Fock Space Prethermalization (FSP)

• FSP is a nonequilibrium regime where the wave function remains confined to a restricted region of Fock space, resulting in a long-lived quasi-steady state.
• Studies using models like the SYK and Floquet circuits demonstrate that FSP arises from constrained Fock-space dynamics and delayed global ergodicity.
• FSP connects dephasing effects, sparse subnetwork formation, and finite-size scaling, offering new experimental diagnostics for probing nonequilibrium quantum dynamics.

Fock Space Prethermalization (FSP) denotes a nonequilibrium regime in which a many-body system develops a long-lived quasi-steady state because the wave function spreads only within a restricted, highly structured region of Fock space, while full ergodic exploration of the Hilbert space is delayed. In this sense, observables can appear nearly stationary even though the state remains far from the uniform Fock-space distribution expected in a fully thermalizing system. In the recent literature, FSP has been formulated in several complementary ways: as constrained spreading near a chaos–integrable crossover in the complex SYK model, as dephasing at fixed mode occupations in an integrable Luttinger liquid, and as sparse-subnetwork dynamics in disorder-free Floquet circuits with approximately conserved domain-wall number (Dieplinger et al., 2023, Smith et al., 2012, Bao et al., 28 Oct 2025).

1. Definition and operational meaning

In the SYK$_4$+SYK$_2$ setting, prethermalization is defined operationally by an intermediate-time plateau of the infinite-temperature on-site density–density correlator,

$$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$

whose value is close to the saturation value of the integrable SYK$_2$ limit, followed only at much later times by decay toward the small ergodic SYK$_4$ value (Dieplinger et al., 2023). In that formulation, the hallmark of FSP is not merely slow relaxation, but the coexistence of quasi-stationary observables with a wave function that remains strongly nonuniform in Fock space.

A central consequence is that FSP is not restricted to the conventional setting of nearly integrable local lattices. In the complex SYK model, it arises in an all-to-all quantum dot with no spatial locality and no appeal to quasi-local conserved quantities; the controlling structure is instead the geometry and connectivity of Fock space itself (Dieplinger et al., 2023). This directly separates FSP from the standard near-integrable prethermal scenario in which relaxation is delayed by approximate conservation laws in real space.

A broader operational picture emerges from other works. In the Luttinger-liquid description of a rapidly split one-dimensional Bose gas, prethermalization occurs because the system dephases at approximately fixed mode occupations $n_k$, so the long-time observables are governed by a diagonal ensemble rather than by true redistribution of occupation numbers (Smith et al., 2012). In the disorder-free Floquet Ising experiment on a superconducting processor, FSP is defined explicitly as a mechanism that “divides the Fock-space network into linearly many sparse sub-networks,” thereby suppressing heating and prolonging the thermalization timescale even at high energy density (Bao et al., 28 Oct 2025). Across these formulations, the shared content is quasi-equilibration inside a restricted Fock-space manifold.

2. Canonical finite-size realization in the complex SYK model

The clearest analytic and numerical realization of finite-size FSP is the complex-fermion SYK quantum dot with both quartic and quadratic all-to-all random couplings,

$$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$

with

$$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$

Here the random couplings are Gaussian with zero mean and variances

$$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$

which ensure an extensive many-body bandwidth (Dieplinger et al., 2023).

The two limiting theories define the relevant endpoints of the crossover. At $\lambda=1$, the pure SYK$_2$0 model has Wigner–Dyson statistics, ETH eigenstates, and fully ergodic infinite-temperature dynamics. At $_2$1, the pure SYK$_2$2 model is integrable, has Poisson level statistics, and its eigenstates are Slater determinants of single-particle orbitals (Dieplinger et al., 2023). Between these limits, a finite-size crossover scale

$_2$3

(with logarithmic corrections $_2$4) separates regimes that are dynamically close to Fock-space localization from those that are extended and RMT-like in the SYK$_2$5 basis (Dieplinger et al., 2023).

The dynamics is studied at infinite temperature and half filling,

$_2$6

using the correlator $_2$7 above, with time evolution computed by a Chebyshev expansion of $_2$8 and stochastic typicality, and long-time limits obtained by exact diagonalization (Dieplinger et al., 2023). Operationally, the thermalization time $_2$9 is defined by a threshold such as

$$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$0

For small but nonzero $$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$1 on the ergodic side $$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$2, $$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$3 exhibits a pronounced prethermal plateau before eventual decay to the ergodic saturation value (Dieplinger et al., 2023).

3. Fock-space geometry, distance, and constrained spreading

The Fock-space formulation is essential. In the SYK analysis, the basis states are occupation-number configurations

$$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$4

restricted to half filling. The natural metric is the Fock-space or Hamming distance

$$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$5

with maximal distance $$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$6 (Dieplinger et al., 2023).

