IEEE Transactions on Pattern Analysis and Machine Intelligence

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

Annual review of condensed matter physics

A possible sighting of Majorana states Nearly 80 years ago, the Italian physicist Ettore Majorana proposed the existence of an unusual type of particle that is its own antiparticle, the so-called Majorana fermion. The search for a free Majorana fermion has so far been unsuccessful, but bound Majorana-like collective excitations may exist in certain exotic superconductors. Nadj-Perge et al. created such a topological superconductor by depositing iron atoms onto the surface of superconducting lead, forming atomic chains (see the Perspective by Lee). They then used a scanning tunneling microscope to observe enhanced conductance at the ends of these chains at zero energy, where theory predicts Majorana states should appear. Science, this issue p. 602; see also p. 547 Scanning tunneling microscopy is used to observe signatures of Majorana states at the ends of iron atom chains. [Also see Perspective by Lee] Majorana fermions are predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin-orbit interaction. With the goal of realizing a one-dimensional topological superconductor, we have fabricated ferromagnetic iron (Fe) atomic chains on the surface of superconducting lead (Pb). Using high-resolution spectroscopic imaging techniques, we show that the onset of superconductivity, which gaps the electronic density of states in the bulk of the Fe chains, is accompanied by the appearance of zero-energy end-states. This spatially resolved signature provides strong evidence, corroborated by other observations, for the formation of a topological phase and edge-bound Majorana fermions in our atomic chains.

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Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor

The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. S. 211-215

Pyrite-structured PdSb2 with the nonsymmorphic crystalline symmetry, has long been predicted to host sixfold-degenerate exotic fermions beyond Dirac and Weyl. Although magnetotransport measurements on PdSb2 have suggested its topologically nontrivial character, the direct spectroscopic evidence still remains absent. By utilizing high-resolution angle-resolved photoemission spectroscopy, we present a systematic study on the low-energy bulk and surface electronic structure of pyrite-structured PdSb2. Through careful comparison with first-principles calculations, we verify the existence of sixfold fermions in this compound, which are formed by three doubly degenerate bands centered at the R point in the Brillouin zone. These bands exhibit parabolic dispersion close to sixfold fermion nodes, in sharp contrast to previously reported sixfold double spin-1 chiral fermions. Furthermore, our findings reveal no protected Fermi arcs in PdSb2, which is compatible with its achiral structure. Our findings suggest that pyrite-structured PdSb2 provides an ideal platform for the investigation of fermions and their potential applications.

Direct observation of sixfold exotic fermions in the pyrite-structured topological semimetal PdSb2

The tensor network (TN) has recently triggered extensive interests in developing machine learning models defined in quantum Hilbert space. Here, we propose a generative TN classification (GTNC) model for supervised machine learning. The strategy is first to map the classical data onto the states in a many-body Hilbert space and then to capture these states with tensor network schemes. We adopt the TN in the form of matrix product states as an example to implement GTNC where the testing images are classified by comparing the fidelities between different states. Our results show that GTNC has a very impressive performance on benchmark datasets in comparison to several well-known machine learning models. The advantage of GTNC is reflected from the facts that the samples are naturally clustering in the many-body Hilbert space, and it relies much less on hyperparameters. These characters make GTNC an adaptive and universal quantum-inspired method, which would have important applications in quantum computation and quantum information.

Generative tensor network classification model for supervised machine learning

The vacuum ultraviolet beamline BL03U with a photon energy range from 7 eV upwards has been constructed at the 3.5 GeV Shanghai Synchrotron Radiation Facility. Equipped with an APPLE-Knot undulator, this beamline is dedicated to angle-resolved photoemission spectroscopy. An energy-resolving power of higher than 4.6 × 104 has been achieved in the photon energy range 21.6–48 eV, which is almost the same as the theoretical estimation.

Performance of the BL03U beamline at SSRF.

The gradient-based optimization method for deep machine learning models suffers from gradient vanishing and exploding problems, particularly when the computational graph becomes deep. In this work, we propose the tangent-space gradient optimization (TSGO) for probabilistic models to keep the gradients from vanishing or exploding. The central idea is to guarantee the orthogonality between variational parameters and gradients. The optimization is then implemented by rotating the parameter vector towards the direction of gradient. We explain and test TSGO in tensor network (TN) machine learning, where TN describes the joint probability distribution as a normalized state |ψ〉 in Hilbert space. We show that the gradient can be restricted in tangent space of 〈ψ|ψ〉=1 hypersphere. Instead of additional adaptive methods to control the learning rate η in deep learning, the learning rate of TSGO is naturally determined by rotation angle θ as η=tanθ. Our numerical results reveal better convergence of TSGO in comparison to the off-the-shelf Adam.

Tangent-space gradient optimization of tensor network for machine learning

Identifying quantum phases and phase transitions is key to understanding complex phenomena in statistical physics. In this work, we propose an unconventional strategy to access quantum phases and phase transitions by visualization based on the distribution of ground states in Hilbert space. By mapping the quantum states in Hilbert space onto a two-dimensional feature space using an unsupervised machine learning method, distinct phases can be directly specified and quantum phase transitions can be well identified. Our proposal is benchmarked on gapped, critical, and topological phases in several strongly correlated spin systems. As this proposal directly learns quantum phases and phase transitions from the distributions of the quantum states, it does not require priori knowledge of order parameters of physical systems, which thus indicates a perceptual route to identify quantum phases and phase transitions particularly in complex systems by visualization through learning.

Zheng-Zhi Sun

Papers

Direct observation of sixfold exotic fermions in the pyrite-structured topological semimetal PdSb2

Generative tensor network classification model for supervised machine learning

Performance of the BL03U beamline at SSRF.

Tangent-space gradient optimization of tensor network for machine learning

Visualizing quantum phases and identifying quantum phase transitions by nonlinear dimensional reduction