Fixed Point Action Research Articles

Machine learning a fixed point action for SU(3) gauge theory with a gauge equivariant convolutional neural network

Fixed point lattice actions are designed to have continuum classical properties unaffected by discretization effects and reduced lattice artifacts at the quantum level. They provide a possible way to extract continuum physics with coarser lattices, thereby allowing one to circumvent problems with critical slowing down and topological freezing toward the continuum limit. A crucial ingredient for practical applications is to find an accurate and compact parametrization of a fixed point action, since many of its properties are only implicitly defined. Here we use machine learning methods to revisit the question of how to parametrize fixed point actions. In particular, we obtain a fixed point action for four-dimensional SU(3) gauge theory using convolutional neural networks with exact gauge invariance. The large operator space allows us to find superior parametrizations compared to previous studies, a necessary first step for future Monte Carlo simulations and scaling studies. Published by the American Physical Society 2024

On the reconstruction problem in quantum gravity

Path integrals and the Wilsonian renormalization group provide two complementary computational tools for investigating continuum approaches to quantum gravity. The starting points of these constructions utilize a bare action and a fixed point of the renormalization group flow, respectively. While it is clear that there should be a connection between these ingredients, their relation is far from trivial. This results in the so-called reconstruction problem. In this work, we demonstrate that the map between these two formulations does not generate non-localities at quadratic order in the background curvature. At this level, the bare action in the path integral and the fixed-point action obtained from the Wilsonian renormalization group differ by local terms only. This conclusion does not apply to theories coming with a physical ultraviolet cutoff or a fundamental non-locality scale.

Finite cutoff CFT's and composite operators

Recently a conformally invariant action describing the Wilson-Fisher fixed point in D=4−ϵ dimensions in the presence of a finite UV cutoff was constructed [44]. In the present paper we construct two composite operator perturbations of this action with definite scaling dimension also in the presence of a finite cutoff. Thus the operator (as well as the fixed point action) is well defined at all momenta 0≤p≤∞ and at low energies they reduce to ∫xϕ2 and ∫xϕ4 respectively. The construction includes terms up to O(ϵ2). In the presence of a finite cutoff they mix with higher order ∫xϕn operators. The dimensions are also calculated to this order and agree with known results.

Slow scrambling and hidden integrability in a random rotor model

We analyze the out-of-time-order correlation functions of a solvable model of a large number, $N$, of $M$-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. We focus on the growth of commutators of operators at a temperature $T$ above the zero temperature quantum critical point separating the spin-glass and paramagnetic phases. In the large $N,~M$ limit, the squared commutators of the rotor fields do not display any exponential growth of commutators, in spite of the absence of any sharp quasiparticle-like excitations in the disorder-averaged theory. We show that in this limit, the problem is integrable and point out interesting connections to random-matrix theory. At leading order in $1/M$, there are no modifications to the critical behavior but an irrelevant term in the fixed-point action leads to a small exponential growth of the squared commutator. We also introduce and comment on a generalized model involving $p$-pair rotor interactions.

Wilson action for the O(N) model

In this paper the fixed-point Wilson action for the critical O(N) model in D=4−ϵ dimensions is written down in the ϵ expansion to order ϵ2. It is obtained by solving the fixed-point Polchinski Exact Renormalization Group equation (with anomalous dimension) in powers of ϵ. This is an example of a theory that has scale and conformal invariance despite having a finite UV cutoff. The energy-momentum tensor for this theory is also constructed (at zero momentum) to order ϵ2. This is done by solving the Ward-Takahashi identity for the fixed point action. It is verified that the trace of the energy-momentum tensor is proportional to the violation of scale invariance as given by the exact RG, i.e., the β function. The vanishing of the trace at the fixed point ensures conformal invariance. Some examples of calculations of correlation functions are also given.

Non-Lorentzian M5-brane theories from holography

M-theory on AdS7 × S4 admits a description where the AdS7 factor is constructed as a timelike Hopf fibration over a non-compact three dimensional complex projective space {tilde{mathrm{mathbb{CP}}}}^3 [1]. We consider the worldvolume theory for M5-branes at a fixed {tilde{mathrm{mathbb{CP}}}}^3 radius which, after reduction along the timelike fibre, is given by an Ω-deformed Yang-Mills theory with eight supercharges. Taking the radius to infinity then induces a classical RG flow. We construct the fixed point action which has an enhanced 24 supercharges and which can be understood as the (2, 0) theory of M5-branes on flat space reduced along a compact null Killing direction.

