CRITICAL BEHAVIOR AND THRESHOLD OF COEXISTENCE OF A PREDATOR-PREY STOCHASTIC MODEL IN A 2D LATTICE

We investigate the critical behavior of a stochastic lattice model describing a predator-prey system. By means of Monte Carlo procedure we simulate the model defined on a regular square lattice and determine the threshold of species coexistence, that is, the critical phase boundaries related to the transition between an active state, where both species coexist and an absorbing state where one of the species is extinct. A finite size scaling analysis is employed to determine the order parameter, order parameter fluctuations, correlation length and the critical exponents. Our numerical results for the critical exponents agree with those of the directed percolation universality class. We also check the validity of the hyperscaling relation and present the data collapse curves.

Critical behavior of a contact process with aperiodic transition rates

We performed Monte Carlo simulations to investigate the steady-state critical behavior of a one-dimensional contact process with an aperiodic distribution of rates of transition. As in the presence of randomness, spatial fluctuations can lead to changes of critical behavior. For sufficiently weak fluctuations, we give numerical evidence to show that there is no departure from the universal critical behavior of the underlying uniform model. For strong spatial fluctuations, the analysis of the data indicates a change of critical universality class.

Critical behavior of the dilute antiferromagnet in a magnetic field

We study the critical behavior of the diluted antiferromagnet in a field with the tethered Monte Carlo formalism. We compute the critical exponents (including the elusive hyperscaling violations exponent θ). Our results provide a comprehensive description of the phase transition and clarify the inconsistencies between previous experimental and theoretical work. To do so, our method addresses the usual problems of numerical work (large tunneling barriers and self-averaging violations).

Critical behavior of three-dimensional disordered Potts models with many states

We study the 3D Disordered Potts Model with p = 5 and p = 6. Our numerical simulations (that severely slow down for increasing p) detect a very clear spin glass phase transition. We evaluate the critical exponents and the critical value of the temperature, and we use known results at lower p values to discuss how they evolve for increasing p. We do not find any sign of the presence of a transition to a ferromagnetic regime.

Critical behavior of one-particle spectral weights in the transverse Ising model

We investigate the critical behavior of the spectral weight of a single quasiparticle, one of the key observables in experiment, for the particular case of the transverse Ising model. Series expansions are calculated for the linear chain and the square and simple cubic lattices. For the chain model, a conjectured exact result is discovered. For the square and simple cubic lattices, series analyses are used to estimate the critical exponents. The results agree with the general predictions of Sachdev [Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 1999)].

Positive solutions of a Neumann problem with competing critical nonlinearities

We present existence results for a Neumann problem involving critical Sobolev nonlinearities both on the right hand side of the equation and at the boundary condition.. Positive solutions are obtained through constrained minimization on the Nehari manifold. Our approach is based on the concentration 'compactness principle of P. L. Lions and M. Struwe.

Nonequilibrium critical scaling from quantum thermodynamics

The emerging field of quantum thermodynamics is contributing important results and insights into archetypal many-body problems, including quantum phase transitions. Still, the question whether out-of-equilibrium quantities, such as fluctuations of work, exhibit critical scaling after a sudden quench in a closed system has remained elusive. Here, we take a novel approach to the problem by studying a quench across an impurity quantum critical point. By performing density matrix renormalization group computations on the two-impurity Kondo model, we are able to establish that the irreversible work produced in a quench exhibits finite-size scaling at quantum criticality. This scaling faithfully predicts the equilibrium critical exponents for the crossover length and the order parameter of the model, and, moreover, implies a new exponent for the rescaled irreversible work. By connecting the irreversible work to the two-impurity spin correlation function, our findings can be tested experimentally.

