This textbook provides ample opportunities for practice and real experimental demonstrations. Conceptual understanding and mastering key techniques are enhanced by rigorous derivations, numerous worked examples, more than 300 exercises, about 150 problems and 16 computer codes. The preface summarizes all of the key concepts and formulas, along with a detailed schedule for teaching. The first three chapters introduce the quantum idea, wave-particle duality, operators and measurement. The Noether theorem is invoked to introduce the Schrödinger equation, followed by applications to infinite and finite quantum wells, quantum tunneling, harmonic oscillators, Heisenberg equation of motion, uncertainty principle, blackbody radiation and photoelectric effect. Chapters 4 and 5 are on angular momentum, the hydrogen atom and time-independent approximate methods. Chapters 6 and 7 are on spin and time-dependent perturbation theory. Chapters 8, 9 and 10 are on molecular orbitals, energy bands, quantum transport, scanning tunneling microscopy, lattice vibrations, Berry phase and quantum computing. The book is intended for a one-semester or one-year course and is also appropriate for researchers in related fields.
The thesis comprises two major themes of quantum statistical dynamics. One is the development of quantum dissipation theory (QDT). It covers the establishment of some basic relations of quantum statistical dynamics, the construction of several nonequivalent complete second-order formulations, and the development of exact QDT. Another is related to the applications of quantum statistical dynamics to a variety of research fields. In particular, unconventional but novel theories of the electron transfer in Debye solvents, quantum transport, and quantum measurement are developed on the basis of QDT formulations. The thesis is organized as follows. In Chapter 1, we present some background knowledge in relation to the afore-mentioned two themes of this thesis. The key quantity in QDT is the reduced density operator ρ(t) ≡ trBρT(t); i.e., the partial trace of the total system and bath composite ρT(t) over the bath degrees of freedom. QDT governs the evolu-tion of reduced density operator, where the effects of bath are treated in a quantum statistical manner. In principle, the reduced density operator contains all dynam-ics information of interest. However, the conventional quantum transport theory is formulated in terms of nonequilibrium Green's function. The newly emerging field of quantum measurement in relation to quantum information and quantum computing does exploit a sort of QDT formalism. Besides the background of the relevant theoretical development, some representative experiments on molecular nanojunctions are also briefly discussed. In Chapter 2, we outline some basic (including new) relations that highlight several important issues on QDT. The content includes the background of nonequi-librium quantum statistical mechanics, the general description of the total compos-ite Hamiltonian with stochastic system-bath interaction, a novel parameterization scheme for bath correlation functions, a newly developed exact theory of driven Brownian oscillator (DBO) systems, and its closely related solvation mode trans-formation of system-bath coupling Hamiltonian in general. The exact QDT of DBO systems is also used to clarify the validity of conventional QDT formulations that involve Markovian approximation. In Chapter 3, we develop three nonequivalent but all complete second-order QDT (CS-QDT) formulations. Two of them are of the conventional prescriptions in terms of time-local dissipation and memory kernel, respectively. The third one is called the correlated driving-dissipation equations of motion (CODDE). This novel CS-QDT combines the merits of the former two for its advantages in both the application and numerical implementation aspects. Also highlighted is the importance of correlated driving-dissipation effects on the dynamics of the reduced system. In Chapter 4, we construct an exact QDT formalism via the calculus on path integrals. The new theory aims at the efficient evaluation of non-Markovian dis-sipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. By adopting exponential-like expansions for bath corre-lation function, hierarchical equations of motion formalism and continued fraction Liouville-space Green's function formalism are established. The latter will soon be used together with the Dyson equation technique for an efficient evaluation of non-perturbative reduced density matrix dynamics. The interplay between system-bath interaction strength, non-Markovian property, and the required level of hierarchy is also studied with the aid of simple spin-boson systems, together with the three proposed schemes to truncate the infinite hierarchy. In Chapter 5, we develop a nonperturbative theory of electron transfer (ET) in Debye solvents. The resulting exact and analytical rate expression is constructed on the basis of the aforementioned continued fraction Liouville-space Green's func-tion formalism, together with the Dyson equation technique. Not only does it re-cover the celebrated Marcus' inversion and Kramers' turnover behaviors, the new theory also shows some distinct quantum solvation effects that can alter the ET mechanism. Moreover, the present theory predicts further for the ET reaction ther-modynamics, such as equilibrium Gibbs free-energy and entropy, some interesting solvent-dependent features that are calling for experimental verification. In Chapter 6, we discuss the constructed QDTs, in terms of their unified math-ematical structure that supports a linear dynamics space, and thus facilitates their applications to various physical problems. The involving details are exemplified with the CODDE form of QDT. As the linear space is concerned, we identify the Schrödinger versus Heisenberg picture and the forward versus backward propaga-tion of the reduced, dissipative Liouville dynamics. For applications we discuss the reduced linear response theory and the optimal control problems, in which the correlated effects of non-Markovian dissipation and field driving