Symmetry Realization and Order Parameters

An exact symmetry of a theory need not leave every state invariant. If the full quantum theory has a global symmetry $G$ and a selected state is fixed by all of $G$, the symmetry is unbroken in that state; if the state preserves only a proper subgroup, the symmetry is spontaneously broken there. By contrast, explicit, approximate, and emergent symmetry describe the dynamics at fixed theory data or in a specified limiting regime. They are not alternative names for a noninvariant state.

An order parameter is a physical observable used to diagnose this distinction. An expectation value changed by at least one element of $G$ is sufficient to show that the state is not $G$-invariant, but one vanishing candidate is never sufficient to prove that the state is symmetric. Robust spontaneous breaking is an infinite-system, phase-selection statement; a classical minimum or a generic finite-volume ground-state vector does not establish it. This page develops that classification for ordinary internal global symmetries and local observables. Detailed limits, Goldstone conclusions, renormalization-group emergence, and gauge theories have separate hypotheses and are handed off below.

Required background. What Is a Symmetry of a QFT? supplies the action on physical states and operators and the distinction between an exact quantum symmetry and a symmetric-looking Lagrangian. Vacua, States, and Representations distinguishes a state from the algebra and Hilbert-space representation used to describe it.

Helpful background. Clustering, Vacuum Assumptions, and Long-Range Correlations provides the long-distance qualifications used when a phase is selected.

Theory symmetry versus state symmetry

Let an ordinary global group $G$ act on the physical operator algebra by automorphisms $\alpha_g$. Calling $G$ an exact symmetry means that this action preserves the complete quantum theory at fixed couplings, sources, boundary conditions, and other defining data. Invariance of a classical action is evidence for such an action, but it is not enough if the regulator, measure, or renormalized Ward identity violates the transformation even without background fields. A ’t Hooft anomaly is different: the global symmetry can remain exact while its consistent gauging is obstructed. Anomalies, Inflow, and Matching develops that distinction.

A state is a normalized positive functional $\omega$ on the operator algebra. Its preserved subgroup is

$$\begin{aligned} H_\omega = \{g\in G\mid {}& \omega\!\left(\alpha_g(X)\right)=\omega(X) \\ &\text{for every physical }X\}. \end{aligned}$$

For a pure vector state in a representation where the symmetry is unitarily implemented, the same condition says that $U(g)|\Omega\rangle$ differs from $|\Omega\rangle$ by at most a phase. The operator-algebra definition is more general and makes the quantifier “every physical $X$” visible.

The two basic realizations of an exact symmetry are now precise:

$$\begin{aligned} H_\omega=G &\Longrightarrow \text{unbroken in }\omega,\\ H_\omega\subsetneq G &\Longrightarrow \text{spontaneously broken in }\omega. \end{aligned}$$

where the second line is used for a selected infinite-volume phase, normally taken to be extremal or pure and to satisfy the appropriate clustering condition. The dynamics remains $G$-symmetric: $G$ maps the chosen state to other states on its orbit. What fails is invariance of this particular state. The state-based distinction and its infinite-volume character are developed in Weinberg 1995, § 19.1, pp. 163–167 and Tong 2019, § 2 opening and § 2.1, pp. 48–55, official PDF.

Five realization labels

The labels answer different questions, so a complete statement names the group, the theory data, the state when relevant, and the scale or limit.

Exact and unbroken. The quantum theory has exact symmetry $G$, and the selected state has $H_\omega=G$.

Exact and spontaneously broken. The quantum theory has exact symmetry $G$, but a selected infinite-volume phase has $H_\omega\subsetneq G$.

Explicitly broken. The fixed theory is not $G$-invariant, although it may retain an exact subgroup $K\subset G$. One may separately ask whether the selected state breaks $K$.

Approximate. The theory is a controlled deformation of an exact-$G$ reference theory. Specified $G$-violating observables must be bounded by a named small parameter in a named regime.

Emergent. The microscopic theory lacks the full group $G_{\mathrm{IR}}$, but long-distance observables approach $G_{\mathrm{IR}}$ covariance with controlled symmetry-breaking corrections.

