Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
This book presents a statistical theory of complex wave scattering and quantum transport in a class of physical systems of current interest having chaotic classical dynamics (e.g., microwave cavities ...
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This book presents a statistical theory of complex wave scattering and quantum transport in a class of physical systems of current interest having chaotic classical dynamics (e.g., microwave cavities and quantum dots) or possessing quenched randomness (e.g., disordered conductors). The emphasis here is on mesoscopic fluctuations of the sample-specific transport. The universal character of the statistical behaviour of these phenomena is revealed in a natural way through a novel maximum-entropy approach (MEA). The latter leads to the most probable distribution for the set of random matrices that describe the ensemble of disordered/chaotic samples, which are macroscopically identical but differ in microscopic details. Here, the Shannon information entropy associated with these random matrices is maximized subject to the symmetries and the constraints which are physically relevant. This non-perturbative information-theoretic approach is reminiscent of, but distinct from, the standard random-matrix theory, and indeed forms the most distinctive feature of the book.Less
This book presents a statistical theory of complex wave scattering and quantum transport in a class of physical systems of current interest having chaotic classical dynamics (e.g., microwave cavities and quantum dots) or possessing quenched randomness (e.g., disordered conductors). The emphasis here is on mesoscopic fluctuations of the sample-specific transport. The universal character of the statistical behaviour of these phenomena is revealed in a natural way through a novel maximum-entropy approach (MEA). The latter leads to the most probable distribution for the set of random matrices that describe the ensemble of disordered/chaotic samples, which are macroscopically identical but differ in microscopic details. Here, the Shannon information entropy associated with these random matrices is maximized subject to the symmetries and the constraints which are physically relevant. This non-perturbative information-theoretic approach is reminiscent of, but distinct from, the standard random-matrix theory, and indeed forms the most distinctive feature of the book.
Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this ...
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This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this chapter include potential scattering in infinite one-dimensional space; Lippmann-Schwinger equation; free Green function; reflection and transmission amplitudes; transfer matrix; T matrix; S matrix and its analytic structure; phase shifts and resonances from the analytic structure of S matrix in complex momentum and complex energy planes; parametrization of the matrices; combination of the S matrices for two scatterers in series; and invariant-imbedding approach for a one-dimensional disordered conductor.Less
This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this chapter include potential scattering in infinite one-dimensional space; Lippmann-Schwinger equation; free Green function; reflection and transmission amplitudes; transfer matrix; T matrix; S matrix and its analytic structure; phase shifts and resonances from the analytic structure of S matrix in complex momentum and complex energy planes; parametrization of the matrices; combination of the S matrices for two scatterers in series; and invariant-imbedding approach for a one-dimensional disordered conductor.
Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter treats the problem of finding the probability distribution of quantities related to quantum transport through a strictly one-dimensional (i.e., 1-channel) and through an N-channel ...
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This chapter treats the problem of finding the probability distribution of quantities related to quantum transport through a strictly one-dimensional (i.e., 1-channel) and through an N-channel quasi-one-dimensional disordered system. It uses the maximum entropy approach wherein the distribution for the random transfer matrix for an elementary building block is determined by maximizing the associated Shannon entropy, subject to the physically relevant constraints of flux conservation, time-reversal symmetry (when relevant), and the Ohmic small length-scale limit. The contents of this chapter include ensemble of transfer matrices; universality classes — the orthogonal and the unitary classes; invariant measure; the Fokker-Planck equation for a disordered one-dimensional conductor; the maximum-entropy ansatz for the building block; construction of the probability density for a system of finite length; the Fokker-Planck equation for a quasi-one-dimensional multi-channel disordered conductor; the diffusion equation for the orthogonal universality class, β = 1; the diffusion equation for the unitary universality class, β = 2; and universal conductance fluctuations in the good-metallic limit.Less
This chapter treats the problem of finding the probability distribution of quantities related to quantum transport through a strictly one-dimensional (i.e., 1-channel) and through an N-channel quasi-one-dimensional disordered system. It uses the maximum entropy approach wherein the distribution for the random transfer matrix for an elementary building block is determined by maximizing the associated Shannon entropy, subject to the physically relevant constraints of flux conservation, time-reversal symmetry (when relevant), and the Ohmic small length-scale limit. The contents of this chapter include ensemble of transfer matrices; universality classes — the orthogonal and the unitary classes; invariant measure; the Fokker-Planck equation for a disordered one-dimensional conductor; the maximum-entropy ansatz for the building block; construction of the probability density for a system of finite length; the Fokker-Planck equation for a quasi-one-dimensional multi-channel disordered conductor; the diffusion equation for the orthogonal universality class, β = 1; the diffusion equation for the unitary universality class, β = 2; and universal conductance fluctuations in the good-metallic limit.
Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter extends the potential scattering theory developed in Chapter 2 to a relatively advanced level with emphasis on quasi-one-dimensional (multi-channel) systems, the associated scattering ...
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This chapter extends the potential scattering theory developed in Chapter 2 to a relatively advanced level with emphasis on quasi-one-dimensional (multi-channel) systems, the associated scattering and transfer matrices, and on how to combine them serially. Both the closed and the open channels are discussed. Scattering by a cavity with an arbitrary number of waveguides (the leads) attached to it is introduced. The Wigner R-matrix theory of two-dimensional scattering is treated in some detail with attention to boundary conditions. A non-trivial exactly soluble example for the two-channel scattering problem is also presented.Less
This chapter extends the potential scattering theory developed in Chapter 2 to a relatively advanced level with emphasis on quasi-one-dimensional (multi-channel) systems, the associated scattering and transfer matrices, and on how to combine them serially. Both the closed and the open channels are discussed. Scattering by a cavity with an arbitrary number of waveguides (the leads) attached to it is introduced. The Wigner R-matrix theory of two-dimensional scattering is treated in some detail with attention to boundary conditions. A non-trivial exactly soluble example for the two-channel scattering problem is also presented.
Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0019
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The ...
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This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.Less
This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.
Jean Zinn-Justin
- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199227198
- eISBN:
- 9780191711107
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227198.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter, within the framework of classical statistical mechanics, discusses a family of models defined on one-dimensional lattices. It studies the simplest local examples: models that involve ...
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This chapter, within the framework of classical statistical mechanics, discusses a family of models defined on one-dimensional lattices. It studies the simplest local examples: models that involve only interactions between nearest neighbours on the lattice. For such models, correlation functions can be calculated by a transfer matrix formalism. The chapter first describes some general properties of transfer matrices in one-dimensional models. This formalism is used to establish various properties of correlation functions, like the thermodynamic or infinite volume limit, the large-distance behaviour of the two-point correlation function, and introduces the very important concept of correlation length. Connected correlation functions, cumulants of the distribution, play a particularly important role. Indeed, these functions satisfy the cluster property, which characterizes their decay at large distance. The transfer matrix formalism is applied to the example of a Gaussian Boltzmann weight, which is studied in detail. The chapter calculates the partition function and correlation functions explicitly, and observes that , the correlation length diverges, making it possible to define a continuum limit. It shows that results of the continuum limit can be reproduced directly by solving a partial differential equation in which all traces of the initial lattice structure have disappeared. Finally, it exhibits a slightly more general class of models which share the same properties: divergent correlation length and continuum limit. Exercises are provided at the end of the chapter.Less
This chapter, within the framework of classical statistical mechanics, discusses a family of models defined on one-dimensional lattices. It studies the simplest local examples: models that involve only interactions between nearest neighbours on the lattice. For such models, correlation functions can be calculated by a transfer matrix formalism. The chapter first describes some general properties of transfer matrices in one-dimensional models. This formalism is used to establish various properties of correlation functions, like the thermodynamic or infinite volume limit, the large-distance behaviour of the two-point correlation function, and introduces the very important concept of correlation length. Connected correlation functions, cumulants of the distribution, play a particularly important role. Indeed, these functions satisfy the cluster property, which characterizes their decay at large distance. The transfer matrix formalism is applied to the example of a Gaussian Boltzmann weight, which is studied in detail. The chapter calculates the partition function and correlation functions explicitly, and observes that , the correlation length diverges, making it possible to define a continuum limit. It shows that results of the continuum limit can be reproduced directly by solving a partial differential equation in which all traces of the initial lattice structure have disappeared. Finally, it exhibits a slightly more general class of models which share the same properties: divergent correlation length and continuum limit. Exercises are provided at the end of the chapter.
M. Pilar HernÁndez
- Published in print:
- 2011
- Published Online:
- January 2012
- ISBN:
- 9780199691609
- eISBN:
- 9780191731792
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199691609.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter reviews the basic building blocks of the regularization of Quantum Field Theories (QFT) on a space-time lattice. It assumes some familiarity with QFT in the continuum. In an introductory ...
