Renormalization Group: Applications in Statistical Physics

These notes provide a concise introduction to important applications of the renormalization group (RG) in statistical physics. After reviewing the scaling approach and Ginzburg-Landau theory for critical phenomena, Wilson's momentum shell RG method is presented, and the critical exponents for the scalar Phi^4 model are determined to first order in an eps expansion about d_c = 4. Subsequently, the technically more versatile field-theoretic formulation of the perturbational RG for static critical phenomena is described. It is explained how the emergence of scale invariance connects UV divergences to IR singularities, and the RG equation is employed to compute the critical exponents for the O(n)-symmetric Landau-Ginzburg-Wilson theory. The second part is devoted to field theory representations of non-linear stochastic dynamical systems, and the application of RG tools to critical dynamics. Dynamic critical phenomena in systems near equilibrium are efficiently captured through Langevin equations, and their mapping onto the Janssen-De Dominicis response functional, exemplified by the purely relaxational models with non-conserved (model A) / conserved order parameter (model B). The Langevin description and scaling exponents for isotropic ferromagnets (model J) and for driven diffusive non-equilibrium systems are also discussed. Finally, an outlook is presented to scale-invariant phenomena and non-equilibrium phase transitions in interacting particle systems. It is shown how the stochastic master equation associated with chemical reactions or population dynamics models can be mapped onto imaginary-time, non-Hermitian `quantum' mechanics. In the continuum limit, this Doi-Peliti Hamiltonian is represented through a coherent-state path integral, which allows an RG analysis of diffusion-limited annihilation processes and phase transitions from active to inactive, absorbing states.Comment: 28 pages; 49th Schladming Theoretical Physics Winter School lecture notes; to appear in Nucl. Phys. B Proc. Suppl. (2012

Conservation Comes of Age

published or submitted for publicatio

Scale invariance and dynamic phase transitions in diffusion-limited reactions

Many systems that can be described in terms of diffusion-limited `chemical' reactions display non-equilibrium continuous transitions separating active from inactive, absorbing states, where stochastic fluctuations cease entirely. Their critical properties can be analyzed via a path-integral representation of the corresponding classical master equation, and the dynamical renormalization group. An overview over the ensuing universality classes in single-species processes is given, and generalizations to reactions with multiple particle species are discussed as well. The generic case is represented by the processes A A + A, and A -> 0, which map onto Reggeon field theory with the critical exponents of directed percolation (DP). For branching and annihilating random walks (BARW) A -> (m+1) A and A + A -> 0, the mean-field rate equation predicts an active state only. Yet BARW with odd m display a DP transition for d <= 2. For even offspring number m, the particle number parity is conserved locally. Below d_c' = 4/3, this leads to the emergence of an inactive phase that is characterized by the power laws of the pair annihilation process. The critical exponents at the transition are those of the `parity-conserving' (PC) universality class. For local processes without memory, competing pair or triplet annihilation and fission reactions k A -> (k - l) A, k A -> (k+m)A with k=2,3 appear to yield the only other universality classes not described by mean-field theory. In these reactions, site occupation number restrictions play a crucial role.Comment: 16 pages, talk given at 2003 German Physical Society Spring Meeting; four figures and style files include

Population oscillations in spatial stochastic Lotka-Volterra models: A field-theoretic perturbational analysis

Field theory tools are applied to analytically study fluctuation and correlation effects in spatially extended stochastic predator-prey systems. In the mean-field rate equation approximation, the classic Lotka-Volterra model is characterized by neutral cycles in phase space, describing undamped oscillations for both predator and prey populations. In contrast, Monte Carlo simulations for stochastic two-species predator-prey reaction systems on regular lattices display complex spatio-temporal structures associated with persistent erratic population oscillations. The Doi-Peliti path integral representation of the master equation for stochastic particle interaction models is utilized to arrive at a field theory action for spatial Lotka-Volterra models in the continuum limit. In the species coexistence phase, a perturbation expansion with respect to the nonlinear predation rate is employed to demonstrate that spatial degrees of freedom and stochastic noise induce instabilities toward structure formation, and to compute the fluctuation corrections for the oscillation frequency and diffusion coefficient. The drastic downward renormalization of the frequency and the enhanced diffusivity are in excellent qualitative agreement with Monte Carlo simulation data.Comment: 36 pages, 7 figures included, to appear in J. Phys. A: Math. Theor. 45 (2012

