- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Hamiltonian Cycles on Ammann-Beenker Tilings

Shobhna Singh1,*, Jerome Lloyd2, and Felix Flicker3

  • 1School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, United Kingdom
  • 2Department of Theoretical Physics, University of Geneva, 24 rue du Général-Dufour, 1211 Genève 4, Switzerland
  • 3School of Physics, Tyndall Avenue, Bristol BS8 1TL, United Kingdom

  • *shobhna862@gmail.com
PDF

Phys. Rev. X 14, 031005 – Published 10 July, 2024

DOI: https://doi.org/10.1103/PhysRevX.14.031005

Abstract

We provide a simple algorithm for constructing Hamiltonian graph cycles (visiting every vertex exactly once) on a set of arbitrarily large finite subgraphs of aperiodic two-dimensional Ammann-Beenker (AB) tilings. Using this result, and the discrete scale symmetry of AB tilings, we find exact solutions to a range of other problems which lie in the complexity class NP-complete for general graphs. These include the equal-weight traveling salesperson problem, providing, for example, the most efficient route a scanning tunneling microscope tip could take to image the atoms of physical quasicrystals with AB symmetries; the longest path problem, whose solution demonstrates that collections of flexible molecules of any length can adsorb onto AB quasicrystal surfaces at density one, with possible applications to catalysis; and the three-coloring problem, giving ground states for the q-state Potts model (q≥3) of magnetic interactions defined on the planar dual to AB, which may provide useful models for protein folding.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

In a “knight’s tour,” the knight chess piece visits every square of a chessboard exactly once before returning to its starting square. This is an example of a Hamiltonian cycle, a loop through a map visiting every stopping point once and only once. Here, we construct Hamiltonian cycles in infinitely large irregular structures describing exotic matter known as quasicrystals. The resulting paths form incredibly complex mazes. Flexible molecules can traverse these mazes to pack perfectly onto quasicrystal surfaces, making quasicrystals unexpected candidates for catalysts, which increase industrial efficiency by lowering the energy of chemical reactions.

The problem of finding Hamiltonian cycles in general settings is so hard that its solution would automatically solve many important open problems in the mathematical sciences. We show that certain quasicrystals provide a special case in which the problem is unexpectedly simple. Consequently, in this setting, we render some seemingly impossible problems tractable. Following the advice of the White Queen from Lewis Carroll’s Through the Looking-Glass, we think of six impossible things, highlighting practical applications from physics to biology to computer science.

Our results open the possibility of translating a wealth of well-studied problems to a new setting. Like chess translated to an exotic new board, we might hope for new and unexpected behaviors.

Article Text

References (144)

