Milk, Coffee’s best mate? Maybe not in the quantum regime!

What happens when you pour cold milk over hot coffee? Given enough time and provided you resist the temptation to swirl the mixture around with a spoon, the milk will completely spread into the coffee and you end up with a delicious mug of coffee albeit a little less warm. The heat from the hot coffee atoms is transferred to the cold atoms of milk until they reach a steady state. However, do start sipping your coffee soon or else the molecules in the coffee will transfer most of their heat to the molecules in the air.

Similarly, if you thrust the end of a metallic rod, which is a good conductor of heat, into a flame it would heat the rod and before long you start feeling it at the other end at your hand. Thus, by some mechanism the heat from a hot region is transferred to the cold region. Also, we know that if by accident a metallic rod we hold touches a live wire, electrocution results!

What is the mechanism by which heat and electricity is conducted along metals? Any student of science would tell you that in a good conductor it is due to the freedom that energy carriers (phonons) and charge carriers (electrons) enjoy, which is just not there in an insulator.

This explanation hardly satiates our curiosity and we often want to know the answer at a deeper level. For that we have to dive into the metal and go underneath the surface. What do we see? We would see a sea of electrons in a regular periodic arrangement of positive ions. In metals, electrons are the predominant carriers of energy and charge. The electrical conductivity is directly proportional to the average length an electron can travel in the metal before it gets scattered by an ion. This is the classical picture, which considers electrons like particles colliding with bigger sized ions, something like in a game of marbles.

However, the classical picture is incomplete at the subatomic scale. Since the ions are positively charged they have an electric field (or in other words a potential) associated with them and since they are arranged in a periodic fashion the electron sees a periodic potential. We know from quantum mechanics that electrons have a wave nature. In fact, all matter has an inherent waviness associated with it: even us humans, although in our case the wavelength would be unimaginably small and unfortunately meaningless. Google “De Broglie wave” sometime. The electronic wave function within some energy bands is spread (or delocalized) over the periodic potential of the positive ions.

One interesting question to ask is what would happen if we make the potential aperiodic, thus introducing disorder into the system? Let’s do just that by adding impurities; for example, substituting some Aluminum ions with Copper ions in a lattice of Aluminum ions. The electron sees an aperiodic potential because the substituted Copper ions will have a different potential. The electronic wave function that was earlier spread out tends to localize around the impurity. In the classical model, this can be thought of as a reduction in the average length that an electron can travel leading to a drop in conductivity. Lets keep adding more impurity just to see what happens: beyond a certain amount of impurity the electronic motion stops altogether resulting in the material becoming an insulator. This means the wave function becomes localized around the disorder. This is known as Anderson localization after its discoverer and Nobel Laureate P. W. Anderson.

Anderson localization is due to wave interference between multiple scattering paths (an easier way to think about this is to imagine that the electron zig zags around the impurity resulting in a net zero motion) and is well understood for particles/waves that do not interact with each other in a disordered (with impurities) medium. One may then ask: We saw that electrons are present in abundance in a metal, and there is every possibility they will interact and if they do what happens when we include these interactions? Further investigations of localization in disordered media in the presence of short-range interactions between quantum particles (electrons, for example) led to the concept of many-body localization (MBL).

Let us try to understand many body localization with a macroscopic analogy. Going back to our earlier example of cold milk and hot coffee, this time instead of pouring milk into the coffee let’s add one drop in the center of the cup. In normal systems our everyday experience of milk and coffee mixing completely leaving no trace of where the drop of milk hit the coffee holds true. However, in an ideal MBL system, even after a very long time the milk drop will still stay where it fell as well as maintain its coldness.

Dibyendu Roy, from the Raman Research Institute, along with his collaborators Rajeev Singh from Bar-Ilan University and Roderich Moessner from Max Plank Institute for the Physics of Complex Systems, have recently published a paper on MBL and have the following to say about their work:

MBL is a recently discovered state of solids. It is an insulator and results from the interplay of disorder and interaction between particles. The nature of MBL in more than one spatial dimension is not entirely understood. We have studied properties of MBL in one-dimensional long-range models and made an analogy to infer features of MBL in higher-dimensional short-range models.

To read this interesting paper please click https://journals.aps.org/prb/pdf/10.1103/PhysRevB.95.094205

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