spontaneous symmetry breaking

Balance a pencil perfectly on its tip. The laws governing it are the same in every horizontal direction — there is no preferred way for it to fall. Yet fall it must, and when it does it picks one direction, breaking the very symmetry the laws respect. A magnet does the same: below the Curie temperature it settles on a definite direction of magnetization, though the underlying physics has no built-in preference. This is spontaneous symmetry breaking — the laws are symmetric, but the realized state is not.

Precisely, spontaneous symmetry breaking occurs when the Hamiltonian (or Lagrangian) of a system possesses a symmetry that its ground state or equilibrium state does not share. The Ising Hamiltonian is exactly invariant under flipping every spin, s to -s, but below T_c the system must choose either overall positive or overall negative magnetization; either choice breaks the symmetry, and the order parameter (here M) measures the breaking. Crucially this requires the thermodynamic limit of infinitely many degrees of freedom: in a finite system, tunneling between the symmetry-related states restores the symmetry on average, and only in the infinite-volume limit do the broken states become genuinely stable and disconnected. When a CONTINUOUS symmetry is broken, Goldstone's theorem guarantees gapless (massless) excitations — the Goldstone modes, such as spin waves in a Heisenberg ferromagnet; when the broken symmetry is a gauge symmetry, those would-be Goldstone modes are eaten and the gauge bosons acquire mass, the Higgs mechanism.

Spontaneous symmetry breaking is one of the great unifying ideas of modern physics, underlying ferromagnetism, superfluidity, superconductivity, crystallization, and the generation of particle masses in the Standard Model. An honest clarification of the word 'spontaneous': the symmetry is not broken by any explicit non-symmetric term in the Hamiltonian. The standard way to make this precise is to apply an infinitesimal external field to select one state, and then take the field to zero AFTER the infinite-volume limit — the two limits do not commute, and that non-commutation is exactly what makes the breaking robust rather than a mere accident.