Renormalization Group Limit Cycle Research Articles

Three-body renormalization group limit cycles based on unsupervised feature learning

Both the three-body system and the inverse square potential carry a special significance in the study of renormalization group limit cycles. In this work, we pursue an exploratory approach and address the question which two-body interactions lead to limit cycles in the three-body system at low energies, without imposing any restrictions upon the scattering length. For this, we train a boosted ensemble of variational autoencoders, that not only provide a severe dimensionality reduction, but also allow to generate further synthetic potentials, which is an important prerequisite in order to efficiently search for limit cycles in low-dimensional latent space. We do so by applying an elitist genetic algorithm to a population of synthetic potentials that minimizes a specially defined limit-cycle-loss. The resulting fittest individuals suggest that the inverse square potential is the only two-body potential that minimizes this limit cycle loss independent of the hyperangle.

Investigation of $$\varXi ^- nn$$ ($$S=-2$$) hypernucleus in low-energy pionless halo effective theory

We study the \(\varXi ^- nn\) (\(S=-2,\,I=3/2,\,J^P={1/2}^+\)) three-body system using low-energy effective field theory (EFT). Due to the acute inadequacy of empirical information in this sector, there exists substantial degree of ambiguity in determining various few-body observables, some of which are expected to yield vital clues to resolving longstanding contentious issues in hypernuclear physics. Moreover, in astrophysical studies, a precise determination of neutron star equation of state (EoS) of putative hyperonic cores relies on essential input from the \(S=-2\) sector. In this obscure current scenario, a pionless EFT analysis provides a systematic model-independent framework for assessing the feasibility of light three-particle-stable bound states, utilizing low-energy universality. Here we take recourse to a simplistic speculation of the three-body system by eliminating the repulsive spin-singlet \(\varXi ^- n\) sub-system, while retaining the predominantly attractive (possibly bound) spin-triplet \(\varXi ^- n\) and the virtual bound spin-singlet nn sub-systems. In particular, a qualitative leading order EFT investigation by introducing a sharp momentum ultraviolet cut-off parameter \(\varLambda _{\mathrm{reg}}\) into the coupled integral equations indicates a discrete scaling behavior akin to a renormalization group limit cycle, thereby suggesting the formal existence of Efimov states in the unitary limit, as \(\varLambda _{\mathrm{reg}}\rightarrow \infty \). Our subsequent non-asymptotic analysis indicates that the three-body binding energy \(B_3\) is sensitively dependent on the cut-off without the inclusion of three-body contact interactions. Furthermore, our analysis reproduces several values of the binding energy \(B_3\sim 3{-}4\) MeV, predicted in context of existing potential models, with the regulator \(\varLambda _{\mathrm{reg}}\) in the range \(\sim 350{-}460\) MeV. Finally, based on these model inputs for \(B_3\), a ballpark estimate of the three-body scattering length in the range \(2.6{-}4.9\) fm, is naively constrained by our EFT analysis. Despite approximations, the resulting Phillips line is expected to yield a robust feature of the halo-bound \(\varXi ^- nn\) system. For pedagogical reasons, using a simple toy model interacting three-bosons system, we highlight in the appendices the typical universal features leading to emergence of RG limit cycle and Efimov states which are amenable to a low-energy EFT formalism.

Renormalization group analysis of graphene with a supercritical Coulomb impurity

We develop a field-theoretic approach to massless Dirac fermions in a supercritical Coulomb potential. By introducing an Aharonov--Bohm solenoid at the potential center, the critical Coulomb charge can be made arbitrarily small for one partial-wave sector, where a perturbative renormalization group analysis becomes possible. We show that a scattering amplitude for reflection of particle at the potential center exhibits the renormalization group limit cycle, i.e., log-periodic revolutions as a function of the scattering energy, revealing the emergence of discrete scale invariance. This outcome is further incorporated in computing the induced charge and current densities, which turn out to have power-law tails with coefficients log-periodic with respect to the distance from the potential center. Our findings are consistent with the previous prediction obtained by directly solving the Dirac equation and can in principle be realized by graphene experiments with charged impurities.

Limit Cycles from the Similarity Renormalization Group

We investigate renormalization group limit cycles within the similarity renormalization group (SRG) and discuss their signatures in the evolved interaction. A quantitative method to detect limit cycles in the interaction and to extract their period is proposed. Several SRG generators are compared regarding their suitability for this purpose. As a test case, we consider the limit cycle of the inverse square potential.

Efimov-like phase of a three-stranded DNA and the renormalization-group limit cycle.