  1. H. S. Chan and K. A. Dill, Compact polymers, Macromolecules 22, 4559 (1989).
  2. P.-J. Flory, Statistical thermodynamics of semi-flexible chain molecules, Proc. R. Soc. A 234, 60 (1956).
  3. H. Li, R. Helling, C. Tang, and N. Wingreen, Emergence of preferred structures in a simple model of protein folding, Science 273, 666 (1996).
  4. H. E. Stanley, Dependence of critical properties on dimensionality of spins, Phys. Rev. Lett. 20, 589 (1968).
  5. H. W. J. Blöte and B. Nienhuis, Fully packed loop model on the honeycomb lattice, Phys. Rev. Lett. 72, 1372 (1994).
  6. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).
  7. L. D. C. Jaubert, M. Haque, and R. Moessner, Analysis of a fully packed loop model arising in a magnetic Coulomb phase, Phys. Rev. Lett. 107, 177202 (2011).
  8. J. d. Cloizeaux and G. Jannink, Polymers in Solution: Their Modelling and Structure (Oxford University Press, New York, 1991).
  9. O. Bodroza-Pantic, B. Pantic, I. Pantic, and M. Bodroza-Solarov, Enumeration of Hamiltonian cycles in some grid graphs, MATCH Commun. Math. Comput. Chem 70, 181 (2013).
  10. G. Jannink, Polymers in Solution: Their Modelling and Structure (Clarendon Press, Oxford, 1990).
  11. B. Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Phys. Rev. Lett. 49, 1062 (1982).
  12. J. L. Jacobsen and J. Kondev, Conformal field theory of the Flory model of polymer melting, Phys. Rev. E 69, 066108 (2004).
  13. T. E. Creighton, Protein folding, Biochem. J. 270, 1 (1990).
  14. K. A. Dill, S. Bromberg, K. Yue, H. S. Chan, K. M. Ftebig, D. P. Yee, and P. D. Thomas, Principles of protein folding—a perspective from simple exact models, Protein Sci. 4, 561 (1995).
  15. H. S. Chan and K. A. Dill, Intrachain loops in polymers: Effects of excluded volume, J. Chem. Phys. 90, 492 (1989).
  16. K. F. Lau and K. A. Dill, A lattice statistical mechanics model of the conformational and sequence spaces of proteins, Macromolecules 22, 3986 (1989).
  17. J. L. Jacobsen, Exact enumeration of Hamiltonian circuits, walks and chains in two and three dimensions, J. Phys. A 40, 14667 (2007).
  18. R. Agarwala, S. Batzoglou, V. Dančík, S. E. Decatur, S. Hannenhalli, M. Farach, S. Muthukrishnan, and S. Skiena, Local rules for protein folding on a triangular lattice and generalized hydrophobicity in the HP model, J. Comput. Biol. 4, 275 (1997).
  19. W. Orr, Statistical treatment of polymer solutions at infinite dilution, Trans. Faraday Soc. 43, 12 (1947).
  20. M. N. Barber and B. W. Ninham, Random and Restricted Walks: Theory and Applications, Vol. 10 (CRC Press, Boca Raton, FL, 1970).
  21. P.-G. De Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, NY, 1979).
  22. K. F. Freed, Renormalization group theory of macromolecules, J. Stat. Phys. 52, 521 (1988).
  23. J. L. Jacobsen, On the universality of fully packed loop models, J. Phys. A 32, 5445 (1999).
  24. J. Kondev, Liouville field theory of fluctuating loops, Phys. Rev. Lett. 78, 4320 (1997).
  25. M. T. Batchelor, J. Suzuki, and C. M. Yung, Exact results for Hamiltonian walks from the solution of the fully packed loop model on the honeycomb lattice, Phys. Rev. Lett. 73, 2646 (1994).
  26. E. H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice, Phys. Rev. Lett. 18, 692 (1967).
  27. J. Suzuki, Evaluation of the connectivity of Hamiltonian paths on regular lattices, J. Phys. Soc. Jpn. 57, 687 (1988).
  28. S. Higuchi, Field theoretic approach to the counting problem of Hamiltonian cycles of graphs, Phys. Rev. E 58, 128 (1998).
