In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics. It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials. In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice. Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity. Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model. In mathematical physics, the Hubbard model has also had an essential role. The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models. In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka's Theorem).

As its name suggests, the 1988 workshop on "Interacting Electrons in Reduced Dimen­ the wide variety of physical effects that are associated with (possibly sions" focused on strongly) correlated electrons interacting in quasi-one- and quasi-two-dimensional mate­ rials. Among the phenomena discussed were superconductivity, magnetic ordering, the metal-insulator transition, localization, the fractional Quantum Hall effect (QHE), Peierls and spin-Peierls transitions, conductance fluctuations and sliding charge-density (CDW) and spin-density (SDW) waves. That these effects appear most pronounced in systems of reduced dimensionality was amply demonstrated at the meeting. Indeed, when concrete illustrations were presented, they typically involved chain-like materials such as conjugated polymers, inorganic CDW systems and organie conductors, or layered materials such as high-temperature copper-oxide superconductors, certain of the organic superconductors, and the QHE samples, or devices where the electrons are confined to a restricted region of sample, e. g. , the depletion layer of a MOSFET. To enable this broad subject to be covered in thirty-five lectures (and ab out half as many posters), the workshop was deliberately focused on theoretical models for these phenomena and on methods for describing as faithfully as possible the "true" behav­ ior of these models. This latter emphasis was especially important, since the inherently many-body nature of problems involving interacting electrons renders conventional effec­ tive single-particle/mean-field methods (e. g. , Hartree-Fock or the local-density approxi­ mation in density-functional theory) highly suspect. Again, this is particularly true in reduced dimensions, where strong quantum fluctuations can invalidatemean-field results.

As its name suggests, the 1988 workshop on "Interacting Electrons in Reduced Dimen­ the wide variety of physical effects that are associated with (possibly sions" focused on strongly) correlated electrons interacting in quasi-one- and quasi-two-dimensional mate­ rials. Among the phenomena discussed were superconductivity, magnetic ordering, the metal-insulator transition, localization, the fractional Quantum Hall effect (QHE), Peierls and spin-Peierls transitions, conductance fluctuations and sliding charge-density (CDW) and spin-density (SDW) waves. That these effects appear most pronounced in systems of reduced dimensionality was amply demonstrated at the meeting. Indeed, when concrete illustrations were presented, they typically involved chain-like materials such as conjugated polymers, inorganic CDW systems and organie conductors, or layered materials such as high-temperature copper-oxide superconductors, certain of the organic superconductors, and the QHE samples, or devices where the electrons are confined to a restricted region of sample, e. g. , the depletion layer of a MOSFET. To enable this broad subject to be covered in thirty-five lectures (and ab out half as many posters), the workshop was deliberately focused on theoretical models for these phenomena and on methods for describing as faithfully as possible the "true" behav­ ior of these models. This latter emphasis was especially important, since the inherently many-body nature of problems involving interacting electrons renders conventional effec­ tive single-particle/mean-field methods (e. g. , Hartree-Fock or the local-density approxi­ mation in density-functional theory) highly suspect. Again, this is particularly true in reduced dimensions, where strong quantum fluctuations can invalidatemean-field results.

In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics. It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials. In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice. Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity. Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model. In mathematical physics, the Hubbard model has also had an essential role. The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models. In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka''s Theorem).

In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics. It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials. In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice. Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity. Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model. In mathematical physics, the Hubbard model has also had an essential role. The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models. In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka's Theorem).

ill the past three decades there has been enonnous progress in identifying the es­ sential role that "nonlinearity" plays in physical systems. Classical nonlinear wave equations can support localized, stable "soliton" solutions, and nonlinearities in quantum systems can lead to self-trapped excitations, such as polarons. Since these nonlinear excitations often dominate the transport and response properties of the systems in which they exist, accurate modeling of their effects is essential to interpreting a wide range of physical phenomena. Further, the dramatic de­ velopments in "deterministic chaos", including the recognition that even simple nonlinear dynamical systems can produce seemingly random temporal evolution, have similarly demonstrated that an understanding of chaotic dynamics is vital to an accurate interpretation of the behavior of many physical systems. As a conse­ quence of these two developments, the study of nonlinear phenomena has emerged as a subject in its own right. During these same three decades, similar progress has occurred in understand­ ing the effects of "disorder". Stimulated by Anderson''s pioneering work on "dis­ ordered" quantum solid state materials, this effort has also grown into a field that now includes a variety of classical and quantum systems and treats "disorder" arising from many sources, including impurities, random spatial structures, and stochastic applied fields. Significantly, these two developments have occurred rather independently, with relatively little overlapping research.

