A quantum Bose-Hubbard model with evolving graph as toy model for emergent spacetime
Alioscia Hamma,
1,
*
Fotini Markopoulou,
1
Seth Lloyd,
2
Francesco Caravelli,
1, 3
Simone Severini,
4
and Klas Markstr¨om
5
1
Perimeter Institute for Theoretical Physics, 31 Caroline St. N, N2L 2Y5, Waterloo ON, Canada
†
2
Massachusetts Institute of Technology, 77 Massachusetts Ave. Cambridge MA 02139
‡
3
University of Waterloo, 200 University Ave W, Waterloo ON, N2L 3G, Canada
§
4
Department of Physics and Astronomy, University College London, WC1E 6BT London, United Kingdom
¶
5
Department of Mathematics and Mathematical Statistics, Ume universitet, S-901 87 Umea, Sweden
We present a toy model for interacting matter and geometry that explores quantum dynamics in a spin system
as a precursor to a quantum theory of gravity. The model has no a priori geometric properties, instead, locality is
inferred from the more fundamental notion of interaction between the matter degrees of freedom. The interaction
terms are themselves quantum degrees of freedom so that the structure of interactions and hence the resulting
local and causal structures are dynamical. The system is a Hubbard model where the graph of the interactions
is a set of quantum evolving variables. We show entanglement between spatial and matter degrees of freedom.
We study numerically the quantum system and analyze its entanglement dynamics. We analyze the asymptotic
behavior of the classical model. Finally, we discuss analogues of trapped surfaces and gravitational attraction in
this simple model.
PACS numbers:
INTRODUCTION
The quest for a quantum theory of gravity involves searching for a microscopic quantum theory whose low energy limit is the
known physics of general relativity, dynamical space-time metrics whose evolution is governed by the Einstein equations. Many
approaches, like loop quantum gravity [1], Causal Dynamical Triangulations (CDT) [2], and spin foams [3], expect that the
quantum theory of gravity becomes manifest at very high energy. That is, quantum analogs of gravitational properties such as
the quantum Hilbert-Einstein action or Lorentz invariance are built into the high energy theory. Alternatively, one could regard
quantum gravity as an emergent phenomenon from the low energy theory of a condensed matter system. In this approach, the
fundamental theory would have no gravity, no diffeomorphism invariance or any other property of general relativity.
In recent years, there has been a growing attention to the notion of gravity as an emergent phenomenon [4]. From Aristotle to
Philip Anderson, a long-standing tradition in physics asserts that ”The whole is more than the sum of its parts” and that ”More
is different”. The emergent approach is concerned with the study of the macroscopic properties of systems with many bodies.