Symmetry Breaking and Emergence

A perturbation can split a multiplet, mix states with different quantum numbers, or leave part of the original symmetry intact. These are useful starting points for understanding symmetry breaking. A different phenomenon occurs when symmetric many-body equations support ordered phases that select a direction. This chapter separates those mechanisms and explains when approximate, hidden, or emergent symmetry helps predict observable behavior.

Helpful background. Symmetry principles supplies the invariance test. Commutators and conservation connects it to dynamics, and degenerate perturbation theory explains how to treat a multiplet.

Enter this chapter

Question	Read next
Which transformations remain symmetries of the full problem?	Exact symmetry and explicit breaking
When do old quantum numbers remain useful?	Approximate symmetry
How does a perturbation split or mix degenerate states?	Degeneracy lifting
What explains an unexpectedly large degeneracy?	Accidental symmetry and hidden symmetry
Which algebra connects different energies?	Dynamical symmetry
How can a symmetric Hamiltonian support an ordered phase?	Spontaneous symmetry breaking and Goldstone modes
Can a low-energy description be more symmetric?	Emergent symmetry
Which features survive all permitted perturbations?	Symmetry-protected structure

Explicit breaking changes the Hamiltonian symmetry

For a specified unitary or antiunitary transformation $S$, exact symmetry means $SHS^{-1}=H$, including the operator domain and boundary conditions. Write

$$H=H_0+\lambda V, \qquad SH_0S^{-1}=H_0.$$

If $SVS^{-1}\ne V$ and $\lambda\ne0$, the added term explicitly breaks this symmetry. Other symmetries may survive. For example, a fixed magnetic field gives a spin Hamiltonian $H=-\gamma B S_z$, which retains rotations about $z$ but not arbitrary spin rotations. Rotating the field together with the spin compares different laboratory configurations.

A central spin–orbit term $\xi(r)\mathbf L\cdot\mathbf S$ provides another instructive case. It breaks independent orbital and spin rotations but preserves simultaneous rotations generated by $\mathbf J=\mathbf L+\mathbf S$. Describing it simply as “breaking rotation symmetry” would miss the conserved total angular momentum.

Weak perturbations can mix states strongly

Away from degeneracy, the leading admixture of an unperturbed state $|b\rangle$ into $|a\rangle$ is controlled by

$$\left|\frac{\lambda\langle b|V|a\rangle}{E_a^{(0)}-E_b^{(0)}}\right|.$$

The numerator must be small compared with the relevant gap. Small $\lambda$ alone is insufficient near a crossing. A two-state example makes this visible:

$$H=E_0I+\frac\delta2\sigma_z+\lambda v\sigma_x, \qquad v\in\mathbb R.$$

Its energies are $E_0\pm\sqrt{\delta^2/4+\lambda^2v^2}$. For $|\lambda v|\ll|\delta|$, the $\sigma_z$ states remain approximately energy eigenstates. At $\delta=0$, any nonzero $\lambda v$ selects equal-weight superpositions. The energy splitting can be tiny while the change of eigenbasis is large.

Within a degenerate subspace with projector $P$, diagonalize $PVP$ rather than taking diagonal expectation values in an arbitrary basis. If $PVP$ is proportional to $P$, there is no first-order splitting; this alone does not exclude splitting at higher order.

Approximate conservation also has a time range. For a time-independent generator $G$ with $[H_0,G]=0$,

$$\frac{d}{dt}\langle G\rangle =\frac{i\lambda}{\hbar}\langle[V,G]\rangle.$$

A small drift can accumulate over a long observation. Likewise, a formerly forbidden transition may acquire a small amplitude through mixing, while a surviving exact symmetry can still force it to vanish. Spin–orbit coupling, for example, preserves parity and cannot by itself remove a parity-forbidden electric-dipole zero.

Conserved algebra and spectrum-generating algebra

The ideal spinless Coulomb spectrum is degenerate across different orbital angular momenta at a fixed principal quantum number. Ordinary rotations explain only the $m$ degeneracy within each $\ell$. The conserved Runge–Lenz vector supplies additional structure; together with angular momentum it yields an $\mathfrak{so}(4)$ algebra on the bound-state energy sectors. Thus a degeneracy that looked accidental relative to rotations has a hidden explanation.

An isotropic oscillator illustrates a different distinction. With $H=\hbar\omega(\sum_i a_i^\dagger a_i+d/2)$,

$$[H,a_i^\dagger a_j]=0, \qquad [H,a_i^\dagger]=\hbar\omega a_i^\dagger.$$

The bilinears move quanta between directions at the same energy. The creation operators change the energy. Both algebras simplify the spectrum, but only the first is a conserved symmetry algebra. The term dynamical symmetry is used in several ways; the commutator states exactly which meaning applies.

Spontaneous breaking concerns ordered phases

A symmetric Hamiltonian can have nonsymmetric states: a localized wavepacket need not have parity. That fact alone is not the many-body phenomenon of spontaneous symmetry breaking.