Topological edge states in two-gap unitary systems: A transfer matrix approach

We construct and investigate a family of two-band unitary systems living on a cylinder geometry and presenting localized edge states. Using the transfer matrix formalism, we solve and investigate in details such states in the thermodynamic limit. Analitycity considerations then suggest the construction of a family of Riemman surfaces associated to the band structure of the system. In this picture, the corresponding edge states naturally wind around non contractile loops, defining by the way a topological invariant associated to each gap of the system.Comment: Accepted version for publication in New Journal of Physic

Bulk-Edge correspondence for two-dimensional Floquet topological insulators

Floquet topological insulators describe independent electrons on a lattice driven out of equilibrium by a time-periodic Hamiltonian, beyond the usual adiabatic approximation. In dimension two such systems are characterized by integer-valued topological indices associated to the unitary propagator, alternatively in the bulk or at the edge of a sample. In this paper we give new definitions of the two indices, relying neither on translation invariance nor on averaging, and show that they are equal. In particular weak disorder and defects are intrinsically taken into account. Finally indices can be defined when two driven sample are placed next to one another either in space or in time, and then shown to be equal. The edge index is interpreted as a quantized pumping occurring at the interface with an effective vacuum.Comment: 28 pages, 5 figures Minor changes, update and addition of some references To appear in Annales Henri Poincar\'

Trajectory characteristics and heating of hypervelocity projectiles having large ballistic coefficients

A simple, approximate equation describing the velocity-density relationship (or velocity-altitude) has been derived from the flight of large ballistic coefficient projectiles launched at high speeds. The calculations obtained by using the approximate equation compared well with results for numerical integrations of the exact equations of motion. The flightpath equation was used to parametrically calculate maximum body decelerations and stagnation pressures for initial velocities from 2 to 6 km/s. Expressions were derived for the stagnation-point convective heating rates and total heat loads. The stagnation-point heating was parametrically calculated for a nonablating wall and an ablating carbon surface. Although the heating rates were very high, the pulse decayed quickly. The total nose-region heat shield weight was conservatively estimated to be only about 1 percent of the body mass

Environmental vs demographic variability in stochastic predator-prey models

In contrast to the neutral population cycles of the deterministic mean-field Lotka--Volterra rate equations, including spatial structure and stochastic noise in models for predator-prey interactions yields complex spatio-temporal structures associated with long-lived erratic population oscillations. Environmental variability in the form of quenched spatial randomness in the predation rates results in more localized activity patches. Population fluctuations in rare favorable regions in turn cause a remarkable increase in the asymptotic densities of both predators and prey. Very intriguing features are found when variable interaction rates are affixed to individual particles rather than lattice sites. Stochastic dynamics with demographic variability in conjunction with inheritable predation efficiencies generate non-trivial time evolution for the predation rate distributions, yet with overall essentially neutral optimization.Comment: 28 pages, 10 figures, Proceedings paper of the STATPHYS 25 conferenc

Perturbative Field-Theoretical Renormalization Group Approach to Driven-Dissipative Bose-Einstein Criticality

The universal critical behavior of the driven-dissipative non-equilibrium Bose-Einstein condensation transition is investigated employing the field-theoretical renormalization group method. Such criticality may be realized in broad ranges of driven open systems on the interface of quantum optics and many-body physics, from exciton-polariton condensates to cold atomic gases. The starting point is a noisy and dissipative Gross-Pitaevski equation corresponding to a complex valued Landau-Ginzburg functional, which captures the near critical non-equilibrium dynamics, and generalizes Model A for classical relaxational dynamics with non-conserved order parameter. We confirm and further develop the physical picture previously established by means of a functional renormalization group study of this system. Complementing this earlier numerical analysis, we analytically compute the static and dynamical critical exponents at the condensation transition to lowest non-trivial order in the dimensional epsilon expansion about the upper critical dimension d_c = 4, and establish the emergence of a novel universal scaling exponent associated with the non-equilibrium drive. We also discuss the corresponding situation for a conserved order parameter field, i.e., (sub-)diffusive Model B with complex coefficients.Comment: 17 pages, 6 figures, to appear in Phys. Rev. X (2014