Using the singular-mode functional renormalization group method, we studied the doping-dependent phase diagram of the two-dimensional extended Hubbard model on square lattice. At half-filling, a variety of states strongly compete, including spin-density wave, charge-density wave, phase separation, as well asd-wave ands-wave superconductivity (SC). Upon electron doping, the system exhibits a strong preference for superconducting states. At slight dopings, the spin-density wave phase is strongly suppressed, andd-wave SC region expands into positiveU,Vregion. At moderate dopings, a region of degeneratedp-wave superconductivities emerge for positiveUand negativeV. Our results demonstrate how doping tunes the dominant fluctuations, suppressing density waves and stabilizing unconventional superconducting states in two dimensions.

Electronic nematicity, the spontaneous breaking of rotational symmetry, has emerged as a key instability in correlated quantum systems. CsTi3Bi5, a kagome metal of the AV3Sb5 (A = K, Rb, Cs) family, hosts rich unconventional electronic phases, yet the origin of its nematicity remains unsettled. Here, we combine polarization-dependent angle-resolved photoemission spectroscopy with functional renormalization group calculations on a fully interacting ab initio model. We reveal an orbital-selective nematic deformation in the low-energy band structure and identify a finite angular momentum (d-wave) Pomeranchuk instability driven by electronic correlations in specific orbital channels and detuning from Van Hove singularities. Our results establish a direct link between orbital selectivity and symmetry-breaking instabilities in CsTi3Bi5, providing a microscopic framework for nematic order in kagome systems.

The recent observation of superconductivity in the vicinity of Fermi surface reconstructed insulating or metallic states has established twisted bilayers of WSe 2 as an exciting platform to study the interplay of strong electron-electron interactions, broken symmetries, and topology. In this work, we use a first-principles, material-specific theoretical treatment that is unbiased with respect to electronic instabilities to study the emergence of electronic ordering in twisted WSe 2 driven by gate-screened Coulomb interactions. We construct exponentially localized moiré Wannier orbitals that faithfully capture the band structure and topology of the system, project the gate-screened Coulomb interaction onto them, and use unbiased functional renormalization group techniques to resolve the momentum and orbital structure of the leading instabilities and the relevant energy scales. We find an interplay between intervalley-coherent antiferromagnetic (IVC-AFM) order and chiral, mixed-parity d / p -wave superconductivity for carrier concentrations near a displacement field and twist-angle-tunable van Hove singularity. Our microscopic approach establishes incommensurate IVC-AFM spin fluctuations as the dominant electronic mechanism driving the formation of superconductivity in θ = 5.08 ° twisted WSe 2 and explains key aspects of recent experiments including the asymmetric density dependence of the spin ordering with respect to the van Hove line, the single- and double-peak structure of the DOS in the ordered (hole-doped) IVC-AFM phase, the emergence of superconductivity as the density is varied across the van Hove line, and the evolution of the displacement field-density phase diagram with twist angles between 3.7 ° … 5 ° . We show that the interplay of electronic correlations and nontrivial quantum geometry in tWSe 2 manifests in orbital-selective order parameters associated to the IVC-AFM and SC states that are detectable by local spectroscopy measurements.

We analyse the critical properties of a weakly diluted (random) Ising model with the long-range interaction decaying with distance x as ∼ x - d - σ in a d-dimensional space. It is known to belong to a new long-range random universality class for certain values of the decay parameter σ. Exploiting the field-theoretic renormalization group approach within the minimal subtraction scheme, we compute the three-loop renormalization group functions. On their basis, with the help of asymptotic series resummation methods, we estimate the correlation length critical exponent ν characterising the new universality class for d = 3 and for those values of σ for which long-range interactions are relevant for the critical behaviour.

Abstract We investigate the effect of an isospin chemical potential ($\mu_{I}$) within the quark-meson model, which approximates quantum chromodynamics (QCD) by modeling low energy phenomena such as chiral symmetry breaking and phase structure under varying conditions of temperature and chemical potential. Using the functional renormalization group (FRG) flow equations, we calculate the phase diagram in the chiral limit within the two-flavor quark-meson model in a finite $\mu_{I}$ with $\rho$ vector meson interactions.
Fluctuation effects significantly decrease the critical chemical potential from the mean-field (MF) value $\mu_{I, MF} > m_\rho$ to lower value, at which point the $\rho$ vector meson condensates alongside the chiral condensate once the isospin chemical potential exceeds the critical value $\mu_{I}^{\text{crit}}$.
This $\rho$ condensation is investigated numerically for different meson coupling strengths. The $\rho$ meson dominated region is delineated from other phases by a second-order phase transition at lower $\mu_{I}$ and a first-order transition at slightly higher $\mu_{I}$. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.

