Symmetry Re-Breaking in Quantum Systems

• Symmetry re-breaking is a nonequilibrium process where a system initially in a symmetry-broken state temporarily loses long-range order and re-selects symmetry through domain coarsening.
• It applies to quantum coarsening, symmetry-aware learning, and constraint programming, illustrating adaptive dynamics and controlled symmetry relaxation.
• Quantitative tools like entanglement asymmetry and normalized scattering measures enable precise tracking of the transient order loss and subsequent re-establishment of symmetry.

Symmetry re-breaking is used in current literature for situations in which an already symmetry-selected or symmetry-constrained system undergoes a further symmetry-selection step. In the most explicit usage, it denotes a nonequilibrium process in which a system prepared in an ordered, symmetry-broken phase temporarily loses global order during Hamiltonian evolution and then re-establishes order by coarsening, so that the late-time magnetization may even have the opposite sign of the initial magnetization (Balducci et al., 23 Jul 2025). Related uses appear in symmetry-aware learning, where a network starts fully equivariant and then readjusts to the smaller symmetry actually present in data (Wang et al., 2023), and in constraint programming, where symmetries acting on symmetry-breaking constraints are themselves used to select different representatives within the same symmetry class (Katsirelos et al., 2010).

1. Definition, scope, and neighboring concepts

In the effective theory of quantum coarsening, symmetry re-breaking is defined by a specific dynamical sequence: the system starts in a globally ordered state with $m>0$, mean-field oscillations drive the order parameter close to zero and sometimes through zero, spatial fluctuations grow and destroy the coherence of the uniform order, the system breaks into domains of both signs, and—because the equilibrium phase is still ordered—the system coarsens again and ultimately selects one of the two symmetry-broken states, with $m_\infty>0$ or $m_\infty<0$ (Balducci et al., 23 Jul 2025). The defining feature is therefore not the initial symmetry breaking alone, but the transient effective loss of long-range order followed by a second symmetry-selection event.

This meaning sits near several related but distinct constructions. “Systems with Symmetry Breaking and Restoration” studies globally ordered phases that contain mesoscopic regions where the competing higher-symmetry phase is locally restored; the framework is based on weighted Hilbert spaces, manifold indicators, and configuration averaging, and the exemplary ferroelectric has paraelectric embryos inside the ferroelectric phase (Yukalov, 2010). “Asymmetric balance in symmetry breaking” concerns the universal unfolding of the pitchfork bifurcation, where spontaneous symmetry-breaking behavior destroyed by one asymmetry can be restored by introducing a second asymmetry of the right kind (Garbin et al., 2019). “Entanglement asymmetry as a probe of symmetry breaking” treats dynamical restoration of a broken global $U(1)$ symmetry after a quantum quench, but at the subsystem level and through a reduced-density-matrix diagnostic rather than through domain re-formation (2207.14693). Taken together, these works place symmetry re-breaking close to symmetry restoration, imperfect bifurcation theory, and subsystem symmetry diagnostics, while keeping its strongest present meaning tied to a second symmetry-selection stage after transient loss of order.

2. Nonequilibrium coarsening and reversal of the order parameter

The 2025 effective theory of quantum coarsening studies the classical limit of the 2D transverse-field Ising model,

$$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$

with classical spins obeying

$$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$

and magnetization

$m=\frac{1}{L^2}\sum_i S_i^z.$

The model has a ferromagnetic phase at small $g$, a paramagnetic phase at larger $g$, and a second-order transition line in the $(g,\varepsilon)$ plane. The work was motivated by a programmable-quantum-simulator experiment showing an apparent speeding up of coarsening near the phase transition and persistent oscillations of the order parameter after quenches within the ordered phase (Balducci et al., 23 Jul 2025).

