田﨑 晴明

We prove two theorems concerning the time evolution in general isolated quantum systems. The theorems are relevant to the issue of the time scale in the approach to equilibrium. The first theorem shows that there can be pathological situations in which the relaxation takes an extraordinarily long time, while the second theorem shows that one can always choose an equilibrium subspace, the relaxation to which requires only a short time for any initial state.

In a system of interacting f = 1 bosons (in the subspace where the total spin in the z direction is vanishing), we prove inequalities for the ground state expectation value of the density of spin-0 bosons. The inequalities imply that the ground state possesses "polar" or "antiferromagnetic" order when the quadratic Zeeman term q is large enough. In the low density limit, the inequalities establish the existence of a sharp transition at q = 0 when q is varied.

We prove basic theorems about the ground states of the S = 1 Bose-Hubbard model. The results are quite universal and depend only on the coefficient U-2 of the spin-dependent interaction. We show that the ground state exhibits saturated ferromagnetism if U-2 < 0, is spin-singlet if U-2 > 0, and exhibits "SU(3)-ferromagnetism" if U-2 = 0, and completely determine the degeneracy in each region. DOI: 10.1103/PhysRevLett.110.130405

Recently, in their attempt to construct steady state thermodynamics (SST), Komatsu, Nakagawa, Sasa, and Tasaki found an extension of the Clausius relation to nonequilibrium steady states in classical stochastic processes. Here we derive a quantum mechanical version of the extended Clausius relation. We consider a small system of interest attached to large systems which play the role of heat baths. By only using the genuine quantum dynamics, we realize a heat conducting nonequilibrium steady state in the small system. We study the response of the steady state when the parameters of the system are changed abruptly, and show that the extended Clausius relation, in which "heat" is replaced by the "excess heat", is valid when the temperature difference is small. Moreover we show that the entropy that appears in the relation is similar to von Neumann entropy but has an extra symmetrization with respect to time-reversal. We believe that the present work opens a new possibility in the study of nonequilibrium phenomena in quantum systems, and also confirms the robustness of the approach by Komatsu et al.

We consider a (small) quantum mechanical system which is operated by an external agent, who changes the Hamiltonian of the system according to a fixed scenario. In particular we assume that the agent (who may be called a demon) performs measurement followed by feedback, i.e., it makes a measurement of the system and changes the protocol according to the outcome. We extend to this setting the generalized Jarzynski relations, recently derived by Sagawa and Ueda for classical systems with feedback. One of the two relations by Sagawa and Ueda is derived here in error-free quantum processes, while the other is derived only when the measurement process involves classical errors. The first relation leads to a second law which takes into account the efficiency of the feedback.

Among various possible routes to extend entropy and thermodynamics to nonequilibrium steady states (NESS), we take the one which is guided by operational thermodynamics and the Clausius relation. In our previous study, we derived the extended Clausius relation for NESS, where the heat in the original relation is replaced by its "renormalized" counterpart called the excess heat, and the Gibbs-Shannon expression for the entropy by a new symmetrized Gibbs-Shannon-like expression. Here we concentrate on Markov processes describing heat conducting systems, and develop a new method for deriving thermodynamic relations. We first present a new simpler derivation of the extended Clausius relation, and clarify its close relation with the linear response theory. We then derive a new improved extended Clausius relation with a "nonlinear nonequilibrium" contribution which is written as a correlation between work and heat. We argue that the "nonlinear nonequilibrium" contribution is unavoidable, and is determined uniquely once we accept the (very natural) definition of the excess heat. Moreover it turns out that to operationally determine the difference in the nonequilibrium entropy to the second order in the temperature difference, one may only use the previous Clausius relation without a nonlinear term or must use the new relation, depending on the operation (i.e., the path in the parameter space). This peculiar "twist" may be a clue to a better understanding of thermodynamics and statistical mechanics of NESS.

