Emergence of helicity ±2 modes (gravitons) from qubit models

We construct two quantum qubit models (or quantum spin models) on three-dimensional lattice in space, L-type model and N-type model. We show that, under a controlled approximation, all   the low energy excitations of the L-type model are described by one set of helicity ±2 modes with ω∝k³$\omega \propto k^3$ dispersion. We also argue that all   the low energy excitations of the N-type model are described by one set of helicity ±2 modes with ω∝k$\omega \propto k$ dispersion. In both model, the low energy helicity ±2 modes can be described by a symmetric tensor field _hμν$h_{\mu \nu }$ in continuum limit, and the gaplessness of the helicity ±2 modes is protected by an emergent linearized diffeomorphism gauge symmetry _hμν→_hμν+_∂μfν+_∂νfμ$h_{\mu \nu }\to h_{\mu \nu }+{\partial }_{\mu }{f}_{\nu }+{\partial }_{\nu }{f}_{\mu }$ at low energies. Thus the linearized quantum gravity emerge from our lattice models  . It turns out that the low energy effective Lagrangian density of the L-type model is invariant under the linearized diffeomorphism gauge transformation. Such a property protects the gapless ω∝k³$\omega \propto k^3$ helicity ±2 modes. In contrast, the low energy effective Lagrangian of the N-type model changes by a boundary term under the linearized diffeomorphism gauge transformation. Such a property protects the gapless ω∝k$\omega \propto k$ helicity ±2 modes. From many-body physics point of view, the ground states of the our two qubit model represent new states of quantum matter, whose low energy excitations are all described by one set of gapless helicity ±2 modes.