This book explains the fundamental concepts and theoretical techniques used to understand the properties of quantum systems having large numbers of degrees of freedom. A number of complimentary approaches are developed, including perturbation theory; nonperturbative approximations based on functional integrals; general arguments based on order parameters, symmetry, and Fermi liquid theory; and stochastic methods.
This book explains the fundamental concepts and theoretical techniques used to understand the properties of quantum systems having large numbers of degrees of freedom. A number of complimentary approaches are developed, including perturbation theory; nonperturbative approximations based on functional integrals; general arguments based on order parameters, symmetry, and Fermi liquid theory; and stochastic methods.
This book provides a comprehensive and pedagogical account of the various methods used in the quantum theory of finite systems, including molecular, atomic, nuclear, and particle phenomena. Covering both background material and advanced topics and including nearly 200 problems, Quantum Theory of Finite Systems has been designed to serve primarily as a text and will also prove useful as a reference in research.The first of the book's four parts introduces the basic mathematical apparatus: second quantization, canonical transformations, Wick theorems and the resulting diagram expansions, and oscillator models. The second part presents mean field approximations and the recently developed path integral methods for the quantization of collective modes. Part three develops perturbation theory in terms of both time-dependent Feynman diagrams and time-independent Goldstone diagrams. A fourth part discusses variational methods based on correlated wavefunctions, including spin correlations. The approximation schemes are formulated for fermions and bosons at eigher zero or non-zero temperature.Although the formalism developed applies to both finite and infinite systems, the book stresses those aspects of the theory that are specific to the description of finite systems. Thus special attention is given to mean field approximations, the ensuing broken symmetries, and the associated collective motions such as rotations. Conversely, some specific features of systems with infinite numbers of degrees of freedom (such as the thermodynamic limit, critical phenomena, and the elimination of ultraviolet divergencies) are deliberately omitted.Jean-Paul Blaizot and Georges Ripka are associated with the Centre d'Etudes Nucleaires de Saclay.
Bloch’s theorem, which is one of the cornerstones of condensed matter physics, has played a fundamental role in the development of many theories, such as the band theory, Fermi liquid theory, and BCS theory. It states that when the system has discrete translational symmetry, the corresponding eigenstate can be labeled by a conserved quantity, or a good quantum number — Crystal momentum. Based on Bloch’s theorem, many important physical quantities can be expressed as the integral over the entire Brillouin zone or the Fermi surface. In realistic macroscopic materials, the translational symmetry is explicitly broken down due to the existence of boundaries. It is natural to ask why we can still use Bloch’s theorem to understand the physical properties of real materials? In traditional textbooks of solid-state physics, a thermodynamic limit argument is used to explain the above question. Since the lattice size of the macroscopic material is very large, e.g. N≈1023 , its asymptotic behaviors can be described by the thermodynamic limit N→∞ . Therefore, if we fix the lattice constant, i.e., a, the system length will extend to infinity, i.e. L=aN→∞ . Since wave functions, e.g. ψ(x)∝eikx , should be bounded at the infinity, e.g. |ψ(x→±∞)|<∞ , the corresponding momentum is restricted to be real numbers. This revives Bloch’s theorem. Physically, for a finite-size system with open boundary conditions, the boundary only acts the role of scattering potential and can be regarded as a perturbation to the Bloch Hamiltonian. Therefore, the eigenstate of the OBC Hamiltonian is a superposition of all the scattered Bloch waves with the same energy. In this sense, one can say that, although the existence of boundary definitely breaks the translational symmetry, Bloch’s theorem is perturbatively broken down, and one can still use Bloch’s theorem to understand the physical properties in real materials. Even for interacting systems, Bloch’s theorem is also approximately preserved. Indeed, in a many-body system, the elementary excitations (or quasiparticles) are referred to the eigenmodes with well-defined energy, momentum, and dispersion relation. This means the excitations of the system are extended Bloch-like states in the bulk, which is consistent with the physical intuition in a disorder-free system. In this paper, we show that all the above arguments are challenged in some Hermitian bosonic systems, in which all the eigenstates, or quasiparticles are localized at the boundary, indicating that Bloch’s theorem is nonperturbatively broken down. This phenomenon generalizes the concept of the non-Hermitian skin effect from dissipative systems to non-dissipative systems, and can be understood as the Hermitian skin effect in general. Here the physical origin of the Hermitian skin effect is that although the bosonic Hamiltonian is Hermitian, its elementary excitations are determined by a non-Hermitian matrix. In order to understand the quasiparticles in such systems, the so-called generalized Brillouin zone theory is necessary. Finally, based on the Bogoliubov theory, the quench dynamic is also studied to illustrate the non-perturbative breakdown of Bloch’s theorem.
"Singlemindedly devoted to its job of educating potential many-particle theorists…deserves to become the standard text in the field." - Physics Today"The most comprehensive textbook yet published in its field and every postgraduate student or teacher in this field should own or have access to a copy." - EndeavorA self-contained, unified treatment of nonrelativistic many-particle systems, this text offers a solid introduction to procedures in a manner that enables students to adopt techniques for their own use. Its discussions of formalism and applications move easily between general theo
Landau theory of Fermi liquids is the foundational concept behind our understanding of conventional metals. Its basic postulate is that the low-energy physics can be described by weakly interacting quasiparticles --- a dressed but effectively free version of electrons formally represented by poles of the single-particle Green's function. However, this concept notoriously fails for strongly interacting systems such as high-Tc cuprate superconductors, and our understanding of properties of these ``non-Fermi'' quantum liquids is far from complete. One well-known candidate theory for non-Fermi liquids is Hertz-Millis theory: a finite-density fermion system coupled to a massless critical boson. In (2+1) dimensions, this theory is strongly interacting at low energies, but as of yet there is no controlled insight in this regime; conventional perturbation theory fails. In this article, the authors show that in the so-called quenched approximation the fermionic spectrum can be computed exactly to all orders in the coupling constant. They show extensive detailed analysis that, in this quenched limit, the nonperturbative low-energy dynamics are indeed that of a non-Fermi liquid. Instead of coherent quasiparticles one finds the emergence of critical fermionic excitations described by branch cuts in the single particle Green's function.
The book reviews several theoretical, mostly exactly solvable, models for selected systems in condensed states of matter, including the solid, liquid, and disordered states, and for systems of few or many bodies, both with boson, fermion, or anyon statistics. Some attention is devoted to models for quantum liquids, including superconductors and superfluids. Open problems in relativistic fields and quantum gravity are also briefly reviewed.The book ranges almost comprehensively, but concisely, across several fields of theoretical physics of matter at various degrees of correlation and at different energy scales, with relevance to molecular, solid-state, and liquid-state physics, as well as to phase transitions, particularly for quantum liquids. Mostly exactly solvable models are presented, with attention also to their numerical approximation and, of course, to their relevance for experiments.