The coarsening sector is described as curvature-driven domain-wall motion. For a roughly circular minority domain of radius $m_\infty>0$0,

$m_\infty>0$1

so the area decreases linearly in time. The areal coarsening speed $m_\infty>0$2 is controlled by two competing effects: deep in the ordered phase, increasing $m_\infty>0$3 strengthens the transverse-field term and speeds up spin motion near domain walls; near the critical line, the domain-wall tension vanishes, so the force driving curvature-induced motion weakens and $m_\infty>0$4 decreases again. The resulting trend is a speed-up followed by slow-down.

For quenches within the ordered phase, the spatially uniform mode is isolated in a mean-field or Lipkin-Meshkov-Glick-like approximation,

$m_\infty>0$5

Because the phase space is the Bloch sphere and the energy is conserved, the motion is regular and produces oscillations in the magnetization. The phase portrait contains a separatrix at

$m_\infty>0$6

where the oscillation frequency has a minimum. This dynamical transition is explicitly distinguished from the thermodynamic critical point $m_\infty>0$7.

Mean field alone does not produce symmetry re-breaking. The decisive ingredient is fluctuation growth. The paper introduces a 2D $m_\infty>0$8 field theory,

$m_\infty>0$9

splits the field into a zero mode and nonzero-$m_\infty<0$0 fluctuations, and shows that after the quench the zero mode can drive the effective fluctuation mass negative: $m_\infty<0$1 When $m_\infty<0$2 becomes small and $m_\infty<0$3, low-$m_\infty<0$4 fluctuations grow exponentially, interrupt the uniform oscillation, and seed a patchwork of domains. Since the equilibrium phase remains ordered, late-time coarsening rebuilds long-range order. The late-time sign is sensitive to how many oscillations occur before coherence is lost, which is why the final magnetization can be opposite in sign to the initial one. In this formulation, symmetry re-breaking is neither a new equilibrium phase transition nor a permanent disordered state; it is a nonequilibrium destruction and reconstruction of order under Hamiltonian dynamics.

3. Re-breaking as adaptive reduction of imposed symmetry in learning systems

A formally different but structurally related usage appears in “Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution” (Wang et al., 2023). There the problem is not post-quench dynamics, but learning the highest level of equivariance consistent with data. Standard group convolution hard-codes exact weight sharing across group elements. Relaxed group convolution replaces the shared kernel by trainable weights $m_\infty<0$5 over group elements,

$m_\infty<0$6

These relaxed weights are initialized identically across $m_\infty<0$7, so the model starts fully equivariant; they then deviate from equality only if the data require it.

The paper’s theoretical bridge to symmetry re-breaking is Proposition 1: if a relaxed group convolutional network is trained from equal initialization to map input $m_\infty<0$8 to output $m_\infty<0$9, then the learned relaxed weights become equivariant to

$U(1)$0

Thus a model can begin with a larger symmetry bias and then re-adjust to the smaller symmetry actually present in the task. The authors explicitly compare the relaxed weights to Landau order parameters, because the trained deviations from uniform weight sharing encode the degree and type of symmetry breaking.

The layerwise symmetry loss is quantified by the equivariance error

$U(1)$1

This quantity is zero before training and grows as weights diverge across group elements. Projecting relaxed weights onto irreducible representations then reveals which symmetry components remain preserved and which are broken.

The empirical case studies exhibit three distinct symmetry-breaking mechanisms. For BaTiO$U(1)$2, relaxed $U(1)$3 group convolution on cubic-to-tetragonal and cubic-to-orthorhombic transitions recovers $U(1)$4 and $U(1)$5, matching the known point groups of the output phases. In turbulent flow, the learned equivariance error decreases as wavenumber increases in the isotropy experiment, while relaxed translation convolution on channel flow data shows the largest symmetry loss near the top and bottom boundaries in the homogeneity experiment. In pendulum dynamics, a 1D relaxed group convolution equivariant to time reflection $U(1)$6 preserves equal weights for a frictionless pendulum but produces diverging weights with friction, detecting time-reversal-symmetry breaking in the governing mapping itself. In this literature, re-breaking is not a transient reordering of domains; it is a controlled relaxation of an initially exact equivariant bias until only the symmetry supported by data remains.