For a finite system with a unique ground state, a unitary symmetry maps that state into the same ray. In a large-system limit, distinct ordered phases can instead persist with the same symmetric equations. An infinitesimal external source can select a phase, and the order of limits matters. For a total order operator $O_N=\sum_{i=1}^N o_i$, use the intensive expectation $m_{N,h}=\langle O_N\rangle_{N,h}/N$ and the source Hamiltonian $H_{N,h}=H_N-hO_N$. One can have

$$\lim_{h\to0^+}\lim_{N\to\infty}m_{N,h}\ne0,$$

while taking $h\to0$ first in a symmetric finite-system equilibrium state gives zero. The source explicitly breaks symmetry during preparation; persistence of order after the appropriate limiting removal is the spontaneous phenomenon. Finite ground-state degeneracy or a long-lived localized state needs further analysis and is not by itself proof of a thermodynamic phase.

For a broken continuous global internal symmetry in the usual local many-body setting, slow variations along the ordered directions can produce gapless Goldstone modes. Discrete breaking does not supply such continuous directions. In nonrelativistic systems, pairs of broken generators can correspond to one mode, so counting one mode per generator is generally wrong. Gauge redundancy and long-range interactions require additional care; the elementary global-symmetry argument cannot simply be copied into them.

Emergence is a statement about scale

A square lattice has discrete rotational symmetry. Nevertheless a nearest-neighbor band with hopping $t>0$ and spacing $a$ has, near its minimum,

$$\begin{aligned} E(\mathbf k)&=-2t[\cos(ak_x)+\cos(ak_y)]\\ &=-4t+ta^2(k_x^2+k_y^2) -\frac{ta^4}{12}(k_x^4+k_y^4)+\cdots. \end{aligned}$$

The leading quadratic term is rotationally invariant. The next term remembers the lattice. At $a|\mathbf k|\ll1$, its relative size is of order $a^2|\mathbf k|^2$. This is emergent low-momentum rotation symmetry, not exact continuous symmetry of the microscopic lattice. Other models need their own argument that symmetry-breaking terms become small at the relevant scale.

Finally, robustness requires specified assumptions. A Kramers pair cannot split under a perturbation preserving time reversal with $\Theta^2=-I$. A Chern number cannot change while its smooth isolated bundle over a closed surface remains defined. These are different protections; neither means that every perturbation preserves every property of the state.

Exercises

1. Let

$$H = \frac{P_x^2}{2m} + \frac12m\omega^2X^2 + \lambda X.$$

Test parity about the original origin. Then find a displaced coordinate in which the potential has an inversion symmetry.

Solution

Under original parity,

$$X\mapsto-X,$$

so the linear term changes sign. For $\lambda\ne0$, parity about $X=0$ is explicitly broken.

For the potential $V(X)=\tfrac12m\omega^2X^2+\lambda X$, complete the square:

$$V(X) = \frac12m\omega^2 \left( X+\frac{\lambda}{m\omega^2} \right)^2 - \frac{\lambda^2}{2m\omega^2}.$$

Define

$$Y = X+\frac{\lambda}{m\omega^2}.$$

The Hamiltonian is invariant under $Y\mapsto-Y$. Thus the original inversion is broken, while a different inversion about the displaced minimum remains exact.

2. In a degenerate two-state subspace, the restricted perturbation is

$$PVP = \begin{pmatrix} 0 & v \\ v^* & 0 \end{pmatrix}.$$

Find the first-order shifts and eigenstates. Explain why arbitrarily small $\lambda$ can produce an order-one basis rotation.

Solution

Write

$$v = |v|e^{i\varphi}.$$

The eigenvalues of $PVP$ are $\pm|v|$, so

$$E_\pm = E^{(0)} \pm \lambda|v| + O(\lambda^2).$$

Normalized eigenvectors can be chosen as

$$|\pm\rangle = \frac{1}{\sqrt2} \left( |1\rangle \pm e^{-i\varphi}|2\rangle \right).$$

These are equal-weight combinations for every nonzero $\lambda$. The splitting tends to zero with $\lambda$, but the preferred eigenbasis need not approach the original arbitrary basis. Degeneracy removes the energy denominator that would suppress mixing.

3. A finite Hamiltonian has a unitary symmetry $U$ and a unique ground state $|\Omega\rangle$. Prove that the ground state carries a definite symmetry phase. Why does this obstruct a literal finite-system spontaneous-breaking claim?

Solution

Because

$$UHU^{-1}=H,$$

one has

$$H(U|\Omega\rangle) = U(H|\Omega\rangle) = E_0U|\Omega\rangle.$$

Thus $U|\Omega\rangle$ is another ground state. Uniqueness implies

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

The finite unique ground state therefore does not select a distinct symmetry-related branch. Spontaneous symmetry breaking becomes sharp when a large-system limit produces stable sectors, vanishing finite-size splittings, and noncommuting source and size limits.

References

• A. Altland and B. Simons, Condensed Matter Field Theory, 3rd ed., Cambridge University Press, 2023.
• P. W. Anderson, “More is different,” Science 177, 393–396, 1972.
• A. Bohm, Y. Ne’eman, and A. O. Barut, eds., Dynamical Groups and Spectrum Generating Algebras, World Scientific, 1988.
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press, 2011.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.