Time evolution of spread complexity in quenched Lipkin-Meshkov-Glick model

TL;DR

This work introduces spread complexity (SC) and spread entropy (S_K) computed from Krylov (Lanczos) bases as robust probes of zero-temperature quantum phase transitions in the infinite-range Lipkin-Meshkov-Glick model under sudden quenches toward criticality. By constructing the Krylov basis and deriving explicit Lanczos coefficients in the thermodynamic-limit mapping to harmonic oscillators, the authors obtain exact expressions for SC and its time evolution, including a new quantitative marker N_eff that tracks how many Krylov elements effectively contribute as criticality is approached. They show that SC grows quadratically with time under critical quenches and remains oscillatory with revivals away from criticality, while spread entropy exhibits oscillations away from criticality but diverges logarithmically at late times for critical quenches. The results demonstrate that SC and N_eff can distinguish the BP and SP phases and provide a robust, model-specific probe of quantum phase transitions in quenched many-body systems, with the LMG-HO mapping offering analytic control and insights into integrable dynamics.

Abstract

We use the spread complexity of a time evolved state after a sudden quantum quench in the Lipkin-Meshkov-Glick (LMG) model prepared in the ground state as a probe of quantum phase transition when the system is quenched towards the critical point. By studying the growth of the effective number of elements of the Krylov basis, those contribute to the spread complexity more than a preassigned cut off, we show how the two phases of the LMG model can be distinguished. We also explore the time evolution of spread entropy after both non-critical and critical quenches. We show that the sum contributing to the spread entropy converges slowly in the symmetric phase of the LMG model compared to that of the broken phase, and for a critical quench, the spread entropy diverges logarithmically at late times.

Figures (14)

• Figure 1: Variation of $\mathcal{F}(t)$ with time in the BP for different post-quench magnetic fields. Here $h_f=0.9$ (green), $h_f=0.95$ (blue) and $h_f=0.99$ (red) respectively. The parameter $\gamma$ has a fixed value $0.1$, and $h_i=0.5$.
• Figure 2: Evolution of $\mathcal{C}_1(t)$ with time in the BP for different post-quench magnetic fields. The parameter values and color coding are the same as those of Fig. \ref{['fig:F_broken']}.
• Figure 3: Plot of the individual contributions to the SC sum in eq. \ref{['SCsum']}. Here $n=9$ (cyan), $n=10$ (red), $n=11$ (blue), $n=12$ (green), $n=30$ (black). The parameter values are $h_i=0.1, h_f=0.99, \gamma_i=\gamma_f=0.1$. For these values of the parameters, the individual contributions towards the total sum start to decrease from $n=10$.
• Figure 4: Plot of $N_{eff}$ with $h_f$ close to the criticality in the BP. The blue dots are numerical data which is fitted with the red curve where the equation of the red curve is given in eq. (\ref{['fit']}). The derivative of $N_{eff}$ with respect to $h_f$ is shown in the inset. We set $\epsilon=0.001$ and $h_i=0.1$.
• Figure 5: Variation of $\mathcal{F}(t)$ with time in the SP of the system for different post-quench magnetic fields. Here $h_f=1.1$ (green), $h_f=1.05$ (blue) and $h_f=1.01$ (red) respectively. The parameter $\gamma$ has a fixed value $0.1$, and $h_i=1.5$.