Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Bulletin of the Australian Mathematical Society

Eigenvalue approximation methods for quantum lattice Hamiltonians

Peter Hornby

Published Jun 1, 1985• 0 citations
Publisher

Abstract

The thesis consists of five major parts or chapters, the first of which is an introduction and review relating the field of investigation to problems in quantum field theory and statistical mechanics.Although the thesis generally deals with eigenvalue bounding techniques, these methods are applied with a view to calculating the critical properties of infinite systems [7].The approach advocated in the thesis intends the use of approximate methods to calculate the thermodynamics of large, but finite lattices.Finite size scaling [3] can then be used to estimate the bulk system thermodynamics, especially critical phenomena, from the approximate large lattice data.The approximations used include projected mean fields [5], for both ground and excited states, correlated product wave functions [6], which are extended to kink states, and blocking approximations [2].Part of this work is motivated by the promise these wave functions have in their application to variationally guided Monte Carlo methods for finite lattices [4], Also, it is shown that the critical indices predicted by the finite size rescaling formalism, are left invariant by the application of an approximate (blocking) renormalisation group to the finite lattices involved.Moreover, this invariance of the finite size rescaling equations is used to define a sequence transformation suitable for improving

DOI
10.1017/s0004972700009424

Related papers

Finite lattice methods in statistical mechanics

Murray T. Batchelor

This thesis is concerned with the finite lattice study of spin models. The underlying theme in Parts I and II is the exploitation of Bethe ansatz equations to provide finite lattice data well beyond the range normally available. In Part I the Bethe ansatz equations for eigenvalues of the eight-vertex model are solved numerically to yield mass gap data on infinitely long strips of up to 512 sites in width. The finite-size corrections, at criticality, to the free energy per site and polarization gap are found to agree with recent studies of the XXZ spin chain. The leading corrections to the finite-size scaling estimates of the critical line and thermal exponent are also found, providing an explanation of the poor convergence seen in earlier studies. Away from criticality, the linear scaling fields are derived exactly in the full parameter space of the spin system, allowing a thorough test of a recently proposed method of extracting linear scaling fields and related exponents from finite lattice data. In Part II, the numerical solutions of the Bethe ansatz equations for the eigen-energies of the XXZ Hamiltonian on very large chains are used to identify, via conformal invariance, the scaling dimensions of various two-dimensional models. With periodic boundary conditions, eight-vertex and Gaussian model operators are found. The scaling dimensions of the Ashkm-Teller and Potts models are obtained by the exact relating of eigenstates of their quantum Hamiltonians to those of the XXZ chain with modified boundary conditions. Eigenstates of the Ashkin-Teller and Potts models with free boundaries are also obtained, allowing an examination of their critical surface properties. In Part HI the critical behaviour of an Ising model with competing firstand third- nearest neighbour interactions on the square lattice is investigated using the finite lattice method. In the ferromagnetic region, the phase boundary is located with an accuracy at least equal to that of alternative methods. In the antiphase region, distinctive structure in the finite lattice estimators is found over an extended temperature range. However, the nature of the transition remains unclear.

