Abstract The previous two chapters were devoted entirely to the concept of multiple scattering, the concept of structure constants, and approximations dealing with the problem of disorder. Since the same formal language was used for both ordered and disordered systems, it is now comparatively easy to discuss spectral densities quite generally. A spectral density is defined in terms of the imaginary part of a representation of a given resolvent. The configuration space representation of the resolvent of the (one-electron) Hamiltonian (see also Chapter 4), for example, yields the (second-order) Dirac density matrix γ(r, r′, ϵ).
Abstract The previous two chapters were devoted entirely to the concept of multiple scattering, the concept of structure constants, and approximations dealing with the problem of disorder. Since the same formal language was used for both ordered and disordered systems, it is now comparatively easy to discuss spectral densities quite generally. A spectral density is defined in terms of the imaginary part of a representation of a given resolvent. The configuration space representation of the resolvent of the (one-electron) Hamiltonian (see also Chapter 4), for example, yields the (second-order) Dirac density matrix γ(r, r′, ϵ).
A detailed account of the density matrix formalism and the resolvent operator theory is given. The statistical mixture of quantum states is described in terms of the density matrix operator, a concept which in an elegant manner allows one to obtain the ensemble average of an observable [a Hermitian operator, which eigenspectrum is complete]. On the basis of an integral equation for the evolution operator, described in the Interaction Representation, the Liouville equation, describing the dynamical evolution of the density matrix operator, is established. A comprehensive analysis of thermal equilibrium in electron systems is presented. For a description of the dynamical time evolution of a system in contact with a heat bath, a reservoir in thermal equilibrium, the den- sity matrix operator approach is shown to be extremely convenient. The Kubo formalism for the time-dependent density matrix is discussed in its most general form. The so-called resolvent propagator formalism is set up starting from a discussion of the analytical part of a Dirac delta function. Having established integral equations for the retarded and advanced scalar propagator operators, the resolvent propagator concept is introduced and analyzed. The resolvent of the Hamiltonian is of particular importance in physics, and the singularities of this resolvent (poles and branch cuts) are identified. We finish by a study of correlated systems in terms of the reduced density matrix operator concept.
We state in the following a structure theory of the density matrices from the standpoint of secind quantization. The difficulties involved in the quantum-mechanical many body problem have so far proved to be so tremendous that as yet some have been obliged to stop before exact treatments and others have retired to phenomenological model theories. There are two main tools to approach the problem, the theory of second quantization and the theory of density matrix. The former, in spite of its generality and compactness, has never revealed us its adaptability to the practical problems of quantum statistical mechanics. In fact, about the exchange effect, namely the statistical correlation of particles, the latter informs us the detailed features of it by application of the group theory, while the former lacks, at present, such an exposition. For a long time a more general and simpler formulation than the current group theoretic method has been desired in the theoretical researches of spectroscopy. A distinguished contribution has been made by P. Jordan (1934) and H. Ostertag (1937) from the standpoint of second quantization. H. Ostertag made use of the theory of hypercomplex ring and introdeced the original concept of the elementary algebra. (Part II) Since the latter part of his formulation, however, seems to be complicated, it will be better, for clearness' sake, to avoid this part. Accordingly the concepts of the elementary algebra and the symmetric algebra will be introduced as a remedy for it. Then we shall complete our theory to include the whole theory of symmetric group. These concepts will serve to clarify and generalize the previous formulations and to unify the several group theoretic standpoints of the spectroscopy. In §1 and §2 the representation of the quantized density of Bose or Fermi particles will reveal itself as the density matrix of Bose or Fermi particles or as the density matrix of many electrons with a certain term of spin multiplets. Indeed, the theory of the statistical operator, or the density matrix of J. von Neumann and P.A.M. Dirac, the equilibrium properties of which have been fully examined by one of the authors (1940), is the natural formalism of dealing with quantum gases. A contribution has been made by W. Kofink (1938) to introduce the intrinsic degrees of freedom of particles. This attempt has resulted in a fair seccess. His formulation consists in making use of the initial stages of the theory of spectroscopy, which will be reexamined in the second section and Part II. In §3, as an example at zero temperature, the Hartree-Fock equation for a prescribed orbital and spin configuration will be derived, and in §4 some properties of the reduced density matrix will be illustrated.