(Quantum-) critical dynamics of magnetoelectric materials

In this thesis the critical dynamics of several magnetoelectric compounds at their phase transition were examined. Mostly measurements of the dielectric properties in the frequency range of below 1 Hz up to 5 GHz were employed to evaluate the critical exponents for both magnetic field and temperature-dependent measurements. Most of the materials that are part of this work show anomalous behavior, especially at very low temperatures where quantum fluctuations are of the order of or even dominate those induced thermally. This anomalous behavior manifests in different forms. In Dy2Ti2O7 we demonstrate the existence of electric dipoles on magnetic monopoles. Here the dynamics at the critical endpoint located at 0.36K and in a magnetic field of 1T parallel to the [111] direction are of special interest. At this critical endpoint the expected critical slowing down of the dynamics could not only not be observed but instead the opposite, critical speeding-up by several orders of magnitude, could be demonstrated. Furthermore, we show that the phase diagram of Dy2Ti2O7 in this field direction can be reproduced solely from the dynamical properties, for example the resonance frequency of the observed relaxation that is connected to the monopole movement. Away from this point of the phase diagram the dynamics are slowing-down with reduction of temperature as one would expect. Additional measurements on Y2Ti2O7, a structurally identical but non-magnetic material, show only slowing down with reduction of temperature and no additional features. A possible explanation for the observed critical speeding-up is a coherent movement of magnetic monopoles close to the critical field that increases the resonance frequency by reducing the damping of the process. LiCuVO4 on the other hand behaves normally at its phase transition as long as the temperature is higher than 0.4 K. In this temperature regime the dynamics show critical slowing-down analogous to classical ferroelectric materials. This analogy extends also towards higher frequencies where the permittivity displays a ‘dispersion’ minimum that is temperature-dependent but of the order of 2 GHz. Below 0.4K the observed behavior changes drastically. Here we found no longer relaxational behavior but instead an excitation with very low energy. This low energy excitation was predicted by theory and is caused by nearly gapless soliton excitations within the 1D Cu2+ chains of LiCuVO4. Finally, in TbMnO3 the dynamics of the phase transition into the multiferroic phase was observed at roughly 27 K, a much higher temperature compared to the other materials. Here the expected critical slowing-down was observed, even though in low-frequency measurements this transition into the ferroelectric phase is overshadowed by the so-called c-axis relaxation. Therefore, only frequencies above 1MHz could be used to determine the critical exponents for both temperatureand magnetic-field-dependent measurements. This was done for both the peak frequency as well as the relaxation strength. In TbMnO3 an electromagnetic soft-mode with small optical weight causes the observed fluctuations, similar to the case of multiferroic MnWO4.

Critical behavior of the three-dimensional Ising spin glass

We have simulated, using parallel tempering, the three-dimensional Ising spin glass model with binary couplings in a helicoidal geometry. The largest lattice (L520) has been studied using a dedicated computer (the SUE machine). We have obtained, measuring the correlation length in the critical region, strong evidence for a second-order finite-temperature phase transition, ruling out other possible scenarios like a KosterlitzThouless phase transition. Precise values for the ν and ƞ critical exponents are also presented.

Non-Fermi-liquid theory for disordered metals near two dimensions

We consider the Finkelstein action describing a system of spin-polarized or spinless electrons in 2+2epsilon dimensions, in the presence of disorder as well as the Coulomb interactions. We extend the renormalization-group analysis of our previous work and evaluate the metal-insulator transition of the electron gas to second order in an epsilon expansion. We obtain the complete scaling behavior of physical observables like the conductivity and the specific heat with varying frequency, temperature, and/or electron density. We extend the results for the interacting electron gas in 2+2epsilon dimensions to include the quantum critical behavior of the plateau transitions in the quantum Hall regime. Although these transitions have a very different microscopic origin and are controlled by a topological term in the action (theta term), the quantum critical behavior is in many ways the same in both cases. We show that the two independent critical exponents of the quantum Hall plateau transitions, previously denoted as nu and p, control not only the scaling behavior of the conductances sigma(xx) and sigma(xy) at finite temperatures T, but also the non-Fermi-liquid behavior of the specific heat (c(v)proportional toT(p)). To extract the numerical values of nu and p it is necessary to extend the experiments on transport to include the specific heat of the electron gas.

Phase diagram of a two-species lattice model with a linear instability

We discuss the properties of a one-dimensional lattice model of a driven system with two species of particles in which the mobility of one species depends on the density of the other. This model was introduced by Lahiri and Ramaswamy (Phys. Rev. Lett., 79, 1150 (1997)) in the context of sedimenting colloidal crystals, and its continuum version was shown to exhibit an instability arising from linear gradient couplings. In this paper we review recent progress in understanding the full phase diagram of the model. There are three phases. In the first, the steady state can be determined exactly along a representative locus using the condition of detailed balance. The system shows phase separation of an exceptionally robust sort, termed strong phase separation, which survives at all temperatures. The second phase arises in the threshold case where the first species evolves independently of the second, but the fluctuations of the first influence the evolution of the second, as in the passive scalar problem. The second species then shows phase separation of a delicate sort, in which long-range order coexists with fluctuations which do not damp down in the large-size limit. This fluctuation-dominated phase ordering is associated with power law decays in cluster size distributions and a breakdown of the Porod law. The third phase is one with a uniform overall density, and along a representative locus the steady state is shown to have product measure form. Density fluctuations are transported by two kinematic waves, each involving both species and coupled at the nonlinear level. Their dissipation properties are governed by the symmetries of these couplings, which depend on the overall densities. In the most interesting case,, the dissipation of the two modes is characterized by different critical exponents, despite the nonlinear coupling.

