Deterministic Cascade of Symmetry Breaking

• The paper demonstrates that each symmetry-breaking transition is dictated by precise RG flows and mean-field gap equations, ensuring a deterministic cascade.
• It details how effective operators in gauge theories, multi-flavor fermionic models, and frustrated lattices sequentially trigger distinct phase transitions.
• The study highlights that both equilibrium and driven systems exhibit universal phase splitting, revealing connections to compositeness scales and exotic ordering phenomena.

A deterministic cascade of symmetry-breaking phase transitions describes a sequence of distinct, stepwise transitions in a many-body system where successive symmetry breakings occur in a uniquely prescribed order, determined by the internal dynamics and feedback between effective couplings and condensed degrees of freedom. Rather than stochastic or accidental multi-criticality, the cascade is dictated by model-specific RG flows, mean-field gap equations, or protocol constraints, such that each later transition is a necessitated outcome of the preceding one. Rigorous examples include multi-quark condensate formation in gauge theories, flavor-group symmetry breakings in multi-flavor fermionic systems, spatial-symmetry cascades in frustrated quantum models, and non-equilibrium cascades emerging in periodically driven collective systems.

1. Cascade Mechanisms in Gauge Theory and Chiral Symmetry Breaking

The “deterministic cascade’’ paradigm was formulated in the context of nonabelian SU(N) gauge theories with $N_f$ Dirac fermions (Fariborz et al., 2016). The starting Lagrangian,

$\mathcal{L}_0 = -\frac14 F^a_{\mu\nu}F^{a,\mu\nu} + \sum_{i=1}^{N_f} \bar\psi_i (i\slashed{\partial} - g \slashed{A}^a T^a)\psi_i\,,$

admits dynamical gluon mass generation at $\mu\lesssim m_A$, after which integrating out the gauge bosons induces effective NJL-type four-fermion operators of the form $(\bar\psi\psi)^2$. The resultant gap equation for the two-quark (diquark) condensate yields a critical coupling, whose satisfaction triggers chiral symmetry breaking (χSB).

Bound-state formation at the first stage produces additional colored degrees of freedom (diquarks and baryons), which must be incorporated in the RG evolution via modified $\beta$-functions with altered multiplicities. Integrating out these bound states yields new four-scalar (“four-quark”) operators driving secondary condensates. The gap equation for the tetraquark condensate prescribes its own critical coupling, which occurs at a higher IR scale and lower flavor count than the first.

Numerical estimates for QCD (SU(3)) show two distinct transitions:

• Two-quark χSB at $N_f\lesssim 3.9$,
• Four-quark χSB at $N_f\lesssim 3.38$, demonstrating that for integer $N_f$, the cascade stops at $N_f=3$ (Fariborz et al., 2016). In composite/preon models, analogous logic sets the flavor-bound for multi-stage strong dynamics, directly relating the vacuum structure to fundamental compositeness scales.

2. Sequential Symmetry Breaking in Multi-Flavor Fermionic Systems

In planar Dirac materials, the deterministic cascade appears as a sequence of $M$ first-order symmetry-breaking transitions within a $\mathrm{U}(M) \times \mathrm{SU}(N)$ invariant four-fermion model (Kanazawa et al., 2021). The large-$N$ solution of the effective potential yields saddle-point equations for $M$ eigenvalues $e_k$, constrained by $\sum_{k=1}^M e_k \approx \lambda$.

Upon varying the parity-breaking flavor-singlet mass parameter, the ground state traverses $M+1$ distinct configurations, with symmetry breakings of the type

$$\mathrm{U}(M) \rightarrow \mathrm{U}(M-k) \times \mathrm{U}(k),\qquad k \in \{0,1,\dots,M\}\,,$$

at $M$ critical masses $m_k$. Each transition corresponds to one eigenvalue switching between potential minima, resulting in a strictly deterministic sequence.

At finite temperature and chemical potential, the deterministic structure persists but phase diagrams enrich with first- and second-order lines, tricritical points, and regions of “exotic” symmetry breaking such as $\mathrm{U}(3)\rightarrow \mathrm{U}(1)^3$, realized when the effective single-particle potential develops three distinct minima. The allocation of eigenvalues to minima, and thus the symmetry breaking pattern, is dictated by explicit parameter dependences of the model, not statistical pairing.

3. Deterministic Cascades in Frustrated Quantum Lattice Models

Frustrated quantum spherical models on $d$-dimensional hypercubic lattices with multineighbor couplings exhibit deterministic cascades of spatial and internal symmetry breakings (Casasola et al., 2021). The Hamiltonian,

$$H = \frac{g}{2}\sum_n \Pi_n^2 - j_1 \sum_{\langle n,m\rangle} S_n S_m + j_2 \sum_{\ll n,m\gg} S_n S_m + j_3 \sum_{\ll n,m\gg_{\mathrm{diag}}} S_n S_m\,,$$

shows a rich phase structure determined by the “frustration” parameter $p=[4j_2+2(d-1)j_3]/j_1$.

At $g=g_c(p,j_2,j_3)$, the system transitions from a disordered gapped phase to ordered gapless phases. Depending on $p$ and the ratios $j_3/j_2$, ordered phases are indexed by sets of pitch vectors $q_c$, giving rise to homogeneous, diagonal, stripe, and antiferromagnetic stripe states, with possible transitions to quantum Lifshitz points ($p=1$).

Transitions—both gapped-gapless and gapless-gapless—occur with symmetry changes in U(1) charge and translations, and, at special parameter values, emergent polynomial shift symmetries define fractonic behavior. The symmetry-breaking sequence is controlled by explicit tuning of model parameters and each phase’s feedback on correlation lengths and excitation spectra.

4. Non-Equilibrium Cascades: Split Criticality in Driven Collective Systems

Driven mean-field Ising models alternating deterministically between two thermal reservoirs ($T_C$, $T_H$) manifest universal splitting of the standard order-disorder transition into two distinct critical points (Forão et al., 14 Dec 2025). For strictly periodic protocols, the Floquet-averaged mean-field equation,

$$m = \frac{1}{2} \left\{ \tanh[\beta_C J m] + \tanh[\beta_H J m] \right\}\,,$$

admits two symmetry-breaking thresholds:

• $T_c^{(1)} = \dfrac{2J}{1/T_C + 1/T_H}$ (continuous transition),
• $T_c^{(2)}:$ the solution of a transcendental saddle-node equation with cubic nonlinearity (discontinuous transition).

The order that sets in first depends strictly on the driving period $\tau$; for slow driving, the two critical points are maximally split, resulting in a true cascade. The phenomenon persists under simultaneous or stochastic protocols and is governed by the competition between non-conservative driving and bath interaction, not stochastic fluctuations.

Protocol Type	Number of Critical Points	Cascade Behavior
Simultaneous	2	Universal split
Stochastic	2	Rate-dependent split
Deterministic	2	$\tau$-dependent

All three protocols confirm universality: the splitting and order-competition are intrinsic to collectively driven systems with multiple baths.

5. Landau–Ginzburg Functional and Order Parameters Across Cascades

At each symmetry-breaking stage $k$ within the cascade, an order parameter $\sigma_k = \langle O_k\rangle$ (with $O_1=\bar\psi\psi$, $O_2=R^\dagger L$, etc.) is introduced and modeled via a Landau–Ginzburg functional,

$$V_k(\sigma_k) = \frac{1}{2} m_k^2 \sigma_k^2 + \frac{1}{4}\lambda_k \sigma_k^4 + \cdots,$$

where $m_k^2$ changes sign exactly at the critical coupling $a_k$. The formation of $\sigma_k \ne 0$ drives RG evolution feeding into the next $\beta$-function, forcing $m_{k+1}^2$ negative at a larger coupling and lower scale. The process is iterative and deterministic: each phase transition seeds the next, with the feedback originating from the prior condensate and its alteration of the RG flow.

A comparable mechanism is found in effective actions for frustrated lattice models and driven collective spin systems, though the specific form of $V(\sigma)$ and the nature of the order parameter depends on microscopic details (e.g., pitch vector modulations, Floquet averages, group-invariant bilinears).

6. Physical Significance, Universality, and Broader Context

The deterministic cascade picture captures a form of “domino logic” in multi-stage symmetry-breaking: the unique feedback and interaction between condensed fields and RG flows necessitate successive phase transitions. For SU(3) QCD, cascading only proceeds for $N_f \le 3$; for larger $N_f$, higher-order condensates do not form, and the sequence truncates (Fariborz et al., 2016). Analogous bounds and correlations emerge in composite models and Dirac fermion systems, where the cascade sets vacuum structure and compositeness scales.

A plausible implication is that deterministic cascades are a generic feature in nonabelian gauge theories, frustrated quantum systems, and non-equilibrium finite-bath models where feedback and criticality are tightly coupled to microscopic or driving protocol parameters. This broadens the landscape of symmetry-breaking phenomena beyond equilibrium, suggesting new avenues in strong-dynamics engineering, quantum critical matter, and the control of non-equilibrium order in collective systems.

7. Summary of Key Deterministic Cascade Phenomena

• Cascade transitions are prescribed by the gap equations and RG flow, rather than random or accidental alignments.
• Each new condensate and symmetry breaking emerges exactly when necessary criteria (beta zero and anomalous dimension) are met, with no extra tuning.
• Multi-stage symmetry breaking is found across QCD, composite models, multi-flavor fermion systems, modulated quantum lattices, and driven spin ensembles.
• Universal splitting of phase transitions is evidenced in both equilibrium and non-equilibrium protocols, with mechanism details contingent upon interaction type, flavor structure, spatial frustration, and protocol timing.
• Exotic breakings (triple-group splits, fractonic phases, etc.) arise when effective potentials admit more than two minima, with system-wide implications for excitation spectra, Goldstone modes, and correlation patterns.

The deterministic cascade thus encapsulates a foundational organizing principle underlying multi-stage spontaneous symmetry breaking in quantum, statistical, and non-equilibrium systems.