Among various possible routes to extend entropy and thermodynamics to nonequilibrium steady states (NESS), we take the one which is guided by operational thermodynamics and the Clausius relation. In our previous study, we derived the extended Clausius relation for NESS, where the heat in the original relation is replaced by its "renormalized" counterpart called the excess heat, and the Gibbs-Shannon expression for the entropy by a new symmetrized Gibbs-Shannon-like expression. Here we concentrate on Markov processes describing heat conducting systems, and develop a new method for deriving thermodynamic relations. We first present a new simpler derivation of the extended Clausius relation, and clarify its close relation with the linear response theory. We then derive a new improved extended Clausius relation with a "nonlinear nonequilibrium" contribution which is written as a correlation between work and heat. We argue that the "nonlinear nonequilibrium" contribution is unavoidable, and is determined uniquely once we accept the (very natural) definition of the excess heat. Moreover it turns out that to operationally determine the difference in the nonequilibrium entropy to the second order in the temperature difference, one may only use the previous Clausius relation without a nonlinear term or must use the new relation, depending on the operation (i.e., the path in the parameter space). This peculiar "twist" may be a clue to a better understanding of thermodynamics and statistical mechanics of NESS.

We introduce and study two classes of Hubbard models with magnetic flux or with spin-orbit coupling, which have a flat lowest band separated from other bands by a non-zero gap. We study the Chern number of the flat bands, and find that it is zero for the first class but can be non-trivial in the second. We also prove that the introduction of on-site Coulomb repulsion leads to ferromagnetism in both the classes. Copyright (C) EPLA, 2010

We introduce and study two classes of Hubbard models with magnetic flux or with spin-orbit coupling, which have a flat lowest band separated from other bands by a non-zero gap. We study the Chern number of the flat bands, and find that it is zero for the first class but can be non-trivial in the second. We also prove that the introduction of on-site Coulomb repulsion leads to ferromagnetism in both the classes. Copyright (C) EPLA, 2010

Recently a novel concise representation of the probability distribution of heat conducting nonequilibrium steady states was derived. The representation is valid to the second order in the "degree of nonequilibrium", and has a very suggestive form where the effective Hamiltonian is determined by the excess entropy production. Here we extend the representation to a wide class of nonequilibrium steady states realized in classical mechanical systems where baths (reservoirs) are also defined in terms of deterministic mechanics. The present extension covers such nonequilibrium steady states with a heat conduction, with particle flow (maintained either by external field or by particle reservoirs), and under an oscillating external field. We also simplify the derivation and discuss the corresponding representation to the full order.

Recently a novel concise representation of the probability distribution of heat conducting nonequilibrium steady states was derived. The representation is valid to the second order in the "degree of nonequilibrium", and has a very suggestive form where the effective Hamiltonian is determined by the excess entropy production. Here we extend the representation to a wide class of nonequilibrium steady states realized in classical mechanical systems where baths (reservoirs) are also defined in terms of deterministic mechanics. The present extension covers such nonequilibrium steady states with a heat conduction, with particle flow (maintained either by external field or by particle reservoirs), and under an oscillating external field. We also simplify the derivation and discuss the corresponding representation to the full order.

It is believed that strong ferromagnetic orders in some solids are generated by subtle interplay between quantum many-body effects and spin-independent Coulomb interactions between electrons. Here we describe our rigorous and constructive approach to ferromagnetism in the Hubbard model, which is a standard idealized model for strongly interacting electrons in a solid.

It is believed that strong ferromagnetic orders in some solids are generated by subtle interplay between quantum many-body effects and spin-independent Coulomb interactions between electrons. Here we describe our rigorous and constructive approach to ferromagnetism in the Hubbard model, which is a standard idealized model for strongly interacting electrons in a solid. © 2008 Springer.

Starting from microscopic mechanics, we derive thermodynamic relations for heat conducting nonequilibrium steady states. The extended Clausius relation enables one to experimentally determine nonequilibrium entropy to the second order in the heat current. The associated Shannon-like microscopic expression of the entropy is suggestive. When the heat current is fixed, the extended Gibbs relation provides a unified treatment of thermodynamic forces in the linear nonequilibrium regime.