Non-Lorentzian RG flows and supersymmetry

We describe a general process where a non-Lorentzian rescaling of a supersymmetric field theory leads to a scale-invariant fixed point action without Lorentz invariance but where the supersymmetry is preserved or even enhanced. We apply this procedure to five-dimensional maximally supersymmetric super-Yang-Mills, leading to a field theory with 24 super(conformal) symmetries. We also apply this procedure to the BLG model with 32 super(conformal) symmetries and ABJM models with 24 super(conformal) symmetries.

On de Sitter solutions in asymptotically safe theories

The availability of scaling solutions in renormalisation group improved versions of cosmology are investigated in the high-energy limit. We adopt -type models of quantum gravity which display an interacting ultraviolet fixed point at shortest distances. Expanding the gravitational fixed point action to very high order in the curvature scalar, we detect a convergence-limiting singularity in the complex field plane. Resummation techniques including Padé approximants as well as infinite order approximations of the effective action are used to maximise the domain of validity. We find that the theory displays near de Sitter solutions as well as an anti-de Sitter solution in the UV whereas real de Sitter solutions, for small curvature, appear to be absent. The significance of our results for inflation, and implications for more general models of quantum gravity are discussed.

Nonexistence of smooth effective one fixed point actions of finite Oliver groups on low-dimensional spheres

According to the work of Laitinen, Morimoto, Oliver and Pawalowski, a finite group $G$ has a smooth effective one fixed point action on some sphere if and only if $G$ is an Oliver group. For some finite Oliver groups $G$ of order up to $216$, and for $G=A_5\times C_n$ for $n=3,5,7$, we present a strategy of excluding of smooth effective one fixed point $G$-actions on low-dimensional spheres.

Most standard spheres have smooth one fixed point actions ofA5. II

Most standard spheres have smooth one fixed point actions ofA5. II

Asymptotic freedom in Hořava-Lifshitz gravity.

We use the Wetterich equation for foliated spacetimes to study the renormalization group flow of projectable Hořava-Lifshitz gravity coupled to n Lifshitz scalars. Using novel results for anisotropic heat kernels, the matter-induced beta functions for the gravitational couplings are computed explicitly. The renormalization group flow exhibits an UV attractive anisotropic Gaussian fixed point where Newton's constant vanishes and the extra scalar mode decouples. This fixed point ensures that the theory is asymptotically free in the large-n expansion, indicating that projectable Hořava-Lifshitz gravity is perturbatively renormalizable. Notably, the fundamental fixed point action does not obey detailed balance.

Asymptotic safety: A simple example

We use the Gross-Neveu model in 2<d<4 as a simple fermionic example for Weinberg's asymptotic safety scenario: despite being perturbatively nonrenormalizable, the model defines an interacting quantum field theory being valid to arbitrarily high momentum scales owing to the existence of a non-Gaussian fixed point. Using the functional renormalization group, we study the UV behavior of the model in both the purely fermionic as well as a partially bosonized language. We show that asymptotic safety is realized at non-Gaussian fixed points in both formulations, the universal critical exponents of which we determine quantitatively. The partially bosonized formulation allows to make contact to the large-Nf expansion where the model is known to be renormalizable to all-orders. In this limit, the fixed-point action as well as all universal critical exponents can be computed analytically. As asymptotic safety has become an important scenario for quantizing gravity, our description of a well-understood model is meant to provide for an easily accessible and controllable example of modern nonperturbative quantum field theory.

New overlap construction of Weyl fermions on the lattice

In a recent article Hasenfratz and von Allmen have suggested a fixed point action for two flavors of Weyl fermions on the lattice with gauge group SU(2). The block-spin transformation they use maps the chiral and vector symmetries of the underlying vector theory onto two equations of the Ginsparg–Wilson (GW) type. We show that an overlap Dirac operator can be constructed which solves both GW equations simultaneously. We discuss the properties of this overlap operator and its projection onto lattice Weyl fermions which seems to be free of artefacts, in particular the projection operators are independent of the gauge field.

