Quenched LMG Model Dynamics

• The quenched LMG model is a paradigmatic framework describing collective spin systems undergoing abrupt parameter changes, revealing rich quantum-critical dynamics.
• It employs analytical mappings and large-scale numerical methods to uncover scaling behaviors, spread complexity, and distinct quench regimes related to phase transitions.
• The model's dynamic probes, including fidelity and spread entropy, offer actionable insights for experimental platforms such as trapped-ion and cavity-QED systems.

The quenched Lipkin–Meshkov–Glick (LMG) model is a paradigmatic setting for examining out-of-equilibrium critical dynamics in collective spin systems with mean-field type interactions. Under abrupt changes (“quenches”) of the control parameters, the LMG model exhibits rich dynamical behavior, including signatures of quantum phase transitions and universality classes that are revealed via time-dependent measures such as spread complexity, spread entropy, fidelity, and fluctuation scaling. These properties have been rigorously analyzed through both analytically tractable mappings and large-scale exact numerics, particularly in the thermodynamic ($N\rightarrow\infty$) limit.

1. Definition and Structure of the LMG Model

The LMG model describes $N$ spin-$\frac{1}{2}$ particles with infinite-range anisotropic $XY$-type couplings subjected to a transverse magnetic field. The standard Hamiltonian (up to an irrelevant constant) is

$$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$

where $J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$ are collective spin operators ($\alpha \in\{x,y,z\}$), $\gamma\in [0,1]$ is the $XY$-anisotropy, $h$ the transverse field, and $N$0 the total spin number.

The model undergoes a second-order quantum phase transition at $N$1 (in appropriate units), separating:

• the symmetric phase (SP) with $N$2,
• the broken phase (BP) with $N$3.

Applying a Holstein–Primakoff and Bogoliubov transformation in the $N$4 limit maps the LMG Hamiltonian in each phase to a single harmonic oscillator: $N$5 The excitation gap closes at $N$6, marking the quantum-critical point (Afrasiar et al., 2022, Campbell, 2016, Titum et al., 2019).

2. Quench Protocols and Dynamical Regimes

The typical quench protocol involves initializing the system in the ground state $N$7 of a pre-quench Hamiltonian $N$8 and, at $N$9, abruptly switching to a post-quench Hamiltonian $\frac{1}{2}$0. The subsequent time evolution is governed by $\frac{1}{2}$1.

Quenches can be classified, particularly when targeting the quantum-critical manifold, as follows (Titum et al., 2019):

• Type I ("effectively thermal"): Initial state deep in the disordered (paramagnetic) phase.
• Type II ("quantum-critical"): Initial state on the $\frac{1}{2}$2-critical line, exhibiting quantum-critical scaling.
• Type III ("genuinely non-equilibrium"): Initial state on the $\frac{1}{2}$3-critical line, associated with divergent fluctuations.

Each class exhibits distinct scaling behaviors in dynamical observables and effective temperatures after the quench.

3. Spread Complexity, Krylov Dynamics, and Entropy

The time-evolved state $\frac{1}{2}$4 can be recast in the Krylov (“Lanczos”) basis, generated by iterative action of $\frac{1}{2}$5 on $\frac{1}{2}$6. The continuous expansion is

$\frac{1}{2}$7

where $\frac{1}{2}$8 form an orthonormal chain constructed via the Lanczos algorithm. The probability weights $\frac{1}{2}$9 encode the occupancy of the $XY$0th Krylov level.

The spread complexity is defined as

$XY$1

measuring the “average” position in Krylov space populated by the evolving state. For practical purposes, the effective number of contributing Krylov levels,

$XY$2

(where $XY$3) serves as a robust measure of Hilbert-space complexity.

The spread entropy is the Shannon entropy in Krylov space,

$XY$4

which provides additional insight into the distribution's width and tail properties, with slow convergence near criticality (Afrasiar et al., 2022).

4. Analytical Results: Scaling, Phase Distinction, and Criticality

For quenches away from the critical point, the harmonic oscillator mapping permits closed-form expressions for spread complexity. For $XY$5,

$XY$6

where $XY$7 are phase-appropriate oscillator frequencies. Thus, $XY$8 exhibits bounded oscillations (frequency $XY$9), with amplitude and period diverging as $$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$0. Correspondingly, $$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$1, with distinct exponents: $$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$2 for $$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$3, $$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$4 for $$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$5.

At criticality ($$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$6), the post-quench Hamiltonian is effectively free ($$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$7). The spread complexity grows as

$$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$8

hence, $$H_{\mathrm{LMG}} = -\frac{2}{N}\left(J_x^2 + \gamma J_y^2\right) - 2 h J_z,$$9 displays unbounded quadratic growth. The spread entropy, for critical quenches, diverges logarithmically in time, $J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$0.

The distinction between symmetric and broken phases is thereby encoded not just in long-time asymptotic values, but in the scaling and growth rates of spread-based measures (Afrasiar et al., 2022).

5. Dynamical Probes: Fidelity, Energetics, and Spectral Signatures

The time-dependent fidelity (Loschmidt echo),

$J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$1

offers an independent dynamical signature. For quenches within the same quantum phase, $J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$2 exhibits high-amplitude, nearly periodic revivals. Quenches across $J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$3 destroy this coherence; $J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$4 demonstrates “dynamical orthogonality,” regularly reaching zero, with this phenomenon scaling towards the critical point with increasing $J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$5.

Analysis of average work, free energy difference, and irreversible work after the quench shows that only the latter develops a cusp at $J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$6, marking the quantum phase transition. The spectral function, i.e., the time-Fourier transform of the overlap, sharpens this picture: within phases, the spectrum is peaked near the ground-state. Crossing $J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$7 introduces a broad spectrum—multiple excited states acquire weight, signaling non-equilibrium access to a dense excitation continuum (Campbell, 2016).

6. Scaling Theory and Universality

The non-equilibrium Keldysh field-theory formalism yields critical exponents and scaling forms for post-quench fluctuations and correlations. Specifically, the connected two-time fluctuations $J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$8 scale as $J_\alpha = \frac{1}{2}\sum_{i=1}^N\sigma_i^\alpha$9, and the dynamical susceptibility as $\alpha \in\{x,y,z\}$0. The resulting universality classes are summarized:

Quench Type	$\alpha \in\{x,y,z\}$1	$\alpha \in\{x,y,z\}$2	$\alpha \in\{x,y,z\}$3
Type I	1/4	1/2	finite
Type II	1/3	1/3	$\alpha \in\{x,y,z\}$4
Type III	1/6	2/3	$\alpha \in\{x,y,z\}$5

Type I corresponds to effective thermalization, Type II to quantum-critical scaling, and Type III to a genuinely non-equilibrium regime with diverging effective temperature ($\alpha \in\{x,y,z\}$6). These findings are supported by exact diagonalization for up to $\alpha \in\{x,y,z\}$7 spins (Titum et al., 2019).

7. Significance and Broader Impact

Analysis of the quenched LMG model, especially through spread complexity and entropy measures, offers versatile non-equilibrium protocols for probing quantum phase transitions without recourse to equilibrium observables or arbitrarily chosen geometric cost functions. The robust scaling, non-analyticities, and universality reconstructed from out-of-equilibrium dynamics provide detailed probes into many-body quantum criticality, thermalization, and the generation of novel non-equilibrium universality classes. The structure and insights gained from these protocols are directly relevant to experimental platforms with collective spin interactions, such as trapped-ion and cavity-QED systems (Afrasiar et al., 2022, Campbell, 2016, Titum et al., 2019).