Starting from microscopic mechanics, we derive thermodynamic relations for heat conducting nonequilibrium steady states. The extended Clausius relation enables one to experimentally determine nonequilibrium entropy to the second order in the heat current. The associated Shannon-like microscopic expression of the entropy is suggestive. When the heat current is fixed, the extended Gibbs relation provides a unified treatment of thermodynamic forces in the linear nonequilibrium regime.

We present the first rigorous example of the Hubbard model in any dimension which exhibits metallic ferromagnetism. The model is a genuine Hubbard model with short-range hopping and on-site Coulomb repulsion, and has many single-electron bands. In the limit where the band gap and the Coulomb repulsion become infinite, we prove that the ground states are completely ferromagnetic and at the same time conducting. © 2007 The American Physical Society.

Starting from and only using classical Hamiltonian dynamics, we prove the maximum work principle in a system where macroscopic dynamical degrees of freedom are intrinsically coupled to microscopic degrees of freedom. Unlike in many of the standard and recent works on the second law, the macroscopic dynamics is not governed by an external action but undergoes the back reaction of the microscopic degrees of freedom. Our theorems cover such physical situations as impact between macroscopic bodies, thermodynamic machines, and molecular motors. Our work identifies and quantifies the physical limitations on the applicability of the second law for small systems.

We present the first rigorous example of the Hubbard model in any dimension which exhibits metallic ferromagnetism. The model is a genuine Hubbard model with short-range hopping and on-site Coulomb repulsion, and has many single-electron bands. In the limit where the band gap and the Coulomb repulsion become infinite, we prove that the ground states are completely ferromagnetic and at the same time conducting.

Starting from and only using classical Hamiltonian dynamics, we prove the maximum work principle in a system where macroscopic dynamical degrees of freedom are intrinsically coupled to microscopic degrees of freedom. Unlike in many of the standard and recent works on the second law, the macroscopic dynamics is not governed by an external action but undergoes the back reaction of the microscopic degrees of freedom. Our theorems cover such physical situations as impact between macroscopic bodies, thermodynamic machines, and molecular motors. Our work identifies and quantifies the physical limitations on the applicability of the second law for small systems.

The present paper reports our attempt to search for a new universal framework in nonequilibrium physics. We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi's original proposal. The construction of SST is based on a careful examination of how the basic notions in thermodynamics should be modified in nonequilibrium steady states. We define all thermodynamic quantities through operational procedures which can be (in principle) realized experimentally. Based on SST thus constructed, we make some nontrivial predictions, including an extension of Einstein's formula on density fluctuation, an extension of the minimum work principle, the existence of a new osmotic pressure of a purely nonequilibrium origin, and a shift of coexistence temperature. All these predictions may be checked experimentally to test SST for its quantitative validity.

The present paper reports our attempt to search for a new universal framework in nonequilibrium physics. We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi's original proposal. The construction of SST is based on a careful examination of how the basic notions in thermodynamics should be modified in nonequilibrium steady states. We define all thermodynamic quantities through operational procedures which can be (in principle) realized experimentally. Based on SST thus constructed, we make some nontrivial predictions, including an extension of Einstein's formula on density fluctuation, an extension of the minimum work principle, the existence of a new osmotic pressure of a purely nonequilibrium origin, and a shift of coexistence temperature. All these predictions may be checked experimentally to test SST for its quantitative validity. © Springer Science+Business Media, LLC 2006.

We study a classical mechanical problem in which a macroscopic ball is reflected by a non-deformable wall. The ball is modeled as a collection of classical particles bound together by an arbitrary potential, and its internal degrees of freedom are initially set to be in thermal equilibrium. The wall is represented by an arbitrary potential which is translation invariant in two directions. We then prove that the final normal momentum can exceed the initial normal momentum at most by O(√mkT), where m is the total mass of the ball, k the Boltzmann constant, and T the temperature. This implies the well-known statement in the title in the macroscopic limit where O(√mkT) is negligible. Our result may be interpreted as a rigorous demonstration of the second law of thermodynamics in a system where a macroscopic dynamics and microscopic degrees of freedom are intrinsically coupled with each other. © 2006 Springer Science+Business Media, Inc.

