Models of Strongly Interacting Quantum Matter

2019 ◽  
pp. 241-420
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Particle Physics ◽  
Magnetic Moments ◽  
Selection Criterion ◽  
Quantum Matter ◽  
Strongly Interacting ◽  
Condensed Matter Theory ◽  
Matter Theory ◽  
Rabi Model ◽  
Fermionic Systems ◽  
The One

This chapter introduces a select number of models of strongly interacting quantum many-particle physics and examines their basic properties. These models represent Bosonic and Fermionic systems as well as systems where magnetic moments, i.e. spins, interact. The main selection criterion has been the existence of a variant of the model that is quantum integrable using Bethe ansatz methods. After studying the Bose fluid, the Landau Fermi liquid, and the one-dimensional concept of the Luttinger liquid, it reviews some of the major models of condensed matter theory, including the Hubbard model describing itinerant magnetism, the Heisenberg model describing localized magnetism, and the Kondo model describing the interaction of a magnetic impurity and band electrons. It also presents the Rabi model and some of its descendants in order to describe the interaction of light and quantum matter in quantum optics.

scholarly journals Supercritical entanglement in local systems: Counterexample to the area law for quantum matter

2016 ◽  
Vol 113 (47) ◽  
pp. 13278-13282 ◽  
Author(s):  
Ramis Movassagh ◽  
Peter W. Shor
Keyword(s):  
Physical Models ◽  
Matching Theory ◽  
Many Body ◽  
Quantum Matter ◽  
Condensed Matter Theory ◽  
Simple Quantum ◽  
Matter Theory ◽  
Many Body Systems ◽  
Classical Computer ◽  
Area Law

Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simultaneously responsible for the difficulty of simulating quantum matter on a classical computer and the exponential speedups afforded by quantum computers. Ground states of quantum many-body systems typically satisfy an “area law”: The amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary. A system that obeys an area law has less entanglement and can be simulated more efficiently than a generic quantum state whose entanglement could be proportional to the total system’s size. Moreover, an area law provides useful information about the low-energy physics of the system. It is widely believed that for physically reasonable quantum systems, the area law cannot be violated by more than a logarithmic factor in the system’s size. We introduce a class of exactly solvable one-dimensional physical models which we can prove have exponentially more entanglement than suggested by the area law, and violate the area law by a square-root factor. This work suggests that simple quantum matter is richer and can provide much more quantum resources (i.e., entanglement) than expected. In addition to using recent advances in quantum information and condensed matter theory, we have drawn upon various branches of mathematics such as combinatorics of random walks, Brownian excursions, and fractional matching theory. We hope that the techniques developed herein may be useful for other problems in physics as well.

Models of Quantum Matter

2019 ◽  
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Hubbard Model ◽  
Bethe Ansatz ◽  
Particle Physics ◽  
Finite Size ◽  
Quantum Spin ◽  
Model Systems ◽  
Scattering Method ◽  
Quantum Matter ◽  
Special Cases ◽  
Rabi Model

This book focuses on the theory of quantum matter, strongly interacting systems of quantum many–particle physics, particularly on their study using exactly solvable and quantum integrable models with Bethe ansatz methods. Part 1 explores the fundamental methods of statistical physics and quantum many–particle physics required for an understanding of quantum matter. It also presents a selection of the most important model systems to describe quantum matter ranging from the Hubbard model of condensed matter physics to the Rabi model of quantum optics. The remaining five parts of the book examines appropriate special cases of these models with respect to their exact solutions using Bethe ansatz methods for the ground state, finite–size, and finite temperature properties. They also demonstrate the quantum integrability of an exemplary model, the Heisenberg quantum spin chain, within the framework of the quantum inverse scattering method and through the algebraic Bethe ansatz. Further models, whose Bethe ansatz solutions are derived and examined, include the Bose and Fermi gases in one dimension, the one–dimensional Hubbard model, the Kondo model, and the quantum Tavis–Cummings model, the latter a model descendent from the Rabi model.

scholarly journals An introduction to kinks in $\varphi^4$-theory

2021 ◽  
Author(s):  
Mariya Lizunova ◽  
Jasper van Wezel
Keyword(s):  
Undergraduate Students ◽  
Particle Physics ◽  
Topological Defects ◽  
Resonant Scattering ◽  
Theoretical Physics ◽  
Topological Classification ◽  
Condensed Matter Theory ◽  
Matter Theory ◽  
Quantitative Understanding ◽  
The Rich

As a low-energy effective model emerging in disparate fields throughout all of physics, the ubiquitous \varphi^4φ4-theory is one of the central models of modern theoretical physics. Its topological defects, or kinks, describe stable, particle-like excitations that play a central role in processes ranging from cosmology to particle physics and condensed matter theory. In these lecture notes, we introduce the description of kinks in \varphi^4φ4-theory and the various physical processes that govern their dynamics. The notes are aimed at advanced undergraduate students, and emphasis is placed on stimulating qualitative insight into the rich phenomenology encountered in kink dynamics. The appendices contain more detailed derivations, and allow enquiring students to also obtain a quantitative understanding. Topics covered include the topological classification of stable solutions, kink collisions, the formation of bions, resonant scattering of kinks, and kink-impurity interactions.