For an initial Fock state $$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$7, the probability to be at distance $$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$8 at time $$\mathcal{C}_i(t)=\left\langle \big| \langle n_i(t)n_i(0)\rangle_\infty - n_{\rm fill}^2 \big| \right\rangle_{\rm dis},$$9 is

$_2$0

To remove trivial shell combinatorics, this is compared with the “thermal” distribution $_2$1, corresponding to a wave function uniformly random over all basis states at fixed filling (Dieplinger et al., 2023). The ratio $_2$2 then measures deviation from uniform Fock-space spreading.

The two limits are sharply distinct. In the integrable SYK$_2$3 limit, the infinite-time profile is strongly biased toward small $_2$4 and follows a stretched exponential,

$_2$5

with $_2$6 a Fock-space correlation length (Dieplinger et al., 2023). In the chaotic SYK$_2$7 limit, by contrast,

$_2$8

which is the ETH/RMT expectation for a structureless wave function in the occupation basis (Dieplinger et al., 2023).

FSP is the intermediate regime in which the wave function has relaxed within a restricted Fock-space region but not yet over the full Fock graph. In the SYK crossover, intermediate-time profiles retain the stretched-exponential SYK$_2$9-like shape inside a shell $_4$0, while probabilities outside that shell remain suppressed relative to $_4$1. Only for $_4$2 does the profile flatten toward the ergodic distribution (Dieplinger et al., 2023). This motivates the formulation of FSP as a Fock-space constrained ergodic regime.

A closely related but dynamically opposite benchmark is provided by dual-unitary Floquet dynamics. In the self-dual kicked Ising model, the generalized inverse participation ratios

$_4$3

approach their Haar-random values exponentially fast in time, and the overlap distribution converges to Porter–Thomas on a timescale independent of system size (Claeys et al., 2024). That result functions as an “anti-FSP” reference point: FSP corresponds precisely to the failure of such rapid, system-size-independent delocalization.

4. Timescales, ETH, and the finite-size character of FSP

The central scaling result in the SYK crossover is

$_4$4

with $_4$5 from numerical fits (Dieplinger et al., 2023). The origin of this form is a matching argument between the crossover scale $_4$6 and the Heisenberg time

$_4$7

Substituting $_4$8 gives the observed exponential dependence on $_4$9 (Dieplinger et al., 2023). The prethermal plateau therefore survives essentially up to Heisenberg time as $n_k$0.

This is why the phenomenon is termed finite-size prethermalization. The plateau exists for finite $n_k$1, is bounded above by $n_k$2, and disappears straightforwardly in the thermodynamic limit at fixed $n_k$3, where the asymptotic state is ergodic (Dieplinger et al., 2023). The long-time Fock-space extension is also captured by the first moment

$n_k$4

whose plateau at large nonzero values near $n_k$5 signals localized-like intermediate dynamics, followed by decay toward zero once full ergodicity is restored (Dieplinger et al., 2023).

The ETH connection can be made more explicit through the inverse participation ratio in a different clean interacting fermion problem. There, exact diagonalization shows that eigenstate-to-eigenstate fluctuations of the momentum occupation $n_k$6 are proportional to the IPR

$n_k$7

in the Fock basis of the noninteracting Hamiltonian, so that Fock-space delocalization is the microscopic mechanism for the onset of eigenstate thermalization (Neuenhahn et al., 2010). In that language, FSP corresponds to an intermediate regime in which $n_k$8 is already decreasing, so local observables can become quasi-stationary, but the wave function is still far from the $n_k$9-type scaling associated with a fully chaotic state (Neuenhahn et al., 2010).

A recurrent misconception is that FSP implies permanent nonergodicity. The SYK analysis shows the opposite for any fixed $$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$0: despite an exponentially long delay, the system ultimately thermalizes, and Fock-space observables approach their ergodic SYK$$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$1 values (Dieplinger et al., 2023). A second misconception is that FSP must rely on spatial locality. The SYK realization shows that it can instead arise from finite-size hybridization structure on Fock space itself (Dieplinger et al., 2023).

5. Related paradigms: dephasing, hidden thermal structure, transport, and fragmentation

One broad paradigm realizes prethermalization through dephasing at fixed Fock-space weights. In the experiment on a coherently split one-dimensional Bose gas, the relative sector is described by a harmonic Luttinger liquid,

$$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$2

so the mode occupations $$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$3 are conserved. Rapid splitting prepares a nonthermal distribution $$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$4, and time evolution dephases phases between Fock components $$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$5 without redistributing the occupations. The resulting quasi-steady state is described by an effective temperature

$$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$6

yet it is not the true final thermal state (Smith et al., 2012). In FSP language, this is prethermalization generated by phase scrambling at fixed Fock-space populations.