We study a classical mechanical problem in which a macroscopic ball is reflected by a non-deformable wall. The ball is modeled as a collection of classical particles bound together by an arbitrary potential, and its internal degrees of freedom are initially set to be in thermal equilibrium. The wall is represented by an arbitrary potential which is translation invariant in two directions. We then prove that the final normal momentum can exceed the initial normal momentum at most by O(root T), where m is the total mass of the ball, k the Boltzmann constant, and T the temperature. This implies the well-known statement in the title in the macroscopic limit where O(root T) is negligible. Our result may be interpreted as a rigorous demonstration of the second law of thermodynamics in a system where a macroscopic dynamics and microscopic degrees of freedom are intrinsically coupled with each other.

We develop a controlled high-temperature expansion for nonequilibrium steady states of the driven lattice gas, the "Ising model" for nonequilibrium physics. We represent the steady state as P(eta)proportional to e(-beta H(eta)-Psi(eta)) and evaluate the lowest order contribution to the nonequilibrium effective interaction Psi(eta). We see that, in dimensions d >= 2, all models with nonsingular transition rates yield the same summable Psi(eta), suggesting the possibility of describing the state as a Gibbs state similar to equilibrium. The models with the Metropolis rule show exceptional behavior.

We develop a controlled high-temperature expansion for nonequilibrium steady states of the driven lattice gas, the "Ising model" for nonequilibrium physics. We represent the steady state as P(eta)proportional to e(-beta H(eta)-Psi(eta)) and evaluate the lowest order contribution to the nonequilibrium effective interaction Psi(eta). We see that, in dimensions d >= 2, all models with nonsingular transition rates yield the same summable Psi(eta), suggesting the possibility of describing the state as a Gibbs state similar to equilibrium. The models with the Metropolis rule show exceptional behavior.

It is believed that strong ferromagnetic orders in some solids are generated by subtle interplay between quantum many-body effects and spin-independent Coulomb interactions between electrons. Here we describe our rigorous and constructive approach to ferromagnetism in the Hubbard model, which is a standard idealized model for strongly interacting electrons in a solid. We introduce a class of Hubbard models in any dimensions which are nonsingular in the sense that both the Coulomb interaction and the density of states (at the Fermi level) are finite. We then prove that the ground states of the models exhibit saturated ferromagnetism, i.e., have maximum total spins. Combined with our earlier results, the present work provides nonsingular models of itinerant electrons with only spin-independent interactions where low energy behaviors are proved to be that of a ''healthy'' ferromagnetic insulator.

The Hubbard model is a 'highly oversimplified model' for electrons in a solid which interact with each other through extremely short-ranged repulsive (CouIomb) interaction. The Hamiltonian of the Hubbard model consists of two parts: H-hop which describes quantum mechanical hopping of electrons, and H-int which describes non-linear repulsive interaction. Either H-hop or H-int alone is easy to analyse, and does not favour any specific order. But their sam H = H-hop + H-int is believed to exhibit various non-trivial phenomena including metal-insulator transition, antiferromagnetism, ferrimagnetism, ferromagnetism, Tomonaga-Luttinger liquid, and superconductivity. It is believed that we can find various interesting 'universality classes' of strongly interacting electron systems by studying the idealized Hubbard model. In the present article we review some mathematically rigorous results relating to the Hubbard model which shed light on the 'physics' of this fascinating model. We mainly concentrate on the magnetic properties of the model in its ground states. We discuss the Lieb-Mattis theorem on the absence of ferromagnetism in one dimension, Koma-Tasaki bounds on the decay of correlations at finite temperatures in two dimensions, the Yamanaka-Oshikawa-Affleck theorem on lour-tying excitations in one dimension, Lieb's important theorem for the half-filled model on a bipartite lattice, Kubo-Kishi bounds on the charge and superconducting susceptibilities of half-filled models at finite temperatures, and three rigorous examples of saturated ferromagnetism due to Nagaoka, Mielke, and Tasaki. We have tried to make the article accessible to non-experts by giving basic definitions and describing elementary materials in detail.

