عنوان

Introduction to tensor network methods :

3030014096

9783030014094

9783030014087

Introduction to tensor network methods :

[Book]

numerical simulations of low-dimensional many-body quantum systems /

Simone Montangero.

Cham, Switzerland :

Springer,

[2018]

1 online resource

Includes bibliographical references and index.

Intro; Preface; Contents; Symbols and Abbreviations; 1 Introduction; Part I The Single Body Problem; 2 Linear Algebra; 2.1 System of Linear Equations; 2.1.1 LU Reduction; 2.2 Eigenvalue Problem; 2.2.1 Power Methods; 2.3 Tridiagonal Matrices; 2.4 Lanczos Methods; 2.5 Exercises; 3 Numerical Calculus; 3.1 Classical Quadrature; 3.2 Gaussian Integration; 3.3 Time-Independent Schrödinger Equation; 3.3.1 Finite Difference Method; 3.3.2 Variational Method; 3.4 Time-Dependent Schrödinger Equation; 3.4.1 Spectral Method; 3.4.2 Split Operator Method; 3.4.3 Partial Differential Equations Solvers

3.5 ExercisesPart II The Many-Body Problem; 4 Numerical Renormalization Group Methods; 4.1 Mean Field Theory; 4.1.1 Quantum Ising Model in Transverse Field; 4.1.2 Cluster Mean Field; 4.2 Real-Space Renormalization Group; 4.3 Density Matrix Renormalization Group; 4.4 Exercises; 5 Tensor Network Methods; 5.1 Tensor Definition; 5.2 Tensor Manipulations; 5.2.1 Index Fusion and Splitting; 5.2.2 Compression; 5.2.3 Tensor Network Differentiation; 5.2.4 Gauging; 5.2.5 Tensor Contraction Complexity; 5.3 Ground States via Tensor Networks; 5.3.1 Mean Field; 5.3.2 Graphical Tensor Notation

5.3.3 Matrix Product States5.3.4 Loop-Free Tensor Networks; 5.3.5 Looped Tensor Networks; 5.4 Time Evolution via Tensor Networks; 5.4.1 Time-Dependent Density Matrix Renormalization Group; 5.4.2 Fidelity-Driven Evolution; 5.4.3 Time-Dependent Variational Principle; 5.5 Measurements; 5.6 Further Developments; 5.7 Software; 5.8 Exercises; 6 Symmetric Tensor Networks; 6.1 Elements of Group Theory; 6.2 Global Pointlike Symmetries; 6.3 Quantum Link Formulation of Gauge Symmetries; 6.4 Lattice Gauge Invariant Tensor Networks; 6.5 Exercises; Part III Applications

7 Many-Body Quantum Systems at Equilibrium7.1 Phase Transitions; 7.2 Quantum-Classical Statistical Correspondence; 7.3 Quantum Phase Transition; 7.3.1 Critical Exponents; 7.4 Entanglement Measures; 7.4.1 Central Charge; 7.4.2 Topological Entanglement Entropy; 7.5 Exercises; 8 Out-of-Equilibrium Processes; 8.1 Adiabatic Quantum Computation; 8.1.1 Adiabatic Theorem; 8.1.2 Applications; 8.2 Kibble-Zurek Mechanism; 8.2.1 Crossover from Quantum to Classical Kibble-Zurek; 8.3 Optimal Control of Many-Body Quantum Systems; 8.4 Other Applications; 8.5 Exercises; A Hardware in a Nutshell for Physicists

A.1 ArchitectureA. 2 Data and Formats; A.3 Memory and Data Processing; A.4 Multiprocessors; A.5 Exercises; B Software in a Nutshell for Physicists; B.1 Correctness; B.2 Numerical Stability; B.3 Accurate Discretization; B.4 Flexibility; B.5 Efficiency; B.6 Exercises; References; ; Index

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This volume of lecture notes briefly introduces the basic concepts needed in any computational physics course: software and hardware, programming skills, linear algebra, and differential calculus. It then presents more advanced numerical methods to tackle the quantum many-body problem: it reviews the numerical renormalization group and then focuses on tensor network methods, from basic concepts to gauge invariant ones. Finally, in the last part, the author presents some applications of tensor network methods to equilibrium and out-of-equilibrium correlated quantum matter. The book can be used for a graduate computational physics course. After successfully completing such a course, a student should be able to write a tensor network program and can begin to explore the physics of many-body quantum systems. The book can also serve as a reference for researchers working or starting out in the field.

Springer Nature

com.springer.onix.9783030014094

9783030014087

9783030014100

Tensor fields.

MATHEMATICS-- Algebra-- Intermediate.

Tensor fields.

MAT-- 002040

PHU

PHU

Montangero, Simone

20200823072944.0

pn

[Book]

Y