An approximate symmetry is meaningful only relative to a comparison. Schematically, if

$$S=S_0+\epsilon\!\int\!\mathrm d^d x\,\mathcal B(x),$$

with $S_0$ exactly $G$-invariant, one must identify a dimensionless measure of $\epsilon$ at the scale of interest and show which predictions receive small $G$-violating corrections. A small bare coefficient can grow under renormalization, so “small in the Lagrangian” does not by itself mean “approximately symmetric at every scale.” Controlled symmetry-breaking perturbations are discussed in Weinberg 1995, § 19.3, pp. 177–181.

Emergence reverses the comparison. The microscopic theory may have only $K$, while its long-distance limit approaches a theory with $G_{\mathrm{IR}}\supset K$. At any finite scale the enlarged symmetry may remain approximate. Establishing it requires an infrared limit, a set of observables, and control of the operators that violate $G_{\mathrm{IR}}$; it does not follow merely because the leading term in an effective action looks symmetric. A bounded critical-point example of symmetry enhancement is described in Zinn-Justin 2021, § 16.6.1, p. 405. The renormalization-group mechanism belongs to the Renormalization and Effective Field Theory volume.

Order parameters, stabilizers, and correlators

Let physical local operators $\mathcal O_i$ transform in a representation $R$ of $G$,

$$\alpha_g(\mathcal O_i) =R(g)_i{}^j\mathcal O_j, \qquad v_i=\omega(\mathcal O_i).$$

The expectation-value vector has stabilizer

$$H_v = \{g\in G\mid R(g)v=v\}.$$

If $g$ preserves the whole state, then it preserves this particular expectation value. Therefore

$$H_\omega\subseteq H_v.$$

This inclusion is the right logical direction. If $R(g)v\neq v$, then $g\notin H_\omega$: a transforming nonzero expectation value proves that the state breaks that transformation. But $H_v$ can be larger than $H_\omega$ because one multiplet may miss symmetry breaking visible in another operator, a composite, or a higher correlation function. Equality requires a diagnostic set rich enough to determine the symmetry of the state. In particular, $v=0$ gives $H_v=G$ automatically and proves nothing by itself about $H_\omega$.

For a $U(1)$-charged operator $X_Q$, invariance of the state would imply

$$\omega(X_Q) =e^{iQ\alpha}\omega(X_Q) \qquad\text{for every }\alpha.$$

Thus a nonzero expectation value with $Q\neq0$ is incompatible with an invariant state. Conversely, $\omega(X_Q)=0$ may follow from symmetry, but it may also occur because the chosen operator is a poor diagnostic.

In a translationally invariant clustering phase, a local order parameter obeys the long-distance check

$$\lim_{|\mathbf x-\mathbf y|\to\infty} \omega\!\left(\mathcal O^\dagger(t,\mathbf x) \mathcal O(t,\mathbf y)\right) =|\omega(\mathcal O)|^2,$$

with the separation and state assumptions stated explicitly. A symmetric mixture of differently oriented phases can instead have $\omega(\mathcal O)=0$ while retaining a nonzero two-point plateau; it then fails this clustering factorization. The plateau reveals ordered phase structure, but it does not make the mixed state itself noninvariant. This is why phase selection, state purity, and the order parameter must be discussed together.

The operator must also be physical. A gauge-variant elementary field expectation value is not, by itself, an order parameter for a physical global symmetry. The distinction is developed on Elitzur’s Theorem and the Gauge-Invariant Higgs Mechanism. Within the present ordinary-global-symmetry scope, order parameters can be elementary or composite; Weinberg 1995, §§ 19.1–19.2, pp. 163–169 gives both the state-based construction and a concrete stabilizer analysis.

Why finite volume does not settle the question

At finite spatial volume, a unique exact ground state commonly respects an exact internal symmetry, and tunneling can combine semiclassically distinct configurations into a symmetric state. A small selector $J$ can expose a different ordered limit:

$$\begin{aligned} v(V,J) &=\langle\mathcal O\rangle_{V,J},\\ \lim_{J\to0^+}\lim_{V\to\infty} v(V,J) &\neq0,\\ \lim_{V\to\infty}\lim_{J\to0} v(V,J) &=0. \end{aligned}$$

The two orders need not agree. The zero on the last line assumes the usual unique symmetric finite-volume condition. Exact or protected degeneracies can occur even at finite volume, so the claim is not that every finite system has a unique symmetric ground state. Rather, robust spontaneous breaking is assigned only after the infinite-system phase and its selection prescription are specified. Finite Volume, Thermodynamic Limits, and Pure Phases develops the selector, quasi-degenerate states, clustering, and ordered limits; here they serve only as a safeguard on the definition.