ill the past three decades there has been enonnous progress in identifying the es­ sential role that "nonlinearity" plays in physical systems. Classical nonlinear wave equations can support localized, stable "soliton" solutions, and nonlinearities in quantum systems can lead to self-trapped excitations, such as polarons. Since these nonlinear excitations often dominate the transport and response properties of the systems in which they exist, accurate modeling of their effects is essential to interpreting a wide range of physical phenomena. Further, the dramatic de­ velopments in "deterministic chaos", including the recognition that even simple nonlinear dynamical systems can produce seemingly random temporal evolution, have similarly demonstrated that an understanding of chaotic dynamics is vital to an accurate interpretation of the behavior of many physical systems. As a conse­ quence of these two developments, the study of nonlinear phenomena has emerged as a subject in its own right. During these same three decades, similar progress has occurred in understand­ ing the effects of "disorder". Stimulated by Anderson's pioneering work on "dis­ ordered" quantum solid state materials, this effort has also grown into a field that now includes a variety of classical and quantum systems and treats "disorder" arising from many sources, including impurities, random spatial structures, and stochastic applied fields. Significantly, these two developments have occurred rather independently, with relatively little overlapping research.

The Sixth Annual Conference of the Center for Nonlinear Studies at the Los Alamos National Laboratory was held May 5-9, 1986, on the topic "Nonlinearity in Condensed Matter: Lessons from the Past and Prospects for the Future. " As conference organizers, we felt that the study of non­ linear phenomena in condensed matter had matured to the point where it made sense to take stock of the numerous lessons to be learned from a variety of contexts where nonlinearity plays a fundamental role and to evaluate the prospects for the growth of this general discipline. The successful 1978 Oxford Symposium on nonlinear (soliton) struc­ ture and dynamics in condensed matter (Springer Ser. Solid-State Sci. , Vol. 8) was held at a time when the ubiquity of solitons was just begin­ ning to be appreciated by the condensed matter community; in subsequent years the soliton paradigm has provided a rather useful framework for in­ vestigating a large number of phenomena, particularly in low-dimensional systems. Nevertheless, we felt that the importance of nonlinearity in wider arenas than "solitonics" merited a significant expansion in the scope of the conference over that of the 1978 symposium. Indeed, many of the lessons are quite general and their potential for cross-fertilization of otherwise poorly connected disciplines was certainly one of the prime motivations for this conference. Thus, while these proceedings contain many contribu­ tions pertaining to soliton behavior in different contexts, the reader will find much more as well, particularly in the later chapters.

In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics. It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials. In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice. Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity. Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model. In mathematical physics, the Hubbard model has also had an essential role. The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models. In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka's Theorem).

As its name suggests, the 1988 workshop on "Interacting Electrons in Reduced Dimen­ the wide variety of physical effects that are associated with (possibly sions" focused on strongly) correlated electrons interacting in quasi-one- and quasi-two-dimensional mate­ rials. Among the phenomena discussed were superconductivity, magnetic ordering, the metal-insulator transition, localization, the fractional Quantum Hall effect (QHE), Peierls and spin-Peierls transitions, conductance fluctuations and sliding charge-density (CDW) and spin-density (SDW) waves. That these effects appear most pronounced in systems of reduced dimensionality was amply demonstrated at the meeting. Indeed, when concrete illustrations were presented, they typically involved chain-like materials such as conjugated polymers, inorganic CDW systems and organie conductors, or layered materials such as high-temperature copper-oxide superconductors, certain of the organic superconductors, and the QHE samples, or devices where the electrons are confined to a restricted region of sample, e. g. , the depletion layer of a MOSFET. To enable this broad subject to be covered in thirty-five lectures (and ab out half as many posters), the workshop was deliberately focused on theoretical models for these phenomena and on methods for describing as faithfully as possible the "true" behav­ ior of these models. This latter emphasis was especially important, since the inherently many-body nature of problems involving interacting electrons renders conventional effec­ tive single-particle/mean-field methods (e. g. , Hartree-Fock or the local-density approxi­ mation in density-functional theory) highly suspect. Again, this is particularly true in reduced dimensions, where strong quantum fluctuations can invalidatemean-field results.

