Contributor info: Prof. Slava Rychkov

Marten Reehorst, Slava Rychkov, Benoit Sirois, Balt C. van Rees

SciPost Phys. 18, 060 (2025) · published 19 February 2025 | Toggle abstract · pdf

We study multiscalar theories with $\text{O}(N) × \text{O}(2)$ symmetry. These models have a stable fixed point in $d$ dimensions if $N$ is greater than some critical value $N_c(d)$. Previous estimates of this critical value from perturbative and non-perturbative renormalization group methods have produced mutually incompatible results. We use numerical conformal bootstrap methods to constrain $N_c(d)$ for $3 ≤ d < 4$. Our results show that $N_c> 3.78$ for $d = 3$. This favors the scenario that the physically relevant models with $N = 2,3$ in $d=3$ do not have a stable fixed point, indicating a first-order transition. Our result exemplifies how conformal windows can be rigorously constrained with modern numerical bootstrap algorithms.

Junchen Rong, Slava Rychkov

SciPost Phys. 16, 040 (2024) · published 1 February 2024 | Toggle abstract · pdf

Classifying perturbative fixed points near upper critical dimensions plays an important role in understanding the space of conformal field theories and critical phases of matter. In this work, we consider perturbative fixed points of N=5 scalar bosons coupled with quartic interactions preserving an arbitrary subgroup $G\subset O(5)$. We perform an exhaustive algorithmic search over the symmetry groups G which are irreducible and satisfy the Landau condition, so that the fixed point can be reached by fine-tuning a single mass term and there is no need to tune the cubic couplings. We also impose stability of the RG flow in the space of quartic couplings, and reality. We thus prove that there exist no new stable fixed points in d=4-$\epsilon$ dimensions beyond the two known ones: namely the O(5) nvariant fixed point and the Cubic(5) fixed point. This work is a continuation of the classification of such fixed points with N=4 scalars by Toledano, Michel, Toledano and Brézin in 1985 [Phys. Rev. B 31, 7171 (1985)].

Bing-Xin Lao, Slava Rychkov

SciPost Phys. 15, 243 (2023) · published 18 December 2023 | Toggle abstract · pdf

We consider the transverse field Ising model in (2+1)D, putting 12 spins at the vertices of the regular icosahedron. The model is tiny by the exact diagonalization standards, and breaks rotation invariance. Yet we show that it allows a meaningful comparison to the 3D Ising CFT on $\mathbb{R}× S^2$, by including effective perturbations of the CFT Hamiltonian with a handful of local operators. This extreme example shows the power of conformal perturbation theory in understanding finite N effects in models on regularized $S^2$. Its ideal arena of application should be the recently proposed models of fuzzy sphere regularization.

Marten Reehorst, Slava Rychkov, David Simmons-Duffin, Benoit Sirois, Ning Su, Balt van Rees

SciPost Phys. 11, 072 (2021) · published 28 September 2021 | Toggle abstract · pdf

Current numerical conformal bootstrap techniques carve out islands in theory space by repeatedly checking whether points are allowed or excluded. We propose a new method for searching theory space that replaces the binary information “allowed”/“excluded” with a continuous “navigator” function that is negative in the allowed region and positive in the excluded region. Such a navigator function allows one to efficiently explore high-dimensional parameter spaces and smoothly sail towards any islands they may contain. The specific functions we introduce have several attractive features: they are well-defined in large regions of parameter space, can be computed with standard methods, and evaluation of their gradient is immediate due to an SDP gradient formula that we provide. The latter property allows for the use of efficient quasi-Newton optimization methods, which we illustrate by navigating towards the 3d Ising island.

Slava Rychkov, Andreas Stergiou

SciPost Phys. 6, 008 (2019) · published 17 January 2019 | Toggle abstract · pdf

Fixed points of scalar field theories with quartic interactions in $d=4-\varepsilon$ dimensions are considered in full generality. For such theories it is known that there exists a scalar function $A$ of the couplings through which the leading-order beta-function can be expressed as a gradient. It is here proved that the fixed-point value of $A$ is bounded from below by a simple expression linear in the dimension of the vector order parameter, $N$. Saturation of the bound requires a marginal deformation, and is shown to arise when fixed points with the same global symmetry coincide in coupling space. Several general results about scalar CFTs are discussed, and a review of known fixed points is given.

Victor Gorbenko, Slava Rychkov, Bernardo Zan

SciPost Phys. 5, 050 (2018) · published 22 November 2018 | Toggle abstract · pdf

We study complex CFTs describing fixed points of the two-dimensional $Q$-state Potts model with $Q>4$. Their existence is closely related to the weak first-order phase transition and walking RG behavior present in the real Potts model at $Q>4$. The Potts model, apart from its own significance, serves as an ideal playground for testing this very general relation. Cluster formulation provides nonperturbative definition for a continuous range of parameter $Q$, while Coulomb gas description and connection to minimal models provide some conformal data of the complex CFTs. We use one and two-loop conformal perturbation theory around complex CFTs to compute various properties of the real walking RG flow. These properties, such as drifting scaling dimensions, appear to be common features of the QFTs with walking RG flows, and can serve as a smoking gun for detecting walking in Monte Carlo simulations. The complex CFTs discussed in this work are perfectly well defined, and can in principle be seen in Monte Carlo simulations with complexified coupling constants. In particular, we predict a pair of $S_5$-symmetric complex CFTs with central charges $c\approx 1.138 \pm 0.021 i$ describing the fixed points of a 5-state dilute Potts model with complexified temperature and vacancy fugacity.

Slava Rychkov, David Simmons-Duffin, Bernardo Zan

SciPost Phys. 2, 001 (2017) · published 10 February 2017 | Toggle abstract · pdf

We discuss the 4pt function of the critical 3d Ising model, extracted from recent conformal bootstrap results. We focus on the non-gaussianity Q - the ratio of the 4pt function to its gaussian part given by three Wick contractions. This ratio reveals significant non-gaussianity of the critical fluctuations. The bootstrap results are consistent with a rigorous inequality due to Lebowitz and Aizenman, which limits Q to lie between 1/3 and 1.