  29. J. Kondev, J. de Gier, and B. Nienhuis, Operator spectrum and exact exponents of the fully packed loop model, J. Phys. A 29, 6489 (1996).
  30. M. Batchelor, H. Blöte, B. Nienhuis, and C. Yung, Critical behaviour of the fully packed loop model on the square lattice, J. Phys. A 29, L399 (1996).
  31. O. Bodroža-Pantić, H. Kwong, and M. Pantić, Some new characterizations of Hamiltonian cycles in triangular grid graphs, Discrete Appl. Math. 201, 1 (2016).
  32. A. Kast, Correlation length and average loop length of the fully packed loop model, J. Phys. A 29, 7041 (1996).
  33. H. Orland, C. Itzykson, and C. de Dominicis, An evaluation of the number of Hamiltonian paths, J. Phys. Lett. 46, 353 (1985).
  34. J. L. Jacobsen and J. Kondev, Field theory of compact polymers on the square lattice, Nucl. Phys. B532, 635 (1998).
  35. J. Kondev and J. L. Jacobsen, Conformational entropy of compact polymers, Phys. Rev. Lett. 81, 2922 (1998).
  36. T. Schmalz, G. Hite, and D. Klein, Compact self-avoiding circuits on two-dimensional lattices, J. Phys. A 17, 445 (1984).
  37. M. Peto, T. Z. Sen, R. L. Jernigan, and A. Kloczkowski, Generation and enumeration of compact conformations on the two-dimensional triangular and three-dimensional fcc lattices, J. Chem. Phys. 127, 07B612 (2007).
  38. A. V. Melkikh and D. K. Meijer, On a generalized Levinthal’s paradox: The role of long- and short-range interactions in complex bio-molecular reactions, including protein and dna folding, Prog. Biophys. Molec. Biol. 132, 57 (2018).
  39. J. Lloyd, S. Biswas, S. H. Simon, S. A. Parameswaran, and F. Flicker, Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings, Phys. Rev. B 106, 094202 (2022).
  40. B. Grünbaum and G. C. Shephard, Tilings and Patterns (W. H. Freeman and Company, New York, 1986).
  41. R. Ammann, B. Grünbaum, and G. C. Shephard, Aperiodic tiles, Discrete Comput. Geom. 8, 1 (1992).
  42. M. Baake and U. Grimm, Mathematical diffraction of aperiodic structures, Chem. Soc. Rev. 41, 6821 (2012).
  43. M. Senechal, Quasicrystals and Geometry (Cambridge University Press, Cambridge, England, 1995).
  44. N. Wang, H. Chen, and K. H. Kuo, Two-dimensional quasicrystal with eightfold rotational symmetry, Phys. Rev. Lett. 59, 1010 (1987).
  45. N. Wang, K. K. Fung, and K. H. Kuo, Symmetry study of the Mn‐Si‐Al octagonal quasicrystal by convergent beam electron diffraction, Appl. Phys. Lett. 52, 2120 (1988).
  46. A. Uri, S. C. de la Barrera, M. T. Randeria, D. Rodan-Legrain, T. Devakul, P. J. D. Crowley, N. Paul, K. Watanabe, T. Taniguchi, R. Lifshitz, L. Fu, R. C. Ashoori, and P. Jarillo-Herrero, Superconductivity and strong interactions in a tunable moiré quasiperiodic crystal, Nature (London) 620, 762 (2023).
  47. K. Viebahn, M. Sbroscia, E. Carter, J.-C. Yu, and U. Schneider, Matter-wave diffraction from a quasicrystalline optical lattice, Phys. Rev. Lett. 122, 110404 (2019).
  48. M. Sbroscia, K. Viebahn, E. Carter, J.-C. Yu, A. Gaunt, and U. Schneider, Observing localization in a 2D quasicrystalline optical lattice, Phys. Rev. Lett. 125, 200604 (2020).
  49. G. M. Sommers, M. J. Gullans, and D. A. Huse, Self-dual quasiperiodic percolation, Phys. Rev. E 107, 024137 (2023).
  50. D. E. Gökmen, S. Biswas, S. D. Huber, Z. Ringel, F. Flicker, and M. Koch-Janusz, Machine learning assisted discovery of exotic criticality in a planar quasicrystal, arXiv:2301.11934.
  51. M. Ciardi, A. Angelone, F. Mezzacapo, and F. Cinti, Quasicrystalline Bose glass in the absence of disorder and quasidisorder, Phys. Rev. Lett. 131, 173402 (2023).
