We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra some standard results concerning the spectral properties exhibited by temperature states of disordered quantum spin systems. We investigate the Arveson spectrum, known to physicists as the set of the Bohr frequencies. We also establish its connection with the Connes and Borchers spectra, and with the associated invariants for such W ∗-dynamical systems which determine the type of von Neumann algebras generated by a temperature state. We prove that all such spectra are independent of the disorder. Such results cover infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice ℤ d , including models exhibiting some standard spin-glass-like behavior. As a natural application, we show that a temperature state can generate only a type $\mathop {\rm {III}}$ von Neumann algebra (with the type $\mathop {\rm {III_{0}}}$ component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type $\mathop {\rm {III_{\lambda }}}$ for some λ∈(0,1], independent of the disorder. All such results are in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The approach pursued in the present paper can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra some standard results concerning the spectral properties exhibited by temperature states of disordered quantum spin systems. We investigate the Arveson spectrum, known to physicists as the set of the Bohr frequencies. We also establish its connection with the Connes and Borchers spectra, and with the associated invariants for such W ∗-dynamical systems which determine the type of von Neumann algebras generated by a temperature state. We prove that all such spectra are independent of the disorder. Such results cover infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice ℤ d , including models exhibiting some standard spin-glass-like behavior. As a natural application, we show that a temperature state can generate only a type $\mathop {\rm {III}}$ von Neumann algebra (with the type $\mathop {\rm {III_{0}}}$ component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type $\mathop {\rm {III_{\lambda }}}$ for some λ∈(0,1], independent of the disorder. All such results are in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The approach pursued in the present paper can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.

Aug 16, 201613 citations

FERMIONS ON THE LATTICE

World Scientific lecture notes in physicsJan 1, 19920 citations

Magnetic-superconducting phase coexistence driven by on-site repulsions in BCS-like models

Kauê Rodrigues Alves

This thesis focuses on the study of the thermodynamics of fermionic systems on lattices, from the point of view of the C*-algebraic formulation of quantum mechanics, and on the application of the so-called "catastrophe theory" to the analysis of the phase diagram of an explicit fermionic model.The first chapter is devoted to the introduction of some of the basic and most important properties of C*-algebras, and also to the study of its representations and its set of states.The second chapter is devoted to the development of some important results of catastrophe theory.The results derived in the chapter allows one to analyze the behavior of the minima of members of a family of functions around a degenerate critical point, and they will be used to study the behavior of the thermodynamic pressure for a given fermionic lattice model.The third chapter presents some of the formalism developed in [4], that concerns the existence of the thermodynamics and equilibrium states of lattice fermi systems subject to a suitable set of long-range interactions (i.e., interactions containing mean-field terms).Finally, in the last chapter, using all the formalisms and results already studied in the previous chapters, the thermodynamics of an explicit BCS-like model is analyzed.With the help of catastrophe theory, it is shown that such model exhibits a coexistence of magnetic and superconducting phases for a suitable choice of parameters, and through a perturbative analysis it is also shown that the coexistence still holds when a small kinetic term is added into the model.This thesis focuses on the study of the thermodynamics of fermionic systems on lattices, from the point of view of the C*-algebraic formulation of quantum mechanics, and on the application of the so-called "catastrophe theory" to the analysis of the phase diagram of an explicit fermionic model.The first chapter is devoted to the introduction of some of the basic and most important properties of C*-algebras, and also to the study of its representations and its set of states.The second chapter is devoted to the development of some important results of catastrophe theory.The results derived in the chapter allows one to analyze the behavior of the minima of members of a family of functions around a degenerate critical point, and they will be used to study the behavior of the thermodynamic pressure for a given fermionic lattice model.The third chapter presents some of the formalism developed in [4], that concerns the existence of the thermodynamics and equilibrium states of lattice fermi systems subject to a suitable set of long-range interactions (i.e., interactions containing mean-field terms).Finally, in the last chapter, using all the formalisms and results already studied in the previous chapters, the thermodynamics of an explicit BCS-like model is analyzed.With the help of catastrophe theory, it is shown that such model exhibits a coexistence of magnetic and superconducting phases for a suitable choice of parameters, and through a perturbative analysis it is also shown that the coexistence still holds when a small kinetic term is added into the model.

Mar 8, 20210 citations

FERMIONS ON THE LATTICE

World Scientific lecture notes in physicsJun 1, 19970 citations

Coexistência de fases magnéticas e supercondutoras em modelos do tipo BCS induzida por repulsão local

Kauê Rodrigues Alves

This thesis focuses on the study of the thermodynamics of fermionic systems on lattices, from the point of view of the C*-algebraic formulation of quantum mechanics, and on the application of the so-called \"catastrophe theory\" to the analysis of the phase diagram of an explicit fermionic model. The first chapter is devoted to the introduction of some of the basic and most important properties of C*-algebras, and also to the study of its representations and its set of states. The second chapter is devoted to the development of some important results of catastrophe theory. The results derived in the chapter allows one to analyze the behavior of the minima of members of a family of functions around a degenerate critical point, and they will be used to study the behavior of the thermodynamic pressure for a given fermionic lattice model. The third chapter presents some of the formalism developed in [4], that concerns the existence of the thermodynamics and equilibrium states of lattice fermi systems subject to a suitable set of long-range interactions (i.e., interactions containing mean-field terms). Finally, in the last chapter, using all the formalisms and results already studied in the previous chapters, the thermodynamics of an explicit BCS-like model is analyzed. With the help of catastrophe theory, it is shown that such model exhibits a coexistence of magnetic and superconducting phases for a suitable choice of parameters, and through a perturbative analysis it is also shown that the coexistence still holds when a small kinetic term is added into the model.This thesis focuses on the study of the thermodynamics of fermionic systems on lattices, from the point of view of the C*-algebraic formulation of quantum mechanics, and on the application of the so-called \"catastrophe theory\" to the analysis of the phase diagram of an explicit fermionic model. The first chapter is devoted to the introduction of some of the basic and most important properties of C*-algebras, and also to the study of its representations and its set of states. The second chapter is devoted to the development of some important results of catastrophe theory. The results derived in the chapter allows one to analyze the behavior of the minima of members of a family of functions around a degenerate critical point, and they will be used to study the behavior of the thermodynamic pressure for a given fermionic lattice model. The third chapter presents some of the formalism developed in [4], that concerns the existence of the thermodynamics and equilibrium states of lattice fermi systems subject to a suitable set of long-range interactions (i.e., interactions containing mean-field terms). Finally, in the last chapter, using all the formalisms and results already studied in the previous chapters, the thermodynamics of an explicit BCS-like model is analyzed. With the help of catastrophe theory, it is shown that such model exhibits a coexistence of magnetic and superconducting phases for a suitable choice of parameters, and through a perturbative analysis it is also shown that the coexistence still holds when a small kinetic term is added into the model.

Mar 8, 20210 citations

Fermions on the lattice

Jan Smit A

Cambridge University Press eBooksSep 19, 20020 citations

\mathop {\rm {III}}

von Neumann algebra (with the type

\mathop {\rm {III_{0}}}

component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type

\mathop {\rm {III_{\lambda }}}

for some λ∈(0,1], independent of the disorder. All such results are in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The approach pursued in the present paper can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.