Up to this point in our discussion of the superfluidity phenomenon, we have deliberately not mentioned quantum mechanics: this is in contrast with the conventional approach that usually starts with the notion of the condensate wavefunction. In previous chapters, we have shown that classical systems, whether they are dynamical or purely statistical, can be in the superfluid state. Thus, an important connection to follow in the quantum case is the development of the classical-field component that then undergoes the superfluid transition. This can be achieved naturally within the coherent states approach advocated by Langer [1]. As we will see shortly, this connection is rather straightforward and, surprisingly, brings about the counterintuitive (from the classical-particle perspective) question: “Do we ever have an insulating quantum solid state in bosonic systems?” In Chapter 9, we answer this question positively.
Up to this point in our discussion of the superfluidity phenomenon, we have deliberately not mentioned quantum mechanics: this is in contrast with the conventional approach that usually starts with the notion of the condensate wavefunction. In previous chapters, we have shown that classical systems, whether they are dynamical or purely statistical, can be in the superfluid state. Thus, an important connection to follow in the quantum case is the development of the classical-field component that then undergoes the superfluid transition. This can be achieved naturally within the coherent states approach advocated by Langer [1]. As we will see shortly, this connection is rather straightforward and, surprisingly, brings about the counterintuitive (from the classical-particle perspective) question: “Do we ever have an insulating quantum solid state in bosonic systems?” In Chapter 9, we answer this question positively.
Introduction In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties. A case in point is a classical solid for which the specific heat follows the Dulong–Petit law, C = 3 Nk B , as T → 0, and the entropy has unphysical behavior since S → –∞. Also, thermal expansion coefficients tend to non-vanishing constants for T → 0 while the third law implies that they must be zero. While the position and momentum of a particle can be specified precisely in classical mechanics, and hence the groundstate of a solid is a perfectly rigid crystal lattice (motionless particles localized at the lattice points), in reality the Heisenberg uncertainty principle forbids such a perfect rigid crystal, even at T → 0, due to zero point motions which ‘smear out’ the particles over some region around these lattice points. This delocalization of quantum-mechanical particles increases as the atomic mass is reduced; therefore, these quantum effects are most pronounced for light atoms like hydrogen in metals, or liquid helium. Spectacular phenomena like superfluidity are a consequence of the quantum nature of the particles and have no classical counterpart at all. Even for heavier atoms, which do not show superfluidity because the fluid–solid transition intervenes before a transition from normal fluid to superfluid could occur, there are genuine effects of quantum nature. Examples include the isotope effects (remember that in classical statistical mechanics the kinetic energy part of the Boltzmann factor cancels out from all averages, and thus in thermal equilibrium no property depends explicitly on the mass of the particles).
In this chapter, we will discuss the conditions under which ground states of quantum systems fail to develop the classical-field component and enter an insulating state. There are several reasons for this to happen. If interparticle interactions are strong and particles are heavy, bosons localize into the crystalline structure. One may argue that a solid with small zero-point fluctuations is essentially a classical state. However, in this case, we employ the classical-particle picture, which, somewhat ironically, cannot be connected to the classical-field picture without invoking quantum mechanics in a non-perturbative way. Bosons may be localized into an insulator also by a deep external lattice potential if their number is commensurate with the lattice period (in a commensurate state, the number of particles per unit lattice cell-the so-called filling factor ν-is integer). For integer ν, an insulating state that does not break any of the lattice symmetries is called a Mott insulator (MI); otherwise, the most common insulating structure is a lattice solid. Finally, an insulating state may result from placing bosons in a strong disordered potential. The generic name for such a state is the Bose glass (BG). The crucial difference between the BG and MI is that the former is compressible, while the latter has a gap for creating particle or hole excitations. Under special circumstances (e.g., in the limit of extremely large [but integer] site occupation numbers and disorder in the interaction potential or hopping amplitudes [the so-called off-diagonal disorder]), the system Hamiltonian becomes particle-hole symmetric. This symmetry leads to a new disordered insulating state, the Mott glass (MG), which has no gap for exciting particles and holes and yet has zero compressibility. Finally, superfluidity and crystalline order may coexist in one and the same material, leading to the supersolid state.