  52. F. Flicker and J. van Wezel, Quasiperiodicity and 2D topology in 1D charge-ordered materials, Europhys. Lett. 111, 37008 (2015).
  53. F. Flicker and J. van Wezel, Natural 1D quasicrystals from incommensurate charge order, Phys. Rev. Lett. 115, 236401 (2015).
  54. Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett. 109, 106402 (2012).
  55. K. A. Madsen, E. J. Bergholtz, and P. W. Brouwer, Topological equivalence of crystal and quasicrystal band structures, Phys. Rev. B 88, 125118 (2013).
  56. M. Gardner, Mathematical games: About the remarkable similarity between the Icosian game and the towers of Hanoi, Sci. Am. 196, 150 (1957).
  57. J. A. Bondy and V. Chvátal, A method in graph theory, Discrete Math. 15, 111 (1976).
  58. R. W. Robinson and N. C. Wormald, Almost all cubic graphs are Hamiltonian, Random Struct. Algorithms 3, 117 (1992).
  59. W. T. Tutte, A theorem on planar graphs, Trans. Am. Math. Soc. 82, 99 (1956).
  60. L. Liu, G. P. T. Choi, and L. Mahadevan, Quasicrystal kirigami, Phys. Rev. Res. 4, 033114 (2022).
  61. F. Flicker, S. H. Simon, and S. A. Parameswaran, Classical dimers on Penrose tilings, Phys. Rev. X 10, 011005 (2020).
  62. R. M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations (Springer, New York, 1972) pp. 85–103.
  63. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W.H. Freeman and Company, New York, 1979).
  64. R. McGrath, J. Smerdon, H. Sharma, W. Theis, and J. Ledieu, The surface science of quasicrystals, J. Phys. Condens. Matter 22, 084022 (2010).
  65. J.-M. Dubois, New prospects from potential applications of quasicrystalline materials, Mater. Sci. Eng. A 294–296, 4 (2000).
  66. J. J. Kweon, H.-I. Kim, S.-h. Lee, J. Kim, and S. K. Lee, Quantitative probing of hydrogen environments in quasicrystals by high-resolution NMR spectroscopy, Acta Mater. 226, 117657 (2022).
  67. R. Diehl, W. Setyawan, and S. Curtarolo, Gas adsorption on quasicrystalline surfaces, J. Phys. Condens. Matter 20, 314007 (2008).
  68. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Elsevier, New York, 2016).
  69. F.-Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982).
  70. Y. Hattori, A. Niikura, A. P. Tsai, A. Inoue, T. Masumoto, K. Fukamichi, H. Aruga-Katori, and T. Goto, Spin-glass behaviour of icosahedral Mg-Gd-Zn and Mg-Tb-Zn quasi-crystals, J. Phys. Condens. Matter 7, 2313 (1995).
  71. Z. Islam, I. R. Fisher, J. Zarestky, P. C. Canfield, C. Stassis, and A. I. Goldman, Reinvestigation of long-range magnetic ordering in icosahedral Tb-Mg-Zn, Phys. Rev. B 57, R11047 (1998).
  72. T. J. Sato, H. Takakura, A. P. Tsai, K. Shibata, K. Ohoyama, and K. H. Andersen, Antiferromagnetic spin correlations in the Zn-Mg-Ho icosahedral quasicrystal, Phys. Rev. B 61, 476 (2000).
  73. A. Goldman, T. Kong, A. Kreyssig, A. Jesche, M. Ramazanoglu, K. W. Dennis, S. L. Bud’ko, and P. C. Canfield, A family of binary magnetic icosahedral quasicrystals based on rare earths and cadmium, Nat. Mater. 12, 714 (2013).
  74. R. Tamura, A. Ishikawa, S. Suzuki, T. Kotajima, Y. Tanaka, T. Seki, N. Shibata, T. Yamada, T. Fujii, C.-W. Wang, M. Avdeev, K. Nawa, D. Okuyama, and T. J. Sato, Experimental observation of long-range magnetic order in icosahedral quasicrystals, J. Am. Chem. Soc. 143, 19938 (2021).
  75. T. Garel and H. Orland, Mean-field model for protein folding, Europhys. Lett. 6, 307 (1988).
  76. T. N. De Silva and V. Sivised, A statistical mechanics perspective for protein folding from q-state Potts model, arXiv:1709.04813.
  77. M. Baake and U. Grimm, Aperiodic Order, Vol. 1 (Cambridge University Press, Cambridge, England, 2013).