One complex scalar across all five cases

Consider a complex scalar in four-dimensional Minkowski spacetime,

$$\begin{aligned} \mathcal L_0 &= \partial_\mu\phi^*\partial^\mu\phi -m^2|\phi|^2 \\ &\quad -\frac{\lambda}{2}|\phi|^4, \qquad \lambda>0. \end{aligned}$$

and the global transformation $\phi\mapsto e^{i\alpha}\phi$. Assume that this transformation survives in the quantum theory and that no quantum Ward-identity violation or other defining datum removes it.

Unbroken realization. For $m^2>0$ in the standard weakly coupled regime, the symmetric state has $\omega(\phi)=0$ and $H_\omega=U(1)$. The vanishing expectation value is consistent with the conclusion, but it is not the proof: invariance is the statement about every physical observable.

Spontaneously broken realization. For $m^2<0$, the classical potential

$$V_0(\phi) =m^2|\phi|^2+\frac{\lambda}{2}|\phi|^4$$

has stationary minima at

$$|\phi|^2=-\frac{m^2}{\lambda}.$$

This circle is only a semiclassical candidate for the quantum vacuum structure. If quantum effects preserve a stable nonzero-radius minimum and an infinite-volume phase is selected, one may have

$$\omega_\theta(\phi) =\frac{v_R}{\sqrt2}e^{i\theta}, \qquad v_R\neq0.$$

Because $\phi$ has charge one, the stabilizer of this expectation value is trivial, so $H_{\omega_\theta}$ is trivial as well. The exact $U(1)$ maps $\omega_\theta$ to other states rather than disappearing from the theory. The classical circle and its quantum qualifications are presented in Tong 2019, § 2.2, pp. 58–61, official PDF; the detailed orbit geometry belongs to Vacuum Orbits and Unbroken Subgroups.

A selector is not the same as a permanent deformation. Add a temporary linear source $J^*\phi+J\phi^*$, take the infinite-volume limit with its orientation fixed, and only then remove $J$. This is a prescription for selecting $\omega_\theta$. If $J$ remains nonzero, however, the fixed theory is explicitly broken; it is no longer an example of spontaneous breaking of exact $U(1)$.

Explicit and approximate $U(1)$. Instead add the physical deformation

$$\Delta\mathcal L =h\phi^N+h^*(\phi^*)^N, \qquad N\geq2.$$

For fixed $h\neq0$, the exact $U(1)$ phase-rotation symmetry is explicitly reduced to the subgroup satisfying $e^{iN\alpha}=1$, namely $\mathbb Z_N$. Assigning $h$ the formal charge $-N$ makes a family of theories transform covariantly, but this spurion convention does not restore $U(1)$ as a symmetry of the one theory with $h$ fixed. Assume throughout that the deformed potential is stable in the regime considered. For $N=4$ with the displayed normalization, a strictly stable quartic polynomial requires $|h|<\lambda/4$; for $N>4$ in four dimensions, the anisotropy is naturally interpreted within an effective theory with a declared cutoff or an appropriate ultraviolet completion, not as an unrestricted renormalizable potential. If the dimensionless effect of $h$ is demonstrably small for specified observables and scales, $U(1)$ is also an approximate symmetry relative to the $h=0$ theory.

The exact residual $\mathbb Z_N$ can itself be spontaneously broken by a selected phase. It is therefore consistent to say both “$U(1)$ is explicitly broken to $\mathbb Z_N$” and “the exact $\mathbb Z_N$ is spontaneously broken,” provided the group and the state are named in each statement.

Possible emergent $U(1)$. If a renormalization-group analysis shows that the $N$-fold anisotropy is irrelevant at a specified infrared fixed point, long-distance observables may approach a larger $U(1)$ phase-rotation symmetry even though the microscopic phase-rotation subgroup is only $\mathbb Z_N$. Other discrete symmetries of the microscopic scalar model are a separate part of its full symmetry group. The enhancement is conditional: the anisotropy need not be irrelevant, and a small microscopic $h$ need not remain harmless in the infrared. The critical dynamics and correction exponents lie outside this page.

