Quantum Bose-Hubbard model with an evolving graph as a toy model for emergent spacetime

Hamma, Alioscia; Markopoulou, Fotini; Lloyd, Seth; Caravelli, Francesco; Severini, Simone; Markström, Klas

doi:10.1103/physrevd.81.104032

Cited by 54 publications

(103 citation statements)

References 38 publications

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“…It is therefore necessary to posit mechanisms by which the graph can relax to some global minimum, which is presumed to have the properties of geometry that we experience. Mechanisms to allow such relaxation include the formation of matter on the graph [2], and equilibration with some external heat bath [3].…”

Section: Introductionmentioning

confidence: 99%

Energetics of the quantum graphity universe

Wilkinson

Greentree

2014

Phys. Rev. D

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Quantum graphity is a background independent model for emergent geometry, in which space is represented as a dynamical graph. The high-energy pre-geometric starting point of the model is usually considered to be the complete graph, however we also consider the empty graph as a candidate pre-geometric state. The energetics as the graph evolves from either of these high-energy states to a low-energy geometric state is investigated as a function of the number of edges in the graph. Analytic results for the slope of this energy curve in the high-energy domain are derived, and the energy curve is determined exactly for small number of vertices N . To study the whole energy curve for larger (but still finite) N , an epitaxial approximation is introduced. This work may open the way to compare predictions from quantum graphity with observations of the early universe, making the model falsifiable.

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Section: Introductionmentioning

confidence: 99%

Energetics of the quantum graphity universe

Wilkinson

Greentree

2014

Phys. Rev. D

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“…In fact the model defined on complex networks might provide a useful mean-field approximation to real disordered granular materials that captures essential features of their heterogeneity. Finally the Hubbard model is a theoretical model that has applications far beyond condensed matter physics [34,35] and investigating its properties on scale-free networks might stimulate further applications to other fields. Here we investigate this model when it is defined on a scale-free network topology.…”

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confidence: 99%

Phase diagram of the Bose-Hubbard model on complex networks

Halu¹,

Ferretti²,

Vezzani³

et al. 2012

EPL

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-Critical phenomena can show unusual phase diagrams when defined in complex network topologies. The case of classical phase transitions such as the classical Ising model and the percolation transition has been studied extensively in the last decade. Here we show that the phase diagram of the Bose-Hubbard model, an exclusively quantum mechanical phase transition, also changes significantly when defined on random scale-free networks. We present a mean-field calculation of the model in annealed networks and we show that when the second moment of the average degree diverges the Mott-insulator phase disappears in the thermodynamic limit. Moreover we study the model on quenched networks and we show that the Mott-insulator phase disappears in the thermodynamic limit as long as the maximal eigenvalue of the adjacency matrix diverges. Finally we study the phase diagram of the model on Apollonian scale-free networks that can be embedded in 2 dimensions showing the extension of the results also to this case.Introduction. -Recently great attention [1, 2] has been addressed to critical phenomena unfolding on complex networks. In this context it has been observed that the topology of the networks might significantly change the phase diagram of dynamical processes. For example when networks have a scale-free degree distribution P (k) ∼ k −λ and the second moment k 2 diverges with the network size, i.e. λ ∈ (2, 3] the Ising model [3][4][5][6], the percolation phase transition [7,8] and the epidemic spreading dynamics on annealed networks [9] are strongly affected. Moreover the spectral properties of the networks drive the epidemic spreading on quenched networks [10,11], the synchronization stability [12,13], the critical behavior of O(N ) models [14,15] and the critical fluctuations of an Ising model on spatial scale-free networks [16]. Quantum critical phenomena also might depend on the topology of the underlying lattice as it has been shown for Bose-Einstein condensation in heterogeneous networks [17]. Although large attention has been devoted to classical critical phenomena on scale-free networks, the behavior of quantum critical phenomena on scale-free networks has just started to be investigated.In particular the Anderson localization [18,19] was studied in complex networks showing that by modulating the clustering coefficient of the

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“…This can be interpreted as the graph being in contact with some heat bath, although when our graph represents space itself the idea of an external heat bath is problematic [13]. We follow previous work [12] and take the heat bath to represent the creation and annihilation of matter on the graph, so that the lowing of the energy of the graph can be interpreted as the creation of matter and the total energy of the graph + matter system is conserved. Since we neglect matter in this work, the evolution of the graph is a non-unitary process.…”

Section: Modelmentioning

confidence: 99%

Geometrogenesis under quantum graphity: Problems with the ripening universe

Wilkinson

Greentree

2015

Phys. Rev. D

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Quantum Graphity (QG) is a model of emergent geometry in which space is represented by a dynamical graph. The graph evolves under the action of a Hamiltonian from a high-energy pre-geometric state to a low-energy state in which geometry emerges as a coarse-grained effective property of space. Here we show the results of numerical modelling of the evolution of the QG Hamiltonian, a process we term "ripening" by analogy with crystallographic growth. We find that the model as originally presented favours a graph composed of small disjoint subgraphs. Such a disconnected space is a poor representation of our universe. A new term is introduced to the original QG Hamiltonian, which we call the hypervalence term. It is shown that the inclusion of a hypervalence term causes a connected lattice-like graph to be favoured over small isolated subgraphs.

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Quantum Bose-Hubbard model with an evolving graph as a toy model for emergent spacetime

Cited by 54 publications

References 38 publications

Energetics of the quantum graphity universe

Energetics of the quantum graphity universe

Phase diagram of the Bose-Hubbard model on complex networks

Geometrogenesis under quantum graphity: Problems with the ripening universe

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