Ferenc Iglói

Strong disorder RG approach of random systems

Abstract There is a large variety of quantum and classical systems in which the quenched disorder plays a dominant role over quantum, thermal, or stochastic fluctuations: these systems display strong spatial heterogeneities, and many averaged observables are actually governed by rare regions. A unifying approach to treat the dynamical and/or static singularities of these systems has emerged recently, following the pioneering RG idea by Ma and Dasgupta and the detailed analysis by Fisher who showed that the Ma–Dasgupta RG rules yield asymptotic exact results if the broadness of the disorder grows indefinitely at large scales. Here we report these new developments by starting with an introduction of the main ingredients of the strong disorder RG method. We describe the basic properties of infinite disorder fixed points, which are realized at critical points, and of strong disorder fixed points, which control the singular behaviors in the Griffiths-phases. We then review in detail applications of the RG method to various disordered models, either (i) quantum models, such as random spin chains, ladders and higher dimensional spin systems, or (ii) classical models, such as diffusion in a random potential, equilibrium at low temperature and coarsening dynamics of classical random spin chains, trap models, delocalization transition of a random polymer from an interface, driven lattice gases and reaction diffusion models in the presence of quenched disorder. For several one-dimensional systems, the Ma–Dasgupta RG rules yields very detailed analytical results, whereas for other, mainly higher dimensional problems, the RG rules have to be implemented numerically. If available, the strong disorder RG results are compared with another, exact or numerical calculations.

Quantum relaxation after a quench in systems with boundaries

We study the time dependence of the magnetization profile, m(l)(t), of a large finite open quantum Ising chain after a quench. We observe a cyclic variation, in which starting with an exponentially decreasing period the local magnetization arrives to a quasistationary regime, which is followed by an exponentially fast reconstruction period. The nonthermal behavior observed at near-surface sites turns over to thermal behavior for bulk sites. In addition to the standard time and length scales a nonstandard time scale is identified in the reconstruction period.

Strong disorder fixed point in absorbing-state phase transitions.

The effect of quenched disorder on nonequilibrium phase transitions in the directed percolation universality class is studied by a strong disorder renormalization group approach and by density matrix renormalization group calculations. We show that for sufficiently strong disorder the critical behavior is controlled by a strong disorder fixed point and in one dimension the critical exponents are conjectured to be exact: beta=(3-sqrt[5])/2 and nu( perpendicular )=2. For disorder strengths outside the attractive region of this fixed point, disorder dependent critical exponents are detected. Existing numerical results in two dimensions can be interpreted within a similar scenario.

Long-range correlations in the nonequilibrium quantum relaxation of a spin chain.

We consider the nonstationary quantum relaxation of the Ising spin chain in a transverse field of strength h. Starting from a homogeneously magnetized initial state the system approaches a stationary state by a process possessing quasi-long-range correlations in time and space, independent of the value of h. In particular, the system exhibits aging (or lack of time-translational invariance on intermediate time scales) although no indications of coarsening are present.

Semiclassical theory for quantum quenches in finite transverse Ising chains

We present a quantitative semi-classical theory for the non-equilibrium dynamics of transverse Ising chains after quantum quenches, in particular sudden changes of the transverse field strength. We obtain accurate predictions for the quench dependent relaxation times and correlation lengths, and also about the recurrence times and quasiperiodicity of time dependent correlations in finite systems with open or periodic boundary conditions. We compare the quantiative predictions of our semi-classical theory (local magnetization, equal time bulk-bulk and surface-to-bulk correlations, and bulk autocorrelations) with the results from exact free fermion calculations and discuss the range of applicability of the semi-classical theory and possible generaliztaions and extensions.

On reduced density matrices for disjoint subsystems

We show that spin and fermion representations for solvable quantum chains lead in general to different reduced density matrices if the subsystem is not singly connected. We study the effect for two sites in XX and XY chains as well as for sublattices in XX and transverse Ising chains.

First- and second-order phase transitions in scale-free networks.

We study first- and second-order phase transitions of ferromagnetic lattice models on scale-free networks, with a degree exponent gamma. Using the example of the q-state Potts model we derive a general self-consistency relation within the frame of the Weiss molecular-field approximation, which presumably leads to exact critical singularities. Depending on the value of gamma, we have found three different regimes of the phase diagram. As a general trend first-order transitions soften with decreasing gamma and the critical singularities at the second-order transitions are gamma dependent.

Partially asymmetric exclusion models with quenched disorder.

We consider the one-dimensional partially asymmetric exclusion process with random hopping rates, in which a fraction of particles (or sites) have a preferential jumping direction against the global drift. In this case, the accumulated distance traveled by the particles, x, scales with the time, t, as x approximately t(1/z), with a dynamical exponent z>0. Using extreme value statistics and an asymptotically exact strong disorder renormalization group method, we exactly calculate z(PW) for particlewise disorder, which is argued to be related as z(SW)=z(PW)/2 for sitewise disorder. In the symmetric case with zero mean drift, the particle diffusion is ultraslow, logarithmic in time.

Inhomogeneous systems with unusual critical behaviour

Abstract The phase transitions and critical properties of two types of inhomogeneous systems are reviewed. In one case, the local critical behaviour results from the particular shape of the system. Here scale-invariant forms like wedges or cones are considered as well as general parabolic shapes. In the other case the system contains defects, either narrow ones in the form of lines or stars, or extended ones where the couplings deviate from their bulk values according to power laws. In each case the perturbation may be irrelevant, marginal or relevant. In the marginal case one finds local exponents which depend on a parameter. In the relevant case unusual stretched exponential behaviour and/or local first-order transitions appear. The discussion combines mean field theory, scaling considerations, conformal transformations and perturbation theory. A number of examples are Ising models for which exact results can be obtained. Some walks and polymer problems are considered, too.

Entanglement Entropy at Infinite-Randomness Fixed Points in Higher Dimensions

The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong-disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite-randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite-randomness fixed point in the diluted transverse Ising model in higher dimensions.