Renormalization Group Trajectories Research Articles

Highq-state clock spin glasses in three dimensions and the Lyapunov exponents of chaotic phases and chaotic phase boundaries

Spin-glass phases and phase transitions for $q$-state clock models and their $q\ensuremath{\rightarrow}\ensuremath{\infty}$ limit the $XY$ model, in spatial dimension $d=3$, are studied by a detailed renormalization-group study that is exact for the $d=3$ hierarchical lattice and approximate for the cubic lattice. In addition to the now well-established chaotic rescaling behavior of the spin-glass phase, each of the two types of spin-glass phase boundaries displays, under renormalization-group trajectories, their own distinctive chaotic behavior. These chaotic renormalization-group trajectories subdivide into two categories, namely as strong-coupling chaos (in the spin-glass phase and, distinctly, on the spin-glass--ferromagnetic phase boundary) and as intermediate-coupling chaos (on the spin-glass--paramagnetic phase boundary). We thus characterize each different phase and phase boundary exhibiting chaos by its distinct Lyapunov exponent, which we calculate. We show that, under renormalization group, chaotic trajectories and fixed distributions are mechanistically and quantitatively equivalent. The phase diagrams of arbitrary even $q$-state clock spin-glass models in $d=3$ are calculated. These models, for all non-infinite $q$, have a finite-temperature spin-glass phase. Furthermore, the spin-glass phases exhibit a universal ordering behavior, independent of $q$. The spin-glass phases and the spin-glass--paramagnetic phase boundaries exhibit universal fixed distributions, chaotic trajectories and Lyapunov exponents. In the $XY$ model limit, our calculations indicate a zero-temperature spin-glass phase.

Universality and symmetry breaking in conformally reduced quantum gravity

The scaling properties of quantum gravity are discussed by employing a class of proper-time regulators in the functional flow equation for the conformal factor within the formalism of the background field method. Renormalization group trajectories obtained by projecting the flow on a flat topology are more stable than those obtained from a projection on a spherical topology. In the latter case the ultraviolet flow can be characterized by a Hopf bifurcation with an ultraviolet attractive limiting cycle. Although the possibility of determining the infrared flow for an extended theory space can be severely hampered due to the conformal factor instability, we present a robust numerical approach to study the flow structure around the non-gaussian fixed point as an inverse-problem strategy. In particular it is shown the possibility of having a spontaneous breaking of the diffeomorphism invariance can be realized with non-local functionals of the volume operator.

COMMENTS ON SCALE INVARIANT BUT NONCONFORMAL SUPERSYMMETRIC FIELD THEORIES

We investigate a possibility of scale invariant but nonconformal supersymmetric field theories from a perturbative approach. The explicit existence of monotonically decreasing a-function that generates beta-functions as a gradient flow provides a strong obstruction for such a possibility at two-loop order. We comment on the "discovery" of scale invariant but nonconformal renormalization group trajectories via a "change of scheme" in (4-ϵ) dimension proposed in literatures.

Scale without conformal invariance at three loops

We carry out a three-loop computation that establishes the existence of scale without conformal invariance in dimensional regularization with the MS scheme in d=4-epsilon spacetime dimensions. We also comment on the effects of scheme changes in theories with many couplings, as well as in theories that live on non-conformal scale-invariant renormalization group trajectories. Stability properties of such trajectories are analyzed, revealing both attractive and repulsive directions in a specific example. We explain how our results are in accord with those of Jack & Osborn on a c-theorem in d=4 (and d=4-epsilon) dimensions. Finally, we point out that limit cycles with turning points are unlike limit cycles with continuous scale invariance.

Running boundary actions, Asymptotic Safety, and black hole thermodynamics

Previous explorations of the Asymptotic Safety scenario in Quantum Einstein Gravity (QEG) by means of the effective average action and its associated functional renormalization group (RG) equation assumed spacetime manifolds which have no boundaries. Here we take a first step towards a generalization for non-trivial boundaries, restricting ourselves to action functionals which are at most of second order in the derivatives acting on the metric. We analyze two examples of truncated actions with running boundary terms: full fledged QEG within the single-metric Einstein-Hilbert (EH) truncation, augmented by a scale dependent Gibbons-Hawking (GH) surface term, and a bi-metric truncation for gravity coupled to scalar matter fields. The latter contains 17 running couplings, related to both bulk and boundary terms, whose beta-functions are computed in the induced gravity approximation. We find that the bulk and the boundary Newton constant, pertaining to the EH and GH term, respectively, show opposite RG running; proposing a scale dependent variant of the ADM mass we argue that the running of both couplings is consistent with gravitational anti-screening. We describe a simple device for counting the number of field modes integrated out between the infrared cutoff scale and the ultraviolet. This method makes it manifest that, in an asymptotically safe theory, there are effectively no field modes integrated out while the RG trajectory stays in the scaling regime of the underlying fixed point. As an application, we investigate how the semiclassical theory of Black Hole Thermodynamics gets modified by quantum gravity effects and compare the new picture to older work on `RG-improved black holes' which incorporated the running of the bulk Newton constant only. We find, for instance, that the black hole's entropy vanishes and its specific heat capacity turns positive at Planckian scales.