Simulating full QCD with the fixed point action

Because of its complex structure the parametrized fixed point action can not be simulated with the available local updating algorithms. We constructed, coded, and tested an updating procedure with $2+1$ light flavors, where the targeted $s$ quark mass is at its physical value while the $u$ and $d$ quarks should produce pions lighter than 300 MeV. In the algorithm a partially global gauge update is followed by several accept/reject steps, where parts of the determinant are switched on gradually in the order of their costs. The trial configuration that is offered in the last, most expensive, stochastic accept/reject step differs from the original configuration by a Metropolis $+$ over-relaxation gauge update over a subvolume of $\ensuremath{\sim}(1.3\text{ }\text{ }\mathrm{fm}{)}^{4}$. The acceptance rate in this accept/reject step is $\ensuremath{\sim}0.4$. The code is optimized on different architectures and is running on lattices with ${L}_{s}\ensuremath{\approx}1.2\text{ }\text{ }\mathrm{fm}$ and 1.8 fm at a resolution of $a\ensuremath{\approx}0.15\text{ }\text{ }\mathrm{fm}$.

The dimension of spheres with smooth one fixed point actions

The article proves that there are smooth one fixed point actions of the alternating group of degree 5 on the 8-dimensional sphere. It follows that a sphere has a smooth one fixed point action of some finite group if and only if the dimension of the sphere is greater than or equal to 6.

New results on cut-off effects in spectroscopy with the fixed point action

Our study on the cut-off effects in quenched light hadron spectroscopy and pion scattering length with the fixed point action is extended by results obtained at a lattice spacing a=0.102 fm in a box of size L=1.8 fm. The cut-off effects are small, but clearly seen as the resolution is increased from a=0.153 fm to a=0.102 fm. In the quark mass region where the errors are small and under control, our results on the APE plot lie close to the extrapolated numbers of the CP-PACS Collaboration.

I=2 pion scattering length with the parametrized fixed point action

We report on the pion-pion scattering length in the I=2 channel using the parametrized fixed point action. Pion masses of 320 MeV were reached in this quenched calculation of the scattering length.

First results from a parametrized Fixed-Point QCD action

We have constructed a new fermion action which is an approximation to the (chirally symmetric) Fixed-Point action, containing the full Clifford algebra with couplings inside a hypercube and paths built from renormalization group inspired fat links. We present an exploratory study of the light hadron spectrum and the energy-momentum dispersion relation.

Classically perfect gauge actions on anisotropic lattices

We present a method for constructing classically perfect anisotropic actions for SU(3) gauge theory based on an isotropic Fixed Point Action. The action is parametrised using smeared (“fat”) links. The construction is done explicitly for anisotropy ξ= a s / a t =2 and 4. The corresponding renormalised anisotropies are determined using the torelon dispersion relation. The renormalisation of the anisotropy is small and the parametrisation describes the true action well. Quantities such as the static quark–antiquark potential, the critical temperature of the deconfining phase transition and the low-lying glueball spectrum are measured on lattices with anisotropy ξ=2. The mass of the scalar 0 ++ glueball is determined to be 1580(60) MeV, while the tensor 2 ++ glueball is at 2430(60) MeV.

CPN-1 Models with a Term and Fixed Point Action

The topological charge distribution P(Q) is calculated for lattice ${\rm CP}^{N-1}$ models. In order to suppress lattice cut-off effects we employ a fixed point (FP) action. Through transformation of P(Q) we calculate the free energy $F(\theta)$ as a function of the $\theta$ parameter. For N=4, scaling behavior is observed for P(Q), $F(\theta)$ as well as the correlation lengths $\xi(Q)$. For N=2, however, scaling behavior is not observed as expected. For comparison, we also make a calculation for the ${\rm CP}^{3}$ model with standard action. We furthermore pay special attention to the behavior of P(Q) in order to investigate the dynamics of instantons. For that purpose, we carefully look at behavior of $\gamma_{\it eff}$, which is an effective power of P(Q)($\sim \exp(-CQ^{\gamma_{\it eff}})$), and reflects the local behavior of P(Q) as a function of Q. We study $\gamma_{\it eff}$ for two cases, one of which is the dilute gas approximation based on the Poisson distribution of instantons and the other is the Debye-H\"uckel approximation of instanton quarks. In both cases we find similar behavior to the one observed in numerical simulations.