Many body physics is concerned with macroscopic systems consisting of a very large number of microscopic particles (N ≈ 10 20 ). In this case the direct dynamical treatment of the complete system is impossible. A quite useful and frequently applied approximation consists in the neglection of terms which are relatively small as the volume of the systems tends to infinity, the particle density remaining constant. For the concise theoretical description of the typical many body phenomena such as phase transitions and spontaneous symmetry breakdown the use of the thermodynamical limit is even necessary, since no existing theory for the finite systems shows these effects. As a general rule one can say that a quantum system behaves like a thermodynamical system only in the infinite-volume limit. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Many body physics is concerned with macroscopic systems consisting of a very large number of microscopic particles (N ≈ 10 20 ). In this case the direct dynamical treatment of the complete system is impossible. A quite useful and frequently applied approximation consists in the neglection of terms which are relatively small as the volume of the systems tends to infinity, the particle density remaining constant. For the concise theoretical description of the typical many body phenomena such as phase transitions and spontaneous symmetry breakdown the use of the thermodynamical limit is even necessary, since no existing theory for the finite systems shows these effects. As a general rule one can say that a quantum system behaves like a thermodynamical system only in the infinite-volume limit. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract 1. GENERALITIES Much of solid state theory and statistical physics is concerned with the properties of macroscopic systems. These are often considered while using the “thermodynamic limit” (system’s volume, Ω, and particle number, N, tending to infinity with n = N /Ω kept constant), which is a convenient mathematical device for obtaining bulk properties. Usually, the system approaches the macroscopic limit once its size is much larger than some correlation length ζ (or, more generally, than all such relevant lengths).
The language of this first section is quite heuristic, as it only aims at a broad presentation of the problems we want to turn our attention to. A rigorous formulation of some of these questions, and of their answers, is reviewed in section 2. A few illustrative models are mentioned briefly in Section 3. Section 4 is entirely devoted to the question raised by the very existence of crystals in a world where the fundamental interactions are believed to be covariant at least under the group of rigid motions in IR 3 . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The basic concepts of statistical mechanics and its application to many-body systems, phase transitions and critical phenomena will be discussed briefly in this chapter. We will be using these ideas frequently in the subsequent lectures. Statistical mechanics describes the thermodynamic behaviour of macroscopic systems from the laws which govern the behaviour of the constituent elements at the microscopic level. The microscopic elements can be atoms, molecules, dipole moments or magnetic moments, etc. A macroscopic system is generally composed of a large number of these elements (of the order of Avogadro number N A ≈ 6.022 × 1023 per mole). Each element may have a large number of internal degrees of freedom associated with different types of motion such as translation, rotation, vibration etc. The constituent elements may interact with the external field applied to the system. There can also be very complex interaction among the constituent elements. The macroscopic properties of a system in the thermodynamic limit is thus determined by the properties of the constituent molecules, their interaction with external field as well as interaction among themselves. For a system of N elements, volume V and density ρ, the thermodynamic limit is defined as lim N → ∞, lim V → ∞ keeping ρ = N/V finite. In this limit, the extensive properties of the system become directly proportional to the size of the system (N or V), while the intensive properties become independent of the size of the system.
Envisioned by Richard Feynman in the early 1980s, quantum simulation has received dramatic impetus thanks to the development of a variety of plateforms able to emulate a wide class of quantum Hamiltonians.During the past decade, most of the quantum simulators have implemented rather well-known models, hence permitting a direct comparison with theoretical calculations and a precise benchmarking of their reliability.The field has now reached a maturity such that one can address difficult problems, which cannot be solved efficiently using classical algorithms.These advances provide unprecedented opportunities to explore previously unreachable fields, test theoretical predictions, and inspire novel approaches.This contribution is an elementary introduction to quantum simulation.We discuss the challenges, define both digital and analog quantum simulators, and list the demanding conditions they require.We also provide a brief account of the contributions gathered in the dossier on Quantum simulation in the Compte rendus Physique of the French Academy of Sciences [1][2][3][4][5][6].The latter completes excellent reviews that appeared previously, see for instance . Universal models and the role of simulations in many-body physicsUnderstanding the behavior of macroscopic quantum systems is a major challenge of modern physics.The basic laws of low-energy physics are by now quite well known at the microscopic, say atomic, scale.Conversely, many fundamental questions remain open, and even debated, about the collective dynamics at the macroscopic scale.By "macroscopic scale", here we mean systems made up of a huge number of constituents, or degrees of freedom, say 10 6 , 10 23 , or even more, as relevant in condensed matter physics or in astrophysics, for instance.Such huge systems cannot be treated exactly, be it a the classical level and, even worse, at the quantum level.Yet, it is the main outcome of the thermodynamic approach that the collective behavior of a macroscopic system can drastically differ from that of its elementary constituents.For instance, the elementary interactions between the H 2 O molecules are fundamentally unchanged when a water bucket turns from the liquid phase to the solid phase at zero degree Celsius.Similarly, the interactions between the microscopic magnetic moments do not show any brutal change when a magnetic material gets magnetized underneath the Curie temperature.Hence, the dramatic effects observed in macroscopic systems are governed by large-scale instabilities, without obvious counterparts at the microscopic level.This observation takes a universal character, summarized in the celebrated motto "More is Different" [15].Such so-called emerging phenomena also appear in quantum systems, where new effects that are impossible in the classical world show up below some critical temperature or at zero temperature when some interaction parameter passes through a critical value.Celebrated examples include the superfluid transition in helium or super-conducting transitions and other metal-insulator transitions in electronic systems.Strikingly enough at first sight, while emerging phenomena only appear in very large scale macroscopic systems, their germs are contained in the mutual interactions of their elementary constituents.Hence, local two-body interactions are sufficient to explain a huge class of phase transitions, such as the liquid-solid and magnetic transitions mentioned above.The Boltzmann statistical approach proved particularly successful in describing this connection between the microscopic and macroscopic scales.To understand emerging phenomena on grounds as fundamental as possible, the programme is now well established: One tries to identify the basic microscopic terms that seem to be relevant and disregards all the other microscopic details.One then elaborates a model, as generic as possible, likely to reproduce the main experimental observations.The most usual examples are the Ising and Heisenberg models for magnetic transitions, or the Hubbard model for metal-insulator transitions [16][17][18].Then all is left to do is to check that the phenomenon of interest indeed emerges from the dynamics of the simplified model.The realization of this programme is nothing but a simulation.It consists in building up a simplified system that mimics the main properties of a real system.
Even if we still face problems in formulating concise theories that explain the resolution of various apparent paradoxes, one may simply assume such a theory to exist, and that it will one day be found. This leads to the CA (Cellular Automaton) Interpretation. We now formulate the rules. The universe has a wave function, which, in terms of the classical degrees of freedom, collapses continuously, yet fully in agreement with Schrödinger’s equation. We formulate how to look at various notoriously ‘quantum mechanical’ phenomena, such as superposition, exponential decay, and the double slit experiment. What can we say about the ‘quantum computer’? These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.