Spontaneous Symmetry Breaking

Spontaneous symmetry breaking is a phenomenon in physics in which the equations governing a system possess a symmetry that is not shared by a particular ground state or equilibrium phase. The laws do not favor that state over its symmetry-related alternatives, but the system realizes one of them. This distinction between symmetric laws and asymmetric states connects magnetism, collective ordering, and elementary-particle masses, making the concept important in both condensed matter physics and particle physics. (damtp.cam.ac.uk)

Symmetry of laws and states

A symmetry is a transformation that leaves the governing physical description unchanged. In quantum field theory, it may leave the Lagrangian invariant while transforming one vacuum into another. In equilibrium statistical mechanics, the corresponding distinction is between the symmetry of the free energy and that of an equilibrium phase. “Broken” therefore does not mean that the underlying equations cease to obey the symmetry. (damtp.cam.ac.uk)

Using the language of group theory, let $G$ denote the symmetry group of the laws and $H$ the subgroup preserving a chosen state. Symmetry breaking occurs when $H$ is smaller than $G$ . Transformations outside $H$ carry the state into symmetry-related alternatives. A suitable order parameter distinguishes these states through a quantity that transforms nontrivially under the broken symmetry. (damtp.cam.ac.uk)

Spontaneous breaking differs from explicit breaking. An external magnetic field, for example, can favor one magnetization direction by introducing an asymmetric term into the governing description. Spontaneous magnetization instead persists when the external field is removed under the appropriate limiting procedure. (damtp.cam.ac.uk)

Magnetic ordering and a simple model

Ferromagnetism provides a standard illustration. In the zero-field Ising model, reversing every spin leaves the interaction energy unchanged. Above the ordering temperature, the equilibrium magnetization vanishes. Below it, two ordered phases with opposite magnetizations become possible. Either phase individually fails to respect the spin-reversal symmetry, although the symmetry still relates the pair. (damtp.cam.ac.uk)

A simple free-energy density for a scalar magnetization $m$ is

f(m)=f_0+a(T)m^2+bm^4,\qquad b>0.

This expression is invariant under $m\mapsto-m$ . For $a(T)>0$ , its minimum occurs at $m=0$ . For $a(T)<0$ , two minima appear:

m=\pm\sqrt{-\frac{a(T)}{2b}}.

Choosing either minimum produces a nonzero order parameter without adding a symmetry-breaking term. This is a mean-field description of a phase transition; fluctuations can change its quantitative predictions, particularly near a critical point. (damtp.cam.ac.uk)

For a complex order parameter, a potential depending only on its magnitude can instead have a continuous circle of minima. Selecting one phase breaks the global phase-rotation symmetry. This construction underlies descriptions of superfluidity and related ordered systems. (damtp.cam.ac.uk)

The thermodynamic limit

The mathematical definition requires care. In a finite zero-field Ising system with symmetry-preserving boundary conditions, the equilibrium average magnetization is zero because configurations with opposite magnetizations have equal statistical weights. A symmetric average can therefore conceal the distinct ordered phases found in an infinite system. (damtp.cam.ac.uk)

A standard definition introduces a small symmetry-breaking field $h$ , takes the thermodynamic limit, and only afterward removes the field:

m_{\mathrm{sp}} =\lim_{h\to0^+}\lim_{V\to\infty}\langle m\rangle_{V,h}.

A nonzero result identifies spontaneous ordering. Reversing these limits generally gives zero in the symmetric finite-volume ensemble. The weak field selects a phase; its continued presence is not required after the infinite-volume limit. This explains why the order of limits is essential rather than merely a calculational detail. (damtp.cam.ac.uk)

Continuous symmetries and collective excitations

Breaking a continuous global symmetry has consequences beyond producing an order parameter. Slowly varying the state along symmetry-related directions costs little energy, giving rise to low-energy collective excitations called Nambu–Goldstone modes. Goldstone’s theorem relates these excitations to the broken continuous symmetry. In relativistic theories satisfying its assumptions, they are massless; discrete symmetry breaking does not require such modes. (damtp.cam.ac.uk)

Fluctuations also constrain whether ordering is possible. The Mermin–Wagner theorem rules out conventional spontaneous breaking of continuous symmetries at nonzero temperature in one- and two-dimensional systems under its short-range-interaction assumptions. These restrictions depend on dimensionality and interaction structure, not simply on whether a classical potential has asymmetric minima. (damtp.cam.ac.uk)

Gauge theories and the Higgs mechanism

In the Standard Model, the Higgs field has a nonzero vacuum expectation value in the usual gauge-fixed description. The Higgs mechanism gives the W and Z bosons their masses while retaining the gauge-invariant structure of the theory. The Higgs boson is the associated physical scalar excitation; its discovery in 2012 provided experimental support for this mechanism. (home.web.cern.ch)

The phrase “spontaneous breaking of gauge symmetry” requires qualification. Local transformations in a gauge theory describe redundancies, rather than physically distinct configurations. Elitzur’s theorem forbids spontaneous breaking of a local gauge symmetry without gauge fixing. A gauge-dependent field expectation value is therefore not, by itself, a gauge-invariant observable. The physical content of the Higgs mechanism must be expressed through observable particle spectra and interactions. (arxiv.org)

Historical development

Yoichiro Nambu transferred ideas associated with superconductivity into subatomic physics, developing spontaneous symmetry breaking as a mechanism for understanding particle phenomena. Subsequent work established its role in modern field theories. In 2008, Nambu received half of the Nobel Prize in Physics for discovering the mechanism of spontaneous broken symmetry in subatomic physics. (nobelprize.org)