•Jan 1, 1987•2 citations

Finite element methods for finite size scaling

Winton Moy

The study of phase transitions and critical phenomena is an area of great interest in science. Liquid to gas, ferromagnetic to paramagnetic, fluid to superfluid, insulator to conductor are a few examples of physical systems exhibiting phase transition and critical phenomena. Classical phase transitions have thermal fluctuations as the main driving force for the transition. In statistical mechanics phase transitions are associated with singularities in the free energy. These singularites only occur in the thermodynamic limit where the volume (V) and particles (N) go to infinity with the density held constant (N/V). By examining the partition function for a finite system, the function is a sum of a set number of terms. The partition function then would be analytical. It is only when an infinite number of terms are added is there a singularity in the partition function. The subject of Finite Size Scaling theory is the relation of the phenomena in a finite systems to the true phase transition of an infinite system. Finite Size Scaling theory provides a numerical method to obtain accurate results for infinite systems simply by studying corresponding small systems. The focus of this thesis is on transitions of a different nature. Quantum phase transitions (transitions that occur at absolute zero) have Heisenberg’s uncertainty principle as the driving force for the transition. In quantum mechanics, the finite size problem occurs when looking at the critical behaviour of the Hamiltonian as a function of a set of parameters. The interest is where the bound states energies become non-analytical. The size of the system is related to the number of elements in a complete basis set used to expand the exact wave function. In this thesis we present the use of the Finite Element Method (a numerical technique commonly used in engineering problems to solve partial differential equations or integral equations). Previous finite size scaling studies in quantum mechanics employed a basis set expansion of the wave function and the problem is solved variationally. Slater type basis sets were used for this purpose. In the effort to apply finite size scaling calculations for larger systems Gaussian type functions were employed however the a larger number of Gaussian functions are needed to obtain accurate results. The use of standard electronic structure packages was also explored, however, the lack of consistently increasing size for basis sets limits the application for finite size scaling purposes. We first apply the finite element element method (FEM) to solve for the short range Yukawa potential and obtained the critical parameters for the model potential. We develop the framework necessary to use FEM for finite size scaling. As mentioned before the 'size' of the system is the number of basis functions needed in the expansion of the wave function. However, using FEM the size of the system is the number of elements used. We then studied the Hulthen potential in comparison to the basis set results and also compare the results to the analytical solutions for the both methods. Finally we explore the use of FEM in combination with Hartree-Fock theory to solve 2 to 4 electron systems. The use of Hartree-Fock theory allows for the implementation of finite size scaling to larger systems. We used the FEM Hartree-Fock to calculate for the critical parameters of 2 to 4 electron systems. Although the results are not exact (as to be expected due to the use of Hartree-Fock theory) we get an estimate to the critical points of these systems. FEM can be combined with other electronic structure theories, which would allow for the study of larger systems exhibiting critical phenomena.

•Jan 1, 2010•0 citations

Finite lattice calculation of the quantum mechanical n-vector model

M. Kolb, Rémi Jullien, Pierre M. Pfeuty

The truncated quantum mechanical n-vector model in one dimension is studied by means of the phenomenological renormalisation group (PRG) and the block renormalisation group (BRG) for general values of n. Particular emphasis is put on the extrapolation of the finite size results using the finite size scaling hypothesis. Accurate estimates of the critical properties can be made for 0<or approximately=n<or approximately=2. The two methods used are critically compared.

