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Complexity of patterns is key information for human brain to differ objects of about the same size and shape. Like other innate human senses, the complexity perception cannot be easily quantified. We propose a transparent and universal machine method for estimating structural (effective) complexity of two-dimensional and three-dimensional patterns that can be straightforwardly generalized onto other classes of objects. It is based on multistep renormalization of the pattern of interest and computing the overlap between neighboring renormalized layers. This way, we can define a single number characterizing the structural complexity of an object. We apply this definition to quantify complexity of various magnetic patterns and demonstrate that not only does it reflect the intuitive feeling of what is “complex” and what is “simple” but also, can be used to accurately detect different phase transitions and gain information about dynamics of nonequilibrium systems. When employed for that, the proposed scheme is much simpler and numerically cheaper than the standard methods based on computing correlation functions or using machine learning techniques.

A bstract We study the quantum phase transition of the (1+1)-dimensional O(3) nonlinear sigma model at finite density using the tensor renormalization group method. This model suffers from the sign problem, which has prevented us from investigating the properties of the phase transition. We investigate the properties of the phase transition by changing the chemical potential μ at a fixed coupling of β . We determine the transition point μ c and the critical exponent ν from the μ dependence of the number density in the thermodynamic limit. The dynamical critical exponent z is also extracted from the scaling behavior of the temporal correlation length as a function of μ .

A bstract We demonstrate that gluon transverse-momentum-dependent parton distribution functions (TMDPDFs) can be extracted from lattice calculations of appropriate Euclidean correlations in large-momentum effective theory (LaMET). Based on perturbative calculations of gluon unpolarized and helicity TMDPDFs, we present a matching formula connecting them and their LaMET counterparts, where the latter are renormalized in a scheme facilitating lattice calculations and converted to the MS ¯ scheme. The hard matching kernel is given up to one-loop level. We also show that the perturbative result is independent of the prescription used for the pinch-pole singularity in the relevant correlations. Our results offer a guidance for the extraction of gluon TMDPDFs from lattice simulations, and have the potential to greatly facilitate perturbative calculations of the hard matching kernel.

All-electron electronic structure methods based on the linear combination of atomic orbitals method with Gaussian basis set discretization offer a well established, compact representation that forms much of the foundation of modern correlated quantum chemistry calculations-on both classical and quantum computers. Despite their ability to describe essential physics with relatively few basis functions, these representations can suffer from a quartic growth of the number of integrals. Recent results have shown that, for some quantum and classical algorithms, moving to representations with diagonal two-body operators can result in dramatically lower asymptotic costs, even if the number of functions required increases significantly. We introduce a way to interpolate between the two regimes in a systematic and controllable manner, such that the number of functions is minimized while maintaining a block-diagonal structure of the two-body operator and desirable properties of an original, primitive basis. Techniques are analyzed for leveraging the structure of this new representation on quantum computers. Empirical results for hydrogen chains suggest a scaling improvement from O(N4.5) in molecular orbital representations to O(N2.6) in our representation for quantum evolution in a fault-tolerant setting, and exhibit a constant factor crossover at 15 to 20 atoms. Moreover, we test these methods using modern density matrix renormalization group methods classically, and achieve excellent accuracy with respect to the complete basis set limit with a speedup of 1-2 orders of magnitude with respect to using the primitive or Gaussian basis sets alone. These results suggest our representation provides significant cost reductions while maintaining accuracy relative to molecular orbital or strictly diagonal approaches for modest-sized systems in both classical and quantum computation for correlated systems.

A bstract We apply the joint threshold and transverse momentum dependent (TMD) factorization theorem to introduce new threshold-TMD distribution functions, including threshold-TMD parton distribution functions (PDFs) and fragmentation functions (FFs). We apply Soft-Collinear Effective Theory and renormalization group methods to carry out QCD evolution for both threshold-TMD PDFs and FFs. We show the universality of threshold-TMD functions among three standard processes, i.e. the Drell-Yan production in pp collisions, semi-inclusive deep-inelastic scattering and back-to-back two hadron production in e + e − collisions. In the end, we present the numerical predictions for different threshold-TMD functions and also transverse momentum distributions at pp , ep , and e + e − collisions.