  78. Legal patches can be defined according to edge matching rules [77], and simply ensure that the starting patch corresponds to a group of tiles that can be found together within the infinite tiling. In practice, usually a single square or rhombus tile is used as the initial patch.

  79. A. Haynes, H. Koivusalo, and J. Walton, A characterization of linearly repetitive cut and project sets, Nonlinearity 31, 515 (2018).
  80. F. Fang, S. Paduroiu, D. Hammock, and K. Irwin, Non-local game of life in 2D quasicrystals, Crystals 8, 416 (2018).
  81. W. T. Tutte and W. T. Tutte, Graph Theory, Vol. 21 (Cambridge University Press, Cambridge, England, 2001).
  82. J. A. Bondy et al., Graph Theory with Applications, Vol. 290 (Macmillan, London, 1976).
  83. G. Chartrand, Introductory Graph Theory (Courier Corporation, Boston, 1977).

  84. Strictly, loops can be open if they connect two points on the boundary. However, these open paths lie entirely in the region Wn\Un so are not relevant to our construction on Un.

  85. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Harcourt College Publishers, New York, 1976).
  86. D. B. Shmoys, J. Lenstra, A. R. Kan, and E. L. Lawler, The Traveling Salesman Problem, Vol. 12 (John Wiley & Sons, Incorporated, New York, 1985).
  87. G. Binnig and H. Rohrer, Scanning tunneling microscopy, Surf. Sci. 126, 236 (1983).
  88. P. K. Hansma and J. Tersoff, Scanning tunneling microscopy, J. Appl. Phys. 61, R1 (1987).
  89. Y. Martin and H. K. Wickramasinghe, Magnetic imaging by “force microscopy” with 1000 Å resolution, Appl. Phys. Lett. 50, 1455 (1987).
  90. U. Hartmann, Magnetic force microscopy, Annu. Rev. Mater. Sci. 29, 53 (1999).

  91. State-of-the-art STM measurements can obtain intraunit-cell data on crystal surfaces; for a 100×100  nm2 region this would exceed 250 000 pixels. A typical measurement takes around 100 ms, so if each pixel is scanned at 100 energies, the result takes around 29 days.