Scale without conformal invariance: An example

We give an explicit example of a model in D = 4 − ϵ space–time dimensions that is scale but not conformally invariant, is unitary, and has finite correlators. The invariance is associated with a limit cycle renormalization group (RG) trajectory. We also prove, to second order in the loop expansion, in D = 4 − ϵ , that scale implies conformal invariance for models of any number of real scalars. For models with one real scalar and any number of Weyl spinors we show that scale implies conformal invariance to all orders in perturbation theory.

Evolution of cosmological perturbations in a renormalization-group-driven inflationary scenario

A gauge-invariant, linear cosmological perturbation theory of an almost homogeneous and isotropic universe with dynamically evolving Newton constant G and cosmological constant $\Lambda$ is presented. The equations governing the evolution of the comoving fractional spatial gradients of the matter density, G and $\Lambda$ are thus obtained. Explicit solutions are discussed in cosmologies, featuring an accelerated expansion, where both G and $\Lambda$ vary according to renormalization group equations in the vicinity of an ultraviolet fixed point. Finally, a similar analysis is carried out in the late universe regime described by the part of the renormalization group trajectory close to the gaussian fixed point.

Renormalization group functional equations

Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories, and to gain insight into the interplay between continuous and discrete rescaling. With minimal assumptions, the methods produce continuous flows from step-scaling {\sigma} functions, and lead to exact functional relations for the local flow {\beta} functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of {\sigma} are sometimes not true fixed points under continuous changes in scale, and zeroes of {\beta} do not necessarily signal fixed points of the flow, but instead may only indicate turning points of the trajectories.

Interacting fermions on the honeycomb bilayer: From weak to strong coupling

Many-body instabilities of the half-filled honeycomb bilayer are studied using weak-coupling renormalization group (RG) as well as strong-coupling expansion. For spinless fermions and assuming parabolic degeneracy, there are four independent four-fermion contact couplings. While the dominant instability depends on the microscopic values of the couplings, the broken symmetry state is typically a gapped insulator with either broken inversion symmetry or broken time-reversal symmetry, with a quantized anomalous Hall effect. Under certain conditions, the dominant instability may appear in the particle-particle (pairing) channel. For some nongeneric fine-tuned initial conditions, weak-coupling RG trajectories flow into the noninteracting fixed point, although generally we find runaway flows which we associate with ordering tendencies. Additionally, a tight-binding model with nearest-neighbor hopping and nearest-neighbor repulsion is studied in weak and strong couplings and in each regime a gapped phase with inversion symmetry breaking is found. In the strong-coupling limit, the ground-state wave function is constructed for vanishing in-plane hopping but finite interplane hopping, which explicitly displays the broken inversion symmetry and a finite difference between the number of particles on the two layers. Finally, we discuss the spin-1/2 case and use Fierz identities to show that the number of independent four-fermion contact couplings is 9. The corresponding RG equations in the spin-1/2 case are also presented, and used to show that, just as in strong coupling, the most dominant weak-coupling instability of the repulsive Hubbard model (at half filling) is an antiferromagnet.

Correlated spin networks in frustrated systems

We introduce a network model for frustrated spin systems based on highly correlated spin fluctuations, to quantify and visualize their ordering. This model shows that networks of strongly correlated but non-contiguous spins exist at low temperatures on a triangular Ising lattice with competing nearest-neighbor interactions. This finding is consistent with chaotic renormalization-group trajectories previously reported for frustrated hierarchical lattices.

A short review on entanglement in quantum spin systems

We review some of the recent progress on the study of entropy of entanglement in many-body quantum systems. Emphasis is placed on the scaling properties of entropy for one-dimensional multi-partite models at quantum phase transitions and, more generally, on the concept of area law. We also briefly describe the relation between entanglement and the presence of impurities, the idea of particle entanglement, the evolution of entanglement along renormalization group trajectories, the dynamical evolution of entanglement and the fate of entanglement along a quantum computation.

