Statistical mechanics of graphity models

Konopka, Tomasz

doi:10.1103/physrevd.78.044032

Cited by 24 publications

(27 citation statements)

References 31 publications

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“…When it comes to quantum phase transitions however, the relative interaction strengths are of the essence. In future work, we would be interested in applying the techniques developed in this paper to Quantum Graphity [18,19], a model of quantum gravity where it is believed that space emerges from spacetime via a quantum phase transition.…”

Section: Introductionmentioning

confidence: 99%

Lieb-Robinson bounds on the speed of information propagation

Prémont‐Schwarz

Hnybida

2010

Phys. Rev. A

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We propose new Lieb-Robinson bounds (bounds on the speed of propagation of information in quantum systems) with an explicit dependence on the interaction strengths of the Hamiltonian. For systems with more than two interactions it is found that the Lieb-Robinson speed is not always algebraic in the interaction strengths. We consider Hamiltonians with any finite number of bounded operators and also a certain class of unbounded operators. We obtain bounds and propagation speeds for quantum systems on lattices and also general graphs possessing a kind of homogeneity and isotropy. One area for which this formalism could be useful is the study of quantum phase transitions which occur when interactions strengths are varied.

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Section: Introductionmentioning

confidence: 99%

Lieb-Robinson bounds on the speed of information propagation

Prémont‐Schwarz

Hnybida

2010

Phys. Rev. A

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“…Other developments of significant interest include [16,17] which show that it is possible to entangle macroscopically separated nanoelectromechanical oscillators of the oscillator chain and that the resulting entanglement is robust to decoherence. Such a system is of great interest for its possible application as a quantum channel and as a tool to investigate the boundary between the classical and quantum worlds.The LRBs have found a more exotic use in the field of emergent gravity, where one wants to study locality, geometry and Lorentz symmetry as emergent phenomena [22,23]. An example of the usefulness of the Lieb-Robinson bounds can be found in [24], where it was shown that in spin systems with emergent electromagnetism [19], the speed of light is also the maximum speed of signals, without imposing from the beginning any Lorentz invariance.…”

mentioning

confidence: 99%

“…The LRBs have found a more exotic use in the field of emergent gravity, where one wants to study locality, geometry and Lorentz symmetry as emergent phenomena [22,23]. An example of the usefulness of the Lieb-Robinson bounds can be found in [24], where it was shown that in spin systems with emergent electromagnetism [19], the speed of light is also the maximum speed of signals, without imposing from the beginning any Lorentz invariance.…”

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confidence: 99%

Lieb-Robinson bounds for commutator-bounded operators

et al. 2010

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We generalise the Lieb-Robinson theorem to systems whose Hamiltonian is the sum of local operators whose commutators are bounded. The principle of locality is at the heart of the foundations of all modern physics. In quantum field theory, the principle of locality is enforced by an exact light cone. Whenever two (bosonic) observables are spacelike separated, they have to commute, so that neither can have any causal influence on the other. In ordinary quantum mechanics, no explicit request for locality is imposed, and it is, in principle, possible to signal between arbitrarily far apart points in an arbitrarily short time. Nevertheless, a simple perturbation analysis shows that such an influence must decay exponentially with the distance between the observables. The seminal work by Lieb and Robinson [1] has made this statement rigourous for nonrelativistic spin systems. In essence, it states that any quantum system whose Hilbert space is composed of a tensor product of local, finite-dimensional Hilbert spaces and whose Hamiltonian is the sum of local operators will have an approximately maximum speed of signals. Here, local just means that every operator has as a support the tensor product of few degrees of freedom. The approximation consists of the fact that outside the effective light cone there is an exponentially decaying tail.Recently, Lieb-Robinson bounds (LRBs) have received renewed interest in both the fields of theoretical condensed matter and quantum information theory [2][3][4][5][6][7][8][9][10][13][14][15][16][17]. In particular, the LRB has been used to prove that a nonvanishing spectral gap implies an exponential clustering in the ground state [6,8,13]. Further developments can be found in [9], where the LRB is used also to argue about the existence of dynamics. The LRB has also been instrumental in the recent extension of the LiebSchultz-Mattis theorem to higher dimensions [11,14]. In [7,12], it has been shown how the Lieb-Robinson bounds can be exploited to find general scaling laws for entanglement. In [2], these techniques have been exploited to characterise the creation of topological order. The locality of dynamics has important consequences on the simulability of quantum spin systems. In [20,21] it has been shown that one-dimensional gapped spin systems can be efficiently simulated. A review of some of the most relevant aspects of the locality of dynamics for * ipremont-schwarz@perimeterinstitute.ca quantum spins systems can be found in [15]. Other developments of significant interest include [16,17] which show that it is possible to entangle macroscopically separated nanoelectromechanical oscillators of the oscillator chain and that the resulting entanglement is robust to decoherence. Such a system is of great interest for its possible application as a quantum channel and as a tool to investigate the boundary between the classical and quantum worlds.The LRBs have found a more exotic use in the field of emergent gravity, where one wants to study locality, geometry and Lorentz symmetry as em...

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“…In this approach, using the average degree of the graph as an order parameter, it was found that as the number of vertices goes to infinity, the critical temperature of the geometrogenic phase transition goes to zero [13]. Another approach has been to estimate the partition function of the model by considering which types of graphs will contribute most [15]. The two main competing families of graphs are homogeneous lattice-like graphs (due to their low energy and consequent high Boltzmann factor) and random graphs (despite being higher in energy, the large number of random graphs means that they contribute significantly).…”

Section: Introductionmentioning

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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Statistical mechanics of graphity models

Cited by 24 publications

References 31 publications

Lieb-Robinson bounds on the speed of information propagation

Lieb-Robinson bounds on the speed of information propagation

Lieb-Robinson bounds for commutator-bounded operators

Geometrogenesis under quantum graphity: Problems with the ripening universe

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