Dynamical glass in weakly non-integrable many-body systems

Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a {\it nonintegrable} perturbation creates a coupling network in action space which can be short- or long-ranged. We analyze the dynamics of observables which turn into the conserved actions in the integrable limit. We compute distributions of their finite-time averages and obtain the ergodization time scale TE on which these distributions converge to δ-distributions. We relate TE∼(σ+τ)2/μ+τ to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations σ+τ dominating the means μ+τ. The Lyapunov time TΛ (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long- and short-range coupling networks by tuning its energy density. For long-range coupling networks TΛ≈σ+τ, which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a {\it dynamical glass}, where TE grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which TΛ≲μ+τ. This is due to the formation of a highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of non-chaotic regions. These structures persist up to the ergodization time TE.Published versio

Dynamical glass in weakly non-integrable Klein-Gordon chains

Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a {\it nonintegrable} perturbation creates a coupling network in action space which can be short- or long-ranged. We analyze the dynamics of observables which turn into the conserved actions in the integrable limit. We compute distributions of their finite-time averages and obtain the ergodization time scale $T_E$ on which these distributions converge to $\delta$-distributions. We relate $T_E \sim (\sigma_\tau^+)^2/\mu_\tau^+$ to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations $\sigma_\tau^+$ dominating the means $\mu_\tau^+$. The Lyapunov time $T_{\Lambda}$ (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long- and short-range coupling networks by tuning its energy density. For long-range coupling networks $T_{\Lambda}\approx \sigma_\tau^+$, which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a {\it dynamical glass}, where $T_E$ grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which $T_{\Lambda} \lesssim \mu_\tau^+$. This is due to the formation of a highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of non-chaotic regions. These structures persist up to the ergodization time $T_E$

Controlling Vortex Lattice Structure of Binary Bose-Einstein Condensates via Disorder Induced Vortex Pinning

We study the vortex pinning effect on the vortex lattice structure of the rotating two-component Bose-Einstein condensates (BECs) in the presence of impurities or disorder by numerically solving the time-dependent coupled Gross-Pitaevskii equations. We investigate the transition of the vortex lattice structures by changing conditions such as angular frequency, the strength of the inter-component interaction and pinning potential, and also the lattice constant of the periodic pinning potential. We show that even a single impurity pinning potential can change the unpinned vortex lattice structure from triangular to square or from triangular to a structure which is the overlap of triangular and square. In the presence of periodic pinning potential or optical lattice, we observe the structural transition from the unpinned vortex lattice to the pinned vortex lattice structure of the optical lattice. In the presence of random pinning potential or disorder, the vortex lattice melts following a two-step process by creation of lattice defects, dislocations, and disclinations, with the increase of rotational frequency, similar to that observed for single component Bose-Einstein condensates. However, for the binary BECs, we show that additionally the two-step vortex lattice melting also occurs with increasing strength of the inter-component interaction

Vortex nucleation in rotating Bose-Einstein condensates with density-dependent gauge potential

We study numerically the vortex dynamics and vortex-lattice formation in a rotating density-dependent Bose-Einstein condensate (BEC), characterized by the presence of nonlinear rotation. By varying the strength of nonlinear rotation in density-dependent BECs, we calculate the critical frequency, $\Omega_{\text{cr}}$, for vortex nucleation both in adiabatic and sudden external trap rotations. The nonlinear rotation modifies the extent of deformation experienced by the BEC due to the trap and shifts the $\Omega_{\text{cr}}$ values for vortex nucleation. The critical frequencies and thereby, the transition to vortex-lattices in an adiabatic rotation ramp, depend on conventional $\textit{s}$-wave scattering lengths through the strength of nonlinear rotation, $\mathit{C}$, such that $\Omega_{\text{cr}}(\mathit{C}>0) < \Omega_{\text{cr}}(\mathit{C}=0) < \Omega_{\text{cr}}(\mathit{C}<0)$. In an analogous manner, the critical ellipticity ($\epsilon_{\text{cr}}$) for vortex nucleation during an adiabatic introduction of trap ellipticity ($\epsilon$) depends on the nature of nonlinear rotation besides trap rotation frequency. The nonlinear rotation additionally affects the vortex-vortex interactions and the motion of the vortices through the condensate by altering the strength of Magnus force on them. The combined result of these nonlinear effects is the formation of the non-Abrikosov vortex-lattices and ring-vortex arrangements in the density-dependent BECs.Comment: 10 pages, 10 figures, Accepted for publication in PR

Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

We investigate the computational performance of various numerical methods for the integration of the equations of motion and the variational equations for some typical classical many-body models of condensed matter physics: the Fermi-Pasta-Ulam-Tsingou (FPUT) chain and the one- and two-dimensional disordered, discrete nonlinear Schr\"odinger equations (DDNLS). In our analysis we consider methods based on Taylor series expansion, Runge-Kutta discretization and symplectic transformations. The latter have the ability to exactly preserve the symplectic structure of Hamiltonian systems, which results in keeping bounded the error of the system's computed total energy. We perform extensive numerical simulations for several initial conditions of the studied models and compare the numerical efficiency of the used integrators by testing their ability to accurately reproduce characteristics of the systems' dynamics and quantify their chaoticity through the computation of the maximum Lyapunov exponent. We also report the expressions of the implemented symplectic schemes and provide the explicit forms of the used differential operators. Among the tested numerical schemes the symplectic integrators $ABA864$ and $SRKN^a_{14}$ exhibit the best performance, respectively for moderate and high accuracy levels in the case of the FPUT chain, while for the DDNLS models $s9\mathcal{ABC}6$ and $s11\mathcal{ABC}6$ (moderate accuracy), along with $s17\mathcal{ABC}8$ and $s19\mathcal{ABC}8$ (high accuracy) proved to be the most efficient schemes.Comment: Accepted for publication in Mathematics in Engineerin