Conformal sector of quantum Einstein gravity in the local potential approximation: Non-Gaussian fixed point and a phase of unbroken diffeomorphism invariance

We explore the nonperturbative renormalization group flow of quantum Einstein gravity (QEG) on an infinite dimensional theory space. We consider ``conformally reduced'' gravity where only fluctuations of the conformal factor are quantized and employ the local potential approximation for its effective average action. The requirement of ``background independence'' in quantum gravity entails a partial differential equation governing the scale dependence of the potential for the conformal factor which differs significantly from that of a scalar matter field. In the infinite dimensional space of potential functions we find a Gaussian as well as a non-Gaussian fixed point which provides further evidence for the viability of the asymptotic safety scenario. The analog of the invariant cubic in the curvature which spoils perturbative renormalizability is seen to be unproblematic for the asymptotic safety of the conformally reduced theory. The scaling fields and dimensions of both fixed points are obtained explicitly and possible implications for the predictivity of the theory are discussed. Spacetime manifolds with ${R}^{d}$ as well as ${S}^{d}$ topology are considered. Solving the flow equation for the potential numerically we obtain examples of renormalization group trajectories inside the ultraviolet critical surface of the non-Gaussian fixed point. The quantum theories based upon some of them show a phase transition from the familiar (low energy) phase of gravity with spontaneously broken diffeomorphism invariance to a new phase of unbroken diffeomorphism invariance; the latter phase is characterized by a vanishing expectation value of the metric.

Bare action and regularized functional integral of asymptotically safe quantum gravity

Investigations of quantum Einstein gravity (QEG) based upon the effective average action employ a flow equation which does not contain any ultraviolet (UV) regulator. Its renormalization group trajectories emanating from a non-Gaussian fixed point define asymptotically safe quantum field theories. A priori these theories are, somewhat unusually, given in terms of their effective rather than bare action. In this paper we construct a functional integral representation of these theories. We fix a regularized measure and show that every trajectory of effective average actions, depending on an IR cutoff only, induces an associated trajectory of bare actions which depend on a UV cutoff. Together with the regularized measure these bare actions give rise to a functional integral which reproduces the prescribed effective action when the UV cutoff is removed. In this way we are able to reconstruct the underlying microscopic (classical) system and identify its fundamental degrees of freedom and interactions. The bare action of the Einstein-Hilbert truncation is computed and its flow is analyzed as an example. Various conceptual issues related to the completion of the asymptotic safety program are discussed.

The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice ℤ3

We consider an Euclidean supersymmetric field theory in ℤ3 given by a supersymmetric Φ4 perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green’s function of a (stable) Lévy random walk in ℤ3. The Green’s function depends on the Lévy-Khintchine parameter $\alpha ={3+\varepsilon \over 2}$ with 0<α<2. For $\alpha ={3\over 2}$ the Φ4 interaction is marginal. We prove for $\alpha -{3\over 2}={\varepsilon \over 2}>0$ sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green’s function of a (weakly) self-avoiding Lévy walk in ℤ3 is a second moment (two point correlation function) of the supersymmetric measure governing this model. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in ℤ3.

Occam’s razor on surfaces: renormalization of microscopic processes

We describe a methodology for the expression of atomistic models of fluctuating interfacesas continuum equations. We begin with formally exact Langevin equations based on theatomistic transition rules, which are regularized to produce stochastic partial differentialequations. Subsequent coarse graining is accomplished by calculating renormalization-group(RG) trajectories from initial conditions determined by the regularized equations. The RGanalysis shows that the morphological manifestation of a given atomistic relaxationmechanism can depend on the length scales and timescales considered as well as on thedimensionality of the fluctuating interface. Even complex surface processes, after amoderate degree of coarse graining, are reduced to low-order stochastic partialdifferential equations. We illustrate these ideas with a model of a growing surfaceunder the competition between the deposition of new material and the subsequentrelaxation through surface diffusion. We conclude with an augmentation of ourdifferential equations with a pinning term to account for lattice effects in theearly stages of growth, where surface electron diffraction oscillations indicatelayer-by-layer growth. The calculation of submonolayer morphology, which is composed ofseparated monolayer islands, provides an illustration of the efficacy of our method.

