Sudden-Quench Protocol in Many-Body Systems

• Sudden-Quench Protocol is defined as an instantaneous change of a system’s Hamiltonian at t=0, producing nonequilibrium states with interference among eigenstates.
• It is widely applied to study relaxation, universal scaling, and prethermalization in various platforms such as Fermi gases, Ising chains, and condensates.
• The protocol enables precise probing of transport properties, work statistics, and operator growth through diverse experimental and computational techniques.

Searching arXiv for recent and canonical papers on sudden-quench protocols to ground the article in cited literature. A sudden-quench protocol is a nonequilibrium preparation in which a control parameter of the Hamiltonian, or more generally of the dynamical generator, is changed instantaneously at a specified time, conventionally $t=0$, so that the pre-quench state is no longer an eigenstate of the post-quench dynamics. In the closed-system setting this is the standard initialization $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$ after an abrupt replacement of $H_<$ by $H$; in ramp formulations it is the limit in which a switching function becomes a Heaviside step, $S(t)\to \theta(t)$ (Caux, 2016, Huang et al., 2019). Across the literature represented here, sudden quenches function both as an idealized theoretical limit and as an experimentally relevant approximation, providing a reference point for questions of relaxation, universality, work statistics, transport, operator growth, prethermalization, and the crossover to adiabatic or continuous driving (Dóra et al., 2010, Kuić et al., 20 Jun 2025, Peluso et al., 19 Feb 2026).

1. Formal definition and mathematical structure

In the canonical closed-system formulation, the system is prepared in a distinguished initial state—typically the ground state of a pre-quench Hamiltonian $H_<$—and then evolved with a different Hamiltonian $H$ after an instantaneous parameter change at $t=0$. The Quench Action review adopts precisely this definition and emphasizes that the initial state must be expanded over the eigenbasis of the post-quench Hamiltonian, with the subsequent dynamics generated by interference and dephasing among many eigenstates (Caux, 2016). In time-dependent interaction ramps this same structure is encoded by

$H(t)=H_0+S(t)V,$

with the sudden limit $S(t)=\theta(t)$, while finite ramp protocols interpolate continuously away from that limit (Huang et al., 2019).

Several papers make the “instantaneous” character operational rather than purely formal. In sudden-quench quantum Otto engines, the unitary work stroke is assumed so rapid that the state has no time to evolve during the stroke, which is represented by $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$0, so that work reduces to a change of Hamiltonian expectation value evaluated in the initial equilibrium state (Watson et al., 28 May 2025). In open-system Pontus-Mpemba protocols, the sudden-quench limit is the direct protocol in which the parameters of the Lindblad generator are abruptly switched from the initial steady-state generator to the final one; in that limit the continuous protocol satisfies $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$1 and $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$2 (Peluso et al., 19 Feb 2026).

This formalization already implies a distinction between sudden quench and adiabatic evolution. The sudden protocol does not assume eigenstate tracking, and in several models its consequences are explicitly non-adiabatic: mode occupations are frozen at the switching event, high-energy sectors can be populated, and late-time states need not coincide with equilibrium states of the post-quench Hamiltonian (Huang et al., 2019, Maghrebi, 2017).

2. Physical realizations across many-body platforms

The protocol has been instantiated in a wide range of systems, with the “quenched” control variable depending on the microscopic context. In a two-component Fermi gas, the interaction is turned on through a switching protocol $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$3, and the sudden-quench limit is $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$4, so $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$5 (Huang et al., 2019). In two initially independent condensates, the quench is the abrupt switching-on of a tunneling term $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$6 at $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$7, converting a gapless relative-phase sector into a gapped one and driving rephasing dynamics (Torre et al., 2012). In the transverse-field Ising chain, the quench is a global sudden change of the transverse field $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$8, with unitary evolution thereafter under $|\Psi(t)\rangle = e^{-iHt}|\Psi(t=0)\rangle$9 (Calabrese et al., 2012). In SYK-based quench dynamics, the Hamiltonian is explicitly piecewise,

$H_<$0

so a local-field ground state is abruptly evolved under an SYK$H_<$1 Hamiltonian (Carrega et al., 2020).

