Unconventional Environments

Abstract

An analysis is given of some systems which do not behave as “conventional environments”. This means that the interaction between lowenergy excitations of the system and some external probe is either (a) not ~ 0(N ^-1/2), where N is the number of excitation degrees of freedom, or (b) not linear in the excitation coordinates, or (c) whilst formally ~O(N ^-1/2), it is infra-red divergent in a way which invalidates perturbation expansions.

The examples of v = 1/2 and Fractional Hall effects are extensively discussed, including much recent work on the new “singular gauge theory” of the FQHE. A comparative discussion is given of the Luttinger liquid with backscattering (solved recently by Prokof’ev), of domain wall tunneling in magnets, and of magnetic grain dynamics in a background spin environment.

It is seen that non-perturbative methods, outside the conventional approach, are necessary to understand these systems. Typically the experimental consequences of coupling to an unconventional environment are rather dramatic. Some of the consequences of the theory we discussed for the above examples with suggestions for experimental tests.

In the last few decades a very comprehensive “many-body” picture has evolved to describe the low-energy physics of condensed matter systems. According to this picture, if we immerse ourselves in a large or even mesoscopic quantum system (solid, liquid or gas) we will be surrounded by an “environmental sea” of excitations (collective modes, single particle excitations, etc) which are dilute at low energies or temperatures. If there axe N degrees of freedom in the system, the couplings ƒ _ij between excitations |i〉 and |j〉 will be ~ O(1/N), and the couplings V _j of |j〉 to an external probe will be ~ O(1/√N). We are all familiar with the canonical examples: ³He (normal, superfluid, solid, and quantum gas), ⁴He superfluid and solid, metals, semiconductors, superconductors, magnets, simple insulators, etc, etc. This is not to say that these environments are necessarily simple- the notorious complexity of the “Ohmic environments”, which was first unravelled in studies of the X-ray edge catastrophe [1] and the Kondo problem [2, 3, 4], demonstrates the contrary. Nevertheless they all have important features in common, which are made quite explicit in the Caldeira-Leggett theory of macroscopic quantum tunneling [5]. In this theory, the system excitations are described as bosonic oscillators ( an idea with a long history [6]) and their couplings to the outside world are not only ~ O(1/√N), but they are linear in the bosonic coordinates. This sort of model is also widely used in studies of “quantum dissipation” [7] and decoherence [8].

I will refer to quantum systems which can be described in this way as “conventional environments”; they are so pervasive that some writers seem to feel that all low-temperature systems can be so described. However, in this article I will describe “unconventional environments”, some way outside the conventional picture I have just sketched. In some cases the departures from conventional behaviour are quite subtle (as in the example of “composite fermions” in the Fractional Quantum Hall effect (FQHE) to be discussed below). In other cases, they can really be very radical, as in the example of spin environments (described briefly here, and in more detail in a companion article in this volume [9]). It is interesting to notice that, thus far, most of the interesting cases of unconventional environments seem to have arisen in magnetic systems of one kind or another.

In what follows I will spend a lot of time on the new “singular gauge theory” of both the FQHE and the v = 1/2 Hall effect. This theory is the result of a long effort on the part of both theorists and experimentalists to understand the FQHE in a way which goes beyond the Laughlin theory. Recent experiments, which give very strong evidence for the existence of a new kind of quasiparticle (the “composite fermion”) have led to a field theory which allows for the first time the calculation of many experimental quantities that were previously beyond the reach of anything but numerical small-size simulations. The situation is somewhat analogous to that in superconductivity, where the BCS theory (the analogue of Laughlin’s theory) revealed the basic structure, but one needed Gor’kov-Nambu-Eliashberg field theory to do realistic calculations, particularly for “strong coupling” superconductors. In the FQHE we are always in a strong-coupling regime, and so one absolutely needs a field theory. However there is a severe price to pay. The composite fermion singular gauge theory involves long-range unscreened interactions between the quasiparticles, which give rise to very unconventional behaviour, still not completely understood. Section I will be devoted to a description of recent progress here.

The FQHE simply represents one example of an “unconventional environment”. Others include the “1-dimensional Luttinger liquid with backscattering”, as well as examples more obviously related to this conference, such as magnetic domain walls interacting with magnons, or “central spins” interacting with their spin environment. There are also obvious connections between this last example and current work on disordered magnetic insulators and on the “quantum spin glass”. Each of these examples has its own story to tell. In section II I will say something about each one, and show how this leads us inevitably to a new “non-perturbative” picture of unconventional environments.

Finally in section III, as an antidote to all this theory, I will say something about how all this relates to experiment. This will be done for all the examples mentioned, but I will concentrate on those examples on which I am currently working (ie., I will only mention in passing the current experimental work on Luttinger liquids).