Renormalization of stochastic lattice models: Epitaxial surfaces

We present the application of a method [C. A. Haselwandter and D. D. Vvedensky, Phys. Rev. E 76, 041115 (2007)] for deriving stochastic partial differential equations from atomistic processes to the morphological evolution of epitaxial surfaces driven by the deposition of new material. Although formally identical to the one-dimensional (1D) systems considered previously, our methodology presents substantial additional technical issues when applied to two-dimensional (2D) surfaces. Once these are addressed, subsequent coarse-graining is accomplished as before by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic models. Our applications are to the Edwards-Wilkinson (EW) model [S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London, Ser. A 381, 17 (1982)], the Wolf-Villain (WV) model [D. E. Wolf and J. Villain, Europhys. Lett. 13, 389 (1990)], and a model with concurrent random deposition and surface diffusion. With our rules for the EW model no appreciable crossover is obtained for either 1D or 2D substrates. For the 1D WV model, discussed previously, our analysis reproduces the crossover sequence known from kinetic Monte Carlo (KMC) simulations, but for the 2D WV model, we find a transition from smooth to unstable growth under repeated coarse-graining. Concurrent surface diffusion does not change this behavior, but can lead to extended transient regimes with kinetic roughening. This provides an explanation of recent experiments on Ge(001) with the intriguing conclusion that the same relaxation mechanism responsible for ordered structures during the early stages of growth also produces an instability at longer times that leads to epitaxial breakdown. The RG trajectories calculated for concurrent random deposition and surface diffusion reproduce the crossover sequences observed with KMC simulations for all values of the model parameters, and asymptotically always approach the fixed point corresponding to the equation proposed by Villain [J. Phys. I 1, 19 (1991)] and by Lai and Das Sarma [Phys. Rev. Lett. 66, 2899 (1991)]. We conclude with a discussion of the application of our methodology to other growth settings.

TRANSIENT SCALING OF THE KINETIC SPHERICAL MODEL

The dynamical equation for the kinetic spherical model of a system of spins that is initially in an arbitrary equilibrium state of a given temperature and an external magnetic field is solved. The properties of the non-equilibrium process which ensues when the temperature and field are suddenly changed are studied. The scaling hypotheses by introducing three characteristic functions are generalized. It is found that the process is scale invariant in the low temperature phase while finite-scale invariant in the high temperature phase. The characteristic functions governing the renormalization group trajectories are obtained. The diagram of renormalization group flow reveals a scaling violating point at 3Tc.

Basins of attraction for renormalization-group trajectories arising in a spin-1 Ising model with competing biquadratic interactions

The current investigation probes frustration that arises in a Blume-Emery-Griffiths system due to competing biquadratic (+/-K) interactions while uniform bilinear and crystal-field interactions are maintained. This competition directly affects the clustering and density of nonmagnetic impurities. All calculations have been conducted using a hierarchical lattice in conjunction with renormalization-group methods. Phase diagrams have been calculated for a series of planes of constant biquadratic coupling while varying the temperature and concentration of annealed vacancies in the system. Each phase diagram reveals three qualitatively unique basins of attraction, each sharing renormalization-group trajectories that flow to common sinks.

Sensitivity of nonrenormalizable trajectories to the bare scale

Working in scalar field theory, we consider renormalization group trajectories which correspond to nonrenormalizable theories, in the Wilsonian sense. An interesting question to ask of such trajectories is, given some fixed starting point in parameter space, how the effective action at the effective scale, Λ, changes as the bare scale (and hence the duration of the flow down to Λ) is changed. When the effective action satisfies Polchinski's version of the exact renormalization group equation, we prove, directly from the path integral, that the dependence of the effective action on the bare scale, keeping the interaction part of the bare action fixed, is given by an equation of the same form as the Polchinski equation but with a kernel of the opposite sign. We then investigate whether similar equations exist for various generalizations of the Polchinski equation. Using nonperturbative, diagrammatic arguments we find that an action can always be constructed which satisfies the Polchinski-like equation under variation of the bare scale. For the family of flow equations in which the field is renormalized, but the blocking functional is the simplest allowed, this action is essentially identified with the effective action at Λ = 0. This does not seem to hold for more elaborate generalizations.

A renormalization-group study of an Ising spin-glass with annealed vacancies

Systems driven and characterized by fluctuations in density and magnetization can be realistically modeled using the Blume–Emery–Griffiths model; a spin-1 Ising model with bilinear, biquadratic, and crystal-field interactions. In this study, renormalization-group techniques are used on an exactly solvable system in which frustration is present due to competing ferromagnetic and antiferromagnetic interactions. Thus, this calculation models a spin-glass system with annealed vacancies. To determine the effects of these competing bilinear interactions, an exactly solvable frustrated hierarchical model has been constructed, similar to those introduced to study spin glasses [S.R. McKay, A.N. Berker, S. Kirkpatrick, Phys. Rev. Lett. 48 (1982) 767]. Phase diagrams have been calculated for a series of planes of constant biquadratic coupling while varying the temperature and concentration of annealed vacancies in the system. In addition, a phase diagram was produced for constant concentration of annealed vacancies as the biquadratic coupling (i.e. clustering bias) was varied. Each phase diagram reveals three qualitatively unique basins of attraction, each corresponding to a phase distinguished by a unique renormalization-group trajectory. The sink of each trajectory is interpreted to determine the nature of each phase: dense paramagnetic, dilute paramagnetic and spin-glass.