Using a high-accuracy variational Monte Carlo approach based on group-convolutional neural networks, we obtain the symmetry-resolved low-energy spectrum of the spin-1/2 Heisenberg model on several highly symmetric fullerene geometries, including the famous C60 buckminsterfullerene. We argue that as the degree of frustration is lowered in large fullerenes, they display characteristic features of incipient magnetic ordering: Correlation functions show high-intensity Bragg peaks consistent with Néel-like ordering, while the low-energy spectrum is organized into a tower of states. Competition with frustration, however, turns the simple Néel order into a noncoplanar one. Remarkably, we find and predict chiral incipient ordering in a large number of fullerene structures.

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Quantum electrodynamics in <math display="inline"><mn>2</mn><mo>+</mo><mn>1</mn></math> dimensions (<math display="inline"><mrow><msub><mrow><mi>QED</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math>) has been proposed as a critical field theory describing the low-energy effective theory of a putative algebraic Dirac spin liquid or of quantum phase transitions in two-dimensional frustrated magnets. We provide compelling evidence that the intricate spectrum of excitations of the elementary but strongly frustrated <math display="inline"><mrow><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mtext>-</mtext><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math> Heisenberg model on the triangular lattice is in one-to-one correspondence to a zoo of excitations from <math display="inline"><mrow><msub><mrow><mi>QED</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math>, in the quantum spin liquid regime. This evidence includes a large manifold of explicitly constructed monopole and bilinear excitations of <math display="inline"><mrow><msub><mrow><mi>QED</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math>, which is thus shown to serve as an organizing principle of phases of matter in triangular lattice antiferromagnets and their low-lying excitations. Moreover, we observe signatures of emergent valence-bond solid (VBS) correlations, which can be interpreted either as evidence of critical VBS fluctuations of an emergent Dirac spin liquid or as a transition from the 120° Néel order to a VBS whose quantum critical point is described by <math display="inline"><mrow><msub><mrow><mi>QED</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math>. Our results are obtained by comparing ansatz wave functions from a parton construction to exact eigenstates obtained using large-scale exact diagonalization up to <math display="inline"><mi>N</mi><mo>=</mo><mn>48</mn></math> sites.

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Topological insulators and superconductors support extended surface states protected against the otherwise localizing effects of static disorder. Specifically, in the Wigner-Dyson insulators belonging to the symmetry classes A, AI, and AII, a band of extended surface states is continuously connected to a likewise extended set of bulk states forming a “bridge” between different surfaces via the mechanism of spectral flow. In this work we show that this mechanism is absent in the majority of non-Wigner-Dyson topological superconductors and chiral topological insulators. In these systems, there is precisely one point with granted extended states, the center of the band, <math display="inline"><mi>E</mi><mo>=</mo><mn>0</mn></math>. Away from it, states are spatially localized, or can be made so by the addition of spatially local potentials. Considering the three-dimensional insulator in class AIII and winding number <math display="inline"><mi>ν</mi><mo>=</mo><mn>1</mn></math> as a paradigmatic case study, we discuss the physical principles behind this phenomenon, and its methodological and applied consequences. In particular, we show that low-energy Dirac approximations in the description of surface states can be treacherous in that they tend to conceal the localizability phenomenon. We also identify markers defined in terms of Berry curvature as measures for the degree of state localization in lattice models, and back our analytical predictions by extensive numerical simulations. A main conclusion of this work is that the surface phenomenology of non-Wigner-Dyson topological insulators is a lot richer than that of their Wigner-Dyson siblings, extreme limits being spectrumwide quantum critical delocalization of all states versus full localization except at the <math display="inline"><mi>E</mi><mo>=</mo><mn>0</mn></math> critical point. As part of our study we identify possible experimental signatures distinguishing between these different alternatives in transport or tunnel spectroscopy.

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Chiral Spin Liquids (CSL) based on spin-1/2 fermionic Projected Entangled Pair States (fPEPS) are considered on the square lattice. First, fPEPS approximants of Gutzwiller-projected Chern insulators (GPCI) are investigated by Variational Monte Carlo (VMC) techniques on finite size tori. We show that such fPEPS of finite bond dimension can correctly capture the topological properties of the chiral spin liquid, as the exact GPCI, with the correct topological ground state degeneracy on the torus. Further, more general fPEPS are considered and optimized (on the infinite plane) to describe the CSL phase of a chiral frustrated Heisenberg antiferromagnet. The chiral modes are computed on the edge of a semi-infinite cylinder (of finite circumference) and shown to follow the predictions from Conformal Field Theory. In contrast to their bosonic analogs the (optimized) fPEPS do not suffer from the replication of the chiral edge mode in the odd topological sector.

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