Abstract We investigate the QCD chiral phase transition at finite temperature and finite baryon density using the functional Renormalization Group (fRG). While conventional fRG studies often employ techniques such as dynamical bosonization to regularize divergences, we instead pursue the weak solution of the fRG equations which allows for non-analytic behavior in the flow to compute the pure fermionic potential $$V_k(\psi ,\bar{\psi })$$ V k ( ψ , ψ ¯ ) within the local potential approximation. This approach enables us to explore the effects of purely quark-level fluctuations on dynamical chiral symmetry breaking without introducing any auxiliary bosonic fields. Based on this framework, we present the resulting chiral phase diagram as a function of temperature and baryon chemical potential.

Scaling behaviors in active model B+ via the functional renormalization group

Altermagnetism, a recently proposed and experimentally confirmed class of magnetic order, features collinear compensated magnetism with unconventional spin-split bands. Here, we show that in a metallic 2D d -wave altermagnet with [ C 2 | | C 4 ] symmetry, secondary instabilities can arise. Using an unbiased functional renormalization group approach, we analyze the weak-coupling instabilities of a 2D Hubbard model with a preexisting altermagnetic order inspired by our electronic structure calculations of realistic material candidates from V 2 X 2 O ( X = Te, Se) family. We identify two distinct spin-density-wave (SDW) states that break the underlying altermagnetic [ C 2 | | C 4 ] symmetry. Additionally, we find spin-fluctuation-induced instabilities leading to a singlet d -wave superconducting state and an unconventional commensurate pair-density-wave (PDW) state with extended s -wave and spin-triplet symmetry. We analyze the pairing mechanism and characterize the excitation spectrum, which exhibits Bogoliubov Fermi surfaces or nodal points depending on the gap size.

Abstract The Kardar–Parisi–Zhang (KPZ) equation is a celebrated non-linear stochastic dynamical equation yielding non-equilibrium universal scaling. It exhibits notorious non-perturbative aspects. The KPZ fixed point is strong-coupling, all the more in d > 1. Strikingly, another, even stronger-coupling fixed point of the KPZ equation, called inviscid Burgers fixed point, has been recently unveiled. These non-pertubative features can be theoretically accessed and studied in a controlled way in all dimensions using the functional renormalisation group. We propose an overview of the related results, which provide a unified picture of the fixed-point structure and associated scaling regimes of the KPZ equation in d = 1 and in higher dimensions.

Using the recent six loop renormalization group functions for Lee-Yang and percolation theory constructed by Schnetz from a scalar cubic Lagrangian, we deduce the ε expansion of the critical exponents for both cases. Estimates for the exponents in three, four, and five dimensions are extracted using two-sided Padé approximants and shown to be compatible with values from other approaches.

Functional renormalization group analysis of the quark-condensation pattern on the Fermi surface: A simple effective-model approach

We study the large-order behavior of the functional renormalization group (FRG). For a model in dimension zero, we establish Borel summability for a large class of microscopic couplings. Writing the derivatives of FRG as contour integrals, we express the Borel transform as well as the original series as integrals. Taking the strong-coupling limit in this representation, we show that all short-ranged microscopic disorders flow to the same universal fixed point. Our results are relevant for FRG in disordered elastic systems.

Functional renormalization group meets computational fluid dynamics: RG flows in a multidimensional field space

Abstract The Kardar–Parisi–Zhang (KPZ) equation is a celebrated non-linear stochastic equation featuring non-equilibrium scaling. Although its statistical properties are well understood in one dimension, a new scaling regime has been reported in recent numerical simulations. This new regime is characterised by a dynamical exponent z = 1, markedly different from the expected one z = 3 / 2 for the KPZ universality class, and it emerges when approaching the inviscid limit. The origin of this scaling has been traced back to the existence of a new fixed point, termed the inviscid Burgers (IB) fixed point, which was uncovered using the functional renormalisation group (FRG). The FRG equations can be solved analytically in the asymptotic regime of vanishing viscosity and large momenta, showing that indeed z = 1 exactly at the IB fixed point. In this work, we set up an advanced method to numerically solve the full FRG flow equations in a certain approximation, which allows us to determine in a unified way the correlation function over the whole range of momenta, not restricted to some particular regime. We analyse the crossover between the different fixed points, and quantitatively determine the extent of the IB regime.