Experimental realizations need not be phrased as direct temperature jumps. In a binary two-dimensional colloidal suspension of superparamagnetic particles at a water-air interface, a sudden increase of a perpendicular magnetic field produces an ultra-fast quench because the coupling

$H_<$2

scales as $H_<$3. The experiment quenches $H_<$4 from $H_<$5 to $H_<$6 at $H_<$7, with a field ramp of about $H_<$8, far shorter than the Brownian time $H_<$9, so the protocol is effectively instantaneous on the structural timescale (0811.1498).

Other realizations are geometric or spacetime-structured rather than purely parametric. In the fractional quantum Hall matrix model, a geometric quench is implemented by suddenly replacing the Euclidean metric $H$0 by a constant unimodular anisotropy metric $H$1, thereby changing the ambient geometry seen by the droplet (Lapa et al., 2018). In a $H$2D conformal-field-theory setting, the quench may be sudden in conformal time rather than in laboratory time: the mass gap is removed along the spacelike hyperbola $H$3, producing a boost-invariant “hyperbolic sudden-quench protocol” (Mitra et al., 2018). In holography, a sudden quench can be realized as a rapid Gaussian pulse in a scalar source,

$H$4

applied to an already phase-separated initial state (Chen et al., 2022).

These examples show that “sudden” is not tied to a single experimental control knob. A plausible implication is that the defining criterion is scale separation between the switching time and the system’s intrinsic response time, not the particular laboratory variable being varied.

3. Dynamical mechanisms activated by sudden quenches

The generic consequence of a sudden quench is that the post-quench state contains coherent weight over many eigenstates of the new dynamics. The Quench Action formalism encodes this through the overlap functional and the associated saddle-point macrostate, while time dependence is reconstructed from excitations around that saddle (Caux, 2016). In nonintegrable or effectively continuum settings, the same basic mechanism is described in terms of dephasing. For an initially localized interacting wave packet, suddenly switching off short-range interactions leaves the external potential unchanged but exposes the pre-existing high-momentum content of the initial state. Because Tan’s relation gives $H$5, the initial state has nonzero overlap with arbitrarily high-momentum states, and if the post-quench Hamiltonian has scattering states, dephasing turns this overlap into real occupation of delocalized states (Doggen et al., 2014).

Ballistic propagation and light-cone structures are another recurrent mechanism. In the transverse-field Ising chain, two-point order-parameter correlators display a sharp front at the Fermi time $H$6, with the asymptotics governed by mode-dependent velocities $H$7 and the Bogoliubov-angle difference $H$8 (Calabrese et al., 2012). In sudden coupling of two condensates, the Coulomb-gas expansion exhibits a light-cone constraint, and the resulting non-perturbative rephasing dynamics is controlled by a universal scaling function rather than by simple harmonic Josephson motion (Torre et al., 2012). In a one-dimensional cold atomic gas, a sudden interaction quench does not preserve an initial soliton as a post-quench eigen-solution; instead it splits into transmitted and reflected packets with amplitudes

$H$9

and the paper stresses that the bulk velocities of these packets differ from the velocities of their peaks (Franchini et al., 2016).

In all-to-all interacting systems, the quench can reveal operator-space dynamics. For SYK$S(t)\to \theta(t)$0, the post-quench commutator algebra preserves operator size and the paper interprets the dynamics as operator hopping. For SYK$S(t)\to \theta(t)$1, by contrast,

$S(t)\to \theta(t)$2

so locality grows under time evolution, and the decay of connected spin-spin correlators is interpreted as a direct signature of operator growth (Carrega et al., 2020).

The papers also show that sudden quenches can reorganize energy flow in less conventional ways. In the hyperbolic quench, the stress tensor is constrained to a form that drives excess energy toward the null lines $S(t)\to \theta(t)$3 while the bulk approaches the vacuum energy density (Mitra et al., 2018). In the holographic first-order-transition problem, the quench excites domain-wall structures that evolve into energy peaks or wells and can transiently approach a critical nucleus (Chen et al., 2022).

4. Universal scaling, crossover structure, and sudden-versus-adiabatic limits

A major theme is that sudden quench often appears as one endpoint of a controlled crossover. For the interaction ramp in a two-component Fermi gas, the sudden limit $S(t)\to \theta(t)$4 and the adiabatic limit $S(t)\to \theta(t)$5 produce sharply different contacts. In the sudden case, rapid oscillations are followed by a stationary nonequilibrium contact satisfying

$S(t)\to \theta(t)$6

whereas in the adiabatic limit

$S(t)\to \theta(t)$7

so the contact grows quadratically during the ramp and saturates to the equilibrium value (Huang et al., 2019). The same paper further identifies a mode-resolved criterion: if $S(t)\to \theta(t)$8, a large-$S(t)\to \theta(t)$9 mode sees the ramp adiabatically, while if $H_<$0, it effectively experiences a sudden quench.