We state in the following a structure theory of the density matrices from the standpoint of secind quantization. The difficulties involved in the quantum-mechanical many body problem have so far proved to be so tremendous that as yet some have been obliged to stop before exact treatments and others have retired to phenomenological model theories. There are two main tools to approach the problem, the theory of second quantization and the theory of density matrix. The former, in spite of its generality and compactness, has never revealed us its adaptability to the practical problems of quantum statistical mechanics. In fact, about the exchange effect, namely the statistical correlation of particles, the latter informs us the detailed features of it by application of the group theory, while the former lacks, at present, such an exposition. For a long time a more general and simpler formulation than the current group theoretic method has been desired in the theoretical researches of spectroscopy. A distinguished contribution has been made by P. Jordan (1934) and H. Ostertag (1937) from the standpoint of second quantization. H. Ostertag made use of the theory of hypercomplex ring and introdeced the original concept of the elementary algebra. (Part II) Since the latter part of his formulation, however, seems to be complicated, it will be better, for clearness' sake, to avoid this part. Accordingly the concepts of the elementary algebra and the symmetric algebra will be introduced as a remedy for it. Then we shall complete our theory to include the whole theory of symmetric group. These concepts will serve to clarify and generalize the previous formulations and to unify the several group theoretic standpoints of the spectroscopy. In §1 and §2 the representation of the quantized density of Bose or Fermi particles will reveal itself as the density matrix of Bose or Fermi particles or as the density matrix of many electrons with a certain term of spin multiplets. Indeed, the theory of the statistical operator, or the density matrix of J. von Neumann and P.A.M. Dirac, the equilibrium properties of which have been fully examined by one of the authors (1940), is the natural formalism of dealing with quantum gases. A contribution has been made by W. Kofink (1938) to introduce the intrinsic degrees of freedom of particles. This attempt has resulted in a fair seccess. His formulation consists in making use of the initial stages of the theory of spectroscopy, which will be reexamined in the second section and Part II. In §3, as an example at zero temperature, the Hartree-Fock equation for a prescribed orbital and spin configuration will be derived, and in §4 some properties of the reduced density matrix will be illustrated.
In the first part of the Thesis we mostly concentrate on spectral properties of strongly correlated systems and on their equilibrium properties. This is accomplished by the general concept of imaginary-time dynamics which we apply to a number of different problems in which different strengths of this approach emerge. In Chapter 1 we introduce the formalism that allows for a connection between the quantum and the classical worlds. The connection is established by means of the imaginary-time quantum evolution which, under certain circumstances, is shown to be equivalent to a classical stochastic process. It is further shown that exact static and spectral properties of correlated systems can be obtained when this mapping is feasible. The relationship between the imaginary-time dynamics in different frameworks such as the path-integral and the perturbative one is also underlined. In Chapter 2 we present a specific implementation of the general ideas previously presented. In particular we introduced an extension to lattice systems of the Reptation Monte Carlo algorithm [30] which benefits of a sampling scheme based on directed updates. Specific improvements over the existing methodologies consist in the unbiased evaluation of the imaginary-time path integrals for bosons and a systematic scheme to improve over the Fixed-node approximation for fermions. Applications to the Hubbard and the Heisenberg models are presented. In Chapter 3 we demonstrate the application of the imaginary-time dynamics to the exact study of spectral properties. Subject of our attention is a highly anharmonic and correlated quantum crystal such as Helium 4 at zero temperature.[33] Concerning this system, we have obtained the first ab-initio complete phonon dispersion in good agreement with neutron spectroscopy experiments. Moreover, we have also studied the density excitations of solid helium in a region of wave-vectors in between the collective (phonon) and the single-particle regimes, where the presence of residual coherence in the dynamics shows analogies between the highly anharmonic crystal and the superfluid phase. In Chapter 4 we introduce a novel method, based on the imaginary-time dynamics, to obtain unbiased estimates of fermionic properties.