Spontaneous Symmetry Breaking Preview

Spontaneous symmetry breaking is the situation where the equations or Hamiltonian have a symmetry, but the physically relevant state, phase, or long-distance description does not display the full symmetry.

The phrase is easy to misuse. In ordinary finite quantum mechanics, a state can fail to be invariant under a symmetry even when the Hamiltonian is symmetric. That alone is not the many-body phenomenon called spontaneous symmetry breaking.

The essential many-body idea is:

\text{symmetric dynamics} \quad \text{plus an appropriate large-system limit} \quad \Longrightarrow \quad \text{stable symmetry-broken phases}.

This page is a preview. It explains the conceptual structure and common traps. Spontaneous Symmetry Breaking is the canonical many-body treatment of source-selected phases, finite-size cat states, discrete quasi-degeneracy, and continuous towers of states. Order Parameters owns the operator and diagnostic dictionary, while Thermodynamic Limit owns the physics-level limiting procedure. Rigorous algebraic theory, superconductivity, magnetism, and QFT vacuum structure retain their own canonical homes.

Exact Symmetry Remains in the Equations

Start with a family of finite systems whose Hamiltonians obey

U(g)H_NU(g)^{-1} = H_N

for every symmetry element $g$ in a group $G$ . The Hamiltonian symmetry is exact at every finite size $N$ .

Spontaneous symmetry breaking is not the statement

U(g)H_NU(g)^{-1}\ne H_N.

That would be explicit symmetry breaking.

Instead, the symmetry remains in the Hamiltonian, while the relevant limiting states preserve only a subgroup $H\subset G$ :

G \longrightarrow H.

The broken generators or broken group directions label transformations that move one symmetry-broken state into another physically distinct state in the same phase.

Why Finite Systems Are Subtle

For a finite system with a unique ground state, an exact symmetry of the Hamiltonian usually implies that the ground state itself is symmetric, up to a phase. If

H\lvert\Omega\rangle = E_0\lvert\Omega\rangle

is nondegenerate and $[H,U(g)]=0$ , then

U(g)\lvert\Omega\rangle

is another ground state with the same energy. Nondegeneracy forces

U(g)\lvert\Omega\rangle = e^{i\phi(g)} \lvert\Omega\rangle.

Thus a finite, nondegenerate ground state cannot literally choose one of several symmetry-related ordered states.

What changes in many-body systems is the large- $N$ or infinite-volume limit. Energy splittings between symmetry-related combinations can vanish with size, local measurements can become insensitive to the symmetric superposition, and tiny symmetry-breaking fields can select one branch.

Order Parameter

An order parameter is an observable or field whose expectation value diagnoses the broken symmetry. If $O$ transforms nontrivially under $G$ , a symmetry-broken state may have

\langle O\rangle\ne0.

For a discrete spin-flip symmetry, a magnetization component can be an order parameter. If a spin-flip operation sends

M_z \mapsto -M_z,

then a phase with

\langle M_z\rangle \ne 0

does not preserve that spin-flip symmetry.

For a system with a continuous $U(1)$ symmetry, a complex order parameter can choose a phase:

\langle\Psi\rangle = |\Psi|e^{i\theta}.

Different values of $\theta$ are related by the original symmetry. A finite symmetric state may have $\langle\Psi\rangle=0$ , while a symmetry-broken thermodynamic-limit state has a chosen phase after the correct limiting procedure.

Order of Limits

A common way to express spontaneous symmetry breaking is to add a small source $h$ that explicitly selects one branch:

H_{N,h} = H_N-hO.

One first takes the large-system limit, then removes the source:

\lim_{h\to0^+} \lim_{N\to\infty} \langle O\rangle_{N,h}.

This can differ from taking $h\to0$ first at fixed finite $N$ :

\lim_{N\to\infty} \lim_{h\to0^+} \langle O\rangle_{N,h}.

The noncommuting limits are not a technical nuisance. They are the mathematical signal that an infinitesimal perturbation can select a stable ordered phase only after the system is large enough.

Discrete Versus Continuous Breaking

Breaking a discrete symmetry can produce multiple ordered phases, domain walls, and near-degenerate finite-size structures. It does not by itself imply a Goldstone mode.

Breaking a continuous global symmetry is different. Under suitable locality and stability assumptions, it leads to low-energy collective excitations. These are Nambu–Goldstone modes. The physical preview is Goldstone Modes Preview, and the compact theorem card is Goldstone Theorem Preview.

The word “global” is important. Gauge redundancy is not broken in the same direct sense as a physical global symmetry. Gauge theories, superconductors, and Higgs-like systems require separate treatment.

Examples and Nonexamples

Useful examples include:

an Ising ferromagnet choosing positive or negative magnetization below a critical temperature;
a Heisenberg ferromagnet choosing a magnetization direction and breaking spin-rotation symmetry;
a neutral superfluid or Bose condensate choosing a phase in an effective $U(1)$ order parameter;
a crystal choosing a lattice of positions and breaking continuous translations to a discrete subgroup.

