* is predicted for irregular block copolymers: N* proportional, variant delta(-8)chi(-8)n(-8)(0)N(3)(e), where delta is the degree of block polydispersity, chi the Flory AB interaction parameter, and n(0) the mean block length. The strongest dynamical effect of AB interactions is predicted for correlated random copolymers near the critical point related to the formation of microdomain superstructures.     

Ising-type critical exponents of the fully frustrated spin-1/2 Heisenberg FM/AF square bilayer at a critical magnetic field

The fully frustrated spin-1/2 Heisenberg FM/AF square bilayer in a magnetic field with the ferromagnetic inter-dimer interaction and the antiferromagnetic intra-dimer interaction is explored by the use of localized many-magnon approach, which allows to connect the original purely quantum Heisenberg spin model on a square bilayer with the effective ferromagnetic Ising model on a simple square lattice. Magnetization and specific heat are investigated exactly at a field-driven phase transition from the singlet-dimer phase towards the fully saturated ferromagnetic phase, which changes from a discontinuous phase transition to a continuous one at a certain critical temperature. The mapping correspondence between the spin-1/2 Heisenberg FM/AF square bilayer and the ferromagnetic Ising square lattice suggests for this special critical point of the spin-1/2 Heisenberg FM/AF square bilayer critical exponents from the standard two-dimensional Ising universality class.     

Order-from-disorder effect in the exactly solved mixed spin-(1/2, 1) Ising model on fully frustrated triangles-in-triangles lattices

The mixed spin-(1/2, 1) Ising model on two fully frustrated triangles-in-triangles lattices is exactly solved with the help of the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the equivalent spin- 1/2 Ising model on a triangular lattice. It is shown that the mutual interplay between the spin frustration and single-ion anisotropy gives rise to various spontaneously ordered and disordered ground states, which differ mainly in an occurrence probability of the non-magnetic spin state of the integer-valued decorating spins. We have convincingly evidenced a possible coexistence of the spontaneous long-range order with a partial disorder within the striking ordered–disordered ground state, which manifests itself through a non-trivial criticality at finite temperatures as well. A rather rich critical behavior including the order-from-disorder effect and reentrant phase transitions with either two or three successive critical points is also found.