[34] By means of this method and of a very accurate variational state, we provide strong evidence for the stability of a saturated ferromagnetic phase in the high-density regime of the two-dimensional infinite-U Hubbard model. By decreasing the electron density, we observe a discontinuous transition to a paramagnetic phase, accompanied by a divergence of the susceptibility on the paramagnetic side. This behavior, resulting from a high degeneracy among different spin sectors, is consistent with an infinite-order phase transition scenario. In Chapter 5 the use of imaginary-time dynamics in the context of finite-temperature response functions is highlighted. As an application, we study an intriguing quantum phase featuring both glassy order and Bose-Einstein condensation. [35] We introduce and validate a model for the role of geometrical frustration in the coexistence of off-diagonal long range order with an amorphous density profile. The exact characterization of the response of the system to an external density perturbation is what allows here to establish the existence of a spin-glass phase. The differences between such a phase and the otherwise insulating "Bose glasses" are further elucidated in the Chapter. In the second part of the Thesis we focus our attention on the dynamics of closed systems out of equilibrium. This is accomplished by both non-stochastic exact methods for the dynamics and the introduction of a novel time-dependent Variational Monte Carlo scheme. In Chapter 6 exact diagonalization schemes and renormalization-based methods for one-dimensional systems are introduced. We identify key phenomenological traits resulting from the many-body correlation in closed systems driven sufficiently away from equilibrium.[31] We provide evidences that the dynamics of interacting lattice bosons away from equilibrium can be trapped into extremely long-lived inhomogeneous metastable states. The slowing down of incoherent density excitations above a threshold energy, much reminiscent of a dynamical arrest on the verge of a glass transition, is identified as the key feature of this phenomenon. In Chapter 7 we present an extension to dynamical properties of the Variational Quantum Monte Carlo method.[32] This is accomplished by introducing a general class of time-dependent variational states which is based on the mapping of the many-body dynamics onto an instantaneous ground-state problem. The application of the method to the experimentally relevant quantum quenches of interacting bosons reveals the accuracy and the reliability of the introduced numerical scheme. We indeed obtain for the first time a consistent variational description of the approach to the equilibrium of local observables and underline the origin of the metastability and glassy behavior previously identified. In the very last part we draw our conclusions and show some possible paths for stimulating future research.
Art is the elimination of the unnecessary. Pablo Picasso Summary Condensed matter is constituted by a huge number of electrons and nuclei interacting with Coulomb potentials. The topic of this chapter is a way to deal with the full, coupled problem by separating it into a part that is tractable and the rest that is approximated. This naturally leads to the appearance of dynamical fields, and to the concept of “quasi-particles” that have the same quantum numbers as non-interacting electrons. The quasi-particles obey equations where the potential and bare interactions in the hamiltonian are replaced by dynamical self-energies and screened interactions that can describe many of the effects of correlation. In this chapter we discuss the intuitive concepts behind this approach in general terms to motivate the more rigorous formulations and approximations used in the Green's function methods of the following chapter and in Part III. The first equations of this book, Eqs. (1.1) and (1.2), express the fundamental theory of matter in terms of electrons and nuclei that interact with Coulomb potentials. For systems with only a few electrons, a solution of the problem can be obtained by exact diagonalization, using configuration interaction methods. For many-electron systems one has to resort to other methods. For example, QMC stochastic simulations (Chs. 22–25) are among the most accurate methods known to calculate certain expectation values, such as equilibrium thermodynamic properties, total energies, the density, and various static correlation functions like those in Ch. 5. Other properties such as excitation spectra are more difficult to access with QMC. Straightforward perturbation theory is, in general, not appropriate since interactions among the electrons are of the same order of magnitude as the independent-particle terms. Hence, we must develop other approaches. One strategy that gives access to equilibrium thermodynamic as well as dynamical excitation properties is to separate the interacting many-body problem into a part that is tractable by some means and the rest of the problem that is dealt with more approximately. The present chapter is devoted to this idea. It is a prelude to many-body Green's function methods; the aim is to provide a unified framework for the developments in Chs. 8–21, along with a qualitative description of the most relevant quantities.