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This chapter reviews the basic building blocks of the regularization of Quantum Field Theories (QFT) on a space-time lattice. It assumes some familiarity with QFT in the continuum. In an introductory section, the path integral formulation is reviewed, focusing on important aspects such as the transfer matrix, the relation of correlation functions and physical observables, the perturbative expansion, and the key issue of renormalization and the Wilsonian renormalization group. It then considers in detail the lattice formulation of scalar, fermion and gauge field theories, paying careful attention to their physical interpretation, and the continuum limit. The difficulty of discretizing chiral fermions is discussed in detail, and various fermion discretizations are described. The strong coupling expansion is introduced in the context of lattice Yang-Mills theory and the criteria for confinement and for the presence of a mass gap are presented. It concludes with a description of Wilson's formulation of lattice QCD and a brief overview of its applications.Less
This chapter reviews the basic building blocks of the regularization of Quantum Field Theories (QFT) on a space-time lattice. It assumes some familiarity with QFT in the continuum. In an introductory section, the path integral formulation is reviewed, focusing on important aspects such as the transfer matrix, the relation of correlation functions and physical observables, the perturbative expansion, and the key issue of renormalization and the Wilsonian renormalization group. It then considers in detail the lattice formulation of scalar, fermion and gauge field theories, paying careful attention to their physical interpretation, and the continuum limit. The difficulty of discretizing chiral fermions is discussed in detail, and various fermion discretizations are described. The strong coupling expansion is introduced in the context of lattice Yang-Mills theory and the criteria for confinement and for the presence of a mass gap are presented. It concludes with a description of Wilson's formulation of lattice QCD and a brief overview of its applications.
Hans-Peter Eckle
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials
Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and ...
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Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.Less
Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.
Keith Slevin and Tomi Ohtsuki
- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199592593
- eISBN:
- 9780191741050
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199592593.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter reviews briefly the theory of the Anderson localisation of electrons. In disordered materials at low temperatures, quantum interference may lead to the suppression of diffusion. If this ...
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This chapter reviews briefly the theory of the Anderson localisation of electrons. In disordered materials at low temperatures, quantum interference may lead to the suppression of diffusion. If this occurs, the material becomes an insulator at zero temperature and zero frequency even though the density of states at the Fermi level is finite. This transition from metal to insulator is called the Anderson transition. Anderson localisation occurs particularly easily in low dimensional systems. After describing very briefly some elements of the theory of Anderson localisation, the chapter focuses on numerical simulations of Anderson localisation using the transfer matrix method, and the analysis and interpretation of the results using finite size scaling. After mentioning other approaches such as diagonalisation, this chapter closes by describing some of the experimental signatures of Anderson localisation.Less
This chapter reviews briefly the theory of the Anderson localisation of electrons. In disordered materials at low temperatures, quantum interference may lead to the suppression of diffusion. If this occurs, the material becomes an insulator at zero temperature and zero frequency even though the density of states at the Fermi level is finite. This transition from metal to insulator is called the Anderson transition. Anderson localisation occurs particularly easily in low dimensional systems. After describing very briefly some elements of the theory of Anderson localisation, the chapter focuses on numerical simulations of Anderson localisation using the transfer matrix method, and the analysis and interpretation of the results using finite size scaling. After mentioning other approaches such as diagonalisation, this chapter closes by describing some of the experimental signatures of Anderson localisation.
JEAN ZINN-JUSTIN
- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0023
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter reviews a few properties, from the point of view of phase transitions, of simple ferromagnetic lattice models. In systems with finite range interactions a transfer matrix can be defined. ...
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This chapter reviews a few properties, from the point of view of phase transitions, of simple ferromagnetic lattice models. In systems with finite range interactions a transfer matrix can be defined. It first examines its properties in a finite volume. In the infinite volume limit, low and high temperature considerations provide convincing evidence of the existence of phase transitions in Ising-like systems. The notion of order parameter is related to cluster properties in the low temperature broken phase. It is shown in a simple example that phase transitions indeed correspond to breaking of ergodicity. The chapter extends the analysis to ferromagnetic systems with continuous symmetries. The appendix contains a brief discussion of quenched disorder.Less
This chapter reviews a few properties, from the point of view of phase transitions, of simple ferromagnetic lattice models. In systems with finite range interactions a transfer matrix can be defined. It first examines its properties in a finite volume. In the infinite volume limit, low and high temperature considerations provide convincing evidence of the existence of phase transitions in Ising-like systems. The notion of order parameter is related to cluster properties in the low temperature broken phase. It is shown in a simple example that phase transitions indeed correspond to breaking of ergodicity. The chapter extends the analysis to ferromagnetic systems with continuous symmetries. The appendix contains a brief discussion of quenched disorder.