Abstract The gradient flow exact renormalization group (GFERG) is a variant of the exact renormalization group of gauge theory that aims to preserve gauge symmetry as manifestly as possible. From an integral representation of the Wilson action in GFERG for the Yang–Mills theory, we explicitly compute the one-loop renormalization group functions that reproduce the correct coefficients. From the correspondence with the gradient flow formalism by Lüscher and Weisz (J. High Energy Phys. 1102, 051 (2011)), we also argue that GFERG reproduces the conventional renormalization group functions in all orders of perturbation theory.

A bstract We analyze the matrix model characterizing the Ising model coupled to Causal Dynamical Triangulations (CDT) from the point of view of the Functional Renormalization Group Equation (FRGE). This model is a dually weighted matrix model, whose Feynman diagrams are in correspondence with discrete triangulations of two-dimensional geometries with a preferred time foliation. In particular, we find the fixed points of the beta-function equations, showing that the number of relevant directions in each case is compatible with the physical interpretation of the CFT living on the fixed points. In addition to recovering the fixed points for topological gravity and pure gravity with a cosmological constant, we find a new fixed point featuring three relevant directions which match the number of primary fields in the Ising CFT.

In the context of gravity the Lagrangian and Hamiltonian formalisms have been developed largely independently, emphasizing renormalization and quantization, respectively. The formalisms use a different methodology to distinguish between gauge and physical degrees of freedom. In this review we analyze the connection between the Asymptotically Safe and Canonical Quantum Gravity approaches. Based on the Hamiltonian formulation, the Canonical Quantum Gravity approach inherently provides a natural framework for defining observables. This serves as the foundation for constructing the generating functional of the n-point correlation functions of physical degrees of freedom. By means of background-independent, non-perturbative renormalization methods well-established in the Lagrangian framework and typically employed in Asymptotic Safety, the resulting generating functional can be handled. In particular, we employ the Functional Renormalization Group to regularize the path integral and to compute the flow connecting the bare theory in the ultraviolet with the effective infrared theory. An important advantage of this approach is that it establishes an explicit, systematic relation between the quantization procedure and the systematics of quantum field theory-based renormalization group methods. More importantly, this synthesis not only bridges canonical and covariant approaches but also paves the way for a consistent and predictive quantum theory of gravity grounded in physically meaningful, gauge-invariant observables.

We study the porous medium equation(PME) in one space dimension in the presence of additive nonconservative white noise and interpreted as a stochastic growth equationfor the height field of an interface. We predict the values of the two growth exponents α and β using the functional renormalization group. Extensive numerical simulations show agreement with the predicted values for these exponents; however, they also show anomalous scaling with an additional local exponent α_{loc} as well as multiscaling originating from broad distributions of local height differences. The stationary measure of the stochastic PME is found to be well described by a random walk model related to a Bessel process. This model allows for several predictions about the multiscaling properties.

In this paper, we develop an approach to calculate the valence-quark quasiparton distribution amplitude (quasi-PDA) and quasiparton distribution function (quasi-PDF) for the pion with a large longitudinal momentum with the functional renormalization group (fRG). This is demonstrated in a low energy effective theory (LEFT) with four-quark scatterings. In the study of the complex structure of quasi-PDA, we introduce a deformed integration contour in the calculations of quasi-PDA or quasi-PDF, which allows us to obtain correct integrals for all momentum fractions. It is found that the pion light-front PDA extrapolated from quasi-PDA based on the large momentum effective theory in the LEFT is comparable with lattice QCD and Dyson-Schwinger equation. This work paves the way to study the PDA and PDF within the fRG approach to first-principles QCD.

We study the full infrared dynamics of 2+1 flavor quantum chromodynamics (QCD) with the functional renormalization group approach. We resolve self-consistently the glue dynamics as well as the dynamics of chiral symmetry breaking. The computation hosts no phenomenological parameter or external input. The only ultraviolet input parameters are the physical ones in QCD: the light and strange quark masses. They are adjusted to the physical ratios of the pion and kaon masses, divided by the pion decay constant. The results for other observables of current first-principles computations are in quantitative agreement with the physical ones. This work completes the series of papers, initiated and furthered in [; ], on dynamical chiral symmetry breaking and the emergence of mesonic bound states within the functional renormalization group. As a first application we discuss the formation of light mesonic bound states. Among other applications such as the phase structure of QCD, the current work paves the way for studying QCD parton distribution functions within the functional renormalization group approach to first-principles QCD.