  92. D. Karger, R. Motwani, and G. D. Ramkumar, On approximating the longest path in a graph, Algorithmica 18, 82 (1997).
  93. J. K. Roberts, Some properties of adsorbed films of oxygen on tungsten, Proc. R. Soc. A 152, 464 (1935).
  94. R. Fowler and G. Rushbrooke, An attempt to extend the statistical theory of perfect solutions, Trans. Faraday Soc. 33, 1272 (1937).
  95. O. J. Heilmann and E. H. Lieb, Theory of monomer-dimer systems, Commun. Math. Phys. 25, 190 (1972).
  96. M. R. Garey, D. S. Johnson, and L. Stockmeyer, Some simplified NP-complete problems, in Proceedings of the Sixth Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, 1974), pp. 47–63.
  97. P. J. Heawood, Map color theorems, Quant. J. Math. 24, 332 (1890).
  98. K. Appel and W. Haken, Every planar map is four colorable, Contemp. Math. 98, xx (1989).
  99. T. Sibley and S. Wagon, Rhombic Penrose tilings can be 3-colored, Am. Math. Mon. 107, 251 (2000).
  100. V. Beffara and H. Duminil-Copin, The self-dual point of the two-dimensional random-cluster model is critical for q≥1, Probab. Theory Relat. Fields 153, 511 (2012).
  101. W. Selke and D. A. Huse, Interfacial adsorption in planar Potts models, Z. Phys. B 50, 113 (1983).
  102. A. D. Sokal and B. Webb, The multivariate Tutte polynomial (alias Potts model), Surveys Combinatorics 327, 173 (2005).
  103. M. Alford, S. Chandrasekharan, J. Cox, and U.-J. Wiese, Solution of the complex action problem in the Potts model for dense QCD, Nucl. Phys. B602, 61 (2001).
  104. F. Graner and J. A. Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model, Phys. Rev. Lett. 69, 2013 (1992).
  105. W. G. Wilson and C. A. Vause, Evidence for universality of the Potts model on the two-dimensional Penrose lattice, Phys. Lett. A 126, 471 (1988).
  106. G. Xiong, Z.-H. Zhang, and D.-C. Tian, Real-space renormalization group approach to the Potts model on the two-dimensional Penrose tiling, Physica (Amsterdam) 265A, 547 (1999).
  107. R. M. Levy, A. Haldane, and W. F. Flynn, Potts Hamiltonian models of protein co-variation, free energy landscapes, and evolutionary fitness, Curr. Opin. Struct. Biol. 43, 55 (2017).
  108. R. B. Potts, Some generalized order-disorder transformations, in Mathematical Proceedings of the Cambridge Philosophical Society (Cambridge University Press, Cambridge, England, 1952), Vol. 48, pp. 106–109.
  109. J. E. Hopcroft and R. M. Karp, An n5/2 algorithm for maximum matchings in bipartite graphs, SIAM J. Comput. 2, 225 (1973).
  110. N. de Bruijn, Algebraic theory of Penrose’s non-periodic tilings of the plane I, Indagationes Mathematicae (Proceedings) 84, 39 (1981).
  111. J. Colclough, github.com/joshcol9232/tiling.
  112. D. Smith, J. S. Myers, C. S. Kaplan, and C. Goodman-Strauss, An aperiodic monotile, arXiv:2303.10798.
  113. D. Smith, J. S. Myers, C. S. Kaplan, and C. Goodman-Strauss, A chiral aperiodic monotile, arXiv:2305.17743.
  114. J. d. Gier, Fully packed loop models on finite geometries, in Polygons, Polyominoes and Polycubes (Springer, New York, 2009), pp. 317–346.
  115. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
  116. R. P. Feynman, Statistical Mechanics: A Set of Lectures (CRC Press, Boca Raton, FL, 2018).
  117. E. H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett. 62, 1201 (1989).
  118. R. Bhola, S. Biswas, M. M. Islam, and K. Damle, Dulmage-Mendelsohn percolation: Geometry of maximally packed dimer models and topologically protected zero modes on site-diluted bipartite lattices, Phys. Rev. X 12, 021058 (2022).
  119. M. Kohmoto and B. Sutherland, Electronic states on a Penrose lattice, Phys. Rev. Lett. 56, 2740 (1986).
  120. A. Koga and H. Tsunetsugu, Antiferromagnetic order in the Hubbard model on the Penrose lattice, Phys. Rev. B 96, 214402 (2017).
  121. E. Day-Roberts, R. M. Fernandes, and A. Kamenev, Nature of protected zero-energy states in Penrose quasicrystals, Phys. Rev. B 102, 064210 (2020).
  122. A. Koga, Superlattice structure in the antiferromagnetically ordered state in the Hubbard model on the Ammann-Beenker tiling, Phys. Rev. B 102, 115125 (2020).
  123. B. Sutherland, Localization of electronic wave functions due to local topology, Phys. Rev. B 34, 5208 (1986).
  124. J. Schirmann, S. Franca, F. Flicker, and A. G. Grushin, Physical properties of the hat aperiodic monotile: Graphene-like features, chirality and zero modes, Phys. Rev. Lett. 132, 086402 (2024).
  125. P. Baniasadi, V. Ejov, J. A. Filar, M. Haythorpe, and S. Rossomakhine, Deterministic “snakes and ladders” heuristic for the Hamiltonian cycle problem, Math. Program. Comput. 6, 55 (2014).
  126. P. Baniasadi, V. Ejov, M. Haythorpe, and S. Rossomakhine, A new benchmark set for traveling salesman problem and Hamiltonian cycle problem, arXiv:1806.09285.
  127. J. Sleegers and D. van den Berg, The hardest Hamiltonian cycle problem instances: the plateau of yes and the cliff of no, SN Comput. Sci. 3, 372 (2022).
  128. K. Helsgaun, An effective implementation of the Lin-Kernighan traveling salesman heuristic, Eur. J. Oper. Res. 126, 106 (2000).
  129. D. L. Applegate, R. E. Bixby, V. Chvátal, and W. J. Cook, The traveling salesman problem, The Traveling Salesman Problem (Princeton University Press, Princeton, NJ, 2011).