A three-stranded DNA with short range base pairings only is known to exhibit a classical analog of the quantum Efimov effect, viz., a three-chain bound state at the two-chain melting point where no two are bound. By using a nonperturbative renormalization-group method for a rigid duplex DNA and a flexible third strand, with base pairings and strand exchange, we show that the Efimov-DNA is associated with a limit cycle type behavior of the flow of an effective three-chain interaction. The analysis also shows that thermally generated bubbles play an essential role in producing the effect. A toy model for the flow equations shows the limit cycle in an extended three-dimensional parameter space of the two-chain coupling and a complex three-chain interaction.

Renormalization Group Limit Cycle for Three-Stranded DNA

We show that there exists an Efimov-like three strand DNA bound state at the duplex melting point and it is described by a renormalization group limit cycle. A nonperturbative renormalization group is used to obtain this result in a model involving short range pairing only. Our results suggest that Efimov physics can be tested in polymeric systems.

Weakly bound molecules trapped with discrete scaling symmetries

When the scattering length is proportional to the distance from the center of the system, two particles are shown to be trapped about the center. Furthermore, their spectrum exhibits discrete scale invariance, whose scale factor is controlled by the slope of the scattering length. While this resembles the Efimov effect, our system has a number of advantages when realized with ultracold atoms. We also elucidate how the emergent discrete scaling symmetry is violated for more than two bosons, which may shed new light on Efimov physics. Our system thus serves as a tunable model system to investigate universal physics involving scale invariance, quantum anomaly, and renormalization group limit cycle, which are important in a broad range of quantum physics.

Below the Breitenlohner-Freedman bound in the nonrelativistic AdS/CFT correspondence

We propose that there is no analogue of the Breitenlohner-Freedman stability bound on the mass of a scalar field in the context of the nonrelativistic AdS/CFT correspondence. Our treatment is based on an equivalence between the field equation of a complex scalar in the AdS/CFT correspondence and the one-dimensional Schroedinger equation with an inverse square potential. We compute the two-point boundary correlation function for m^2<m_{BF}^2 and discuss its relation to renormalization group limit cycles and the Efimov effect in quantum mechanics. The equivalence also helps to elucidate holographic renormalization group flows and calculations in the global coordinates for Schroedinger spacetime.

Functional renormalization group and conformal invariance in cold atoms

We present an application of a field-theoretic functional renormalization group technique to fewbody (vacuum) physics of nonrelativistic atoms near a Feshbach resonance. We consider and compare a onecomponent bosonic system with a two-component fermionic system, focusing on the scale-free unitarity limit of an infinite scattering length. While the two-body problem is consistent with nonrelativistic conformal symmetry for the two considered systems, a conformal anomaly, manifested through a renormalization group limit cycle with the Efimov parameter s0 ≈ 1.006, is found for bosons in the three-body problem.

Few-body effects in cold atoms and limit cycles

Physical systems with a large scattering length have universal properties independent of the details of the interaction at short distances. Such systems can be realized in experiments with cold atoms close to a Feshbach resonance. They also occur in many other areas of physics such as nuclear and particle physics. The universal properties include a geometric spectrum of three-body bound states (so-called Efimov states) and log-periodic dependence of low-energy observables on the physical parameters of the system. This behavior is characteristic of a renormalization group limit cycle. We discuss universality in the three- and four-body sectors and give an overview of applications in cold atoms.

Limit cycles of effective theories

A simple example is used to show that renormalization group limit cycles of effective quantum theories can be studied in a new way. The method is based on the similarity renormalization group procedure for Hamiltonians. The example contains a logarithmic ultraviolet divergence that is generated by both real and imaginary parts of the Hamiltonian matrix elements. Discussion of the example includes a connection between asymptotic freedom with one scale of bound states and the limit cycle with an entire hierarchy of bound states.

More on the infrared renormalization group limit cycle in QCD

We present a detailed study of the recently conjectured infrared renormalization group limit cycle in QCD using chiral effective field theory. It was conjectured that small increases in the up and down quark masses can move QCD to the critical trajectory for an infrared limit cycle in the three-nucleon system. At the critical quark masses, the binding energies of the deuteron and its spin-singlet partner are tuned to zero and the triton has infinitely many excited states with an accumulation point at the three-nucleon threshold. We exemplify three parameter sets where this effect occurs at next-to-leading order in the chiral counting. For one of them, we study the structure of the three-nucleon system in detail using both chiral and contact effective field theories. Furthermore, we investigate the matching of the chiral and contact theories in the critical region and calculate the influence of the limit cycle on three-nucleon scattering observables.

