Quantum Phase Transitions from Topology in Momentum Space

Григорий Ефимович Воловик

Many quantum condensed-matter systems, and probably the quantum vacuum of our Universe, are strongly correlated and strongly interacting fermionic systems, which cannot be treated perturbatively. However, physics which emerges in the low-energy does not depend on the complicated details of the system and is relatively simple. It is determined by the nodes in the fermionic spectrum, which are protected by topology in momentum space (in some cases, in combination with the vacuum symmetry). Here we illustrate this universality on some examples of quantum phase transitions, which can occur between the vacua with the same symmetry but with diferent topology in momentum space. The quantum phase transitions between the fully gapped states with different momentum-space topology are also discussed.

Ralf Schützhold

Angelo C. M. Carollo, Davide Valenti, Bernardo Spagnolo

Yanming Che, Clemens Gneiting, Tao Liu +1 more

The discovery of topological features of quantum states plays an important role in modern condensed matter physics and various artificial systems. Due to the absence of local order parameters, the detection of topological quantum phase transitions remains a challenge. Machine learning may provide effective methods for identifying topological features. In this work we show that the unsupervised manifold learning can successfully retrieve topological quantum phase transitions in momentum and real space. Our results show that the Chebyshev distance between two data points sharpens the characteristic features of topological quantum phase transitions in momentum space, while the widely used Euclidean distance is in general suboptimal. Then a diffusion map or isometric map can be applied to implement the dimensionality reduction, and to learn about topological quantum phase transitions in an unsupervised manner. We demonstrate this method on the prototypical Su-Schrieffer-Heeger (SSH) model, the Qi-Wu-Zhang (QWZ) model, and the quenched SSH model in momentum space, and further provide implications and demonstrations for learning in real space, where the topological invariants could be unknown or hard to compute. The interpretable good performance of our approach shows the capability of manifold learning, when equipped with a suitable distance metric, in exploring topological quantum phase transitions.

Linhu Li, Shu Chen

We study topological properties of phase transition points of topological quantum phase transitions by assigning a topological invariant defined on a closed circle or surface surrounding the phase transition point in the parameter space of momentum and transition driving parameter. By applying our scheme to the Su-Schrieffer-Heeger model and Haldane model, we demonstrate that the topological phase transition can be well characterized by the defined topological invariant of the transition point, which reflects the change of topological invariants of topologically different phases across the phase transition point.

Chon‐Fai Kam, Wei-Min Zhang, D.H. Feng