Journal of Physics A Mathematical and General•Dec 1, 1982•10 citations

On numerical methods in quantum spin systems

Dmitrii Kochkov

This thesis is focused on the application and development of numerical methods for studying quantum many- body spin systems. Part I sets the stage for research projects. In chapter 1, I go over major milestones in the field of frustrated magnetism, how it fits into a larger category of condensed matter physics, and recent developments that provide context for the research projects presented in part II. Chapters 2 3 provide the reader with necessary concepts in physics of phase transitions and numerical methods that are often employed when studying ground states of quantum many body systems. Part II focuses on the research projects carried out during my PhD research. In chapter 4 I present a machine-learning inspired method for solving quantum many body problems. This method addresses issues of representation of variational wave functions and their optimization. In chapter 5, I present a numerical study on the embedding of an exactly solvable point in a phase diagram of an extended Heisenberg model on a kagome lattice. Chapter 6 studies a quantum Heisenberg model on the stuffed honeycomb lattice. Phase transitions in quantum many body systems manifest themselves as a reorganization of low energy spectra of the system. They come in a variety of forms ranging from symmetry breaking transitions, where a local order parameter is formed and to topological phase transitions where the change of the ground state can be attributed to non-local changes in the entanglement patterns. In chapter 2, I review classic models studied in the context of frustrated magnetism and briefly go over the formulation of Landau symmetry breaking theory and common order parameters. The breakdown of Landau symmetry-breaking theory for topological phase transition is discussed using the example of Resonance Valence Bond (RVB) state, which serves as a qualitative exemplification of spin liquid phases. I discuss means of detecting phase transitions with a focus on methods that are applicable in a numerical setting. Chapter 3 presents elements of the numerical machinery that the rest of the thesis employs and builds upon. I discuss traditional trade-offs and provide rough metrics used for analysis of strengths and drawbacks of different approaches. Then I focus on Exact Diagonalization (ED) and Variational Monte Carlo (VMC) methods that are used in later chapters. In chapter 4, I describe the application of machine learning techniques to the variational approach to the quantum many body problem. I introduce Computational Graph States (CGS), a new class of variational wave functions based on the dataflow programming paradigm. The use of computational graphs makes composing new wave functions easy due to its modular structure. Automatic differentiation allows us to obtain efficient derivatives with respect to variational parameters automatically. We also develop a novel optimization scheme, Supervised Wave-function Optimization. It combines ideas of supervised learning and imaginary time evolution. Our method resolves a number of technical challenges and allows seamless optimization of complicated variational models. During the exploration process we constructed a new class of system size invariant model wave functions that can be trained on one system size and generalized to other system sizes with high accuracy. In chapter 5, we study the phase diagram of the kagome antiferromagnet in the J2, Jz plane, which corresponds to second nearest neighbor interactions and spin anisotropy. We report an exactly solvable point with macroscopic degeneracy at J2 = 0, Jz = −0.5 and show that many phases emanate from around this point. In addition, we find a previously unknown transition within the spin-liquid region suggesting a possibility of two distinct spin liquids. In chapter 6, we consider a Heisenberg model on the stuffed honeycomb lattice which interpolated between the triangular and honeycomb lattices. Using exact diagonalization we compute the phase diagram of this model. We find that for this model, the triangular lattice J1-J2 spin liquid region is extended to a larger domain that has lower symmetries than that of the triangular lattice. We perform a variational study of projected BCS wave functions which we directly compare to exact ground states on 36 site cluster as well as consider variational energies from larger system sizes. We argue that construction of symmetric states is essential to obtain good variational energies and give evidence that Dirac spin-liquid state is likely to characterize the spin liquid phase.

•May 2, 2019•0 citations

Machine learning and high-performance computing in numerical simulation of quantum many-body systems

Hongwei Chen

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 $J_1-J_2$ 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 $\mathtt{cuBLAS}$.--Author's abstract

•Jan 1, 2024•0 citations

Phase Transitions and Critical Phenomena: Renormalisation Group Method

Deepak Kumar

These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.

