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Topological phase transitions constitute a new class of quantum critical phenomena. They cannot be described within the usual framework of the Landau theory since, in general, the different phases cannot be distinguished by an order parameter, neither can they be related to different symmetries. In most cases, however, one can identify a diverging length at these topological transitions. This allows us to describe them using a scaling approach and to introduce a set of critical exponents that characterize their universality class. Here we consider some relevant models of quantum topological transitions associated with well-defined critical exponents that are related by a quantum hyperscaling relation. We extend to these models a finite-size scaling approach based on techniques for calculating the Casimir force in electromagnetism. This procedure allows us to obtain universal Casimir amplitudes at their quantum critical points. Our results verify the validity of finite-size scaling in these systems and confirm the values of the critical exponents obtained previously.

We develop a supervised machine learning algorithm that is able to learn topological phases for finite condensed matter systems in real lattice space. The algorithm employs diagonalization in real space together with any supervised learning algorithm to learn topological phases through an eigenvector-ensembling procedure. We combine our algorithm with decision trees to successfully recover topological phase diagrams of Su-Schrieffer-Heeger (SSH) models from lattice data in real space and show how the Gini impurity of ensembles of lattice eigenvectors can be used to retrieve a topological signal detailing how topological information is distributed along the lattice. The discovery of local Gini topological signals from the analysis of data from several thousand SSH systems illustrates how machine learning can advance the research and discovery of new quantum materials with exotic properties that may power future technological applications such as quantum computing.

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A proposal to study topological models beyond the standard topological classification and that exhibit breakdown of Lorentz invariance is presented. The focus of the investigation relies on their anisotropic quantum critical behavior. We study anisotropic effects on three-dimensional (3D) topological models, computing their anisotropic correlation length critical exponent $$\nu$$ ν obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point. A generalized Weyl semimetal model with broken time-reversal symmetry is introduced and studied using a modified Dirac equation. An approach to characterize topological surface states in topological insulators when applied to Fermi arcs allows to capture the anisotropic critical exponent $$\theta =\nu _{x}/\nu _{z}$$ θ = ν x / ν z . We also consider the Hopf insulator model, for which the study of the topological surface states yields unusual values for $$\nu$$ ν and for the dynamic critical exponent z. From an analysis of the energy dispersions, we propose a scaling relation $$\nu _{\bar{\alpha }}z_{\bar{\alpha }}=2q$$ ν α ¯ z α ¯ = 2 q and $$\theta =\nu _{x}/\nu _{z}=z_{z}/z_{x}$$ θ = ν x / ν z = z z / z x for $$\nu$$ ν and z that only depends on the Hopf insulator Hamiltonian parameters p and q and the axis direction $$\bar{\alpha }$$ α ¯ . An anisotropic quantum hyperscaling relation is also obtained.