Dario Benedetti

A bstract: The multicritical generalizations of the Lee-Yang universality class arise as renormalization-group fixed points of scalar field theories with complex iφ ^{2 n +1} interaction, n ∈ ℕ, just below their upper critical dimension. It has been recently conjectured that their continuation to two dimensions corresponds to the non-unitary conformal minimal models ℳ(2 , 2 n + 3). Motivated by that, we revisit the functional renormalization group approach to complex 𝒫𝒯-symmetric scalar field theories in the Local Potential Approximation, without or with wavefunction renormalization (LPA and LPA’ respectively), aiming to explore the fate of the iφ ^{2 n +1} theories from their upper critical dimension to two dimensions. The iφ ^{2 n +1} fixed points are identified using a perturbative expansion of the functional fixed-point equation near their upper critical dimensions, and they are followed to lower dimensions by numerical integration of the full equation. A peculiar feature of the complex 𝒫𝒯-symmetric potentials is that the fixed points are characterized by real but negative anomalous dimensions η , and in low dimension d , this can lead to a change of sign of the scaling dimensions ∆ = ( d − 2 + η ) / 2, thus requiring a novel analysis of the analytical properties of the functional fixed-point equations. We are able to follow the Lee-Yang universality class ( n = 1) down to two dimensions, and numerically determine the scaling dimension of the fundamental field as a function of d . On the other hand, within the LPA’, multicritical Lee-Yang fixed points with n > 1 cannot be continued to d = 2 due to the existence of unexpected non-perturbative fixed points that annihilate with the iφ ^{2 n +1} fixed points.

We investigate the one-dimensional Ising model with long-range interactions decaying as 1/r^{1+s} 1 / r 1 + s . In the critical regime, for 1/2 ≤ s ≤ 1 1 / 2 ≤ s ≤ 1 , this system realizes a family of nontrivial one-dimensional conformal field theories (CFTs), whose data vary continuously with s s . For s;gt1 s > 1 the model has instead no phase transition at finite temperature, as in the short-range case. In the standard field-theoretic description, involving a generalized free field with quartic interactions, the critical model is weakly coupled near s=1/2 s = 1 / 2 but strongly coupled in the vicinity of the short-range crossover at s=1 s = 1 . We introduce a dual formulation that becomes weakly coupled as s \to 1 s → 1 . Precisely at s=1 s = 1 , the dual description becomes an exactly solvable conformal boundary condition of the two-dimensional free scalar. We present a detailed study of the dual model and demonstrate its effectiveness by computing perturbatively the CFT data near s=1 s = 1 , up to next-to-next-to-leading order in 1-s 1 − s , by two independent approaches: (i) standard renormalization of our dual field-theoretic description and (ii) the analytic conformal bootstrap. The two methods yield complete agreement.

A bstract: We study a class of nonlocal conformal field theories in two dimensions which are obtained as deformations of the Virasoro minimal models. The construction proceeds by coupling a relevant primary operator ϕ _r,s of the m -th minimal model to a generalized free field, in such a way that the interaction term has scaling dimension 2 − δ . Flowing to the infrared, we reach a new class of CFTs that we call long-range minimal models . In the case r = s = 2, the resulting line of fixed points, parametrized by δ , can be studied using two perturbative expansions with different regimes of validity, one near the mean-field theory end, and one close to the long-range to short-range crossover. This is due to a straightforward generalization of an infrared duality which was proposed for the long-range Ising model ( m = 3) in 2017. We find that the large- m limit is problematic in both perturbative regimes, hence nonperturbative methods will be required in the intermediate range for all values of m . For the models based on ϕ _1,2 , the situation is rather different. In this case, only one perturbative expansion is known but it is well behaved at large m . We confirm this with a computation of infinitely many anomalous dimensions at two loops. Their large- m limits are obtained from both numerical extrapolations and a method we develop which carries out conformal perturbation theory using Mellin amplitudes. For minimal models, these can be accessed from the Coulomb gas representations of the correlators. This method reveals analytic expressions for some integrals in conformal perturbation theory which were previously only known numerically.

We study the 1D Ising model with long-range interactions decaying as 1 / r 1 + s . The critical model corresponds to a family of 1D conformal field theories whose data depend nontrivially on s in the range 1 / 2 ≤ s ≤ 1 . The model is known to be described by a generalized free field with quartic interaction, which is weakly coupled near s = 1 / 2 but strongly coupled near the short-range crossover at s = 1 . We propose a dual description that becomes weakly coupled at s = 1 . At s = 1 , our model becomes an exactly solvable conformal boundary condition for the 2D free scalar. We perform a number of consistency checks of our proposal and calculate the perturbative conformal field theory data around s = 1 analytically using both (1) our proposed field theory and (2) the analytic conformal bootstrap. Our results show complete agreement between the two methods. Published by the American Physical Society 2025

We correct the computation of one Feynman diagram in the three-loop beta functions for the long-range quartic multi-scalar model, originally presented in Benedetti et al (2020 J. Phys. A: Math. Theor. 53 445008) (arXiv:2007.04603). The correction requires the use of a different method than in the original paper, and we give here full details about the method. We then report the updated numerics for critical exponents of the Ising model, vector model, cubic model and bifundamental model. Mathematica files for the numerical evaluation of the corrected diagram are provided in ancillary.

Quantum long-range models at zero temperature can be described by fractional Lifshitz field theories, that is, anisotropic models whose actions are short range in time and long range in space. In this paper, we study the renormalization of fractional Lifshitz field theories with weakly relevant cubic or quartic self-interactions. Their nontrivial infrared fixed points exhibit Lifshitz scale invariance and we compute the lowest-order corrections to the dynamic critical exponent.