In quantum theory, both quantum mechanics and quantum field theory, symmetries of the action or Hamiltonian play a huge role. Symmetries can be either continuous and local, like gauge symmetry in quantum field theory, or discrete and global. In the case of quantum field theory, phase transitions occur when the parameters of the theory (temperature) change, violating the original symmetries. The properties of an effective theory in different phases turn out to be fundamentally different. The simplest example illustrating this is the 2D Ising model, in which a second-order phase transition occurs. In the high-temperature phase, the average magnetization is zero, $Z_2$ symmetry is preserved, while in the low-temperature phase, the $Z_2$ symmetry is broken and the average magnetization is nonzero.

In this paper, we study discrete quantum gravity defined on a 4D irregular lattice (simplicial complex). This theory is a mathematical model of the following physical idea: at extremely small scales, space-time exhibits granularity.

The model includes gravitational degrees of freedom as well as a Dirac fields $\Psi_{\cal V}$ and $\Psi^{\dagger}_{\cal V}$. The bare action of the theory has a global discrete $Z_4$ symmetry. The fermionic part of this action, which is bilinear relative to Dirac fields, can be rewritten in terms of effective fermionic variables $\Phi_{\cal V}$ and $\Phi^{\dagger}_{\cal V}$. These are linear combinations of the Dirac field and its conjugate at each vertex of the lattice. It can be said that the effective Dirac variables are analogous to the quasiparticle operators in superconductivity theory, constructed using Bogolyubov transformations.

A peculiarity of the construction of effective fermion operators in the lattice theory of gravity is the following fact: "particle" operators $\Phi_{\cal V}$ are invariant under $Z_4$ transformations, but their Hermitian conjugate operators $\Phi^{\dagger}_{\cal V}$ are transformed according to the simplest non-trivial representation of the group $Z_4$.

In the model under study, high-temperature expansion is correct at the highest temperatures.

As in the Ising model, it is established in this way that at the highest temperatures a $Z_4$-symmetrical phase occurs. As the temperature decreases, the $Z_4$ symmetry is broken down to its $Z_2$ symmetry subgroup. The order parameter here is the contribution to the lattice action, which transforms into the Hilbert-Einstein action in the long-wave limit. This contribution preserves $Z_2$-symmetry, but it breaks the $Z_4$-symmetry.

As the temperature decreases further, another phase transition occurs, breaking the $Z_2$-symmetry. The corresponding order parameter is a tetrad, the mean of which becomes nonzero. Spacetime emerges.

S.N. Vergeles
JETP Letters 123, issue 3 (2026)