Mathematical Analysis and Computation for Quantum Systems

Transcription

1 Mathematical Analysis and Computation for Quantum Systems Peking University January 4-6, 2019

2 Mathematical Analysis and Computation for Quantum Systems Peking University, January 4-6, 2019 Venue: Room 77201, Jingchunyuan 78, BICMR, Peking University Scientific Committee: Wei Cai (Southern Methodist University ) Shi Jin (Shanghai Jiao Tong University) Organization Committee: Jianfeng Lu (Duke University) Sihong Shao (Peking University) Zhennan Zhou, (Peking University) List of Speakers: 1. Guillaume Bal (University of Chicago) 2. Victor Batista (Yale University) 3. Wei Cai (Southern Methodist University) 4. Yongyong Cai (Beijing Computational Science Research Center) 5. Zhenning Cai (National University of Singapore) 6. Lihui Chai (Sun Yat-sen University) 7. Chenjie Fan (University of Chicago) 8. Di Fang (University of Wisconsin-Madison) 9. Jiequn Han (Princeton University)

3 10. Jialin Hong (Chinese Academy of Sciences) 11. Shi Jin (Shanghai Jiao Tong University) 12. Yingzhou Li (Duke University) 13. Zhendong Li (California Institute of Technology) 14. Jian Liu (Peking University) 15. Liu Liu (The University of Texas at Austin ) 16. Wenjian Liu (Shandong University) 17. William H. Miller (University of California, Berkeley) 18. Thomas F. Miller (California Institute of Technology) 19. Qian Niu (The University of Texas at Austin ) 20. Haobin Wang (University of Colorado Denver) 21. Jia Yin (National University of Singapore) 22. Yi Zhu (Tsinghua University) Contact: Xiaoni Tan, xntan@bicmr.pku.edu.cn Website:

4 Friday (Jan 4) Schedule Moderator: 8:25-8:30 Opening Remark (To be determined) 8:30-9:15 Talk 1 9:15-10:00 Talk 2 Shi Jin (Shanghai Jiao Tong University) Yi Zhu (Tsinghua University) 10:00-10:30 Coffee Break 10:30-11:15 Talk 3 11:15-12:00 Talk 4 William H. Miller (University of California, Berkeley) Victor Batista (Yale University) Semiclassical computational methods for quantum dynamics with band-crossing and uncertainty Topological edge states in honeycomb photonic materials Classical Molecular Dynamics Simulations of Electronically Non- Adiabatic Processes Tensor train algorithms for quantum dynamics simulations of excited state nonadiabatic processes and quantum control 12:00-14:00 Lunch Break (Lunch boxes provided) Moderator: 14:00-14:45 Talk 5 14:45-15:30 Talk 6 Wenjian Liu (Shandong University) Lihui Chai (Sun Yat-sen University) ici With Selection And Perturbation Frozen Gaussian Approximation for the Dirac equation in semi-classical regime 15:30-16:00 Coffee Break (Group Photo, Approximate number 30) 16:00-16:45 Talk 7 16:45-17:30 Talk 8 17:30-18:15 Talk 9 Yongyong Cai (Beijing Computational Science Research Center) Jiequn Han (Princeton University) Liu Liu (The University of Texas at Austin ) Ground states of spinor Bose-Einstein condensates Solving Many-Electron Schr\"odinger Equation Using Deep Neural Networks Gaussian wave packet transform based numerical scheme for the semiclassical Schrödinger equation with random inputs 18:30 Conference Banquet (Venue: to be determined)

