I do this research in 2024/09-present. PI: Prof. Hong-Yan Shih and Yi-Ping Huang, Academia Sinica
Motivation
Lattice dynamical systems in two dimensions typically exhibit a broader range of behaviors compared to their one-dimensional counterparts. Many fascinating non-equilibrium steady states (NESS) can only be observed or are more easily realized in two-dimensional systems. As a result, we are motivated to develop a two-dimensional version of VMPS specifically for these dynamical systems.
It is essential to note that the theoretical foundations of VMPS for the SIS model differ significantly from those of quantum many-body systems. Consequently, the formulation of a two-dimensional VMPS may also diverge from the approaches used in quantum contexts.
To effectively transition from the one-dimensional VMPS to a two-dimensional version for many-body quantum systems, we must first clarify the physical interpretation of VMPS as it pertains to dynamical systems. A solid understanding of this physical framework will be crucial in guiding the formulation of our two-dimensional approach.
Remarks on the Summer Project
Compute the Target Eigenstate
We employ the Lanczos algorithm to compute the leading eigenvalue and its corresponding eigenstate. Additionally, it is possible to calculate other eigenvalues and their associated eigenstates. By transforming the Hamiltonian into the infinitesimal Markov generator and applying the appropriate diagonalization method, we can obtain the desired eigenstate.
Rare Events Simulation
Simulating rare events using the Monte Carlo method presents significant challenges due to their infrequent occurrence, which can result in unreliable outcomes and lengthy computational times. In contrast, the VMPS approach offers a robust alternative, as it is capable of computing the exact distribution of the model. This level of precision makes VMPS particularly valuable for investigating rare events within dynamical systems, enabling a more accurate understanding of their behavior and implications. By leveraging VMPS, researchers can effectively explore these infrequent occurrences and gain deeper insights into the underlying dynamics.
Convergence Behaviors of VMPS and Physical Picture
We introduce three properties of the infinitesimal Markov generator to compute the target eigenstate. Moreover, these properties can provide insights into the physical interpretation of this algorithm. Since we know that the eigenvalues for the infinitesimal Markov generator are less than or equal to zero, the only relevant model as time approaches infinity is the eigenstate with an eigenvalue of zero. In the SIS model, this algorithm can be interpreted as a process that neglects the fast modes, which are not crucial for the NESS.
However, this physical picture cannot fully explain the convergence behaviors. As shown in Figure 1, the convergence in the active phase is more erratic compared to that in the absorbing phase.
Figure 1: The convergence behaviors of the absorbing and active phases of the SIS model simulated by VMPS.
Additionally, unlike the VMPS for the transverse Ising model near the critical point, the SIS model exhibits a much slower convergence rate close to the critical point, as depicted in Figure 2.
Figure 2: The convergence behaviors at the critical point of the SIS model simulated by VMPS.
Therefore, the transition from the traditional VMPS for quantum many-body systems to the VMPS for dynamical systems is not straightforward.
Research Method
In this research, we employ the Density Matrix Renormalization Group (DMRG) method to compute quantum many-body systems. Our objective is to redefine the density matrix within the SIS model and develop the DMRG framework specifically for it. This implementation facilitates comparisons between results obtained from DMRG and the Monte Carlo method. Given that some concepts in quantum mechanics do not apply to classical systems, certain physical quantities remain undefined at this stage. We plan to investigate these aspects in subsequent steps.
To analyze convergence behaviors, we start with Monte Carlo simulations that illustrate the time evolution of the active and absorbing phases. Up to this point, we have focused solely on the average of the model's trajectories, neglecting the details of individual trajectories. Our next step involves analyzing the time series for both phases and computing their differences. According to VMPS theory, we compute the eigenstate corresponding to the eigenvalue 0, which represents the NESS. By applying the Fast Fourier Transform (FFT), we can determine the Fourier spectrum of the NESS, confirming it is indeed the eigenstate with eigenvalue 0. It is important to note that noise from the FFT has not been addressed, and its variability may differ across phases even with the same number of samples. Additionally, Monte Carlo simulations at the critical point indicate that, in contrast to the transverse Ising model, the SIS model's critical point exhibits significant variations in trajectory phases. Consequently, the time series will vary across phases, potentially aiding in the characterization of these states. We will conduct Monte Carlo simulations under various parameter values to compile a comprehensive dataset for further analysis.
Expected Results
Our aim is to identify the statistical quantities that influence the phenomena observed in the VMPS results. The VMPS analysis indicates that convergence rates slow significantly near the critical point, suggesting that specific statistical quantities reaching their maximum values at this juncture may impact convergence dynamics. Furthermore, as the parameter λ increases, the convergence behaviors become increasingly erratic, complicating predictions. By systematically comparing different values of λ, we hope to uncover insights into how these statistical quantities interact with system dynamics.
After determining the relevant statistical quantities, we will speculate on their roles in phase transitions, which will provide a foundation for refining the algorithm from a physical perspective. This revised algorithm will then be applied to the SIS model, allowing for meaningful comparisons with results from the VMPS analysis. Ultimately, this comprehensive approach aims to deepen our understanding of phase transitions and the SIS model's behavior in various contexts, as well as to elucidate the factors affecting convergence behaviors in VMPS.