Hidetoshi Nishimori and Gerardo Ortiz
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577224
- eISBN:
- 9780191722943
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577224.003.0011
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter introduces a few typical numerical methods used in modern studies of phase transitions and critical phenomena in spin systems. The first section describes the stochastic dynamics of a ...
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This chapter introduces a few typical numerical methods used in modern studies of phase transitions and critical phenomena in spin systems. The first section describes the stochastic dynamics of a generic system with discrete degrees of freedom following the master equation. This section serves as a theoretical basis for the Monte Carlo method that includes the heat bath and Metropolis algorithms of configuration updates. Another useful numerical technique is the transfer matrix method, described in the last section, and which is applied for numerically exact evaluation of the free energy and related physical quantities.Less
This chapter introduces a few typical numerical methods used in modern studies of phase transitions and critical phenomena in spin systems. The first section describes the stochastic dynamics of a generic system with discrete degrees of freedom following the master equation. This section serves as a theoretical basis for the Monte Carlo method that includes the heat bath and Metropolis algorithms of configuration updates. Another useful numerical technique is the transfer matrix method, described in the last section, and which is applied for numerically exact evaluation of the free energy and related physical quantities.
Cord A. Müller and Dominique Delande
- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199603657
- eISBN:
- 9780191729515
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199603657.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
In the presence of disorder, classical transport is usually diffusive. This chapter deals with the effect of interference between multiply scattered waves on transport properties. Interference may ...
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In the presence of disorder, classical transport is usually diffusive. This chapter deals with the effect of interference between multiply scattered waves on transport properties. Interference may lead to reduced diffusive transport—this is known as weak localization—or to complete inhibition of transport—which is known as Anderson localization or strong localization. Key parameters are the dimension of the system and the strength of the disorder. After introductory sections on a transfer-matrix description of 1D transport, scaling theory of localization, and key numerical and experimental results, a general microscopic theory of transport in disordered systems is presented, with emphasis on experimental realizations with cold atomic gases. Simple examples are the propagation of light in a disordered medium, for which we show results from a live coherent backscattering expriment, and the propagation of atomic matter waves in an effective disordered potential created by an optical speckle. Finally, the dynamical localization transition of the kicked rotor, as observed with cold atoms, is discussed.Less
In the presence of disorder, classical transport is usually diffusive. This chapter deals with the effect of interference between multiply scattered waves on transport properties. Interference may lead to reduced diffusive transport—this is known as weak localization—or to complete inhibition of transport—which is known as Anderson localization or strong localization. Key parameters are the dimension of the system and the strength of the disorder. After introductory sections on a transfer-matrix description of 1D transport, scaling theory of localization, and key numerical and experimental results, a general microscopic theory of transport in disordered systems is presented, with emphasis on experimental realizations with cold atomic gases. Simple examples are the propagation of light in a disordered medium, for which we show results from a live coherent backscattering expriment, and the propagation of atomic matter waves in an effective disordered potential created by an optical speckle. Finally, the dynamical localization transition of the kicked rotor, as observed with cold atoms, is discussed.
Jean Zinn-Justin
- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199227198
- eISBN:
- 9780191711107
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227198.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter focuses on the concept of phase transition — a concept that is far from being trivial in the sense that a phase transition requires the interaction of an infinite number of degrees of ...
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This chapter focuses on the concept of phase transition — a concept that is far from being trivial in the sense that a phase transition requires the interaction of an infinite number of degrees of freedom. It first solves exactly a particular model in the limit in which the number of space dimensions becomes infinite. In this limit, the model exhibits a behaviour that the analysis presented in the following chapters will identify as quasi-Gaussian or mean-field like. It then discusses in general terms the existence of phase transitions as a function of the space dimension. It emphasizes the difference between models with discrete and continuous symmetries in dimension two. Exercises are provided at the end of the chapter.Less
This chapter focuses on the concept of phase transition — a concept that is far from being trivial in the sense that a phase transition requires the interaction of an infinite number of degrees of freedom. It first solves exactly a particular model in the limit in which the number of space dimensions becomes infinite. In this limit, the model exhibits a behaviour that the analysis presented in the following chapters will identify as quasi-Gaussian or mean-field like. It then discusses in general terms the existence of phase transitions as a function of the space dimension. It emphasizes the difference between models with discrete and continuous symmetries in dimension two. Exercises are provided at the end of the chapter.
