PIF: Quantum Monte Carlo Methods for Non-Equilibrium Dynamics of Interacting Quantum Many-Body Systems

在这个项目中，研究了量子多体系统动力学的新的计算和理论方法。其核心思想是，在虚时间中使用本质上任意演化协议的量子猝灭可以使用现有量子蒙特卡罗（QMC）算法的修改版本在数值上进行。本文提出了两种算法：(i)在非平衡态QMC （NEQMC）模拟中，计算系统在虚时间内经过哈密顿演化后的最终状态。（ii）准绝热QMC （QAQMC）方法使用大量演化哈密顿算子的乘积（而不是标准的指数时间演化算子）同时获得整个时间路径上的期望值。利用这些方法的关键是对虚时信息如何与实时动态相关联的理论见解。具体地，研究了在量子临界点附近作为淬灭速度（或在非线性协议情况下的广义速度）的函数的标度行为。还得到了表征非平衡动力学的磁化率，包括表征状态空间的几何张量。这些理论思想是与QMC算法并行发展的。考虑了几种测试方法并演示其用途的应用，包括横向场Ising模型和海森堡型自旋模型中的量子相变，以及量子自旋玻璃和其他无序系统。在量子计算和量子系统的绝热或准绝热演化的其他应用的背景下感兴趣的量子退火协议也进行了研究。此外，在量子系统背景下发展的一些思想也应用于经典的多体动力学，例如，获得计算动态临界指数的改进方法。当代理论物理学最大的挑战之一是如何计算（预测）由大量相互作用的微观粒子组成的系统的动态演化。这种系统的例子包括固体中的电子或被限制的原子“云”冷却到超冷的温度。这些粒子遵循量子力学定律，这使得计算它们的行为极其困难，特别是关于它们在时间上的演变（量子动力学）。即使使用今天的超级计算机，也没有普遍适用的数值算法可以在合理的时间内解决这类问题（计算时间通常随着系统中粒子的数量呈指数级增长）。这个项目是围绕着一个想法，部分规避这些问题的某些类别的重要量子系统。利用将复平面上的时间维度“旋转”到虚时间轴上的数学技巧，可以修改某些著名的量子系统平衡态模拟算法，以求解作为虚时间函数的演化。在开发这些计算工具的同时，也进行了相关的理论工作，以获得实时间动力学和虚时间动力学之间的精确关系。正在研究的系统是感兴趣的，例如，在量子磁性（电子尺度上的磁性）和超冷原子领域，可以预期的是，结果也可以产生广泛的影响，超出这些系统（因为量子动力学问题在物理学中非常普遍），例如，在量子计算（目前研究基于量子力学原理的计算机）。参与该项目的研究生接受前沿科学计算和理论物理方面的培训。

In this project, novel computational and theoretical approaches to the dynamics of quantum many-body systems are investigated. The central idea is that quantum quenches using essentially arbitrary evolution protocols in imaginary time can be carried out numerically using modified versions of existing quantum Monte Carlo (QMC) algorithms. We develop two algorithms: (i) In non-equilibrium QMC (NEQMC) simulations, the final state of a system after a Hamiltonian evolution in imaginary time is computed. (ii) In the quasi-adiabatic QMC (QAQMC) method, expectation values along a full time-path are obtained simultaneously, using a product of a large number of evolving Hamiltonians (instead of the standard exponential time evolution operator). Key to the utility of these approaches are theoretical insights into how the imaginary-time information is related to real-times dynamics. Specifically, scaling behaviors as a function of a quench velocity (or a generalized velocity in the case of nonlinear protocols) in the neighborhood of a quantum-critical point are investigated. Susceptibilities characterizing non-equilibrium dynamics are also obtained, including the geometric tensor characterizing the state space. These theoretical ideas are developed in parallel with the QMC algorithms. Several applications to test the methods and demonstrate their uses are considered, including quantum phase transitions in transverse-field Ising models and Heisenberg-type spin models, as well as quantum spin glasses and other disordered systems. Quantum annealing protocols of interest in the context of quantum computations and other applications of adiabatic or quasi-adiabatic evolution of quantum systems are also investigated. In addition, some of the ideas developed within the context of quantum systems are also applied to classical many-body dynamics, e.g., to obtain improved ways of computing the dynamic critical exponent.One of the greatest challenges of contemporary theoretical physics is how to compute (predict) the dynamical evolution of systems of a large (macroscopic) number of interacting microscopical particles. Examples of such systems include the electrons in a solid or confined "clouds" of atoms cooled to ultra-cold temperature. These particles obey the laws of quantum mechanics, which makes computations of their behaviors extremely difficult, especially as regards their evolution in time (quantum dynamics). Even with today's supercomputers, there are no generally applicable numerical algorithms useful for solving this class of problems within reasonable time (the computation time typically scaling exponentially with the number of particles in the system). This project is centered around an idea to partially circumvent these problems for some classes of important quantum systems. Using a mathematical trick of "rotating" the time dimension in the complex plane to the imaginary-time axis, certain well known simulation algorithms for quantum systems in equilibrium can be modified to solve the evolution as a function of imaginary time. In parallel with the development of these computational tools, related theoretical work is conducted in order to obtain precise relationships between real and imaginary time dynamics. The systems under investigation are of interest, e.g., in the fields of quantum magnetism (magnetism at the electronic scale) and ultra-cold atoms, and it can be expected that the results can have impact also broadly beyond these systems (as the quantum dynamics problem is of very general interest in physics), e.g., in quantum computation (currently researched computers based on quantum-mechanical principles). The graduate students involved in the project receive training in cutting-edge scientific computation and theoretical physics.