Systematic approaches of numerical calculations to quantum critical phenomena

Vladislav Chertenkov, Щур Лев Николаевич

Abstract Supervised machine learning is successfully applied to the study of critical phenomena and allows us to obtain a numerical estimate of the phase transition temperature and the correlation length exponent. We discuss the influence of possible systematic errors, as well as statistical errors, on the accuracy of such numerical estimates. Errors in the training and testing phases are considered separately. It is important to note that the analysis of functions obtained based on network predictions in the testing phase uses finite-size corrections, which also affect the accuracy of the estimation results.

L. De Cesare, Dimo I. Uzunov

Masuo Suzuki, Makoto Katori

A new method is proposed to study critical phenomena, which is a unification of Weiss' mean-field theory and Fisher's finite-size scaling theory. A systematic series of clusters is considered and generalized mean-fields are applied to the boundary spins of such clusters. From the size-dependence of the coefficients of mean-field singularities systematically obtained, the static critical exponents α, β and γ, and critical slowing down exponent Δ can be estimated. The critical point T C can be also evaluated from the size-dependence of mean-field critical points obtained in a systematic series of clusters. Some applications to the two- and three-dimensional Ising models are given explicitly to obtain T C =(2.24(0.036)) J / k B and γ=1.7(0.05) in the two-dimensional square lattice in our preliminary calculations.

Matthias Vojta C, R. Bulla, Fabian Güttge +1 more

We discuss a particular source of error in the numerical renormalization group (NRG) method for quantum impurity problems, which is related to a renormalization of impurity parameters due to the bath propagator. At any step of the NRG calculation, this renormalization is only partially taken into account, leading to systematic variation in the impurity parameters along the flow. This effect can cause qualitatively incorrect results when studying quantum-critical phenomena, as it leads to an implicit variation in the phase transition's control parameter as function of the temperature and thus to an unphysical temperature dependence of the order-parameter mass. We demonstrate the mass-flow effect for bosonic impurity models with a power-law bath spectrum, $J(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$, namely, the dissipative harmonic oscillator and the spin-boson model. We propose an extension of the NRG to correct the mass-flow error. Using this, we find unambiguous signatures of a Gaussian critical fixed point in the spin-boson model for $s<1/2$, consistent with mean-field behavior as expected from quantum-to-classical mapping.

Giovanni Battimelli, Giovanni Ciccotti, Pietro Greco