5 Saturday (Jan 5) Moderator: 8:30-9:15 Talk 10 9:15-10:00 Talk 11 Wei Cai (Southern Methodist University) Jialin Hong (Chinese Academy of Sciences) 10:00-10:30 Coffee Break Fast integral equation methods for quantum dots embedded in layered materials Stochastic Symplecticity and Ergodicity of Numerical Methods for Stochastic Nonlinear Schroedinger Equation 10:30-11:15 Talk 12 11:15-12:00 Talk 13 Guillaume Bal (University of Chicago) Chenjie Fan (University of Chicago) Topological Insulators and obstruction to localization On well posedness of Stochastic mass critical NLS 12:00-14:00 Lunch Break (Lunch boxes provided) Moderator: 14:00-14:45 Talk 14 14:45-15:30 Talk 15 Thomas F. Miller (California Institute of Technology) Haobin Wang (University of Colorado Denver) 15:30-16:00 Coffee Break 16:00-16:45 Talk 16 16:45-17:30 Talk 17 Jian Liu (Peking University) Zhenning Cai (National University of Singapore) Classical and Machine-Learning Methods for Quantum Simulation Multilayer Multiconfiguration Time- Dependent Hartree Theory Trajectory-Based Approaches for Quantum Statistics and Quantum Dynamics Inchworm Monte Carlo method for open quantum systems 17:30-18:15 Talk 18 Zhendong Li (California Institute of Technology) Recent developments of tensor network states for quantum chemistry 18:30 Conference Dinner (Venue: to be determined)

6 Sunday (Jan 6) Moderator: 8:30-9:15 Talk 19 9:15-10:00 Talk 20 Qian Niu (The University of Texas at Austin ) Yingzhou Li (Duke University) General relativity of Bloch electrons Coordinate descent full configuration interaction 10:00-10:30 Coffee Break 10:30-11:15 Talk 21 11:15-12:00 Talk 22 Di Fang (University of Wisconsin-Madison) Jia Yin (National University of Singapore) A diabatic surface hopping algorithm based on time perturbation theory and semiclassical analysis Numerical methods and comparison for the Dirac and nonlinear Dirac equation 12:00-14:00 Lunch Break (Lunch boxes provided) Afternoon Free Discussion

7 Abstract 1. Semiclassical computational methods for quantum dynamics with bandcrossing and uncertainty Shi Jin (Shanghai Jiao Tong University) Abstract: Band-crossing is a quantum dynamical behavior that contributes to important physics and chemistry phenomena such as quantum tunneling, Berry connection, chemical reaction etc. In this talk, we will discuss some recent works in developing semiclassical methods for band-crossing in surface hopping. For such systems we will also introduce an "asymptotic-preserving" method that is accurate uniformly for all wave numbers, including the problem with random uncertain band gaps. 2. Topological edge states in honeycomb photonic materials Yi Zhu (Tsinghua University) Abstract: Recent progresses in topological materials stimulate rigorous mathematical analysis and numerical computation in such fields. This talk focuses on recent advances in the analysis and computation of topological edge states in honeycomb photonic materials. Specifically, I shall present rigorous justifications for the existence of Dirac points, edge states in Maxwell's system with honeycomb symmetries. These are hallmarks of topological materials. Then by studying the reduced envelope equation, I shall explain the topological propagation of electromagnetic waves. At the end, an efficient and highly accurate numerical scheme will be introduced to compute such edge states for the purpose of real applications. 3. Classical Molecular Dynamics Simulations of Electronically Non-Adiabatic Processes William H. Miller (University of California, Berkeley) Abstract: A recently described symmetrical quasi-classical (SQC) windowing methodology for classical trajectory simulations has been applied to the Meyer- Miller (MM) model for the electronic degrees of freedom in electronically nonadiabatic dynamics. The approach treats nuclear and electronic degrees of freedom (DOF) equivalently (i.e., by classical mechanics, thereby retaining the simplicity of standard molecular dynamics), providing ``quantization'' of the electronic states through the symmetrical quasi-classical (SQC) windowing model. The approach is seen to be capable of treating extreme regimes of strong and weak coupling between the electronic states, as well as accurately describing coherence effects in the electronic DOF (including the de-coherence of such effects caused by coupling to the nuclear DOF). It is able to provide the full