Texts and readings in physical sciences•Jan 1, 2001•0 citations
The thesis consists of five major parts or chapters, the first of which is an introduction and review relating the field of investigation to problems in quantum field theory and statistical mechanics.Although the thesis generally deals with eigenvalue bounding techniques, these methods are applied with a view to calculating the critical properties of infinite systems [7].The approach advocated in the thesis intends the use of approximate methods to calculate the thermodynamics of large, but finite lattices.Finite size scaling [3] can then be used to estimate the bulk system thermodynamics, especially critical phenomena, from the approximate large lattice data.The approximations used include projected mean fields [5], for both ground and excited states, correlated product wave functions [6], which are extended to kink states, and blocking approximations [2].Part of this work is motivated by the promise these wave functions have in their application to variationally guided Monte Carlo methods for finite lattices [4], Also, it is shown that the critical indices predicted by the finite size rescaling formalism, are left invariant by the application of an approximate (blocking) renormalisation group to the finite lattices involved.Moreover, this invariance of the finite size rescaling equations is used to define a sequence transformation suitable for improving
This thesis is concerned with the finite lattice study of spin models. The underlying theme in Parts I and II is the exploitation of Bethe ansatz equations to provide finite lattice data well beyond the range normally available. In Part I the Bethe ansatz equations for eigenvalues of the eight-vertex model are solved numerically to yield mass gap data on infinitely long strips of up to 512 sites in width. The finite-size corrections, at criticality, to the free energy per site and polarization gap are found to agree with recent studies of the XXZ spin chain. The leading corrections to the finite-size scaling estimates of the critical line and thermal exponent are also found, providing an explanation of the poor convergence seen in earlier studies. Away from criticality, the linear scaling fields are derived exactly in the full parameter space of the spin system, allowing a thorough test of a recently proposed method of extracting linear scaling fields and related exponents from finite lattice data. In Part II, the numerical solutions of the Bethe ansatz equations for the eigen-energies of the XXZ Hamiltonian on very large chains are used to identify, via conformal invariance, the scaling dimensions of various two-dimensional models. With periodic boundary conditions, eight-vertex and Gaussian model operators are found. The scaling dimensions of the Ashkm-Teller and Potts models are obtained by the exact relating of eigenstates of their quantum Hamiltonians to those of the XXZ chain with modified boundary conditions. Eigenstates of the Ashkin-Teller and Potts models with free boundaries are also obtained, allowing an examination of their critical surface properties. In Part HI the critical behaviour of an Ising model with competing firstand third- nearest neighbour interactions on the square lattice is investigated using the finite lattice method. In the ferromagnetic region, the phase boundary is located with an accuracy at least equal to that of alternative methods. In the antiphase region, distinctive structure in the finite lattice estimators is found over an extended temperature range. However, the nature of the transition remains unclear.
The study of phase transitions and critical phenomena is an area of great interest in science. Liquid to gas, ferromagnetic to paramagnetic, fluid to superfluid, insulator to conductor are a few examples of physical systems exhibiting phase transition and critical phenomena. Classical phase transitions have thermal fluctuations as the main driving force for the transition. In statistical mechanics phase transitions are associated with singularities in the free energy. These singularites only occur in the thermodynamic limit where the volume (V) and particles (N) go to infinity with the density held constant (N/V). By examining the partition function for a finite system, the function is a sum of a set number of terms. The partition function then would be analytical. It is only when an infinite number of terms are added is there a singularity in the partition function. The subject of Finite Size Scaling theory is the relation of the phenomena in a finite systems to the true phase transition of an infinite system. Finite Size Scaling theory provides a numerical method to obtain accurate results for infinite systems simply by studying corresponding small systems. The focus of this thesis is on transitions of a different nature. Quantum phase transitions (transitions that occur at absolute zero) have Heisenberg’s uncertainty principle as the driving force for the transition. In quantum mechanics, the finite size problem occurs when looking at the critical behaviour of the Hamiltonian as a function of a set of parameters. The interest is where the bound states energies become non-analytical. The size of the system is related to the number of elements in a complete basis set used to expand the exact wave function. In this thesis we present the use of the Finite Element Method (a numerical technique commonly used in engineering problems to solve partial differential equations or integral equations). Previous finite size scaling studies in quantum mechanics employed a basis set expansion of the wave function and the problem is solved variationally. Slater type basis sets were used for this purpose. In the effort to apply finite size scaling calculations for larger systems Gaussian type functions were employed however the a larger number of Gaussian functions are needed to obtain accurate results. The use of standard electronic structure packages was also explored, however, the lack of consistently