4. Diagnostics and quantitative measures

Several recent works provide quantitative diagnostics for breaking, restoration, and persistence of symmetry, and these diagnostics are directly relevant to any analysis of re-breaking.

The most developed subsystem diagnostic is entanglement asymmetry,

$U(1)$7

introduced for a quantum system with conserved charge $U(1)$8 (2207.14693). It satisfies $U(1)$9 and vanishes iff $$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$0. In a quench of an XX spin chain from an initially broken global $$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$1 state, the asymmetry decays with late-time behavior $$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$2, larger subsystems restore symmetry more slowly, and stronger initial symmetry breaking can restore faster, producing a quantum Mpemba effect. This is a subsystem restoration result rather than a domain-coarsening re-breaking result, but it provides a precise language for tracking when symmetry is effectively lost or regained.

A later study applies entanglement asymmetry to non-symmetric random circuits and non-symmetric Hamiltonian quenches (Yu et al., 23 Jan 2025). In the random-circuit setting, subsystem $$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$3 symmetry is restored at late times even though the circuit contains symmetry-breaking gates, while in the non-symmetric Hamiltonian quench the subsystem remains symmetry-broken at late times because the effective subsystem Hamiltonian itself breaks $$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$4. In both settings the early-time growth of entanglement asymmetry exhibits overshooting, unlike charge variance. This establishes that transient excess symmetry breaking can occur even when the long-time limit differs sharply between restoration and persistence.

A complementary observation-based construction is the normalized scattering measure

$$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$5

which quantifies how much a target system breaks a symmetry represented by $$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$6 through symmetry-sector-resolved intensity measurements (Fernandez-Corbaton, 2017). For continuous symmetries generated by $$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$7, the local small-parameter behavior is

$$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$8

This replaces the binary question of whether a symmetry is broken by a quantitative measure of how strongly different symmetry sectors are coupled.

Diagnostic	Definition or role	Representative source
Entanglement asymmetry	Subsystem measure of symmetry breaking in reduced density matrices	(2207.14693)
Early-time EA overshoot	Distinguishes transient symmetry breaking from monotone charge-variance relaxation	(Yu et al., 23 Jan 2025)
Equivariance error	Layerwise measure of deviation from exact weight sharing in relaxed group convolution	(Wang et al., 2023)
Observation-based measure $$H[g] = - \sum_{\langle ij\rangle} S_i^z S_j^z - g \sum_i S_i^x,$$9	Intensity-based normalized distance between $$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$0 and $$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$1	(Fernandez-Corbaton, 2017)

These diagnostics do not define symmetry re-breaking by themselves, but they provide the quantitative infrastructure needed to distinguish transient restoration, persistent explicit breaking, and secondary symmetry selection.

5. Algorithmic and geometric reinterpretations

In constraint programming, symmetry re-breaking appears as a formal property of symmetry-breaking constraints themselves. “Symmetries of Symmetry Breaking Constraints” proves that if $$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$2 is a sound or complete set of symmetry-breaking constraints for a symmetry group $$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$3, then for any $$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$4, the transformed set $$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$5 is also sound or complete (0909.3276). Different symmetries of the same symmetry-breaking set pick different representatives in each symmetry class. This observation underlies two methods: model restarts, in which search is periodically restarted with a different symmetry of the symmetry-breaking constraints, and dynamically posting symmetry-breaking constraints only when the current branching decisions force that choice.

The companion account centered on model restarts states the principle even more directly: any problem symmetry acting on a set of symmetry-breaking constraints can itself be used to break symmetry, and different transformed constraint sets select different representatives from each symmetry class (Katsirelos et al., 2010). In this usage, symmetry re-breaking does not mean additional physical symmetry loss. It means reapplying the symmetry group at the level of the orbit-selection mechanism.