A bstract We use renormalization group methods to study composite operators existing at a boundary of an interacting conformal field theory. In particular we relate the data on boundary operators to short-distance (near-boundary) divergences of bulk two-point functions. We further argue that in the presence of running couplings at the boundary the anomalous dimensions of certain composite operators can be computed from the relevant beta functions and remark on the implications for the boundary (pseudo) stress-energy tensor. We apply the formalism to a scalar field theory in d = 3 − 𝜖 dimensions with a quartic coupling at the boundary whose beta function we determine to the first non-trivial order. We study the operators in this theory and compute their conformal data using 𝜖 − expansion at the Wilson-Fisher fixed point of the boundary renormalization group flow. We find that the model possesses a non-zero boundary stress-energy tensor and displacement operator both with vanishing anomalous dimensions. The boundary stress tensor decouples at the fixed point in accordance with Cardy’s condition for conformal invariance. We end the main part of the paper by discussing the possible physical significance of this fixed point for various values of 𝜖.

The Haldane model is a paradigmatic 2 d lattice model exhibiting the integer quantum Hall effect. We consider an interacting version of the model, and prove that for short-range interactions, smaller than the bandwidth, the Hall conductivity is quantized, for all the values of the parameters outside two critical curves, across which the model undergoes a ‘topological’ phase transition: the Hall coefficient remains integer and constant as long as we continuously deform the parameters without crossing the curves; when this happens, the Hall coefficient jumps abruptly to a different integer. Previous works were limited to the perturbative regime, in which the interaction is much smaller than the bare gap, so they were restricted to regions far from the critical lines. The non-renormalization of the Hall conductivity arises as a consequence of lattice conservation laws and of the regularity properties of the current–current correlations. Our method provides a full construction of the critical curves, which are modified (‘dressed’) by the electron–electron interaction. The shift of the transition curves manifests itself via apparent infrared divergences in the naive perturbative series, which we resolve via renormalization group methods.

Abstract We consider the production of a pair of heavy quarksQQ̅ ̅in association with a generic colour singlet system V at lepton colliders, and present the first analytic calculation of the two-loop soft function differential in the total momentum of the real radiation. The calculation is performed by reducing the relevant Feynman integrals into a canonical basis of master integrals by means of integration-by-parts identities. The resulting integrals are then evaluated by solving a system of differential equations in the kinematic invariants, whose boundary conditions are determined analytically with some care due to the presence of Coulomb singularities. The fully differential soft function is expressed in terms of Goncharov polylogarithms. This result is an essential ingredient for a range of N3LL resummations for key collider observables at lepton colliders, such as theQQ̅ ̅Vproduction cross section at threshold and observables sensitive to the total transverse momentum of the radiation in heavy-quark final states. Moreover, it constitutes the complete final-final dipole contribution to the fully differential soft function needed for the description ofQQ̅ ̅Vproduction at hadron colliders, which plays an important role in the LHC physics programme.

A bstract We consider a QED d +1 , d = 1 , 3 lattice model with emergent Lorentz or chiral symmetry, both when the interaction is irrelevant or marginal. While the correlations present symmetry breaking corrections, we prove that the Adler-Bardeen (AB) non-renormalization property holds at a non-perturbative level even at finite lattice: all radiative corrections to the anomaly are vanishing. The analysis uses a new technique based on the combination of non-perturbative regularity properties obtained by exact renormalization Group methods and Ward Identities. The AB property, essential for the renormalizability of the standard model, is therefore a robust feature imposing no constraints on possible symmetry breaking terms, at least in the class of lattice models considered.

It is proposed that we use the term “approximation” for inexact description of a target system and “idealization” for another system whose properties also provide an inexact description of the target system. Since systems generated by a limiting process can often have quite unexpected—even inconsistent—properties, familiar limit processes used in statistical physics can fail to provide idealizations but merely provide approximations.