Useful nonexamples include:

a single spin in the state $|+z\rangle$ while the Hamiltonian is zero;
one localized wave packet in a symmetric finite double well;
an explicitly applied magnetic field selecting an axis;
a gauge choice that makes a field component look nonzero.

The nonexamples may be excellent approximations or useful pictures. They are not, by themselves, spontaneous symmetry breaking in the many-body sense.

Relation to Explicit Breaking

Explicit breaking and spontaneous breaking often appear together in practice. A tiny external field can select a branch of an ordered phase. The field is explicit breaking; the stable ordered response in the large-system limit is the spontaneous-breaking phenomenon.

If a continuous symmetry is explicitly but weakly broken, the would-be Goldstone mode can acquire a small gap. Such excitations are often called pseudo-Goldstone modes. The weak-breaking language is an approximate symmetry statement, while the source term itself is explicit breaking. The exact terminology and counting rules depend on the physical system.

What This Preview Does Not Cover

This preview does not derive:

phase transitions or critical exponents;
rigorous thermodynamic-limit constructions;
the full Goldstone theorem;
topological defects;
superconductivity and the Anderson–Higgs mechanism;
chiral symmetry breaking in QCD;
algebraic QFT superselection and inequivalent representations.

Those subjects require many-body, statistical, or field-theoretic machinery. Quantum Phase Transitions develops general zero-temperature criticality and gap scaling, while the Transverse-Field Ising Model owns that chain’s finite-size symmetry sectors and thermodynamic ordered phase. The purpose here is to keep the symmetry-language distinction clean.

Common Mistakes

Saying “the state is not symmetric” and calling that spontaneous symmetry breaking in a finite system.
Forgetting the thermodynamic-limit or infinite-volume assumption.
Confusing a tiny explicit field that selects a branch with the spontaneous phenomenon itself.
Applying Goldstone theorem to a broken discrete symmetry.
Treating gauge redundancy as an ordinary global symmetry.
Assuming a finite symmetric ground state and a symmetry-broken thermodynamic state are contradictory.
Using an order parameter without stating how it transforms.

Cross-Links

References

P. W. Anderson, “Plasmons, gauge invariance, and mass,” Physical Review 130, 439-442, 1963.
J. Goldstone, “Field theories with superconductor solutions,” Il Nuovo Cimento 19, 154-164, 1961.
J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Physical Review 127, 965-970, 1962.
Y. Nambu, “Quasi-particles and gauge invariance in the theory of superconductivity,” Physical Review 117, 648-663, 1960.
P. Coleman, Introduction to Many-Body Physics, Cambridge University Press, 2015.
A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
S. Weinberg, The Quantum Theory of Fields, Volume II, Cambridge University Press, 1996.

Exercises

Why does a unique finite-system ground state usually preserve an exact symmetry?

Assume $[H,U]=0$ and $H|\Omega\rangle=E_0|\Omega\rangle$ , with $E_0$ nondegenerate.

Solution

Because $[H,U]=0$ ,

H(U|\Omega\rangle) = UH|\Omega\rangle = E_0(U|\Omega\rangle).

Thus $U|\Omega\rangle$ is also a ground state. If the ground state is nondegenerate, it must be proportional to $|\Omega\rangle$ :

U|\Omega\rangle = e^{i\phi}|\Omega\rangle.

The state is symmetric as a ray. A literal choice among distinct symmetry-related ground states requires degeneracy or a limiting procedure.

Explain the order of limits in a source-selected order parameter.

Why can

\lim_{h\to0^+}\lim_{N\to\infty}\langle O\rangle_{N,h}

differ from

\lim_{N\to\infty}\lim_{h\to0^+}\langle O\rangle_{N,h}?

Solution

At finite $N$ , removing the source first can restore the exact symmetry and force a symmetry-odd order parameter to vanish. Taking $N\to\infty$ first can make different ordered branches stable and separated by an effectively infinite barrier. Then an infinitesimal source can select one branch, and the order parameter can remain nonzero after the source is sent to zero.

Does breaking a discrete spin-flip symmetry imply a Goldstone mode?

Solution

No. Goldstone modes are tied to broken continuous global symmetries under appropriate assumptions. A discrete symmetry has separated alternatives, not a continuous family of nearby states. Discrete symmetry breaking can produce domain walls or nearly degenerate finite-size states, but not a Goldstone mode merely from discreteness.

Identify explicit versus spontaneous breaking.

A tiny magnetic field $h$ is applied to a ferromagnet and then sent to zero after the thermodynamic limit. Which part is explicit breaking, and which part signals spontaneous breaking?

Solution

The finite $h$ term in the Hamiltonian is explicit symmetry breaking: it directly selects a magnetization direction. If, after taking the thermodynamic limit, the magnetization remains nonzero as $h\to0^+$ , that persistent ordered response is the spontaneous symmetry-breaking signal.