It is believed that strong ferromagnetic interactions in some solids are generated by subtle interplay between quantum many-body effects and spin-independent Coulomb interactions between electrons. It is a challenging problem to verify this scenario in the Hubbard model, which is an idealized model for strongly interacting electrons in a solid. Nagaoka's ferromagnetism is a well-known rigorous example of ferromagnetism in the Hubbard model. It deals with the limiting situation in which there is one fewer electron than in the half-filling and the on-site Coulomb interaction is infinitely large. There are relatively new rigorous examples of ferromagnetism in Hubbard models called flat-band ferromagnetism. Flat-band ferromagnetism takes place in carefully prepared models in which the lowest bands tin the single-electron spectral are "flat." Usually, these two approaches are regarded as two complimentary routes to ferromagnetism in the Hubbard model. In the present paper we describe Nagaoka's ferromagnetism and flat-band ferromagnetism in detail, giving all the necessary background as well as complete (but elementary) mathematical proofs. By studying an intermediate model called the long-range hopping model, we also demonstrate that there is indeed a deep relation between these two seemingly different approaches to ferromagnetism. We further discuss some attempts to go beyond these approaches. We briefly discuss recent rigorous example of ferromagnetism in the Hubbard model which has neither infinitely large parameters nor completely flat bands. We give preliminary discussion regarding possible experimental realizations of the (nearly-)flat-band ferromagnetism. Finally, we focus on some theoretical attempts to understand metallic ferromagnetism. We discuss three artificial one-dimensional models in which the existence of metallic ferromagnetism can be easily proved. We have tried to make the present paper as self-contained as possible, keeping in mind readers who are new to the field. Although the present paper is written as a review, it contains some material which appears for the first time.

Derivation of the canonical (or Boltzmann) distribution based only on quantum dynamics is discussed. Consider a closed system which consists of a mutually interacting subsystem and a heat bath, and assume that the whole system is initially in a pure state (which can be far from equilibrium) with small energy fluctuation. Under the “hypothesis of equal weights for eigenstates,” we derive the canonical distribution in the sense that, at sufficiently large and typical time, the (instantaneous) quantum mechanical expectation value of an arbitrary operator of the subsystem is almost equal to the desired canonical expectation value. We present a class of examples in which the above derivation can be rigorously established without any unproven hypotheses. © 1998 The American Physical Society.

Derivation of the canonical (or Boltzmann) distribution based only on quantum dynamics is discussed. Consider a closed system which consists of a mutually interacting subsystem and a heat bath, and assume that the whale system is initially in a pure state (which can be far from equilibrium) with small energy fluctuation. Under the "hypothesis of equal weights for eigenstates," we derive the canonical distribution in the sense that, at sufficiently large and typical time, the (instantaneous) quantum mechanical expectation value of an arbitrary operator of the subsystem is almost equal to the desired canonical expectation value. We present a class of examples in which the above derivation can be rigorously established without any unproven hypotheses.