Ferromagnetic transition and specific heat of Pr(0.6)Sr(0.4)MnO(3)

The critical properties of orthorhombic Pr(0.6)Sr(0.4)MnO(3) single crystals were investigated by a series of static magnetization measurements along the three different crystallographic axes as well as by specific heat measurements. A careful range-of-fitting-analysis of the magnetization and susceptibility data obtained from the modified Arrott plots shows that Pr(0.6)Sr(0.4)MnO(3) has a very narrow critical regime. Nevertheless, the system belongs to the three-dimensional (3D) Heisenberg universality class with short-range exchange. The critical exponents obey Widom scaling and are in excellent agreement with the single scaling equation of state M(H,epsilon) = vertical bar epsilon vertical bar(beta) f(+/-)(H/vertical bar epsilon vertical bar((beta+gamma)); with f(+) for T > T(c) and f(-) for T < T(c). A detailed analysis of the specific heat that account for all relevant contributions allows us to extract and analyze the contribution related to the magnetic phase transition. The specific heat indicates the presence of a linear electronic term at low temperatures and a prominent contribution from crystal field excitations of Pr. A comparison with data from literature for PrMnO(3) shows that a Pr-Mn magnetic exchange is responsible for a sizable shift in the lowest lying excitation.

Quantum fidelity for one-dimensional Dirac fermions and two-dimensional Kitaev model in the thermodynamic limit

We study the scaling behavior of the fidelity (F) in the thermodynamic limit using the examples of a system of Dirac fermions in one dimension and the Kitaev model on a honeycomb lattice. We show that the thermodynamic fidelity inside the gapless as well as gapped phases follow power-law scalings, with the power given by some of the critical exponents of the system. The generic scaling forms of F for an anisotropic quantum critical point for both the thermodynamic and nonthermodynamic limits have been derived and verified for the Kitaev model. The interesting scaling behavior of F inside the gapless phase of the Kitaev model is also discussed. Finally, we consider a rotation of each spin in the Kitaev model around the z axis and calculate F through the overlap between the ground states for the angle of rotation eta and eta + d eta, respectively. We thereby show that the associated geometric phase vanishes. We have supplemented our analytical calculations with numerical simulations wherever necessary.

Quantum fidelity for one-dimensional Dirac fermions and two-dimensional Kitaev model in the thermodynamic limit

We study the scaling behavior of the fidelity (F) in the thermodynamic limit using the examples of a system of Dirac fermions in one dimension and the Kitaev model on a honeycomb lattice.We show that the thermodynamic fidelity inside the gapless as well as gapped phases follow power-law scalings, with the power given by some of the critical exponents of the system. The generic scaling forms of F for an anisotropic quantum critical point for both the thermodynamic and nonthermodynamic limits have been derived and verified for the Kitaev model. The interesting scaling behavior of F inside the gapless phase of the Kitaev model is also discussed. Finally, we consider a rotation of each spin in the Kitaev model around the z axis and calculate F through the overlap between the ground states for the angle of rotation η and η + dη, respectively. We thereby show that the associated geometric phase vanishes. We have supplemented our analytical calculations with numerical simulations wherever necessary

Continuous phase transition in mixed-valent manganite Nd0.6Pb0.4MnO3

The critical behaviour has been investigated in single crystalline Nd0.6Pb0.4MnO3 near the paramagnetic to ferromagnetic transition temperature (TC) by static magnetic measurements. The values of TC and the critical exponents β, γ and δ are estimated by analysing the data in the critical region. The exponent values are very close to those expected for 3D Heisenberg ferromagnets with short-range interactions. Specific heat measurements show a broad cusp at TC (i.e., exponent α<0) being consistent with Heisenberg-like behaviour.