A geometrical analogue is proposed in “Symmetry breaking in geometry,” where a geometry $$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$6 is reduced to a subgeometry $$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$7 by adjoining an absolute configuration $$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$8 whose stabilizer is $$\partial_t \vec{S} = -\big(g\hat{x} + \hat{z}\sum_{j\in\partial i} S_j^z\big)\wedge \vec S,$$9 (Fuchs et al., 2022). The paper explicitly discusses iterated reductions, including chains such as projective geometry $m=\frac{1}{L^2}\sum_i S_i^z.$0 Lie sphere geometry $m=\frac{1}{L^2}\sum_i S_i^z.$1 Möbius geometry $m=\frac{1}{L^2}\sum_i S_i^z.$2 Euclidean, hyperbolic, spherical, or similarity geometry. The supplied discussion identifies this as the closest mathematical analogue to repeated symmetry breaking or “re-breaking”: the same object can first be viewed in a higher-symmetry ambient geometry and then reduced in different ways depending on the chosen absolute data.

6. Adjacent physical phenomena, ambiguities, and limits of the term

Not every additional symmetry loss is described as symmetry re-breaking. Re-based superconductors provide a useful counterexample. Zero-field $m=\frac{1}{L^2}\sum_i S_i^z.$3SR studies show time-reversal-symmetry breaking below $m=\frac{1}{L^2}\sum_i S_i^z.$4 in a family that includes pure Re and several Re$m=\frac{1}{L^2}\sum_i S_i^z.$5 compounds, while many systems remain fully gapped and compatible with $m=\frac{1}{L^2}\sum_i S_i^z.$6-wave-like superconductivity; the review emphasizes that TRSB is not tied in any straightforward way to inversion symmetry and correlates strongly with the presence and amount of rhenium (Shang et al., 2021). Here the issue is broken TRS in the superconducting state, not a transient loss and re-establishment of order.

A more structurally related case is frustrated multiband superconductivity in Re$m=\frac{1}{L^2}\sum_i S_i^z.$7Hf, where superconductivity breaks $m=\frac{1}{L^2}\sum_i S_i^z.$8 gauge symmetry at $m=\frac{1}{L^2}\sum_i S_i^z.$9 and the condensate also breaks time-reversal symmetry because multiple inter-band nesting channels on an odd number of Fermi-surface sheets make a globally consistent real sign pattern impossible (Manda et al., 2022). The resulting $g$0 state is an additional symmetry breaking internal to the superconducting phase. This is close in spirit to a secondary symmetry-selection event, but the paper describes it as frustration-driven TRS breaking rather than as symmetry re-breaking.

The boundary between physical secondary symmetry selection and spurious symmetry lowering is also important. “Artificial Symmetry Breaking by Self-Interaction Error” shows that semilocal density functionals can spuriously localize charge and break the full cyclic symmetry $g$1 of one-electron multicenter systems, while Hartree–Fock remains symmetry-preserving and exact for the model (Hou et al., 25 Jun 2025). The same paper gives a real-material example in $g$2 in ZnO, where SCAN lowers the symmetry to $g$3 while HSE preserves the expected $g$4 symmetry. Similarly, “Symmetry breaking in a turbulent environment” shows that multiplicative noise in turbulence changes the critical behavior of symmetry-breaking transitions and leads to anomalous exponents, but this is a new nonequilibrium criticality rather than a second breaking event after restoration (Alexakis et al., 2021).

These distinctions delimit the term. In present usage, symmetry re-breaking is most precise when a symmetry-broken or symmetry-constrained system first passes through an intermediate stage that restores, suppresses, or suspends the earlier order or bias, and then selects symmetry again. A plausible implication is that the term is most useful when the intermediate stage is itself dynamically or operationally identifiable—through coarsening, fluctuation growth, learnable departures from equivariance, or explicit re-selection among symmetry-class representatives—rather than when one merely observes a further broken symmetry.