are shown to be important. In Chapter 7, we turn to quantum transport, i.e. electric current through molecular or mesoscopic systems under finite applied voltage. By viewing the nonequilibrium transport setup as a quantum open system, we develop a reduced-density-matrix approach to quantum transport. The resulting current is explicitly expressed in terms of the molecular reduced density matrix by tracing out the degrees of freedom of the electrodes at finite bias and temperature. We propose a conditional quantum master equation theory, which is an extension of the conven-tional (or unconditional) QDT by tracing out the well-defined bath subsets indi-vidually, instead of the entire bath degrees of freedom. Both the current and the noise spectrum can be conveniently analyzed in terms of the conditional reduced density matrix dynamics. By far, the QDT (including the conditional one) has only been exploited in a second-order form. A self-consistent Born approximation for the system-electrode coupling is further proposed to recover all existing nonlinear current-voltage behaviors including the nonequilibrium Kondo effect. Transport theory based on the exact QDT formalism will be developed in future. In Chapter 8, we study the quantum measurement of a qubit with a quantum-point-contact detector. On the basis of a unified quantum master equation (a form of QDT), we study the measurement-induced relaxation and dephasing of the qubit . Our treatment pays particular attention on the detailed-balance relation, which is a consequence of properly accounting for the energy exchange between the qubit and detector during the measurement process. We also derive a conditional quantum master equation for quantum measurement in general, and study the readout characteristics of the qubit measurement. Our theory is applicable to the quantum measurement at arbitrary voltage and temperature. A number of remarkable new features are found and highlighted in concern with their possible relevance to future experiments. In Chapter 9, we discuss the further development of QDT, aiming at an effi-cient evaluation of many-electron systems. This will be carried out by reducing the many-particle (Fermion or Boson) QDT to a single-particle one by exploring, e.g. the Wick's contraction theorem. It also results in a time-dependent density func-tional theory (TDDFT) for transport through complex large-scale (e.g. molecules) systems. Primary results of the TDDFT-QDT are reported. In Chapter 10, we summary the thesis, and comment and remark on the future work on both the theoretical and application aspects of QDT.
Since its very beginning, quantum mechanics has been developed to deal with systems on the atomic or sub-atomic scale. For many decades, there has been no reason to think about its application to macroscopic systems. Actually, macroscopic objects have even been used to show how bizarre quantum effects would appear if quantum mechanics were applied beyond its original realm. This is, for example, the essence of the so-called Schrödinger cat paradox (Schrödinger, 1935). How-ever, due to recent advances in the design of systems on the meso- and nanoscopic scales, as well as in cryogenic techniques, this situation has changed drastically. It is now quite natural to ask whether a specific quantum effect, collectively involving a macroscopic number of particles, could occur in these systems. In this book it is our intention to address the quantum mechanical effects that take place in properly chosen or built “macroscopic” systems. Starting from a very naïve point of view, we could always ask what happens to systems whose classical dynamics can be described by equations of motion equivalent to those of particles (or fields) in a given potential (or potential energy density). These can be represented by a generalized “coordinate” φ( r , t ) which could either describe a field variable or a “point particle” if it is not position dependent, φ( r , t ) = φ( t ).
This PhD thesis is concerned with the investigation of Extended Hubbard Models on one-dimensional lattices (chains); the approach used to such investigation is that of determining exact results. The extended Hubbard models are models of correlated electrons on lattices; they are the extensions of a model proposed by Hubbard in \\cite{HUB} to explain the metal-insulator transition in
d-transition compounds as the effect of electronic correlations, usually neglected in band theories. Later on, a need for extensions of such model was first felt within the context of high-temperature superconductivity; indeed, the different behaviour of doped cuprates with respect to conventional superconductors suggested that a microscopic theory of this phenomenon could be formulated in terms of electron-electron interaction, without explicitly invoking the phonon coupling, like in BCS theory. Although an exhaustive microscopic formulation for cuprate superconductivity has not been achieved yet, the Extended Hubbard Models have been in the meanwhile applied to other problems in condensed matter; they are actually considered the reference paradigm for treating all those systems in which electronic correlations cannot be neglected. \\\\As observed above, this thesis focuses in particular on 1-D chains: indeed in last years the progress in technological synthesization and small-scale observations has shown that there are many classes of materials exhibiting a strong one-dimensional character, such as organic (super)conductors, linear-chain polymers, quantum wires, quantum-dot chains, one-dimensional cuprates. Many experimental results confirm that such materials exhibit a quite rich behaviour, with different phases, charge-spin separation, as well as a strong electronic correlations. However, a sufficient understanding of these phenomena on a theoretical point of view has not been reached yet. This is due to the fact the usual theoretical methods can be shown to fail in D=1, and theoreticians are thus forced to develop new schemes for approaching the problem of many interacting fermions in such dimension. The latter therefore represents a fascinating challenge. Some progress