As we all know, single microscopic particles such as electrons or atoms do not behave according to the familiar classical (Newtonian) physics which describes the everyday world, but rather according to the prescriptions of quantum mechanics, the outcome is often bizarre and counterintuitive. When a macroscopic number of such particles (say ∼ 10) is assembled under conditions where the effects of quantum mechanics are important, the results can be even more spectacular; they include macroscopic phenomena such as superfluidity and superconductivity, which are completely impossible to explain in classical terms. Most, though not all of these exotic phenomena, are a result of a phenomenon which I shall refer to generically as “quantum condensation,” in which a macroscopic number of particles or pairs of particles—a nonzero fraction of all particles present—occupy a single quantum state; this phenomenon is traditionally referred to as Bose-Einstein condensation (BEC) when it occurs in a system of bosons such as liquid He, and as Cooper pairing when it occurs, for pairs of particles, in a system of fermions such as the electrons in a superconducting metal. As a result of quantum condensation, the characteristically quantum-mechanical behavior which we are used to seeing at the level of single electrons is as it were amplified by the cooperation of condensed particles and can manifest itself at the macroscopic level. The present essay is intended as a general introduction to the subject of quantum condensation and its consequences, as manifested in a variety of physical systems. I will try to review the fundamental physical ideas, usually without de-
Abstract The present chapter introduces the statistical mechanics and computational physics of identical bosons. Thus, after studying, in Chapter 3, the manifestation of Heisenberg’s uncertainty principle at finite temperature for a single quantum particle or several distinguishable quantum particles, we now follow identical particles to low temperature, where they lose their individual characters and enter a collective state of matter characterized by Bose condensation and superfluidity.We mostly focus in this chapter on the modeling of noninteracting (ideal) bosons. Our scope is thus restricted, by necessity, because quantum systems are more complex than the classical models of earlier chapters. However, ideal bosons are of considerably greater interest than other noninteracting systems because they can have a phase transition.
In the previous two chapters, we have seen two statistical descriptions of plasmas, and more generally, of complex systems where collective phenomena take place. These two approaches have been developed in a classic framework, that is to say without taking into account principles emanating from quantum mechanics. So, it is natural to ask the following questions: does such a description necessarily have to remain classic or is it possible to extend it to the framework of quantum mechanics? We will see in this chapter that such an extension is indeed possible.
In the previous chapters, the focus was on systems of particles following the law of classical mechanics. Of course, this was mainly done within the framework of classical mechanics but a quantum description of individual (discretised) states was also used, while assuming distinguishability of the particles (for instance, particles located at the nodes of a lattice (spin system, quantum harmonic oscillators,) ). The present chapter as well as Chapters 7 and 8 are dedicated to the study of statistical physics in a fully quantum framework. The first obstacle encountered by a quantum approach is the impossibility to define a quantum phase space (in which the ergodic hypothesis would hold for instance). This problem is inherent to quantum mechanics: because of Heisenberg’s uncertainty principle, one cannot define both the momentum and the position of a particle. Besides, statistical physics considers systems made of a large number of particles, a priori in mutual interaction. In classical physics, model situations for which the interactions between the constituents of the system are neglected can be studied. In quantum mechanics, the indistinguishability of identical particles makes the study of the problem in the absence of interaction already difficult: even for non-interacting particles, the symmetrisation (or antisymmetrisation) of the wave function of https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_25.tif "/> identical particles induces correlations which have non-trivial thermodynamical effects such as the existence of a “Fermi sea” for fermions (see Section 8.1 ) or the Bose–Einstein condensation phenomenon for bosons (see Section 7.4 ). The present chapter addresses these two points consecutively.