Solving the Schrodinger equation and finding excited states for quantum mechanical many-body systems is a fundamental problem. If this problem is formulated in terms of integrals, the dimensionality of the configuration space is often too high to perform the integrations required directly. This is because the volume of configuration space increases exponentially with dimension quickly, making its complete exploration impossible. Monte Carlo methods provide a way to estimate these integrals by statistically sampling a subset of configuration space, and these methods provide 1/√ N convergence regardless of dimension. For many problems involving the quantum mechanics of molecules there exists a large time scale separation between the high-frequency internal vibrations, and low-frequency intermolecular motions. This separation motivates the rigid-body approximation which freezes internal degrees of freedom in order to study intermolecular effects. The reduction in the dimensionality of configuration space further increases the range of accessible problems. This thesis is devoted to the construction and implementation of algorithms which incorporate the rigid-body approximation into existing Monte Carlo methods for solving the quantum mechanical many-body problem. Monte Carlo estimators are constructed as averages over samples drawn from some probability distribution. If this distribution is known in closed form the samples can be generated by application of the Metropolis algorithm. In most cases the distribution is not known in closed form or even representable with a finite number of variables. If this distribution is the dominant eigenstate of some known operator then a stochastic implementation of the power method can be used to generate the required samples. For the quantum mechanical problem this operator can be taken to be the imaginary-time evolution operator and its application can be represented in terms of random walks. For rigid bodies, this method involves the implementation of rotational Brownian motion. The use of quaternion algebra to implement this rotational motion enhances simplicity, performance, and numerical stability. The method described above can be generalized to investigate excited state properties using correlation function Monte Carlo which is a Monte Carlo implementation of the Rayleigh-Ritz variational method. This generalization requires the construction of a trial subspace which is then subjected to the variational method resulting in a generalized eigenvalue problem. It is important to represent optimized trial states accurately and efficiently. For this purpose, we write trial states as functions of invariant polynomials of the interparticle distances. By reducing the number of free interparticle distances, the rigid-body approximation greatly simplifies
Solving the Schrodinger equation and finding excited states for quantum mechanical many-body systems is a fundamental problem. If this problem is formulated in terms of integrals, the dimensionality of the configuration space is often too high to perform the integrations required directly. This is because the volume of configuration space increases exponentially with dimension quickly, making its complete exploration impossible. Monte Carlo methods provide a way to estimate these integrals by statistically sampling a subset of configuration space, and these methods provide 1/√ N convergence regardless of dimension. For many problems involving the quantum mechanics of molecules there exists a large time scale separation between the high-frequency internal vibrations, and low-frequency intermolecular motions. This separation motivates the rigid-body approximation which freezes internal degrees of freedom in order to study intermolecular effects. The reduction in the dimensionality of configuration space further increases the range of accessible problems. This thesis is devoted to the construction and implementation of algorithms which incorporate the rigid-body approximation into existing Monte Carlo methods for solving the quantum mechanical many-body problem. Monte Carlo estimators are constructed as averages over samples drawn from some probability distribution. If this distribution is known in closed form the samples can be generated by application of the Metropolis algorithm. In most cases the distribution is not known in closed form or even representable with a finite number of variables. If this distribution is the dominant eigenstate of some known operator then a stochastic implementation of the power method can be used to generate the required samples. For the quantum mechanical problem this operator can be taken to be the imaginary-time evolution operator and its application can be represented in terms of random walks. For rigid bodies, this method involves the implementation of rotational Brownian motion. The use of quaternion algebra to implement this rotational motion enhances simplicity, performance, and numerical stability. The method described above can be generalized to investigate excited state properties using correlation function Monte Carlo which is a Monte Carlo implementation of the Rayleigh-Ritz variational method. This generalization requires the construction of a trial subspace which is then subjected to the variational method resulting in a generalized eigenvalue problem. It is important to represent optimized trial states accurately and efficiently. For this purpose, we write trial states as functions of invariant polynomials of the interparticle distances. By reducing the number of free interparticle distances, the rigid-body approximation greatly simplifies