scholarly journals Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)

2018 ◽  
Author(s):  
Johannes Hauschild ◽  
Frank Pollmann
Keyword(s):  
Matrix Product ◽  
Many Body ◽  
Density Matrix Renormalization ◽  
Tensor Networks ◽  
Condensed Matter Theory ◽  
Tensor Network ◽  
Matter Theory ◽  
Particle Number Conservation ◽  
Many Body Systems ◽  
The One

Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in condensed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [1]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations.

scholarly journals An Alternative Promotion Time Cure Model with Overdispersed Number of Competing Causes: An Application to Melanoma Data

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1815
Author(s):  
Diego I. Gallardo ◽  
Mário de Castro ◽  
Héctor W. Gómez
Keyword(s):  
Estimation Method ◽  
Selection Criterion ◽  
Likelihood Method ◽  
Simulation Studies ◽  
Cure Rate ◽  
Nested Models ◽  
Proposed Model ◽  
The Em Algorithm ◽  
Competing Causes ◽  
The One

A cure rate model under the competing risks setup is proposed. For the number of competing causes related to the occurrence of the event of interest, we posit the one-parameter Bell distribution, which accommodates overdispersed counts. The model is parameterized in the cure rate, which is linked to covariates. Parameter estimation is based on the maximum likelihood method. Estimates are computed via the EM algorithm. In order to compare different models, a selection criterion for non-nested models is implemented. Results from simulation studies indicate that the estimation method and the model selection criterion have a good performance. A dataset on melanoma is analyzed using the proposed model as well as some models from the literature.

scholarly journals Gauss law, minimal coupling and fermionic PEPS for lattice gauge theories

2020 ◽  
Author(s):  
Patrick Emonts ◽  
Erez Zohar
Keyword(s):  
Gauge Theories ◽  
Point Of View ◽  
Lattice Gauge ◽  
Hamiltonian Lattice ◽  
Tensor Network ◽  
Network Methods ◽  
Matter Theory ◽  
Lattice Gauge Theories ◽  
Points Of View ◽  
The One

In these lecture notes, we review some recent works on Hamiltonian lattice gauge theories, that involve, in particular, tensor network methods. The results reviewed here are tailored together in a slightly different way from the one used in the contexts where they were first introduced. We look at the Gauss law from two different points of view: for the gauge field, it is a differential equation, while from the matter point of view, on the other hand, it is a simple, explicit algebraic equation. We will review and discuss what these two points of view allow and do not allow us to do, in terms of unitarily gauging a pure-matter theory and eliminating the matter from a gauge theory, and relate that to the construction of PEPS (Projected Entangled Pair States) for lattice gauge theories.

Quantum Many-Particle Systems and Second Quantization

2019 ◽  
pp. 5-44
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Quantum Mechanics ◽  
Particle Physics ◽  
Mathematical Formulation ◽  
Particle Systems ◽  
Quantum Field ◽  
Second Quantization ◽  
Creation And Annihilation Operators ◽  
Quantum Matter ◽  
Interacting Systems ◽  
The Many

Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of many particles. It focuses on the foundations of quantum many-particle physics, the many-particle Hilbert spaces to describe large assemblies of interacting systems composed of Bosons or Fermions, which lead to the versatile formalism of second quantization as a convenient and eminently practical language ubiquitous in the mathematical formulation of the theory of many-particle systems of quantum matter. The main objects in which the formalism of second quantization is expressed are the Bosonic or Fermionic creation and annihilation operators that become, in the position basis, the quantum field operators.

scholarly journals Fracton phases of matter

2020 ◽  
Vol 35 (06) ◽  
pp. 2030003 ◽  
Author(s):  
Michael Pretko ◽  
Xie Chen ◽  
Yizhi You
Keyword(s):  
Elasticity Theory ◽  
Bound States ◽  
Spin Liquids ◽  
Exotic Particles ◽  
The Past ◽  
Condensed Matter Theory ◽  
Open Questions ◽  
Introductory Material ◽  
Matter Theory ◽  
New Type

Fractons are a new type of quasiparticle which are immobile in isolation, but can often move by forming bound states. Fractons are found in a variety of physical settings, such as spin liquids and elasticity theory, and exhibit unusual phenomenology, such as gravitational physics and localization. The past several years have seen a surge of interest in these exotic particles, which have come to the forefront of modern condensed matter theory. In this review, we provide a broad treatment of fractons, ranging from pedagogical introductory material to discussions of recent advances in the field. We begin by demonstrating how the fracton phenomenon naturally arises as a consequence of higher moment conservation laws, often accompanied by the emergence of tensor gauge theories. We then provide a survey of fracton phases in spin models, along with the various tools used to characterize them, such as the foliation framework. We discuss in detail the manifestation of fracton physics in elasticity theory, as well as the connections of fractons with localization and gravitation. Finally, we provide an overview of some recently proposed platforms for fracton physics, such as Majorana islands and hole-doped antiferromagnets. We conclude with some open questions and an outlook on the field.

Sign in / Sign up

Export Citation Format

Share Document