increasing size for basis sets limits the application for finite size scaling purposes. We first apply the finite element element method (FEM) to solve for the short range Yukawa potential and obtained the critical parameters for the model potential. We develop the framework necessary to use FEM for finite size scaling. As mentioned before the 'size' of the system is the number of basis functions needed in the expansion of the wave function. However, using FEM the size of the system is the number of elements used. We then studied the Hulthen potential in comparison to the basis set results and also compare the results to the analytical solutions for the both methods. Finally we explore the use of FEM in combination with Hartree-Fock theory to solve 2 to 4 electron systems. The use of Hartree-Fock theory allows for the implementation of finite size scaling to larger systems. We used the FEM Hartree-Fock to calculate for the critical parameters of 2 to 4 electron systems. Although the results are not exact (as to be expected due to the use of Hartree-Fock theory) we get an estimate to the critical points of these systems. FEM can be combined with other electronic structure theories, which would allow for the study of larger systems exhibiting critical phenomena.
This thesis is focused on the application and development of numerical methods for studying quantum many- body spin systems. Part I sets the stage for research projects. In chapter 1, I go over major milestones in the field of frustrated magnetism, how it fits into a larger category of condensed matter physics, and recent developments that provide context for the research projects presented in part II. Chapters 2 3 provide the reader with necessary concepts in physics of phase transitions and numerical methods that are often employed when studying ground states of quantum many body systems. Part II focuses on the research projects carried out during my PhD research. In chapter 4 I present a machine-learning inspired method for solving quantum many body problems. This method addresses issues of representation of variational wave functions and their optimization. In chapter 5, I present a numerical study on the embedding of an exactly solvable point in a phase diagram of an extended Heisenberg model on a kagome lattice. Chapter 6 studies a quantum Heisenberg model on the stuffed honeycomb lattice. Phase transitions in quantum many body systems manifest themselves as a reorganization of low energy spectra of the system. They come in a variety of forms ranging from symmetry breaking transitions, where a local order parameter is formed and to topological phase transitions where the change of the ground state can be attributed to non-local changes in the entanglement patterns. In chapter 2, I review classic models studied in the context of frustrated magnetism and briefly go over the formulation of Landau symmetry breaking theory and common order parameters. The breakdown of Landau symmetry-breaking theory for topological phase transition is discussed using the example of Resonance Valence Bond (RVB) state, which serves as a qualitative exemplification of spin liquid phases. I discuss means of detecting phase transitions with a focus on methods that are applicable in a numerical setting. Chapter 3 presents elements of the numerical machinery that the rest of the thesis employs and builds upon. I discuss traditional trade-offs and provide rough metrics used for analysis of strengths and drawbacks of different approaches. Then I focus on Exact Diagonalization (ED) and Variational Monte Carlo (VMC) methods that are used in later chapters. In chapter 4, I describe the application of machine learning techniques to the variational approach to the quantum many body problem. I introduce Computational Graph States (CGS), a new class of variational wave functions based on the dataflow programming paradigm. The use of computational graphs makes composing new wave functions easy due to its modular structure. Automatic differentiation allows us to obtain efficient derivatives with respect to variational parameters automatically. We also develop a novel optimization scheme, Supervised Wave-function Optimization. It combines ideas of supervised learning and imaginary time evolution. Our method resolves a number of technical challenges and allows seamless optimization of complicated variational models. During the exploration process we constructed a new class of system size invariant model wave functions that can be trained on one system size and generalized to other system sizes with high accuracy. In chapter 5, we study the phase diagram of the kagome antiferromagnet in the J2, Jz plane, which corresponds to second nearest neighbor interactions and spin anisotropy. We report an exactly solvable point with macroscopic degeneracy at J2 = 0, Jz = −0.5 and show that many phases emanate from around this point. In addition, we find a previously unknown transition within the spin-liquid region suggesting a possibility of two distinct spin liquids. In chapter 6, we consider a Heisenberg model on the stuffed honeycomb lattice which interpolated between the triangular and honeycomb lattices. Using exact diagonalization we compute the phase diagram of this model. We find that for this model, the triangular lattice J1-J2 spin liquid region is extended to a larger domain that has lower symmetries than that of the triangular lattice. We perform a variational study of projected BCS wave functions which we directly compare to exact ground states on 36 site cluster as well as consider variational energies from larger system sizes. We argue that construction of symmetric states is essential to obtain good variational energies and give evidence that Dirac spin-liquid state is likely to characterize the spin liquid phase.
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−J2J_1-J_2J1​−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\mathtt{cuBLAS}cuBLAS.--Author's abstract
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.