Common failure modes

“The Lagrangian is invariant, so the quantum symmetry is exact.” One must also check the quantum measure, regulator, renormalized operator relations, and boundary data. A quantum Ward-identity violation can remove the proposed symmetry, whereas a ’t Hooft anomaly can leave the global symmetry exact while obstructing its gauging.

“A circle of classical minima is a vacuum manifold of the quantum theory.” The classical potential identifies candidate configurations. Quantum stability and an infinite-volume state-selection prescription are additional requirements.

“The order parameter vanishes, so the symmetry is unbroken.” A single expectation value supplies only one diagnostic. Another local composite, a higher correlator, or a nonlocal observable may detect the realization.

“The stabilizer of one expectation value is the preserved group.” In general $H_\omega\subseteq H_v$. Equality requires a complete enough family of diagnostics.

“A small breaking coefficient gives an approximate symmetry.” Smallness must be dimensionless and tied to a scale and observable class. Renormalization can amplify a symmetry-breaking perturbation.

“An enlarged symmetry of the leading infrared formula is emergent.” Emergence requires a controlled limiting statement and suppressed symmetry-breaking corrections, not just a convenient truncation.

“A gauge-fixed scalar expectation value breaks gauge symmetry.” Gauge redundancy is not an ordinary physical global symmetry, and a gauge-variant expectation value is not a standalone physical order parameter.

Classify the realization

These questions are for self-study and are not graded.

1. In the scalar model, classify (a) the $h=0$ selected state with $\omega(\phi)\neq0$, (b) the theory at fixed $h\neq0$, and (c) the same fixed-$h$ theory if its selected state also fails to preserve the residual subgroup.
2. Let a local operator of integer charge $q\neq0$ have a nonzero expectation value in a theory with exact $U(1)$. What can be concluded about the preserved subgroup from this one diagnostic?

Check

1. In (a), the theory has exact $U(1)$ while the selected infinite-volume state spontaneously breaks it. In (b), fixed $h$ explicitly breaks $U(1)$ to exact $\mathbb Z_N$; $U(1)$ may additionally be approximate only after a controlled error estimate. In (c), the exact $\mathbb Z_N$ is spontaneously broken in the selected state, even though the larger $U(1)$ remains explicitly broken.
2. The expectation value is fixed by $e^{iq\alpha}=1$, so its stabilizer is $\mathbb Z_{|q|}$. Therefore $H_\omega\subseteq\mathbb Z_{|q|}$, and every transformation outside $\mathbb Z_{|q|}$ is certainly broken. Equality cannot be inferred from this operator alone because other observables may break part or all of $\mathbb Z_{|q|}$.

What this classification enables

Given a symmetry claim, first ask whether the complete fixed quantum theory has the proposed group. Then specify the state or limiting regime. Finally choose physical diagnostics and report what they prove, including the inclusion $H_\omega\subseteq H_v$ rather than silently replacing it by equality.

The next pages separate the consequences that this definition alone does not supply:

• Finite Volume, Thermodynamic Limits, and Pure Phases constructs selected phases and fixes the order of limits.
• Vacuum Orbits and Unbroken Subgroups develops the orbit and $G/H$ geometry.
• Goldstone’s Theorem: Hypotheses and Pole Argument states the assumptions needed before spontaneous breaking implies a gapless pole; Goldstone Counting, Low-Dimensional Obstructions, and Spacetime Exceptions treats counting and failures.
• Explicit Breaking and Pseudo-Goldstone Modes derives controlled mass and Ward-identity consequences of a small breaking.
• Elitzur’s Theorem and the Gauge-Invariant Higgs Mechanism replaces gauge-variant language by physical gauge-invariant diagnostics.
• Developed critical emergence, thermal realization, and microscopic phase structure belong respectively to Renormalization and Effective Field Theory, Thermal and Nonequilibrium QFT, and Many-Body QFT and Quantum Matter.

References

• Tong, David. The Standard Model: 2 Broken Symmetries. Part III lecture notes. Cambridge: University of Cambridge, Department of Applied Mathematics and Theoretical Physics, 2019. Official PDF accessed August 2, 2026. Official course page. Official PDF.
• Weinberg, Steven. The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge: Cambridge University Press, 1995. DOI.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford: Oxford University Press, 2021. DOI.