8 electronic density matrix from the one ensemble of trajectories, and the SQC windowing methodology correctly describes detailed balance (unlike the traditional Ehrenfest approach). Calculations can be (equivalently) carried out in the adiabatic or a diabatic representation of the electronic states, and most recently it has been shown that a modification of the canonical equations of motion in the adiabatic representation eliminates (without approximation) the need for second-derivative coupling terms. 4. Tensor train algorithms for quantum dynamics simulations of excited state nonadiabatic processes and quantum control Victor S. Batista (Yale University) Abstract: We introduce the tensor-train split-operator Fourier transform (TT- SOFT) algorithm for simulations of multidimensional nonadiabatic quantum dynamics [J. Chem. Theory Comput. 13: (2017)]. TT-SOFT is essentially the grid-based SOFT method implemented in dynamically adaptive tensor-train representations. In the same spirit of all matrix product states, the tensor-train format enables the representation, propagation, and computation of observables of multidimensional wave functions in terms of the grid-based wavepacket tensor components, bypassing the need of actually computing the wave function in its full-rank tensor product grid space. We demonstrate the accuracy and efficiency of the TT-SOFT method as applied to propagation of 24-dimensional wave packets, describing the S1/S2 interconversion dynamics of pyrazine after UV photoexcitation to the S2 state. Our results show that the TTSOFT method is a powerful computational approach for simulations of quantum dynamics of polyatomic systems since it avoids the exponential scaling problem of full-rank grid-based representations. The development of ultrafast laser technology has also opened the possibility to control ultrafast reaction dynamics in excited electronic states. Thus, we report a Floquet theoretical study of quantum control of the ultrafast cis-trans photoisomerization dynamics of rhodopsin [J. Chem. Theory Comput. 14(3): (2018)]. The predicted light-induced potentials, including light-induced conical intersections, can open new reaction channels or modify the product yields of existing pathways. The nonadiabatic dynamics is described by a 3-state 2-dimensional wave-packet, coupled to a bath of 23 vibrational modes, evolving according to an empirical model Hamiltonian with frequencies and excited-state gradients parameterized to reproduce the observed resonance Raman excitations of rhodopsin. We analyze the effect of different control pulses on the photoisomerization dynamics, including changes in pulse duration and intensity. We interpret the results in terms of 'dressed states' and we exploit the Floquet description where the effect of control pulses is naturally decoupled along the different channels. Results obtained with 300 fslong pulses suggest that it should be possible to delay the excited-state isomerization for hundreds of femtoseconds. Our findings are thus particularly

9 relevant to the development of ultrafast optical switches based on visual pigments. 5. ici With Selection And Perturbation Wenjian Liu (Shandong University) Abstract: According to when the static and dynamic components of electron correlation are treated, the available wave function-based correlation methods can be classified into three families, viz., "static-then-dynamic", "dynamic-thenstatic", and "static-dynamicstatic (SDS)". Herewith we report a restricted SDS framework, which employs the same number (Np; the number of target states) of primary, secondary and external states for describing the static, dynamic, and again static components of correlation. That is, the secular equation to be diagonalized is of dimension 3Np, irrespective of the numbers of correlated electrons and orbitals. Even the lowest-order realizations of this seemingly restricted SDS framework, i.e., SDSPT2 and SDSCI, are already very accurate for classic test problems of variable degeneracies, whereas a high-order realization, i.e., ici (iterative Configuration Interaction), can converge monotonically and quickly to full CI from above, even when a rather poor reference is taken as the start. Interestingly, the micro-iteration of ici can be reformulated as an iterative Vector Interaction (ivi) method for exterior or interior roots of large matrices. In this lecture, we will introduce a configuration-driven unitary group approach for selection of configurations in ici, so as to make the latter as efficient as possible. 6. Frozen Gaussian Approximation for the Dirac equation in semi-classical regime Lihui Chai (Sun Yat-sen University) Abstract: In this talk, we introduce the Frozen Gaussian Approximation (FGA) for the Dirac equation in the semi-classical regime. Unlike the strictly hyperbolic system studied in [J. Lu and X. Yang, Comm. Pure Appl. Math., 65, , 2012], the Dirac equation possesses eigenfunction spaces of multiplicity two, which demands more delicate expansions for deriving the amplitude equations in FGA. Moreover, we prove that the nonrelativistic limit of the FGA for the Dirac equation is the FGA of the Schr\"odinger equation, which shows that the nonrelativistic limit is asymptotically preserved after one applies FGA as the semiclassical approximation. Numerical experiments including Klein-Paradox are presented to illustrate the method, and confirm part of the analytical results. 7. Ground states of spinor Bose-Einstein condensates Yongyong Cai (Beijing Computational Science Research Center)