Hans-Peter Eckle
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials
This chapter extends the algebraic Bethe ansatz to the quantum Tavis–Cummings model, an N atom generalization of the Jaynes–Cummings model to describe the strong interaction between light and quantum ...
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This chapter extends the algebraic Bethe ansatz to the quantum Tavis–Cummings model, an N atom generalization of the Jaynes–Cummings model to describe the strong interaction between light and quantum matter. In the case of the quantum Tavis–Cum- mings model there is no underlying vertex model to suggest the constituent building blocks of the algebraic Bethe ansatz approach, e.g.like the L-matrix or ultimately the transfer matrix. The algebraic Bethe ansatz is then first applied to the Tavis–Cummings Hamiltonian with an added Stark term using a conjecture for the transfer matrix. The original Tavis–Cummings model and its algebraic Bethe ansatz are obtained in the limit of vanishing Stark term, which requires considerable care.Less
This chapter extends the algebraic Bethe ansatz to the quantum Tavis–Cummings model, an N atom generalization of the Jaynes–Cummings model to describe the strong interaction between light and quantum matter. In the case of the quantum Tavis–Cum- mings model there is no underlying vertex model to suggest the constituent building blocks of the algebraic Bethe ansatz approach, e.g.like the L-matrix or ultimately the transfer matrix. The algebraic Bethe ansatz is then first applied to the Tavis–Cummings Hamiltonian with an added Stark term using a conjecture for the transfer matrix. The original Tavis–Cummings model and its algebraic Bethe ansatz are obtained in the limit of vanishing Stark term, which requires considerable care.
Hans-Peter Eckle
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials
Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fundamental postulate and outlines the structure of the theory using the most important of the various ...
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Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fundamental postulate and outlines the structure of the theory using the most important of the various statistical ensembles, the microcanonical, the canonical, and the grand canonical ensemble and their corresponding thermodynamic potential, the internal energy, the Helmholtz free energy, and the grand canonical potential. The notions of temperature, pressure, and chemical potential are obtained and it introduces the laws of thermodynamics, the Gibbs entropy, and the concept of the partition function. It also discusses quantum statistical mechanics using the density matrix and as applied to non-interacting Bosonic and Fermionic quantum gases, the former showing Bose–Einstein condensation. The mean-field theory of interacting magnetic moments and the transfer matrix to exactly solve the Ising model in one dimension serve as applications.Less
Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fundamental postulate and outlines the structure of the theory using the most important of the various statistical ensembles, the microcanonical, the canonical, and the grand canonical ensemble and their corresponding thermodynamic potential, the internal energy, the Helmholtz free energy, and the grand canonical potential. The notions of temperature, pressure, and chemical potential are obtained and it introduces the laws of thermodynamics, the Gibbs entropy, and the concept of the partition function. It also discusses quantum statistical mechanics using the density matrix and as applied to non-interacting Bosonic and Fermionic quantum gases, the former showing Bose–Einstein condensation. The mean-field theory of interacting magnetic moments and the transfer matrix to exactly solve the Ising model in one dimension serve as applications.
Alexey V. Kavokin, Jeremy J. Baumberg, Guillaume Malpuech, and Fabrice P. Laussy
- Published in print:
- 2017
- Published Online:
- August 2017
- ISBN:
- 9780198782995
- eISBN:
- 9780191826221
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198782995.003.0002
- Subject:
- Physics, Atomic, Laser, and Optical Physics
In this chapter we introduce the basic characteristics of light modes in free space and in different kinds of optically confined structures including Bragg mirrors, planar microcavities, pillars and ...
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In this chapter we introduce the basic characteristics of light modes in free space and in different kinds of optically confined structures including Bragg mirrors, planar microcavities, pillars and spheres. We describe the powerful transfer matrix method that allows for solution of Maxwell’s equations in multilayer structures. We discuss the polarisation of light and mention different ways it is modified including the Faraday and Kerr effects, optical birefringence, dichroism, and optical activity.Less
In this chapter we introduce the basic characteristics of light modes in free space and in different kinds of optically confined structures including Bragg mirrors, planar microcavities, pillars and spheres. We describe the powerful transfer matrix method that allows for solution of Maxwell’s equations in multilayer structures. We discuss the polarisation of light and mention different ways it is modified including the Faraday and Kerr effects, optical birefringence, dichroism, and optical activity.
JEAN ZINN-JUSTIN
- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0037
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Many numerical calculations, like Monte Carlo or transfer matrix calculations, are performed with systems in which the size in several or all dimensions is finite. To extrapolate the results to the ...
