  The quantum Mpemba effect is the counter-intuitive non-equilibrium phenomenon
wherein the dynamic restoration of a broken symmetry occurs more rapidly when
the initial state exhibits a higher degree of symmetry breaking. The effect has
been recently discovered theoretically and observed experimentally in the
framework of global quantum quenches, but so far it has only been investigated
in one-dimensional systems. Here we focus on a two-dimensional free-fermion
lattice employing the entanglement asymmetry as a measure of symmetry breaking.
Our investigation begins with the ground state analysis of a system featuring
nearest-neighbor hoppings and superconducting pairings, the latter breaking
explicitly the $U(1)$ particle number symmetry. We compute analytically the
entanglement asymmetry of a periodic strip using dimensional reduction, an
approach that allows us to adjust the extent of the transverse size, achieving
a smooth crossover between one and two dimensions. Further applying the same
method, we study the time evolution of the entanglement asymmetry after a
quench to a Hamiltonian with only nearest-neighbor hoppings, preserving the
particle number symmetry which is restored in the stationary state. We find
that the quantum Mpemba effect is strongly affected by the size of the system
in the transverse dimension, with the potential to either enhance or spoil the
phenomenon depending on the initial states. We establish the conditions for its
occurrence based on the properties of the initial configurations, extending the
criteria found in the one-dimensional case.