10 Abstract: The remarkable experimental achievement of Bose-Einstein condensation (BEC) in 1995 has drawn significant research interests in understanding the ground states and dynamics of trapped cold atoms. Different from the single component BEC, spinor BEC possesses the spin degree of freedom and exhibits rich phenomenon. In the talk, we will introduce some mathematical results for ground states of spin-1,2 BECs, and a practical imaginary time propagation method for numerical simulation with several different projection strategies. 8. Solving Many-Electron Schr\"odinger Equation Using Deep Neural Networks Jiequn Han (Princeton University) Abstract: We introduce a new family of trial wave-functions based on deep neural networks to solve the many-electron Sch\"rodinger equation. The Pauli exclusion principle is dealt with explicitly to ensure that the trial wave-functions are physical. The optimal trial wave-function is obtained through variational Monte Carlo and the computational cost scales quadratically with the number of electrons. The algorithm does not make use of any prior knowledge such as atomic orbitals. Yet it is able to represent accurately the ground-states of the tested systems, including He, H2, Be, B, LiH, and a chain of 10 hydrogen atoms. This opens up new possibilities for solving large-scale many-electron Schr\"odinger equation. 9. Gaussian wave packet transform based numerical scheme for the semiclassical Schrödinger equation with random inputs Liu Liu (The University of Texas at Austin) Abstract: In this talk, we will discuss about the semiclassical Schrödinger equation with random inputs. We first show that the semiclassical Schrödinger equation produces O(ε) oscillations in the random variable space. With the Gaussian wave packet transform, the original Schrödinger equation is mapped to an ODE system for the wave packet parameters coupled with a PDE for the quantity w in rescaled variables. Then we show that the w equation does not produce ε dependent oscillations, and thus it is more amenable for numerical simulations. We propose multi-level sampling strategy in implementing the Gaussian wave packet transform, where in the most costly part of simulating the w equation, it is sufficient to use ε independent samples. Last, we will provide extensive numerical tests and numerical experiments to demonstrate the efficiency and properties of our algorithm. This is a joint work with Shi Jin, Giovanni Russo and Zhennan Zhou.

11 10. Fast integral equation methods for quantum dots embedded in layered materials Wei Cai (Southern Methodist University) Abstract: In this talk, we will present a volume integral equation method for studying the interaction of quantum dots embedded in layered semi-conductor materials. A fast multipole method is developed for the Schrodinger equation which is based on a new multipole expansion for the Green's function of the layered material as well as the translation from multipole to local expansion. Numerical results of the new FMM method for 3-D layered material will be presented. (Joint work with Wenzhong Zhang, Bo Wang). 11. Stochastic Symplecticity and Ergodicity of Numerical Methods for Stochastic Nonlinear Schroedinger Equation Jialin Hong (Chinese Academy of Sciences) Abstract: In this talk we present a review on stochastic symplecticity (multisymplecticity) and ergodicity of numerical methods for stochastic nonlinear Schroedinger (NLS) equation. The equation considered is charge conservative and has the multi-symplectic conservation law. Based a stochastic version of variational principle, we show that the phase flow of the equation, considered as an evolution equation, preserves the symplectic structure of the phase space. We give some symplectic integrators and multi-symplectic methods for the equation. By constructing control system and invariant control set, it is proved that the symplectic integrator, based on the central difference scheme, possesses a unique invariant measure on the unit sphere. Furthermore, by using the midpoint scheme, we get a full discretization which possesses the discrete charge conservation law and the discrete multi-symplectic conservation law. Utilizing the Poisson equation corresponding to the finite dimensional approximation, the convergence error between the temporal average of the full discretization and the ergodic limit of the symplectic method is derived (In collaboration with Dr. Chuchu Chen, Dr. Xu Wang and Dr. Liying Zhang). 12. Topological Insulators and obstruction to localization Guillaume Bal (University of Chicago) Abstract: Topological insulators(tis) are materials characterized by topological invariants. One of their remarkable features is the asymmetric transport observed at the interface between materials in different topological phases. Such transport is itself described by a topological invariant, and as such protected against random perturbations. This immunity makes TIs extremely promising for many engineering applications and actively researched. In this talk, we present a partial differential model for such TIs, introduce a topology based on indices of Fredholm operators, and analyze the influence of