The Luttinger-liquid analysis gives a complementary spatial and momentum-space crossover. After a finite-time interaction ramp, the single-particle density matrix at late times has a sudden-quench exponent $H_<$1 at large distances and an adiabatic exponent $H_<$2 at short distances, with the crossover scale set by $H_<$3 (Dóra et al., 2010). In momentum space the same structure appears near $H_<$4: sufficiently close to the Fermi point the singularity is SQ-like, while farther away the equilibrium adiabatic exponent is recovered. This suggests that “sudden” and “adiabatic” can coexist within a single post-quench state, resolved by wavelength.

Sudden quenches at criticality can generate their own dynamical universality classes. In the quench of the critical propagation velocity $H_<$5 in the $H_<$6 model, the late prethermal state is controlled by a dynamical fixed point distinct from the equilibrium Wilson-Fisher point, and the critical exponents depend continuously on $H_<$7. The correlation-length exponent is

$H_<$8

while the short-time aging exponent is

$H_<$9

(Maghrebi, 2017). In the sudden-coupling condensate problem, weak coupling implies the scaling form

$H$0

which ties short-time and long-time dynamics to the same universal function (Torre et al., 2012).

Another, more recent, universality statement arises in finite-time path field theory for local magnetic-field quenches in Ising and XY chains. There, a broad class of non-sudden switching functions with specified analyticity properties becomes asymptotically equivalent to the sudden-quench result at long times, because protocol-specific pole contributions decay exponentially in time. The paper states that this holds order by order in perturbation theory and therefore after resummation (Kuić et al., 20 Jun 2025). By contrast, the smooth tanh quenches in lattice Ising and Kitaev models show that once the quench rate approaches the lattice scale, continuum fast-quench scaling saturates and the response approaches the abrupt-quench value (Das et al., 2017). Together these results indicate that “sudden-quench universality” is observable-dependent and asymptotic rather than absolute.

5. Observables, diagnostics, and computational frameworks

Sudden quenches are analyzed through a broad set of observables, each sensitive to different aspects of nonequilibrium structure. In the two-component Fermi gas, the key observables are the total energy and the large-$H$1 tail of the momentum distribution,

$H$2

which identifies the time-dependent Tan contact. Because the contact pseudo-potential has no intrinsic short-distance scale, perturbation theory produces ultraviolet divergences in kinetic and interaction energies individually, and the paper resolves this by renormalizing the bare coupling $H$3 through the physical scattering length $H$4 and introducing a subtraction term in the total-energy shift (Huang et al., 2019).

Order-parameter correlators after sudden quenches admit exact or asymptotically exact treatments in integrable spin chains. In the transverse-field Ising chain, the two principal methods are a determinant/Pfaffian approach and a form-factor expansion in the low-density quantity $H$5, both of which yield analytic asymptotics for one- and two-point functions in ordered, disordered, critical, and cross-phase quenches (Calabrese et al., 2012). In the noisy Ising chain, the work distribution after a sudden field quench exhibits a delta peak at the ground-state energy difference and a continuum above the two-quasiparticle threshold $H$6; static Gaussian disorder smooths the threshold singularity, whereas dynamical Gaussian white noise preserves the square-root edge structure but makes its spectral weight time dependent (Marino et al., 2013).

Loschmidt echo and fidelity constitute another major diagnostic family. In local magnetic-field quenches of Ising and XY chains, the Loschmidt amplitude is

$H$7

and finite-time path methods express its perturbative expansion in terms of projected retarded functions on a finite interval (Kuić et al., 20 Jun 2025). In the geometric quench of fractional quantum Hall matrix models, the fidelity is obtained exactly as

$H$8

identifying the oscillation period with the spin-2 gap $H$9 (Lapa et al., 2018).