A complementary static line of work identifies a “hidden thermal structure” in Fock space. For an ideal Fermi gas, coarse-grained occupations

$$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$7

maximize the combinatorial entropy

$$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$8

under particle-number and energy constraints, producing the typical occupation fraction

$$\mathcal{H}=\lambda\,\mathcal{H}_{\text{SYK}_4}+(1-\lambda)\,\mathcal{H}_{\text{SYK}_2}, \qquad \lambda\in[0,1],$$9

This shows that an overwhelming number of Fock states share an observable-resolved Fermi–Dirac limit shape $$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$0 (Tian et al., 2018). A plausible implication is that some FSP plateaus can be interpreted as dynamics constrained to, or dephasing within, coarse-grained Fock-space manifolds that already possess thermal structure for appropriate observables.

In disordered systems, the language of probability transport and fragmentation sharpens the dynamical picture. For a disordered transverse-field Ising chain written as a tight-binding model on a Fock-space hypercube, the probability distribution

$$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$1

shows strongly inhomogeneous, multifractal intermediate-time structure even in the ergodic phase, while in the MBL phase the inhomogeneity persists indefinitely (Creed et al., 2022). In a distinct disordered interacting-fermion quench problem, the accessible phase space is organized into “potential-energy shells,” and at strong disorder these shells decay into disconnected fragments; the fragment containing the initial state then controls long-lived non-ergodic relaxation (Modak et al., 22 Oct 2025). These results suggest that FSP can arise not only from approximate integrability, but also from multifractal transport or shell fragmentation on the Fock-space graph.

6. Experimental realizations and diagnostics

The most direct explicit realization of FSP on hardware is the periodically driven Ising chain implemented on 72 superconducting qubits. There the Floquet unitary is engineered so that strong Ising interactions and small perturbations make the total domain-wall number $$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$2 approximately conserved. The Fock-space network then separates into $$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$3 sparse sub-networks labeled by even $$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$4, and only spin flips that preserve local domain-wall structure are resonant at leading order (Bao et al., 28 Oct 2025). This yields long-lived Fock-space confinement, suppressed heating, and FSP-based discrete time-crystalline order persisting over 120 cycles for generic initial Fock states (Bao et al., 28 Oct 2025).

The principal observables in that experiment are the Fock-space wave-packet distribution

$$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$5

its mean position

$$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$6

its width

$$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$7

and the domain-wall distribution

$$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$8

Finite-size scaling across $$\mathcal{H}_{\text{SYK}_4}=\sum_{i,j,k,l=1}^{N} J_{ijkl}\,c_i^\dagger c_j^\dagger c_k c_l, \qquad \mathcal{H}_{\text{SYK}_2}=\sum_{i,j=1}^N t_{ij}\,c_i^\dagger c_j .$$9 reveals size-independent crossover behavior in both domain-wall and Fock-space dynamics, and the eigenstates of the Floquet unitary cluster by average domain-wall number $$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$0 in the FSP regime (Bao et al., 28 Oct 2025).

A broader experimental toolkit for FSP-like questions is provided by the 24-qubit two-dimensional Bose–Hubbard/XY experiment that directly measures Fock-space wave-packet propagation. There the many-body problem is mapped to an Anderson-like model on a graph of $$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$1 Fock states, and the radial distribution

$$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$2

is tracked experimentally, together with the displacement $$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$3, width $$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$4, and Bhattacharyya coefficient $$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$5 relative to the ergodic radial distribution $$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$6 (Yao et al., 2022). Although that work does not use the term FSP, it suggests a practical diagnostic program: FSP corresponds to long-lived confinement of $$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$7 away from $$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$8, saturation of $$\langle t_{ij}^2 \rangle_{\rm dis}=\frac{J^2}{64N}, \qquad \langle J_{ijkl}^2 \rangle_{\rm dis}=\frac{J^2}{2N^3},$$9 and $\lambda=1$0 at non-ergodic values, and eventual drift only on much longer timescales.

Taken together, these developments establish FSP as a unifying language for several nonequilibrium mechanisms. In one class, the wave function is temporarily trapped in a stretched-exponential or multifractal region of Fock space near a localization–delocalization crossover; in another, it dephases at fixed mode occupations; in another still, Floquet dynamics creates sparse sub-networks labeled by approximately conserved domain-wall number. The common structure is a separation between local or coarse-grained equilibration and global Fock-space ergodization. Whether the resulting plateau is transient, exponentially long, or asymptotically stable depends on the underlying mechanism: finite-size hybridization in the SYK crossover, exact or approximate integrability in harmonic mode problems, or fragmentation-like constraints in driven and disordered systems (Dieplinger et al., 2023)