Universality and weak-ergodicity breaking in quantum quenches

TL;DR

This work shows that weak ergodicity breaking in the large-$n$ limit of local $O(n)$ models during quantum quenches arises from lattice discretization, via a resonance at the upper edge of the quasiparticle band. In the quantum-field-theory limit, the dynamics thermalize to a Generalized Gibbs Ensemble, with a self-consistent GGE mass $m^2_{\rm gge}$ governing mode occupations and a spectrum $\Omega_k=\sqrt{m^2_{\rm gge}+\omega_k^2}$. The authors reconcile discrepancies in scaling behavior by accounting for the UV lattice cutoff, revealing dynamical critical behavior in the thermal universality class and predicting Higgs-like persistent oscillations tied to band-edge physics for finite lattice spacing. The results underscore the role of lattice effects in dictating non-ergodic features of quantum quenches and provide a concrete, controllable framework for GGE thermalization in the $O(n)$ model at large $n$.

Abstract

Sudden quenches in quantum many-body systems often lead to dynamical evolutions that unveil surprising physical behaviors. In this work, we argue that the emergence of weak ergodicity breaking following quantum quenches in certain local many-body systems is a direct consequence of lattice discretization. To support this claim, we investigate the out-of-equilibrium dynamics of quantum $O(n)$ models on a lattice. In doing so, we also revisit two puzzling results in the literature on quantum $O(n)$ models, concerning universal scaling and equilibration, and demonstrate how these apparent contradictions can be resolved by properly accounting for lattice effects.

Figures (4)

• Figure 1: $(a)$ Sketch of the the quantum $O(n)$ model for $r<0$: the model consist of a chain of locally coupled bosons kept in place by a quartic potential. $(b)$ Sketch of the large-$n$ limit of the model: in this regime the model can be described as gas of phonons scattering through the time-dependent, self consistent mass. $(c)$ Sketch of the quench dynamics: starting from the ground state, the system is driven out of equilibrium by a sudden quench of the bare mass $r^{-} \rightarrow r$ (here $r < 0$).
• Figure 2: (a) Out-of-equilibrium phase diagram. Blue dots represent the long-time average of the order parameter, $\phi = \eta_0 / \sqrt{N}$ (right $y$-axis), while red squares denote the squared mass, $m^2$ (left $y$-axis), as functions of the final bare mass $r$. For $r < r_\mathrm{gge}^c$, the system exhibits a finite order parameter ($\phi > 0$) and a vanishing mass ($m^2 = 0$), whereas for $r > r_\mathrm{gge}^c$ the order parameter vanishes ($\phi = 0$) and the mass remains finite ($m^2 > 0$). The gray dashed line marks the GGE prediction for the dynamical critical point $r = r_\mathrm{gge}^c$. The numerical results for the averaged squared mass (red squares) are also in excellent agreement with the GGE prediction \ref{['eq:mgge']} (green solid line). The insets display the scaling behavior of the averaged order parameter (blue dots, left inset) and of the averaged squared mass (red dots, right inset) as functions of $\delta r = |r - r_\mathrm{gge}^c|$ in a log–log scale near the transition. The extracted critical exponents are consistent with the spherical-model universality class in $d = 3$: $\overline{m^2} \sim |\delta r|^{2\nu}$ with $\nu = 1/(d-2) = 1$, and $\overline{\phi} \sim |\delta r|^{\beta}$ with $\beta = 1/2$. Deviations from the power-law scaling of the squared mass at very small $\delta r$ arise from finite-size effects. The system size is $L = 5a\times 10^3$, the lattice spacing is $a = 3$ the initial value of the bare mass is $r^{-} = 1>r_\mathrm{gge}^c$, $\lambda = 1$. (b) Time evolution of the squared mass $m^2(t)$ following a sudden quench from the disordered phase ($r^- = 1$) to the ordered phase ($r = -10$) in $d = 3$, for different lattice spacings $a$ (curves in different shades). The system size is $L = 8a\times 10^3$, with $\lambda = 1$. In the field-theory limit ($a \to 0$), the dynamics converge to the GGE prediction (green dashed line). For finite lattice spacing, $m^2(t)$ displays persistent, undamped oscillations around the GGE value. (c) Time evolution of the order parameter $\phi(t)$ for the same quench and system parameters as in panel (b). For finite lattice spacing, $\phi(t)$ displays fast, undamped oscillations superimposed to the standard oscillations around the potential minimum in the broken phase. The insets in panels (b) and (c) show the corresponding phase-space trajectories in $(m, \dot{m})$ and $(\phi, \dot{\phi})$ space, respectively.
• Figure 3: $(a)$ Jacobi quasiparticle picture: each microscopic degree of freedom oscillates between the inversion points $u_l^{\pm}$ of the potential $V(u)$ for $u >0$. The negative zero $u =-\mu$ of $V(u)$ physically represent the effective GGE mass $m_{\rm gge}^2$. As the $u_l^{\pm}$ are bounded by the bare frequencies $\omega_l^2$, as the spectrum becomes continuous for $N \rightarrow \infty$, the first $\mathcal{N}-1$ Jacobi quasiparticles acquire a well-defined energy, while the oscillation $\mathcal{N}$-th quasiparticle beyond the edge of the spectral band has a finite support, leading to the emergence of a Higgs mode with persistent oscillations. $(b)$ The set of action variables $\lbrace J_l, \ell_l \rbrace$ which constraints the dynamics on invariant tori (see inset) can be computed from the complex plane singularities of $f(u) = \sqrt{(u+\mu) V(u)}$, as integrals around its branch cuts and residues at the poles $\lbrace \omega^2_l \rbrace$ respectively. In the QFT limit ($a \rightarrow 0$), this allows the direct computation of the spectrum of the model and a proof of the thermalization to the GGE predictions.
• Figure S.1: Panels (a) and (b) show, respectively, the scaling of the long-time average of the squared mass $m^2$ and of the order parameter $\phi = \eta_0 / \sqrt{N}$ as functions of $\delta r = |r - r_\mathrm{gge}^c|$ close to the dynamical transition (log–log scale), for different values of the lattice spacing $a$. The numerical results for $m^2$ obtained from the equations of motion (red squares) are in excellent agreement with the GGE prediction (solid lines). The system size is $L = 5a\times 10^3$, $\lambda = 1$. As discussed in the main text, the scaling behavior consistent with the spherical-model universality class in $d=3$, $\overline{m^2}\sim (r-r_\mathrm{gge}^c)^{2}$, $\overline{\phi}\sim (r_\mathrm{gge}^c-r)^{1/2}$ emerges within the range of numerically accessible $\delta r$ only for sufficiently large lattice spacings ($a > 1$). Conversely, for smaller lattice spacings ($a < 1$), the numerical data display an effective scaling $\overline{m^2}\sim (r-r_\mathrm{gge}^c)$, $\overline{\phi}\sim (r_\mathrm{gge}^c-r)^{1/4}$ in agreement with previous findings in Refs. sciolla2013quantumweidinger2017dynamical.