12 random perturbations on transport. We confirm that topology is an obstruction to Anderson localization, a hallmark of wave propagation in strongly heterogeneous media in the topologically trivial case. We quantify what is or is not protected topologically and show that, for instance, a quantized amount of transmission is protected while back-scattering, a practical nuisance, is not. 13. On well posedness of Stochastic mass critical NLS Chenjie Fan (University of Chicago) Abstract: We will discuss the similarity and difference between deterministic and stochastic NLS. Different notions (or possible formulations) of local solutions will also be discussed. We will also present a global well posedness result for stochastic mass critical NLS. Joint work with Weijun Xu (Oxford) 14. Classical and Machine-Learning Methods for Quantum Simulation Thomas F. Miller (California Institute of Technology) Abstract: A focus of my research is to develop simulation methods that reveal the mechanistic details of quantum mechanical reactions that are central to biological, molecular, and heterogenous catalysis. The nature of this effort is three-fold: we work from the foundation of quantum statistical mechanics and semiclassical dynamics to develop methods that significantly expand the scope and reliability of condensed-phase quantum dynamics simulation; we develop quantum embedding and machine learning methods that improve the description of molecular interactions and electronic properties; and we apply these methods to understand complex chemical systems. The talk will focus on recent developments and applications of Feynman path integral methods for the description of non-adiabatic chemical dynamics, including proton-coupled electron-transfer and long-ranged electron transfer in protein systems. Additionally, we will describe a machine-learning approach to predicting the electronic structure results on the basis of simple molecular orbitals properties, yielding striking accuracy and transferability across chemical systems at low computational cost. 15. Multilayer Multiconfiguration Time-Dependent Hartree Theory Haobin Wang (University of Colorado Denver) Abstract: The multilayer multi-configuration time-dependent Hartree (ML- MCTDH) theory is a particularly powerful tool to simulate quantum dynamics for large systems. This rigorous quantum approach is based on an efficient representation of the functional in a time-dependent variational calculation, which recursively expand the wave function in several (or many) dynamically contracted layers. Mathematically, the expansion is equivalent to hierarchical

13 Tucker tensor decomposition, although the latter usually employs a binary tree structure. A special skewed tree form is called tensor train in mathematics and matrix product states in physics. In this talk I will outline the derivation of the theory, the scaling of the method, and application of the theory to some model reactions that have interesting many-body quantum effects. 16. Trajectory-Based Approaches for Quantum Statistics and Quantum Dynamics Jian Liu (Peking University) Abstract: Unraveling nuclear quantum effects (zero point energy, tunneling, decoherence, coherence, etc.) in dynamic, spectroscopic and thermodynamic properties of complex systems has presented challenging frontiers in modern physical chemistry. We will present an introduction to the theories and to our recent progress. The talk consists of two parts: One is on an efficient unified theoretical scheme for performing path integral molecular dynamics for obtaining thermodynamic properties, the other is on three unified frameworks in the phase space formulation of quantum mechanics for developing trajectory-based methods for studying dynamic and spectroscopic properties. 17. Inchworm Monte Carlo method for open quantum systems Zhenning Cai (National University of Singapore) Abstract: We investigate in this work a recently proposed diagrammatic quantum Monte Carlo method --- the inchworm Monte Carlo method --- for open quantum systems. We establish its validity rigorously based on resummation of Dyson series. Moreover, we introduce an integro-differential equation formulation for open quantum systems, which illuminates the mathematical structure of the inchworm algorithm. This new formulation leads to an improvement of the inchworm algorithm by introducing classical deterministic time-integration schemes. The numerical method is validated by applications to the spin-boson model. 18. Recent developments of tensor network states for quantum chemistry Zhendong Li (California Institute of Technology) Abstract: In this talk I will discuss our recent developments of tensor network states for quantum chemistry: (1) As an extension of ab initio density matrix renormalization group (DMRG), we developed a method named spin-projected matrix product states (SP-MPS), which merges the ideas of DMRG and spin-projections in traditional quantum chemistry. The resulting method allows to map out the low-energy states of challenging open-shell transition metal complexes more efficiently.