  130. The SLH algorithm was run on an Intel Core i7-1165G7 machine with 4 cores, 2.80 GHz clock speed, and 32 GB RAM.

  131. F. L. Metz, I. Neri, and T. Rogers, Spectral theory of sparse non-Hermitian random matrices, J. Phys. A 52, 434003 (2019).
  132. P. Webley and D. Danaci, CO2 capture by adsorption processes, in Carbon Capture and Storage, RSC Energy and Environment Series No. 26, edited by M. Bui and N. Dowell (The Royal Society of Chemistry, London, 2020), pp. 106–167.
  133. D. Louzguine-Luzgin and A. Inoue, Formation and properties of quasicrystals, Annu. Rev. Mater. Res. 38, 403 (2008).
  134. Z. Papadopolos, G. Kasner, J. Ledieu, E. J. Cox, N. V. Richardson, Q. Chen, R. D. Diehl, T. A. Lograsso, A. R. Ross, and R. McGrath, Bulk termination of the quasicrystalline fivefold surface of Al70Pd21Mn9, Phys. Rev. B 66, 184207 (2002).
  135. Z. Papadopolos, P. Pleasants, G. Kasner, V. Fournée, C. J. Jenks, J. Ledieu, and R. McGrath, Maximum density rule for bulk terminations of quasicrystals, Phys. Rev. B 69, 224201 (2004).
  136. R. McGrath, J. Ledieu, E. J. Cox, and R. D. Diehl, Quasicrystal surfaces: structure and potential as templates, J. Phys. Condens. Matter 14, R119 (2002).
  137. R. McGrath, J. A. Smerdon, H. R. Sharma, W. Theis, and R. Ledieu, The surface science of quasicrystals, J. Phys. Condens. Matter 22, 084022 (2010).
  138. M. Pagliaro, C. Della Pina, F. Mauriello, and R. Ciriminna, Catalysis with silver: From complexes and nanoparticles to morals and single-atom catalysts, Catal. Chem. 10, 1343 (2020).
  139. D. A. Rabson, N. D. Mermin, D. S. Rokhsar, and D. C. Wright, The space groups of axial crystals and quasicrystals, Rev. Mod. Phys. 63, 699 (1991).
  140. O. Ovdat and E. Akkermans, Breaking of continuous scale invariance to discrete scale invariance: A universal quantum phase transition, in Fractal Geometry and Stochastics VI, edited by Uta Freiberg, Ben Hambly, Michael Hinz, and Steffen Winter (Birkhäuser, Cham, 2021), p. 209.
  141. S. Biswas and S. A. Parameswaran, Discrete scale invariant fixed point in a quasiperiodic classical dimer model, arXiv:2302.07879.
  142. E. Cockayne and S. Hedetniemi, Optimal domination in graphs, IEEE Trans. Circuits Syst. 22, 855 (1975).
  143. E. J. Cockayne and S. T. Hedetniemi, Towards a theory of domination in graphs, Networks 7, 247 (1977).
  144. M. Yannakakis, The node deletion problem for hereditary properties, Computer Science Laboratory, Princeton University, Report No. TR-240, 1978.

Outline

Information

Phys. Rev. X 14, 031005– Published 10 July, 2024
  • Received 13 March 2023
  • Revised 25 January 2024
  • Accepted 1 May 2024
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation