Dynamical State Freezing in Many-Body Systems

• Dynamical state freezing is characterized by nearly invariant system observables due to emergent conservation laws, interference effects, and kinetic constraints under periodic driving.
• It is quantitatively measured using tools like survival probabilities, Lyapunov exponents, entropy growth, and return fidelities to identify critical and topological transitions.
• The phenomenon spans applications in quantum lattice models and classical systems, revealing nonthermal, metastable states via mechanisms such as Hilbert space fragmentation and absorbing transitions.

Dynamical state freezing refers to a collection of sharply defined, model- and context-dependent phenomena where the evolution of a many-body system under specified dynamics—classical or quantum—is suppressed or “frozen” on timescales that far exceed naive expectations, either due to emergent conservation laws, Hilbert space fragmentation, dynamical constraints, or targeted periodic (Floquet) driving. The effect can be quantified by observables, survival probabilities, Lyapunov exponents, entropy growth, return fidelities, order parameters, or resource monotones, and encompasses an array of critical, topological, and nonequilibrium steady-state freezing transitions. Below is a comprehensive review of the key types, mechanisms, and implications of dynamical state freezing across quantum and classical systems.

1. Conceptual Foundations and Quantitative Characterizations

Dynamical state freezing arises when one or more physically relevant observables (magnetization, local charges, correlation functions) or the quantum state itself exhibits negligible evolution away from its initial value, even under strong, time-dependent driving or in the presence of interactions and/or disorder. In closed quantum settings driven periodically, it is typically formalized using overlap fidelities

$$P(t) = |\langle\psi(0)|\psi(t)\rangle|^2, \qquad Q = \lim_{T\to\infty}\frac{1}{T}\int_0^T P(t')dt'$$

so that $Q\approx 1$ signals absolute or near-absolute freezing, while $Q\ll 1$ marks strong evolution (Sasidharan et al., 2023). Analogously, in classical or stochastic contexts, freezing may be defined as a vanishing fraction of “active” degrees of freedom or as reaching an absorbing state with permanently suppressed fluctuations (Reichhardt et al., 2014).

Physical mechanisms motivating freezing include: (1) destructive quantum interference (Floquet-integrable systems); (2) emergent conservation laws or approximate integrability; (3) Hilbert-space fragmentation via kinetic constraints or gauge symmetries; (4) phase-space trapping and configurational entropy loss in frustrated systems (Wang et al., 2023, Cepas et al., 2012, Hart et al., 2022, Hedges, 2024).

2. Dynamical Freezing in Coherently Driven Quantum Lattice Models

The canonical quantum setting involves periodically driven (Floquet) systems, such as Ising or XY chains, Kitaev honeycomb models, or multilayer graphene. For the 1D transverse-field Ising chain, mapping to decoupled two-level systems under a periodic drive $A_0\cos(\omega t)\sigma_z$ yields effective suppression of dynamics at resonance, specifically when the drive parameters make the zeroth-order Bessel function vanish: $J_0(2A_0/\omega)=0$ (Sasidharan et al., 2023, Bhattacharyya et al., 2011, Hegde et al., 2013, Kar et al., 2016). At these “freezing points,” the system’s evolution operator becomes (nearly) proportional to the identity in each two-level subspace, so observables such as magnetization are locked to their initial values for arbitrary times. Experimentally, such non-monotonic freezing dynamics have been realized in driven Ising chains via NMR, and the effect is robust to finite system size and is dictated purely by drive parameters (Hegde et al., 2013).

Recent generalizations to four-band (multiband) integrable systems, such as AB-stacked bilayer graphene subjected to a periodic interlayer bias $V(t) = V'+V_0\cos(\omega t)$, show that freezing persists only if a static bias $V'\neq 0$ is present (Sasidharan et al., 2023). At zeros of $J_0(2V_0/\omega)$, the effective Hamiltonian commutes with the chosen initial state, and $Q\rightarrow 1$ is achieved. Absence of a bias lifts the freezing, a “switching” effect. Experimental schemes using ultracold atoms in optical lattices have been proposed to implement and observe this multiband freezing.

Floquet-driven models with two modulated parameters (“two-rate” protocols) enable controlled approaches toward freezing in both integrable and non-integrable settings. For instance, pairing $h_1(t) = A\cos(\omega_1 t)$ and $h_2(t)=B\cos(r\omega_1 t)$ drives produces near-perfect return fidelities along lines $r = \omega_2/\omega_1$ where path interference eliminates defect production—demonstrated analytically and numerically in XY chains, Kitaev models, and the Bose-Hubbard model (Kar et al., 2016).

3. Emergent Conservation, Hilbert-Space Fragmentation, and Absorbing Freezing

In models with kinetic or constraint-induced fragmentation, dynamical freezing is associated with the system’s division into exponentially many disjoint Krylov (dynamically connected) subsectors. In the one-dimensional particle-conserving East model, hopping is permitted only if a facilitating particle is present to the left within a range $r$, defining a hierarchy of fragmentation controlled by particle density $n$ (Wang et al., 2023). For $n < n_c = 1/(r+1)$, the system decomposes into exponentially many isolated subsectors, most of which are dynamically frozen: a macroscopic fraction of sites never evolve away from their initial occupation ($n_F>0$). At $n>n_c$, the system thermalizes diffusively within a giant sector ($n_F=0$), and a sharp freezing transition with exactly computable exponents separates the regimes.

In extended quantum spin models such as the 2D transverse-field Ising model deep in its ferromagnetic phase, emergent U(1) and one-form $\mathbb Z_2$ constraints induce “Hilbert-space shattering” (Hart et al., 2022). Configurations (e.g., domain wall densities) partition the full Hilbert space into exponentially many disconnected subspaces, supporting strong dynamical freezing for typical low-entropy initial conditions. The shattering transition—between “strong” (frozen) and “weak” (thermalizing) sectors—occurs with a sharp signature as a function of symmetry sector, driven by a nucleation mechanism with non-trivial rate-function scaling.

Classical and stochastic systems also realize dynamical freezing via absorbing transitions. For instance, in 2D active-matter systems of self-propelled disks, increasing the particle density above a critical value $\phi_c$ drives the system into a dynamically jammed “frozen” state where fluctuations vanish, clusters drift rigidly, and the time to reach the absorbing state diverges as a power law upon approaching $\phi_c$—characteristic of the directed percolation universality class (Reichhardt et al., 2014). Similar mechanisms operate in random organization of colloids under oscillatory shear.

4. Thermodynamic Freezing and Dynamical Phase Transitions

In equilibrium dynamical systems and statistical mechanics, “freezing” denotes a genuine phase transition at zero (or finite) temperature. For a dynamical system $(X,T)$, a freezing phase transition in potential $\phi$ occurs when, beyond some critical inverse temperature $\beta_0$, the equilibrium states of $\beta\phi$ stabilize and cease varying for all $\beta>\beta_0$ (Hedges, 2024). This is characterized by the pressure function $P_\phi(\beta)$ achieving a linear “slant” asymptote, and the set of freezing equilibrium states coincides with those maximizing $\int\phi d\mu$ plus the maximal entropy among those. Rigorous results show that, under upper semi-continuity of entropy (e.g., expansive shifts, symbolic dynamics), any ergodic measure can be obtained as a freezing state for some potential, and the set of freezing potentials is dense in $C(X)$. This dynamical “order-disorder” transition structurally mirrors glass and quasicrystal freezing in classical systems (Hedges, 2024).

In geometrically frustrated magnets without quenched disorder, such as the kagome antiferromagnet, lowering the temperature below a freezing threshold $T_f$ defined by the dynamics of closed-loop moves traps the system in one of exponentially many disconnected metastable clusters (“traps”). The corresponding configurational entropy loss at $T_f$ and the associated two-step spin autocorrelation ($\tau_\beta \ll \tau_\alpha$) are hallmarks of thermodynamic glass-type freezing, well-matched to experimental observations in kagome compounds (Cepas et al., 2012).

5. Floquet Thermalization, Prethermal Plateaux, and Instanton Dynamics Near Freezing

Recent work applies flow-renormalization group (fRG) and Magnus/Floquet expansions to analyze late-time dynamics under periodic driving in the presence of freezing points. For driven systems with local interactions, the Floquet Hamiltonian exhibits emergent approximate conservation laws at discrete values of $A/\Omega$ set by zeros of Bessel functions (Mukherjee et al., 2024). These parameter regimes support a “prethermal plateau” where memory of initial conditions, suppressed entanglement growth, and approximate charge conservation persist over exponentially long times $t_\text{pre} \sim \exp[C \Omega]$. The onset of full thermalization is described by instanton-like events in the flow, where the system undergoes nonperturbative jumps between prethermal and thermal fixed points, accompanied by spectral folding and entanglement surges. At freezing points, this delayed thermalization is further exaggerated, yielding an even larger time window of freezing-style dynamics.

Exactly solvable models, such as coupled SYK quantum-dot systems, support similar conclusions: at drive parameters commensurate with energy differences between sectors, the leading-order Floquet Hamiltonian decouples dynamically, pinning local occupation numbers and entanglement entropy, with relaxation timescales growing as a power of the drive frequency (Guo et al., 2024).

6. Freezing of Quantum Correlations, Coherence, and Resource Monotones

Dynamical freezing phenomena are not restricted to observables but extend to quantum resource theories—coherence, asymmetry, and discord. Measure-independent freezing of quantum coherence refers to the situation where all coherence quantifiers remain constant under strictly incoherent quantum channels if and only if the relative entropy of coherence is frozen (Yu et al., 2016). A parallel result holds for asymmetry: under G-covariant noise, all asymmetry measures are frozen if and only if the relative entropy of asymmetry is preserved during the dynamics (Zhang et al., 2016). These single-scalar conditions provide necessary and sufficient certificates of total immunity to dissipation for broad classes of initial states and noisy dynamics.

Quantum discord—the measure of quantum correlations beyond entanglement—can be dynamically frozen up to a finite “freeze time” by dynamical decoupling protocols in noisy environments. Beyond this point, discord decays; sophisticated pulse sequences can extend the freeze window by suppressing the effective dephasing rate (Singh et al., 2016). These analytic and experimental results reinforce the notion that dynamical state freezing supplies a deep link between dynamical constraints, resource status, and quantum information preservation.

7. Perturbations, Melting, and Slow Relaxation out of Frozen States

The resilience and decay mechanism for dynamically frozen states is highly sensitive to system details and perturbations. Introduction of noise, phase randomness, or disorder can “melt” frozen states, but generically the thawing is slow. In integrable driven Ising chains, imprinted phase noise causes melted magnetization to decay in a stretched-exponential (Kohlrausch) fashion—not the simple exponential expected of a single timescale but rather reflecting a hierarchical spectrum of quasi-conserved modes (Roychowdhury et al., 2024). In the Discrete NonLinear Schrödinger model, exponentially slow relaxation of tall breathers is attributed to an emergent adiabatic invariant, and full equipartition or ergodicity is thus effectively arrested over astronomically long timescales, both at positive and negative effective temperature (Iubini et al., 2018).

The nature of slow relaxation, the distribution of time constants, and the associated spatial or configuration-space structures (e.g., domain walls, frozen clusters, glassy traps) are critical both for understanding the intrinsic limits of freezing and for leveraging these effects in experimental and technological contexts (e.g., quantum metrology (Lu et al., 30 Jul 2025), entanglement control (Gangopadhay et al., 2024)).

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Overall, dynamical state freezing delineates a unifying conceptual and phenomenological thread across quantum and classical systems: the emergence or engineering of dynamical constraints via symmetry, topology, drive protocols, or combinatoric structure, leading to dramatically long-lived memory of initial conditions, critical transitions, and wide-ranging implications for nonequilibrium physics, resource theory, and quantum control (Sasidharan et al., 2023, Wang et al., 2023, Reichhardt et al., 2014, Hart et al., 2022, Hedges, 2024, Iubini et al., 2018, Mukherjee et al., 2024, Lu et al., 30 Jul 2025, Guo et al., 2024, Cepas et al., 2012, Yu et al., 2016, Zhang et al., 2016, Singh et al., 2016, Roychowdhury et al., 2024, Gangopadhay et al., 2024).