How compactness curbs entanglement growth in bosonic systems

Abstract

Zero modes, understood here as degrees of freedom with vanishing confining frequency, play a central role in the nonequilibrium dynamics of bosonic systems. In Gaussian models, however, they lead to an unbounded, logarithmic growth of entanglement entropy. We show that this divergence is not an intrinsic property of zero modes themselves, but arises specifically for non-compact zero modes. Their non-compact configuration space allows unbounded spreading in position space, while their continuous spectra enable indefinite dephasing in momentum space. By contrast, compact zero modes in compact bosonic systems behave fundamentally differently: Spreading and dephasing are eventually halted, so that compactness caps the entanglement entropy at a finite value, making its dynamical role most transparent in the presence of a zero mode. We demonstrate this mechanism in a minimal setting by comparing two coupled harmonic oscillators with two coupled quantum rotors. We then show that the same physics persists in many-body systems by contrasting an N-site compact rotor chain with the non-compact harmonic chain. Finally, we relate these insights to ultra-cold-atom realizations of compact quantum field theories. In particular, we clarify when a compact free-boson (Tomonaga-Luttinger liquid) description is required and when the commonly used non-compact massless Klein-Gordon model breaks down. Even when the initial state is accurately captured by a non-compact Gaussian description, compactness ultimately governs the late-time quench dynamics, curbing entanglement growth rather than allowing a dynamical divergence.

Figures (9)

• Figure 1: Schematic overview. (a) Quantum field theory setting and quench protocol. Consider a system initially prepared in a low-temperature thermal state of the sine–Gordon quantum field theory on a finite spatial interval with Neumann boundary conditions. For a large on-site potential, phase fluctuations are suppressed and the initial state is well described by a quadratic, non-compact approximation—the massive Klein–Gordon theory. We then suddenly quench the on-site potential to zero, causing the center-of-mass to become a zero mode, which subsequently travels freely around the circle. While the early-time dynamics are still captured by a non-compact description with an unbounded zero mode and continuous spectrum, the description of late-time behavior requires the compact theory, where the zero mode lives on a bounded domain and has a discrete spectrum. (b) Minimal models. Two coupled quantum rotors (CR, compact; see Sec. \ref{['section:Sec3']}) and two coupled harmonic oscillators (CHO, non-compact; see Sec. \ref{['section:Sec2']}) provide the simplest realizations of the compact and non-compact theories in (a); their lattice generalizations form rotor and harmonic chains (see Sec. \ref{['section:Sec5']}). (c) Effect of compactness on entanglement growth. From the initial harmonic regime (0), the zero mode spreads in position space (equivalently dephases in momentum space) (1), where compact and non-compact dynamics coincide, until compactness sets in (2). Thereafter spreading continues, yielding logarithmic entanglement growth in the non-compact case (3a), but halts in the compact case, leading to saturation (3b).
• Figure 2: Momentum-space structure of the pre-quench ground state in the two-site coupled rotor (CR) model. (a) Joint probability density $|c(p_1,p_2;0)|^2$ in the truncated angular-momentum basis $p_n\in\{-M,\ldots,M\}$, showing strong localization around $(p_1,p_2)=(0,0)$. (b) Marginal distribution $f(p_1;0)$ on a semi-logarithmic scale together with a Gaussian fit, demonstrating rapid decay of high-momentum components. (c) Time evolution of selected momentum occupations $f(p_1;t)$, showing that localization in momentum space is preserved under the post-quench dynamics. Parameters: $\omega^2=5$, $\kappa=10$.
• Figure 3: Dynamics after a zero-frequency quench in the two-site coupled-rotor (CR) model. (a) Single-site entanglement entropy $S_1(t)$. At early times the CR dynamics coincide with those of the non-compact coupled harmonic oscillator (CHO), before compactness becomes relevant. While the CHO entropy grows logarithmically, the CR entropy saturates at a finite value, remaining below the generalized Gibbs ensemble (GGE) prediction, and exhibits oscillations. The inset shows $\langle\cos(\sqrt{2}x_{\pm})\rangle$, revealing collapse–revival dynamics of the center-of-mass mode $(+)$ and an effectively frozen relative mode $(-)$. (b) Magnitude of the reduced single-site density matrix in position space, $|\rho^{\rm CR}_1(x_1,x_1';t)|$, at selected times. (c) Magnitude of the reduced single-site density matrix in momentum space, $|\rho^{\rm CR}_1(p_1,p_1';t)|$, at the same times as in (b). Parameters: $\kappa=100$, $\omega^2=10$.
• Figure 4: Dynamics after a zero-frequency quench in the two-site coupled-rotor (CR) model for different parameter regimes. Top panels: Single-site entanglement entropy $S_1(t)$ for the CR (solid) and the corresponding coupled harmonic oscillators (CHO, dashed). Bottom panels: Expectation values $\langle\cos(\sqrt{2}x_{\pm})\rangle$. (a) Regime with $\omega^2 \gg 2\kappa \gg 1$ (here $\omega^2 = 100$, $\kappa = 10$), where the initial state and the early-time dynamics are still well described by the Gaussian harmonic approximation, but the relative mode is no longer frozen. (b) Intermediate regime with $\omega^2 \sim \kappa \sim 1$ (here $\omega^2 = 1.5$, $\kappa = 0.5$), where the initial state remains approximately Gaussian but the relative mode contributes substantially; these parameters match those used in Fig. \ref{['fig:Fig5']}. (c) Regime with a very weak trap and strong coupling (here $\omega^2 = 0.1$, $\kappa = 100$), where compactness is already visible in the initial state.
• Figure 5: Entanglement dynamics in the coupled-rotor (CR) chain. Half-chain entanglement entropy $S^{\rm CR}_{N/2}(t)$ for several system sizes $N$ as a function of $t$. At early times the entropy grows sub-extensively in $N$, while at later times compactness curbs the growth and induces saturation at a finite value. For comparison, the corresponding harmonic chain (CHO, dashed lines) exhibits unbounded entanglement growth. Parameters: $\omega^2=1.5$, $\kappa=0.5$. The MPS simulations were performed with an angular momentum cutoff $M = 6$, a truncation error of $\epsilon = 10^{-6}$ and a maximal bond dimension of $\chi = 1024$.