Coleman’s Theorem

Coleman’s Theorem states that in certain two-dimensional quantum field theories with continuous symmetries, spontaneous symmetry breaking cannot occur at finite temperature. In simpler terms, while in higher dimensions a system can choose a specific state that breaks a symmetry (like a magnet aligning in one direction), in two dimensions, thermal fluctuations are too strong and prevent this from happening. As a result, the system remains symmetric, and no long-range order develops. This theorem highlights the unique behaviors of low-dimensional systems, emphasizing that phase transitions involving symmetry breaking are fundamentally different in two-dimensional settings.