has been made and many aspects are nowadays more clear; nevertheless, due to the lack of traditional techniques, the need for a comparison with exact results is perhaps more urgent in D=1 than in higher dimensions. The aim of this thesis is to provide some results to the research in this field. The present work is organized as follows: the two first chapters are introductory, while chapters 3 and 4 mostly contain the author's original contribution. More explicitly, {\\bf chapter 1} is an introduction to the Hubbard models (both the ordinary and the Extended) and is meant for the reader who is non-familiar with these models. We tried to emphasize the motivation that historically led to their introduction and the physical contexts in which they are applied, not necessarily in one-dimension. \\\\{\\bf Chapter 2} is instead devoted to general problematics of one dimensional physics. We have tried to stress that there is a number of physically interesting materials that can be regarded as actual 1-D systems; we give a brief account of the difficulties leading the usual theoretical approaches, such as mean-field and Fermi Liquid, to fail in D=1. While briefly describing the important results of the proposed alternative formulations (Luttinger Liquid, Renormalization Group), we also point out the points that are still unsatisfactory, and emphasize the importance that exactly integrable models have in this field. \\\\Matching the subjects of chapter 1 and 2, the subsequent ones contain the results of our investigation of exactly solvable Extended Hubbard model on 1-D chains. \\\\We wish to emphasize to the reader non familiar with integrable systems that the literature on this subject is often strongly concerned with the mathematical aspects of the involved methods, leaving the determination of the physical features of the derived integrable models as a secondary issue. On the contrary, we believe that both aspects of the research deserve a relevant role in the study. For these reasons, while chapter 3 contains new results about the general formulation of integrability for fermionic systems, chapter 4 is devoted to the investigation of the physical features for some models whose integrability has been proved in the previous chapter. \\\\More explicitly, in {\\bf chapter 3} (apart from a preliminary introduction of sec.3.1) the theory of integrable {\\it fermionic} systems has been developed within the frame of the Quantum Inverse Scattering Method (QISM). The latter is a technique for obtaining integrable models, traditionally formulated with success for spin systems; its more recent application to electron systems was leaving many unsatisfactory aspects. To overcome this weaknesses, in sec.3.2 we both clarify the theory of matrix representation for fermionic operators, and describe a method for obtaining integrable models of fermions (straightforwardly readable in terms of fermion operators), starting from the the solutions of a
mathbbC-number equation (known as Yang-Baxter Equation). In sec.3.3 we first work out a general method of search for solutions of such equation; secondly we explicitly find 96 solutions, corresponding to 96 integrable Extended Hubbard models (most of which are new) for which a set of mutually commuting operators is proved to exist. In sec.3.4 we formulate a general theorem concerning the symmetries of the integrable models in QISM, and widely discuss the symmetries and supersymmetries for the class of extended Hubbard Models.\\\\ {\\bf Chapter 4} is devoted to the investigation of the physical properties of the integrable Extended Hubbard models found in chapter 3. We introduce a technique, called Sutherland's species technique, that, when applicable, yields an effective reduction of the Hilbert space and considerably simplifies the problem of diagonalizing the Hamiltonian. In fact, such technique is proved to be applicable to all the 96 models found above, and is used to determine the exact ground state phase diagram and the thermodynamics of the models. In particular we both recover in a quite straightforward way the phase diagram of a known model, and that of two new models (Model 2 and Model 3) that were not known before. \\\\The phase diagram exhibit structured forms with respect to the filling (density of electrons) and the on-site Coulomb repulsion (
U); the pressure and the compressibility in the ground state are also calculated. Model 2 exhibits a superconducting phase particularly enhanced towards positive values of
U, as well as a band-controlled insulator-superconductor transition at half filling; model 3 is the first example of exactly solved model with an asymmetric shape with respect to half-filling, due to the absence of particle-hole invariance of the Hamiltonian; a filling controlled metal-insulator transition is found to occur for this model. Generalizations to the other cases are also discussed. \\\\As to the thermodynamics, we compute the exact partition function of Model 3, which we believe to be particular worth of interest due to the above asymmetry. We then derive the exact specific heat of the model as a function of the temperature, which exhibits an interesting two-peak behaviour as well as a nearly universal crossing point. These results are discussed and compared with similar ones present in the literature. The chapters are of course linked each to another according to the above scheme; nevertheless, it has been a purpose of the author to make them as independently readable as possible. In particular, the reader who is more interested in the physical aspects of the models rather than in the general formalism of fermionic QISM can skip chapter 3 (at least in a first-sight reading) and pass directly to chapter 4; apart from some references to previous formulas the understanding should not be affected.\\\\{\\it \\noindent \\underline{Notational Remark:}} The customary symbol to denote the electron filling is
n; this has in fact been used throughout the two first introductory chapters. Nevertheless, in order to avoid possible misunderstanding with other extensive quantities, in chapter 3~and~4~the~symbol~has been changed into
rho