Final answer:
Phonons, which are quantized vibrational modes in materials, do exist in 2D solids despite the limitation on long-range order imposed by the Mermin-Wagner theorem. This is because phonons are a result of short-range order and quantized vibrational energy levels in the lattice, which can occur independently of the symmetry breaking that the theorem addresses.
Step-by-step explanation:
The question you've asked relates to whether phonons (quantized modes of vibrations, often referred to as quasiparticles, that transport sound and heat) exist in two-dimensional (2D) solids, given that the Mermin-Wagner theorem suggests that spontaneous symmetry breaking is not possible in systems with short-range interactions in dimensions lower than three. While the Mermin-Wagner theorem does indicate that long-range order and associated spontaneous symmetry breaking cannot occur in 1D and 2D systems at finite temperatures due to thermal fluctuations, this does not altogether prevent the existence of phonons. Phonons are collective excitations that arise due to the quantization of vibrational modes in a lattice, and they can exist in materials of any dimensionality.
In 2D materials, phonons do indeed exist and play a significant role in their thermal and mechanical properties. While true long-range order (as required for Goldstone bosons in spontaneous symmetry breaking) isn't sustained, these materials can still support short-range order and exhibit quantized vibrational energy levels, resulting in phononic excitations. Thus, phonons can be observed in 2D materials despite the restrictions implied by the Mermin-Wagner theorem on spontaneous long-range order and symmetry breaking.