14 (2) Moving towards utilizing more powerful tensor network states, recently we presented a new framework to solve the many-electron Schrodinger equation by combining real-space discretization with projected entangled pair states (PEPS). The major challenge here is how to treat long-range Coulomb interaction efficiently. I will talk about how this problem can be overcome by approximating the Coulomb interaction as a linear combination of auxiliary tensor network operators with small bond dimensions. 19. General relativity of Bloch electrons Qian Niu (The University of Texas at Austin ) Abstract: Semiclassical dynamics of Bloch electrons in a crystal under slowly varying deformation is developed in the geometric language of a lattice bundle. Berry curvatures and gradients of energy are introduced in terms of lattice covariant derivatives, with the corresponding connections given by the gradient and rate of strain. A number of physical effects are discussed: an effective post- Newtonian gravity at band bottom, polarization induced by spatial gradient of strain, orbital magnetization induced by strain rate, and electron energy stress tensor. 20. Coordinate descent full configuration interaction Yingzhou Li (Duke University) Abstract: Full configuration interaction (FCI) problem is a large leading eigenvalue problem with problem size scaled combinatorially as the number of electrons increases. Based on a non-convex optimization formula of the leading eigenvalue problem, we propose coordinate descent full configuration interaction (CoordFCI) to solve the FCI problem to chemical accuracy efficiently. CoordFCI is a greedy coordinate-wise descent method with compression on the variational wavefunction. In order to demonstrate the efficiency of CoordFCI, we present numerical results on small molecules as well as strongly correlated chromium dimer. Also, a binding curve of nitrogen dimer is illustrated. 21. A diabatic surface hopping algorithm based on time perturbation theory and semiclassical analysis Di Fang (University of Wisconsin-Madison) Abstract: Surface hopping algorithms are popular tools to study dynamics of the quantum-classical mixed systems. In this talk, we will present a surface hopping algorithm in diabatic representations, based on time dependent perturbation theory and semiclassical analysis. The algorithm can be viewed as a Monte Carlo sampling algorithm on the semiclassical path space for piecewise deterministic path with stochastic jumps between the energy surfaces. The algorithm is

15 validated numerically and it shows good performance in both weak coupling and avoided crossing regimes. 22. Numerical methods and comparison for the Dirac and nonlinear Dirac equation Jia Yin (National University of Singapore) Abstract: In this talk, we present and compare several numerical methods for solving the Dirac and nonlinear Dirac equation. The methods include frequently used finite difference time domain (FDTD) methods, and two methods with Fourier spectral discretization in space, combined with the exponential wave integrator and time-splitting technique respectively for temporal derivatives. For Dirac equation in the absence of magnetic potential, we further introduce a fourth-order compact time-splitting (S4c) Fourier pseudo-spectral method by using a double commutator. This method is more efficient than many other standard fourth-order splitting methods. Moreover, the surprising superresolution property for time-splitting methods in solving Dirac or nonlinear Dirac equation without external magnetic potentials in the nonrelativistic limit regime is presented and analyzed.