Other diagnostics probe information scrambling and thermodynamic performance. In quench-driven scrambling, the commutator-squared

$t=0$0

can display an early-time exponential regime $t=0$1, and the paper argues that when a Lyapunov exponent exists it is set by the quench-generated energy scale or the corresponding effective temperature (Aramthottil et al., 2021). In sudden-quench Otto engines with Hamiltonian $t=0$2, the work of one stroke is

$t=0$3

the net cycle work is

$t=0$4

and the generalized efficiency obeys the universal bound

$t=0$5

(Watson et al., 28 May 2025). This formulation makes the stroke itself a sudden quench and identifies interparticle correlations as the sole determinant of net work.

Methodologically, sudden-quench research draws on an unusually diverse toolkit: overlap-based variational constructions in integrable Bethe-ansatz systems, determinant asymptotics in free-fermion chains, Keldysh kinetic equations in noisy settings, Ermakov equations for exact oscillator dynamics, hydrodynamic reductions to KdV-type equations, exact $t=0$6 disentangling in matrix models, and fully nonlinear numerical evolution in holography (Caux, 2016, Ghosh et al., 2017, Franchini et al., 2016, Chen et al., 2022).

6. Stationary states, prethermalization, and conceptual limits

A recurrent misconception is that an instantaneous quench necessarily drives the system directly to equilibrium under the final Hamiltonian. The papers surveyed here repeatedly show otherwise. In the two-component Fermi gas, the sudden-quench total energy is conserved after the jump because the second-order correction vanishes, yet the contact settles to a stationary value $t=0$7, not to the equilibrium contact of the final Hamiltonian (Huang et al., 2019). In the localized-wave-packet problem, the long-time density is described by a diagonal ensemble

$t=0$8

and delocalization depends on whether scattering states exist in the post-quench spectrum, not on thermalization (Doggen et al., 2014). In the transverse-field Ising chain, stationary correlators in the disordered phase are consistent with generalized Gibbs ensemble physics rather than ordinary thermal relaxation (Calabrese et al., 2012).

Prethermalization provides an intermediate notion of stationarity. In the noisy quantum Ising chain, the system first dephases toward the generalized Gibbs ensemble of the clean sudden quench, then loses coherences exponentially under noise, and only later heats toward the infinite-temperature state as occupations approach $t=0$9 (Marino et al., 2013). In the quench of the critical propagation velocity, a long-lived prethermal critical state is governed by a dynamical fixed point distinct from the equilibrium one, and the short-time approach is controlled by the quantum-aging exponent $H(t)=H_0+S(t)V,$0 (Maghrebi, 2017). These results indicate that sudden quench can produce robust intermediate regimes with universal structure even when asymptotic thermalization is absent or delayed.

The protocol also has well-defined limitations. In contact-interaction problems, the sudden limit amplifies ultraviolet issues unless couplings are renormalized in terms of physical scattering data (Huang et al., 2019). In finite Hilbert spaces, exponential growth in scrambling diagnostics is transient rather than indefinitely sustained (Aramthottil et al., 2021). In smooth-quench families, sudden behavior emerges only after the quench rate reaches the ultraviolet cutoff scale, so continuum fast-quench formulas cannot simply be extrapolated to zero duration (Das et al., 2017). In continuous Pontus-Mpemba protocols, the sudden limit is only a benchmark: it is fast, but the optimal speed-up can occur at intermediate switching times rather than at strictly instantaneous switching (Peluso et al., 19 Feb 2026).

A further conceptual caution concerns the relation between sudden quench and “instantaneous cooling.” The colloidal magnetic-field experiment realizes a virtually instantaneous cooling only because the field ramp is much shorter than the Brownian diffusion time (0811.1498). The hyperbolic quench in $H(t)=H_0+S(t)V,$1D critical systems achieves rapid vacuum preparation not by a globally instantaneous laboratory-time jump, but by a spacelike quench geometry that expels excess energy toward $H(t)=H_0+S(t)V,$2 (Mitra et al., 2018). This suggests that “suddenness” should be understood relative to the effective dynamical frame of the problem.

Taken together, these works establish the sudden-quench protocol as a unifying baseline for nonequilibrium many-body physics. Its defining idealization—instantaneous switching at $H(t)=H_0+S(t)V,$3—is simple, but the resulting phenomena are not: stationary states can be nonthermal, universal behavior can coexist with strong protocol dependence, and the same abrupt perturbation can generate dephasing, ballistic fronts, delocalization, operator growth, critical nuclei, or correlation-powered work extraction depending on the structure of the post-quench spectrum and the observable under consideration.