On the limit cycle for the 1/ r2 potential in momentum space

The renormalization of the attractive 1/ r 2 potential has recently been studied using a variety of regulators. In particular, it was shown that renormalization with a square well in position space allows multiple solutions for the depth of the square well, including, but not requiring a renormalization group limit cycle. Here, we consider the renormalization of the 1/ r 2 potential in momentum space. We regulate the problem with a momentum cutoff and absorb the cutoff dependence using a momentum-independent counterterm potential. The strength of this counterterm is uniquely determined and runs on a limit cycle. We also calculate the bound state spectrum and scattering observables, emphasizing the manifestation of the limit cycle in these observables.

Renormalization-group limit cycle for the1∕r2potential

Previous work has shown that if an attractive $1∕{r}^{2}$ potential is regularized at short distances by a spherical square-well potential, renormalization allows multiple solutions for the depth of the square well. The depth can be chosen to be a continuous function of the short-distance cutoff $R$, but it can also be a log-periodic function of $R$ with finite discontinuities, corresponding to a renormalization-group (RG) limit cycle. We consider the regularization with a $\ensuremath{\delta}$-shell potential. In this case, the coupling constant is uniquely determined to be a log-periodic function of $R$ with infinite discontinuities, and an RG limit cycle is unavoidable. In general, a regularization with an RG limit cycle is selected as the correct renormalization of the $1∕{r}^{2}$ potential by the conditions that the cutoff radius $R$ can be made arbitrarily small and that physical observables are reproduced accurately at all energies much less than ${\ensuremath{\hbar}}^{2}∕m{R}^{2}$.

Renormalization group limit cycles and field theories for ellipticS-matrices

The renormalization group (RG) for maximally anisotropicsu(2) current interactions in two dimensions is shown to be cyclic at one-loop level. Thefermionized version of the model exhibits spin–charge separation of the four-fermioninteractions and has symmetry. It is proposed that theS-matrices for these theories are the elliptic Zamolodchikov and Mussardo–PenatiS-matrices.The S-matrix parameters are related to Lagrangian parameters by matching the period of theRG. All models exhibit two characteristic signatures of an RG limit cycle: periodicity of theS-matrix as a function of energy and the existence of an infinite number of resonance polessatisfying Russian doll scaling.

Universality, marginal operators, and limit cycles

The universality of renormalization group limit cycle behavior is illustrated with a simple discrete Hamiltonian model. A non-perturbative renormalization group equation for the model is soluble analytically at criticality and exhibits one marginal operator (made necessary by the limit cycle) and an infinite set of irrelevant operators. Relevant operators are absent. The model exhibits an infinite series of bound state energy eigenvalues. This infinite series approaches an exact geometric series as the eigenvalues approach zero - also a consequence of the limit cycle. Wegner's eigenvalues for irrelevant operators are calculated generically for all choices of parameters in the model. We show that Wegner's eigenvalues are independent of location on the limit cycle, in contrast with Wegner's operators themselves, which vary depending on their location on the limit cycle. An example is then used to illustrate numerically how one can tune the initial Hamiltonian to eliminate the first two irrelevant operators. After tuning, the Hamiltonian's bound state eigenvalues converge much more quickly than otherwise to an exact geometric series.

An infrared renormalization group limit cycle in QCD.

We use effective field theories to show that small increases in the up and down quark masses would move QCD very close to the critical renormalization group trajectory for an infrared limit cycle in the three-nucleon system. We conjecture that QCD can be tuned to the critical trajectory by adjusting the quark masses independently. At the critical values of the quark masses, the binding energies of the deuteron and its spin-singlet partner would be tuned to zero and the triton would have infinitely many excited states with an accumulation point at the 3-nucleon threshold. The ratio of the binding energies of successive states would approach a universal constant that is close to 515.

Limit cycles in quantum theories.

Renormalization group limit cycles and more chaotic behavior may be commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience based on classical models with critical behavior, where fixed points are far more common. We discuss the simplest quantum model Hamiltonian identified so far that exhibits a renormalization group with both limit cycle and chaotic behavior. The model is a discrete Hermitian matrix with two coupling constants, both governed by a nonperturbative renormalization group equation that involves changes in only one of these couplings and is soluble analytically.

Renormalization group limit cycles and first-order phase transitions in superconductors and abelian Higgs models

Renormalization group methods are used to investigate the thermodynamic behaviour of a multicomponent abelian Higgs model. Interpreted as a Ginzburg-Landau-type model for superconductivity, the model interpolates, via the replica trick, between superconductors with and without quenched random impurities. For a range of values of the numbers of complex order parameter components and of replicas, the renormalization group trajectories exhibit a stable focus, surrounded by an unstable limit cycle. By evaluating the free energy and the equation of state, we find that flows within the limit cycle correspond to a type of critical behaviour with oscillatory modulations. Outside the limit cycle, there is a region of the parameter space in which runaway of the trajectories may be interpreted in terms of a fluctuation-induced first-order phase transition, but in a third region, no clear interpretation is possible.