As its name suggests, the 1988 workshop on "Interacting Electrons in Reduced Dimen­ the wide variety of physical effects that are associated with (possibly sions" focused on strongly) correlated electrons interacting in quasi-one- and quasi-two-dimensional mate­ rials. Among the phenomena discussed were superconductivity, magnetic ordering, the metal-insulator transition, localization, the fractional Quantum Hall effect (QHE), Peierls and spin-Peierls transitions, conductance fluctuations and sliding charge-density (CDW) and spin-density (SDW) waves. That these effects appear most pronounced in systems of reduced dimensionality was amply demonstrated at the meeting. Indeed, when concrete illustrations were presented, they typically involved chain-like materials such as conjugated polymers, inorganic CDW systems and organie conductors, or layered materials such as high-temperature copper-oxide superconductors, certain of the organic superconductors, and the QHE samples, or devices where the electrons are confined to a restricted region of sample, e. g. , the depletion layer of a MOSFET. To enable this broad subject to be covered in thirty-five lectures (and ab out half as many posters), the workshop was deliberately focused on theoretical models for these phenomena and on methods for describing as faithfully as possible the "true" behav­ ior of these models. This latter emphasis was especially important, since the inherently many-body nature of problems involving interacting electrons renders conventional effec­ tive single-particle/mean-field methods (e. g. , Hartree-Fock or the local-density approxi­ mation in density-functional theory) highly suspect. Again, this is particularly true in reduced dimensions, where strong quantum fluctuations can invalidatemean-field results.

In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics. It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials. In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice. Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity. Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model. In mathematical physics, the Hubbard model has also had an essential role. The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models. In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka''s Theorem).

In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics. It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials. In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice. Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity. Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model. In mathematical physics, the Hubbard model has also had an essential role. The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models. In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka's Theorem).

ill the past three decades there has been enonnous progress in identifying the es­ sential role that "nonlinearity" plays in physical systems. Classical nonlinear wave equations can support localized, stable "soliton" solutions, and nonlinearities in quantum systems can lead to self-trapped excitations, such as polarons. Since these nonlinear excitations often dominate the transport and response properties of the systems in which they exist, accurate modeling of their effects is essential to interpreting a wide range of physical phenomena. Further, the dramatic de­ velopments in "deterministic chaos", including the recognition that even simple nonlinear dynamical systems can produce seemingly random temporal evolution, have similarly demonstrated that an understanding of chaotic dynamics is vital to an accurate interpretation of the behavior of many physical systems. As a conse­ quence of these two developments, the study of nonlinear phenomena has emerged as a subject in its own right. During these same three decades, similar progress has occurred in understand­ ing the effects of "disorder". Stimulated by Anderson''s pioneering work on "dis­ ordered" quantum solid state materials, this effort has also grown into a field that now includes a variety of classical and quantum systems and treats "disorder" arising from many sources, including impurities, random spatial structures, and stochastic applied fields. Significantly, these two developments have occurred rather independently, with relatively little overlapping research.

ill the past three decades there has been enonnous progress in identifying the es­ sential role that "nonlinearity" plays in physical systems. Classical nonlinear wave equations can support localized, stable "soliton" solutions, and nonlinearities in quantum systems can lead to self-trapped excitations, such as polarons. Since these nonlinear excitations often dominate the transport and response properties of the systems in which they exist, accurate modeling of their effects is essential to interpreting a wide range of physical phenomena. Further, the dramatic de­ velopments in "deterministic chaos", including the recognition that even simple nonlinear dynamical systems can produce seemingly random temporal evolution, have similarly demonstrated that an understanding of chaotic dynamics is vital to an accurate interpretation of the behavior of many physical systems. As a conse­ quence of these two developments, the study of nonlinear phenomena has emerged as a subject in its own right. During these same three decades, similar progress has occurred in understand­ ing the effects of "disorder". Stimulated by Anderson's pioneering work on "dis­ ordered" quantum solid state materials, this effort has also grown into a field that now includes a variety of classical and quantum systems and treats "disorder" arising from many sources, including impurities, random spatial structures, and stochastic applied fields. Significantly, these two developments have occurred rather independently, with relatively little overlapping research.

The Sixth Annual Conference of the Center for Nonlinear Studies at the Los Alamos National Laboratory was held May 5-9, 1986, on the topic "Nonlinearity in Condensed Matter: Lessons from the Past and Prospects for the Future. " As conference organizers, we felt that the study of non­ linear phenomena in condensed matter had matured to the point where it made sense to take stock of the numerous lessons to be learned from a variety of contexts where nonlinearity plays a fundamental role and to evaluate the prospects for the growth of this general discipline. The successful 1978 Oxford Symposium on nonlinear (soliton) struc­ ture and dynamics in condensed matter (Springer Ser. Solid-State Sci. , Vol. 8) was held at a time when the ubiquity of solitons was just begin­ ning to be appreciated by the condensed matter community; in subsequent years the soliton paradigm has provided a rather useful framework for in­ vestigating a large number of phenomena, particularly in low-dimensional systems. Nevertheless, we felt that the importance of nonlinearity in wider arenas than "solitonics" merited a significant expansion in the scope of the conference over that of the 1978 symposium. Indeed, many of the lessons are quite general and their potential for cross-fertilization of otherwise poorly connected disciplines was certainly one of the prime motivations for this conference. Thus, while these proceedings contain many contribu­ tions pertaining to soliton behavior in different contexts, the reader will find much more as well, particularly in the later chapters.