This thesis focuses on the development, application, and high-performance implementation of numerical methods for simulating quantum many-body systems. In the first chapter, I provide an overview of several most successful conventional approaches for simulating quantum many-body systems, along with their current limitations. Then, the basic properties of the quantum spin model, which will be the subject of study in the following chapters, and its ground state are briefly summarized. Additionally, I discuss the background and the motivations of the research projects carried out during my PhD research. Chapter 2 introduces two numerical algorithms that are extensively used in this thesis, which are exact diagonalization (ED) and variational Monte Carlo. I outline the process of constructing a Hamiltonian matrix and employing iterative methods for exact diagonalization. Subsequently, I introduce the mechanism of the four important components in the ``standard" variational Monte Carlo that are widely used in the most recent research. In Chapter 3, we propose a symmetry-projected variational ansatz in the form of linear combinations of simple restricted Boltzmann machines. This construction allows one to explore states outside of the original variational manifold and increase the representation power with moderate computational effort. Besides allowing one to restore spatial symmetries, an expansion in terms of Krylov states using a Lanczos recursion offers a solution that can further improve the quantum state accuracy. We illustrate these ideas with an application to the Heisenberg J1−J2 model on the square lattice, a paradigmatic problem under debate in condensed matter physics,and achieve state-of-the-art accuracy in the representation of the ground state. In Chapter 4, we present a variational approach based on a Chebyshev expansion of the spectral function, and each Chebyshev polynomial is represented by a neural network quantum state. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the "Chebyshev wave functions." We present results for the one-dimensional and two-dimensional Heisenberg models on the square lattice and compare them with those obtained by other methods in the literature. In Chapter 5, we describe a method to carry out thermodynamic simulations of quantum many-body models using typicality and variational representations of quantum states. Minimally entangled typical thermal states (METTS) are a construction that allows one to solve for the imaginary time evolution of quantum many-body systems. By using wave functions that are weakly entangled, one can take advantage of efficient representations in the form of matrixproduct states. We generalize these ideas to arbitrary variational wave functions and we focus, as an illustration, on the particular case of restricted Boltzmann machines. We demonstrate the application of the method to quantum spin systems in one and two spatial dimensions. In Chapter 6, we study the properties of the impurity band in heavily doped nonmagnetic semiconductors using the Jacobi-Davidson algorithm and the supervised deep learning method. The disorder averaged inverse participation ratio and Thouless number calculation show us the rich structure inside the impurity band. A convolutional neural network model, which is trained to distinguish the extended/localized phase of the Anderson model with high accuracy, shows us the results in good agreement with the conventional approach. Together, we find that there are three mobility edges in the impurity band for a specific on-site impurity potential, which means the presence of the extended states while filling the impurity band. In the final chapter, Chapter 7, we discuss the implementation of architecture-aware optimizations for spin dynamics simulations on x86 CPUs and Nvidia GPUs. The time cost of evaluating the spin-spin correlation functions in real space increases quadratically as the number of spins N, leading to significant computational effort, making the simulation of large spin systems very challenging. We propose to use a highly optimized general matrix multiply (GEMM) subroutine to calculate the dynamical spin-spin correlation function that can achieve near-optimal hardware utilization. Furthermore, we fuse the element-wise operations in the calculation of S(q,t) into the in-house GEMM kernel to reduce memory latency. On Nvidia GPU, we perform the on-the-fly calculation in the epilogue of the GEMM subroutine to avoid saving intermediate results on global memory, which significantly reduces memory usage and achieves better performance than cuBLAS.--Author's abstract
Quantum Monte Carlo (QMC) is one of the most promising methods for solving quantum many-body problems. QMC combines known analytical properties of wave functions, results from other methods, and stochastic techniques, into a powerful tool for investigation of the electronic structure of real materials. Furthermore, the method has a favorable scaling ( O[N3] in the number of electrons), perfect scalability on parallel architectures, and can be applied to a wide range of systems. Our work has focused both on expanding the capabilities of QMC for calculation of quantities beyond energies, such as interatomic forces, and on application of the method to challenging problems in nanostructure materials research. For the calculation of interatomic forces we have implemented a finite difference correlated sampling method. The correlated sampling enables us to avoid the problem of statistical noise in evaluating the energy differences and allows us to obtain forces with an average 1% error. We have analyzed the effect of the method's main approximations on the estimate of the forces, both by theoretical arguments and by performing a variety of tests on small systems. Furthermore, we have used this newly developed method for estimating equilibrium geometries in a problem where other methods have failed. QMC enabled us to treat the electron correlation effects with high accuracy and to provide accurate predictions in a number of interesting applications. In particular, for carbon rings of C4N+2 stoichiometry we have resolved a long-standing problem regarding the competition between the aromaticity and second order Jahn-Teller effects. We have found that the dominant mechanism at small and intermediate sizes, which stabilizes the bond angle and bond length alternated geometries, is the secondorder Jahn-Teller effect while the aromatic isomer is always found to be a transition state. These high accuracy calculations which have involved multi-reference wave functions, extensive variational optimizations, and the use of massively parallel platforms demonstrate clearly the unique capabilities of the QMC approach.