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Many numerical calculations, like Monte Carlo or transfer matrix calculations, are performed with systems in which the size in several or all dimensions is finite. To extrapolate the results to the in infnite system, it is thus necessary to have some idea about how the infinite size limit is reached. In particular, in a system in which the forces are short range no phase transition can occur in a finite volume, or in a geometry in which the size is infinite only in one dimension. This indicates that the infinite size extrapolation is somewhat non-trivial. This chapter presents an analysis of the problem in the case of second order phase transitions, in the framework of the N-vector model. It first establishes the existence of a finite size scaling, extending RG arguments to this new situation. It then distinguishes between the finite volume geometry and the cylindrical geometry in which the size is finite in all dimensions except one. It explains how to modify the methods used in the case of in infinite systems to calculate the new universal quantities appearing in finite size effects, for example, in d = 4 - ε or d = 2 + ε dimensions. Special properties of the commonly used periodic boundary conditions are emphasized. Finally, both static and dynamical finite size effects are described.Less
Many numerical calculations, like Monte Carlo or transfer matrix calculations, are performed with systems in which the size in several or all dimensions is finite. To extrapolate the results to the in infnite system, it is thus necessary to have some idea about how the infinite size limit is reached. In particular, in a system in which the forces are short range no phase transition can occur in a finite volume, or in a geometry in which the size is infinite only in one dimension. This indicates that the infinite size extrapolation is somewhat non-trivial. This chapter presents an analysis of the problem in the case of second order phase transitions, in the framework of the N-vector model. It first establishes the existence of a finite size scaling, extending RG arguments to this new situation. It then distinguishes between the finite volume geometry and the cylindrical geometry in which the size is finite in all dimensions except one. It explains how to modify the methods used in the case of in infinite systems to calculate the new universal quantities appearing in finite size effects, for example, in d = 4 - ε or d = 2 + ε dimensions. Special properties of the commonly used periodic boundary conditions are emphasized. Finally, both static and dynamical finite size effects are described.
Robert H. Swendsen
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199646944
- eISBN:
- 9780191775123
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199646944.003.0030
- Subject:
- Physics, Condensed Matter Physics / Materials
The Ising model provides an extremely useful example for the investigation of phase transitions. This chapter provides both an introduction to the properties of the Ising model and an overview of the ...
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The Ising model provides an extremely useful example for the investigation of phase transitions. This chapter provides both an introduction to the properties of the Ising model and an overview of the complex phenomena exhibited at phase transitions in general. The one-dimensional model is solved exactly using transfer matrices. Models in higher dimensions are treated in the mean field approximation.Less
The Ising model provides an extremely useful example for the investigation of phase transitions. This chapter provides both an introduction to the properties of the Ising model and an overview of the complex phenomena exhibited at phase transitions in general. The one-dimensional model is solved exactly using transfer matrices. Models in higher dimensions are treated in the mean field approximation.
Robert H. Swendsen
- Published in print:
- 2019
- Published Online:
- February 2020
- ISBN:
- 9780198853237
- eISBN:
- 9780191887703
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198853237.003.0031
- Subject:
- Physics, Condensed Matter Physics / Materials, Theoretical, Computational, and Statistical Physics
Chapter 17 presented one example of a phase transition, the van der Waals gas. This chapter provides another, the Ising model, a widely studied model of phase transitions. We first give the solution ...
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Chapter 17 presented one example of a phase transition, the van der Waals gas. This chapter provides another, the Ising model, a widely studied model of phase transitions. We first give the solution for the Ising chain (one-dimensional model), including the introduction of the transfer matrix method. Higher dimensions are treated in the Mean Field Approximation (MFA), which is also extended to Landau theory. The Ising model is deceptively simple. It can be defined in a few words, but it displays astonishingly rich behavior. It originated as a model of ferromagnetism in which the magnetic moments were localized on lattice sites and had only two allowed values.Less
Chapter 17 presented one example of a phase transition, the van der Waals gas. This chapter provides another, the Ising model, a widely studied model of phase transitions. We first give the solution for the Ising chain (one-dimensional model), including the introduction of the transfer matrix method. Higher dimensions are treated in the Mean Field Approximation (MFA), which is also extended to Landau theory. The Ising model is deceptively simple. It can be defined in a few words, but it displays astonishingly rich behavior. It originated as a model of ferromagnetism in which the magnetic moments were localized on lattice sites and had only two allowed values.