Whether spin-independent Coulomb interaction in an electron system can be the origin of ferromagnetism has been an open problem for a long time. Recently, a ''constructive'' approach to this problem has been developed, and the existence of ferromagnetism in the ground states of certain Hubbard models was established rigorously. A special feature of these Hubbard models is that their lowest bands (in the corresponding single-electron problems) are completely flat. Here we study models obtained by adding small but arbitrary translation-invariant perturbation to the hopping Hamiltonian of these Bat-band models. The resulting models have nearly flat lowest bands. We prove that the ferromagnetic state is stable against a single-spin flip provided that Coulomb interaction U is sufficiently large. (It is easily found that the same state is unstable against a single-spin flip if U is small enough.) We also prove upper and lower bounds for the dispersion relation of the lowest energy eigenstate with a single flipped spin, which bounds establish that the model has ''healthy'' spin-wave excitation. It is notable that the (local) stability of ferromagnetism is proved in nonsingular Hubbard models, in which we must overcome competi tion between the kinetic energy and the Coulomb interaction. We also note that this is one of the very few rigorous and robust results which deal with truly nonperturbative phenomena in many-electron systems. The local stability strongly suggests that the Hubbard models with nearly fat bands have ferromagnetic ground states. We believe that the present models can be studied as paradigm models for (insulating) ferromagnetism in itinerant electron systems.

Whether spin-independent Coulomb interaction in an electron system can be the origin of ferromagnetism has been an open problem for a long time. Recently, a "constructive" approach to this problem has been developed, and the existence of ferromagnetism in the ground states of certain Hubbard models was established rigorously. A special feature of these Hubbard models is that their lowest bands (in the corresponding single-electron problems) are completely flat. Here we study models obtained by adding small but arbitrary translation-invariant perturbation to the hopping Hamiltonian of these flat-band models. The resulting models have nearly flat lowest bands. We prove that the ferromagnetic state is stable against a single-spin flip provided that Coulomb interaction U is sufficiently large. (It is easily found that the same state is unstable against a single-spin flip if U is small enough.) We also prove upper and lower bounds for the dispersion relation of the lowest energy eigenstate with a single flipped spin, which bounds establish that the model has "healthy" spin-wave excitation. It is notable that the (local) stability of ferromagnetism is proved in nonsingular Hubbard models, in which we must overcome competition between the kinetic energy and the Coulomb interaction. We also note that this is one of the very few rigorous and robust results which deal with truly non-perturbative phenomena in many-electron systems. The local stability strongly suggests that the Hubbard models with nearly flat bands have ferromagnetic ground states. We believe that the present models can be studied as paradigm models for (insulating) ferromagnetism in itinerant electron systems.

We present the first rigorous examples of nonsingular Hubbard models which exhibit ferromagnetism at zero temperature. The models are defined in arbitrary dimensions, and are characterized by finite-range hoppings, dispersive bands, and finite on-site Coulomb interaction U. The picture, which goes back to Heisenberg, that sufficiently large Coulomb interaction can revert Pauli paramagnetism into ferromagnetism has been confirmed in concrete examples.

Recently certain Hubbard models with flat lowest bands were proved to exhibit ferromagnetism. Here we study perturbed models with nearly flat bands. We prove that the ferromagnetic state is stable against a single-spin flip for sufficiently large Coulomb interaction U, but is unstable for small U > 0. This is the first time that the (local) stability of ferromagnetism is proved in nonsingular Hubbard models, in which we must overcome competition between the kinetic energy and the Coulomb interaction,

We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite-volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. Typical examples are the Heisenberg antiferromagnet with a Neel order and the Hubbard model with a (superconducting) off-diagonal long-range order. In the corresponding finite system, the symmetry breaking is usually ''obscured'' by ''quantum fluctuation'' and one gets a symmetric ground state with a long-range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose excitation energy is not more than of order N-1, where N denotes the number of sites in the lattice. Here we study the situation where the broken symmetry is a continuous one. For a particular set of states (which are orthogonal to the ground state and with each other), we prove bounds for their energy expectation values. The bounds establish that there exist ever-increasing numbers of low-lying eigenstates whose excitation energies are bounded by a constant times N-1. A crucial feature of the particular low-lying states we consider is that they can be regarded as finite-volume counterparts of the infinite-volume ground states. By forming linear combinations of these low-lying states and the (finite-volume) ground state and by taking infinite-volume limits, we construct infinite-volume ground states with explicit symmetry breaking. We conjecture that these infinite-volume ground states are ergodic, i.e., physically natural. Our general theorems not only shed light on the nature of symmetry breaking in quantum many-body systems, but also provide indispensable information for numerical approaches to these systems. We also discuss applications of our general results to a variety of interesting examples. The present paper is intended to be accessible to readers without background in mathematical approaches to quantum many-body systems.