Since the form of PN is the same as that occurring in the statistical mechanics of the classical gas (replace f 2 ( r ij ) by exp[- V ( r ij /kT ]), we can use the same integration techniques that are used in the classical problem. W. L. McMillan, 1965 Summary Building on the random walk methods developed in the previous chapter, we show how to compute properties of many-body trial wavefunctions using a random walk. This method, called variational Monte Carlo, is the simplest stochastic quantum many-body technique. Whereas mean-field methods are usually limited to single determinants, variational Monte Carlo can treat any correlated trial function, as long as its values are computable. We discuss how to optimize such trial wavefunctions, how to compute their momentum distribution, how to use non-local pseudopotentials, how to compute excited states, and how to correct for the finite size of the simulation cell. Deterministic quantum methods have difficulties. For example, the Hartree–Fock method assumes the wavefunction is a single Slater determinant, neglecting correlation. If one expands as a sum of determinants, it is very difficult to have the results size-consistent since the number of determinants needed will grow exponentially with the system size. As we have seen, the DMFT method introduced in Ch. 16 assumes locality. In Ch. 6 we discussed general properties of many-body wavefunctions. Using Monte Carlo methods, we can directly incorporate correlation into a wavefunction, without having to make any further approximations other than the form of the correlation factors. In many cases the energy and other properties are very close to the exact results. Some of the usual restrictions on the form of the many-body wavefunction are not an issue in variational Monte Carlo. The most important generalization of the HF wavefunction is to put correlation directly into the wavefunction via the “Jastrow” factor. At next order, one can use the “backflow” wavefunction, in which correlation is also built into the determinant. The variational Monte Carlo method (VMC) was first used by McMillan [44] to calculate the ground-state properties of superfluid 4 He. One of the key problems at that time was whether the observed superfluid properties were a consequence of Bose condensation.
Quantum Averages and Statistical Samplings In this chapter, we discuss the general framework in which the variational Monte Carlo methods are defined and few important implementations for interacting systems of bosons and fermions on the lattice. The main advantage of considering this approach relies on the variational principle that has been shown in section 1.4: the energy of a given quantum state is always bounded from below by the exact ground-state one, giving us the route to obtain the best possible solution to the problem. In most cases, a suitable parametrization of the variational wave function allows us to consider a wide range of different quantum phases (e.g., metals, superconductors, and insulators). By performing an optimization of the parameters, we can reach the lowest-energy state, which is expected to capture the correct ground-state behavior. Therefore, variational wave functions represent a flexible and valuable approach to get important insights into the low-energy properties of models that cannot be solved by exact methods. By contrast, the main limitation of this approach is the fact that it is based on a given Ansatz , which may contain a relevant bias that cannot be removed within the chosen parametrization. We also mention that variational wave functions can be easily defined and treated for a wide class of models, irrespective of the range of interactions and the dimension of the local (i.e., single-site) Hilbert space, which can be even infinite and does not need the use of an uncontrolled cutoff to work with a finite-dimensional space. In this respect, variational Monte Carlo is better than other methods, like densitymatrix renormalization group or tensor-network methods in which the complexity dramatically increases with both the range of the interaction and the dimension of the local Hilbert space (White, 1992; Schollwöck, 2005, 2011). Let us start by describing the general framework in which variational Monte Carlo methods are defined. First of all, we fix a complete basis set in the Hilbert space, in which (for simplicity) the states are taken to be orthogonal and normalized such that: Then, any quantum state can be written as: In turn, the expectation value of an operator over a given variational wave function takes the following form: The main problem in evaluating the expectation value is that the number of configurations in the sum is exponentially large with the number of particles.
Convergent hierarchies of theories for calculating many-body vibrational ground and excited-state wave functions, such as M{o}ller-Plesset perturbation theory or coupled cluster theory, tend to rely on matrix-algebraic manipulations of large, high-dimensional arrays of anharmonic force constants, tasks which require large amounts of computer storage space and which are very difficult to implement in a parallel-scalable fashion. On the other hand, existing quantum Monte Carlo (QMC) methods for vibrational wave functions tend to lack robust techniques for obtaining excited-state energies, especially for large systems. By exploiting analytical identities for matrix elements of position operators in a harmonic oscillator basis, we have developed stochastic implementations of the size-extensive vibrational self-consistent field (MC-XVSCF) and size-extensive vibrational M{o}ller-Plesset second-order perturbation (MC-XVMP2) theories which do not require storing the potential energy surface (PES). The programmable equations of MC-XVSCF and MC-XVMP2 take the form of a small number of high-dimensional integrals evaluated using Metropolis Monte Carlo techniques. The associated integrands require independent evaluations of only the value, not the derivatives, of the PES at many points, a task which is trivial to parallelize. However, unlike existing vibrational QMC methods, MC-XVSCF and MC-XVMP2 can calculate anharmonic frequencies directly, rather than as a small difference between two noisy total energies, and do not require user-selected coordinates or nodal surfaces. MC-XVSCF and MC-XVMP2 can also directly sample the PES in a given approximation without analytical or grid-based approximations, enabling us to quantify the errors induced by such approximations.
A new Monte Carlo method for computing excited state properties of quantum systems is introduced. It is a generalization of the transient estimate method used for fermion Green’s function Monte Carlo and of subspace projection methods used for computing eigenstates of matrices. The time dependent autocorrelation function of a vector of trial functions is calculated for a random walk generated by the imaginary-time Schrödinger equation and estimates of energy levels are determined by the eigenvalues of the matrix of correlation functions. This method is especially useful for treating states with the same symmetry as it automatically keeps higher states orthogonal to lower states. The estimated energy converges to the exact eigenvalue with a rate which decreases with increasing excitation energy, thus limiting the method to relatively low-lying states. The method is zero variance in the sense that as better trial functions are introduced, the statistical error decreases to zero. The method has a nontrivial bias which is analyzed. As an illustration, the eigenvalues of a double well are computed.