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Abstract The interplay of correlated spatial modulation and symmetry breaking leads to quantum critical phenomena intermediate between those of the clean and randomly disordered cases. By performing a detailed analytic and numerical case study of the quasi-periodically (QP) modulated transverse field Ising chain, we provide evidence for the conjectures of reference (Crowleyet al2018Phys. Rev. Lett.120175702) regarding the QP-Ising universality class. In the generic case, we confirm that the logarithmic wandering coefficientwgoverns both the macroscopic critical exponents and the energy-dependent localisation length of the critical excitations. However, for special values of the phase difference Δ between the exchange and transverse field couplings, the QP-Ising transition has different properties. For Δ = 0, a generalised Aubry–André duality prevents the finite energy excitations from localising despite the presence of logarithmic wandering. For Δ such that the fields and couplings are related by a lattice shift, the wandering coefficientwvanishes. Nonetheless, the presence of small couplings leads to non-trivial exponents and localised excitations. Our results add to the rich menagerie of quantum Ising transitions in the presence of spatial modulation.
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Efficient optimization of natural resonance theory weightings and bond orders by gram‐based convex programming
We describe the formal algorithm and numerical applications of a novel convex quadratic programming (QP) strategy for performing the variational minimization that underlies natural resonance theory (NRT). The QP algorithm vastly improves the numerical efficiency, thoroughness, and accuracy of variational NRT description, which now allows uniform treatment ofallreference structures at the high level of detail previously reserved only for leading “reference” structures, with little or no user guidance. We illustrate overall QPNRT search strategy, program I/O, and numerical results for a specific application to adenine, and we summarize more extended results for a data set of 338 species from throughout the organic, bioorganic, and inorganic domain. The improved QP‐based implementation of NRT is a principal feature of the newly releasedNBO 7.0program version. © 2019 Wiley Periodicals, Inc.
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- Award ID(s):
- 1740707
- PAR ID:
- 10460459
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Journal of Computational Chemistry
- Volume:
- 40
- Issue:
- 23
- ISSN:
- 0192-8651
- Page Range / eLocation ID:
- p. 2028-2035
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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