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Quantum Graphity: a model of emergent locality

Tomasz Konopka, Fotini Markopoulou, Simone Severini

TL;DR

This work proposes a background-independent quantum model (quantum graphity) in which degrees of freedom live on the edges of the complete graph $K_N$, and a permutation-invariant Hamiltonian drives a transition from a highly connected, nonlocal high-temperature phase to a low-energy phase with emergent locality. The core mechanism combines a valence-enforcing term with a loop term that favors short cycles, leading to a conjectured ground state with hexagonal (3-regular) lattice structure and thermodynamic stability against local perturbations; extensions to additional edge degrees of freedom enable Levin-Wen string-net condensation and an emergent $U(1)$ gauge theory on the emergent background. The paper also reformulates the dynamics in graph-theoretic terms and discusses extensions, perturbations, and comparisons to other graph processes, highlighting open questions about the precise ground state, the nature of the high-to-low transition, and potential connections to continuum gravity. Overall, the model provides a concrete route to geometrogenesis and emergent locality, bridging quantum gravity ideas with condensed-matter style Hamiltonians and suggesting avenues to incorporate matter and gauge structure on dynamical lattices.

Abstract

Quantum graphity is a background independent model for emergent locality, spatial geometry and matter. The states of the system correspond to dynamical graphs on N vertices. At high energy, the graph describing the system is highly connected and the physics is invariant under the full symmetric group acting on the vertices. We present evidence that the model also has a low-energy phase in which the graph describing the system breaks permutation symmetry and appears to be ordered, low-dimensional and local. Consideration of the free energy associated with the dominant terms in the dynamics shows that this low-energy state is thermodynamically stable under local perturbations. The model can also give rise to an emergent U(1) gauge theory in the ground state by the string-net condensation mechanism of Levin and Wen. We also reformulate the model in graph-theoretic terms and compare its dynamics to some common graph processes.

Quantum Graphity: a model of emergent locality

TL;DR

This work proposes a background-independent quantum model (quantum graphity) in which degrees of freedom live on the edges of the complete graph , and a permutation-invariant Hamiltonian drives a transition from a highly connected, nonlocal high-temperature phase to a low-energy phase with emergent locality. The core mechanism combines a valence-enforcing term with a loop term that favors short cycles, leading to a conjectured ground state with hexagonal (3-regular) lattice structure and thermodynamic stability against local perturbations; extensions to additional edge degrees of freedom enable Levin-Wen string-net condensation and an emergent gauge theory on the emergent background. The paper also reformulates the dynamics in graph-theoretic terms and discusses extensions, perturbations, and comparisons to other graph processes, highlighting open questions about the precise ground state, the nature of the high-to-low transition, and potential connections to continuum gravity. Overall, the model provides a concrete route to geometrogenesis and emergent locality, bridging quantum gravity ideas with condensed-matter style Hamiltonians and suggesting avenues to incorporate matter and gauge structure on dynamical lattices.

Abstract

Quantum graphity is a background independent model for emergent locality, spatial geometry and matter. The states of the system correspond to dynamical graphs on N vertices. At high energy, the graph describing the system is highly connected and the physics is invariant under the full symmetric group acting on the vertices. We present evidence that the model also has a low-energy phase in which the graph describing the system breaks permutation symmetry and appears to be ordered, low-dimensional and local. Consideration of the free energy associated with the dominant terms in the dynamics shows that this low-energy state is thermodynamically stable under local perturbations. The model can also give rise to an emergent U(1) gauge theory in the ground state by the string-net condensation mechanism of Levin and Wen. We also reformulate the model in graph-theoretic terms and compare its dynamics to some common graph processes.

Paper Structure

This paper contains 15 sections, 58 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Graph Models
  3. Graph theory preliminaries
  4. Quantum mechanics preliminaries
  5. Hamiltonian
  6. Valence term
  7. Closed Paths
  8. Interaction Terms
  9. Large Graph Example
  10. Finding the ground state
  11. Graphs Above the Ground State
  12. Statistical Mechanics
  13. Extensions with More Degrees of Freedom
  14. Comparison to Other Graph Dynamics
  15. Discussion

Figures (5)

  • Figure 1: Interaction moves on graphs. (a) Exchange moves preserve the valence of each vertex. For convenience the move between the center and the left is called type I and the move between the center and the right is called type II. (b) Other moves can add or subtract edges, changing the valence of some nodes. This move is called type III.
  • Figure 2: Sample $3$-regular graphs: (a) hexagonal lattice, (b) braided line, and (c) braided tree.
  • Figure 3: The value of the loop energy per vertex, in units of $g_B$, for some sample graphs including the flat hexagonal lattice as a function of cutoff length $L$. The parameter $r$ is set to $7.3$.
  • Figure 4: (a) The hexagonal lattice with a defect of type I. (b) The plot shows the energy differences at points in the hexagonal lattice relative to the reference lattice. The vertical axis is in units of $g_B$.
  • Figure 5: (a) The hexagonal lattice with a defect of type II. (b) The corresponding energy difference plot. The vertical axis is in units of $g_B$.