Symmetry-protected topological Hopf insulator and its generalizations

Topological phases of matter are defined by their nontrivial patterns of ground-state quantum entanglement, which is irremovable so long as the excitation gap and the protecting symmetries, if any, are maintained. Recent studies on noninteracting electrons in crystals have unveiled a peculiar variety of topological phases, which harbors nontrivial entanglement that can be dissolved simply by the the addition of entanglement-free, but charged, degrees of freedom. Such topological phases have a weaker sense of robustness than their conventional counterparts, and are therefore dubbed "fragile topological phases." In this work, we show that fragile topology is a general concept prevailing beyond systems of noninteracting electrons. Fragile topological phases can generally occur when a system has a U(1) charge conservation symmetry, such that only particles with one sign of the charge are physically allowed (e.g. electrons but not positrons). We demonstrate that fragile topological phases exist in interacting systems of both fermions and of bosons. arXiv:1809.02128v2 [cond-mat.str-el]

Section: Comparison With Previous Definitionsmentioning

confidence: 90%

Fragile topological phases in interacting systems

Else

Watanabe

2019

“…In this case, the topological invariant of the post-quench order parameter is characterized by the Hopf number. These rings in the entanglement spectrum are analogous to the boundary Fermi rings in the Hopf insulators [22][23][24] .…”

Section: Introductionmentioning

confidence: 85%

“…This classification requires the number of bands to be large. On the other hand, in fewband models, the classification of the single particle Hamiltonian can also be obtained from the structure of the single particle Hamiltonian directly [22][23][24][29][30][31] . In particular, we can parametrize these few-band models with a finite set of functions {f k,α }.…”

Section: Quench Protocolmentioning

confidence: 99%

Topology and entanglement in quench dynamics

Chang

2018

We classify the topology of the quench dynamics by homotopy groups. A relation between the topological invariant of a post-quench order parameter and the topological invariant of a static Hamiltonian is shown in d + 1 dimensions (d = 1, 2, 3). The mid-gap states in the entanglement spectrum of the post-quench states reveal their topological nature. When a trivial quantum state under a sudden quench to a Chern insulator, the mid-gap states in the entanglement spectrum form rings. These rings are analogous to the boundary Fermi rings in the Hopf insulators. Finally, we show a post-quench order parameter in 3+1 dimensions can be characterized by the second Chern number. The number of Dirac cones in the entanglement spectrum is equal to the second Chern number. arXiv:1804.10625v2 [cond-mat.mes-hall]

“…We obtain that on a bipartite (in our case cubic) lattice, hopfions are characterized by an integer topological invarariant, whereas on a non-bipartite lattice (we consider an example of stacked triangular lattice), dimer configurations are characterized by Z 2 invariant. We remark, that this reasoning gave us a hint that on a bipartite lattice, dimer configurations can be characterized by an exact Hopf number [27][28][29][30][31][32][33][34][35][36], which we later verified analytically [37]. However, on a non-bipartite lattice, the neural network is the only known way to obtain the topological classification of dimer configurations.…”

Section: Introductionmentioning

confidence: 82%

Probing topological properties of a three-dimensional lattice dimer model with neural networks

Bednik

2019

Machine learning methods are being actively considered as a new tool of describing many body physics. However, so far, their capabilities has been only demonstrated in previously studied models, such as e.g. Ising model. Here, we consider a simple problem, demonstrating that neural networks can be successfully used to give new insights in statistical physics. Specifically, we consider 3D lattice dimer model, which consists of sites forming a lattice and bonds connecting the neighboring sites, in such a way that every bond can be either empty or filled with a dimer, and the total number of dimers ending at one site is fixed to be one. Dimer configurations can be viewed as equivalent if they are connected through a series of local flips, i.e. simultaneous 'rotation' of a pair of parallel neighboring dimers. It turns out that the whole set of dimer configurations on a given 3D lattice can be split into distinct topological classes, such that dimer configurations belonging to different classes are not equivalent. In this paper we identify these classes by using neural networks. More specifically, we train the neural networks to distinguish dimer configurations from two known topological classes, and after it, we test them on dimer configurations from unknown topological classes. We demonstrate that 3D lattice dimer model on a bipartite lattice can be described by an integer topological invariant ('Hopf number'), whereas lattice dimer model on a non-bipartite lattice is described by Z2 invariant. Thus, we demonstrate that neural networks can be successfully used to identify new topological phases in condensed matter systems, whose existence can be later verified by other (e.g. analytical) techniques. arXiv:1902.01845v2 [cond-mat.dis-nn]