We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite-volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. Typical examples are the Heisenberg antiferromagnet with a Néel order and the Hubbard model with a (superconducting) off-diagonal long-range order. In the corresponding finite system, the symmetry breaking is usually "obscured" by "quantum fluctuation" and one gets a symmetric ground state with a long-range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose excitation energy is not more than of order N-1, where N denotes the number of sites in the lattice. Here we study the situation where the broken symmetry is a continuous one. For a particular set of states (which are orthogonal to the ground state and with each other), we prove bounds for their energy expectation values. The bounds establish that there exist ever-increasing numbers of low-lying eigenstates whose excitation energies are bounded by a constant times N-1. A crucial feature of the particular low-lying states we consider is that they can be regarded as finite-volume counterparts of the infinite-volume ground states. By forming linear combinations of these low-lying states and the (finite-volume) ground state and by taking infinite-volume limits, we construct infinite-volume ground states with explicit symmetry breaking. We conjecture that these infinite-volume ground states are ergodic, i.e., physically natural. Our general theorems not only shed light on the nature of symmetry breaking in quantum many-body systems, but also provide indispensable information for numerical approaches to these systems. We also discuss applications of our general results to a variety of interesting examples. The present paper is intended to be accessible to readers without background in mathematical approaches to quantum many-body systems. © 1994 Plenum Publishing Corporation.

Recently certain Hubbard models with flat lowest bands were proved to exhibit ferromagnetism. Here we study perturbed models with nearly flat bands. We prove that the ferromagnetic state is stable against a single-spin flip for sufficiently large Coulomb interaction U, but is unstable for small U&gt 0. This is the first time that the (local) stability of ferromagnetism is proved in nonsingular Hubbard models, in which we must overcome competition between the kinetic energy and the Coulomb interaction. © 1994 The American Physical Society.

Whether spin-independent Coulomb interaction can be the origin of a realistic ferromagnetism in an itinerant electron system has been an open problem for a long time. Here we study a class of Hubbard models on decorated lattices, which have a special property that the corresponding single-electron Schrodinger equation has N(d)-fold degenerate ground states. The degeneracy N(d) is proportional to the total number of sites absolute value of LAMBDA. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to rho0 = N(d)/(2 Absolute value of LAMBDA), and paramagnetism when the filling factor is sufficiently small. An important feature of the present work is that it provides examples of three dimensional itinerant electron systems which are proved to exhibit ferromagnetism in a finite range of the electron filling factor.

Whether spin-independent Coulomb interaction can be the origin of a realistic ferromagnetism in an itinerant electron system has been an open problem for a long time. Here we study a class of Hubbard models on decorated lattices, which have a special property that the corresponding single-electron Schrodinger equation has N(d)-fold degenerate ground states. The degeneracy N(d) is proportional to the total number of sites absolute value of LAMBDA. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to rho0 = N(d)/(2 Absolute value of LAMBDA), and paramagnetism when the filling factor is sufficiently small. An important feature of the present work is that it provides examples of three dimensional itinerant electron systems which are proved to exhibit ferromagnetism in a finite range of the electron filling factor.

We prove the bound m(s)(beta) greater-than-or-equal-to square-root 3 sigma(beta) between the spontaneous staggered magnetization m(s)(beta) and the long range order parameter sigma(beta) in the quantum Heisenberg antiferromagnets. The same bound has been known for the Heisenberg ferromagnets from the works of Griffiths, and of Dyson, Lieb, and Simon, but extensions to the antiferromagnets had been lacking for more than two decades. When combined with the results on nonvanishing sigma(beta) from the Dyson-Lieb-Simon method, our bound proves the existence of symmetry breaking in the physically natural equilibrium states and ground states obtained by applying an infinitesimal symmetry breaking field.