Introduction: Parallel structures

In physics, we see appearance of certain structures, models or concepts in multitudinous settings and at different scales, even though their physical interpretation would be different within each context. A simple harmonic oscillator and the two-level system are such prototypical quantum mechanical models that occur in various contexts from atomic physics to superconducting systems . As Sidney Coleman says ”The ca- reer of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction”. It captures certain fundamental manifestations of quantum mechanics such as existence of

vacuum, occurrence of discrete spectrum, transition to classical picture and so on. The Ising model is also such a model at a higher level of sophistication and captures many-body physics. It is one of the prototypical modes for a class of classical and quantum phase transitions, predicting the critical exponents without the requirement of details and only considering the dimension and symmetry of the model. We have various such constructs that are of immense value in understanding phenomena across broad scales, such as the Dirac equation, conformal field theories, Luttinger liquids etc.

There is also the emergence of very simple single particle quantum mechanics in complex many-body systems. Such examples involve phase dynamics in Josephson junctions, quantum optics and quasiparticle dynamics in lowest Landau level. All these examples take the form of ‘single-particle’ Schrodinger equations in certain interesting potentials such as the Mathieu potential, Poschl-Teller potential for example. There are also themes that historically appeared first in high energy physics, but are now part of mainstream condensed matter physics. Dirac monopoles, Majorana fermions, Skyrmions and anomalies are just a few examples to quote. Bear in mind that these structures are now accepted as the description of these systems rather than as treating them as mere analogies to things that happened to appear first historically. The emergence of certain bare structures of fundamental quantum mechanics at the level of condensed matter systems that form the basis of standard experiments in labs, gives a tremendous opportunity to probe those structures in depth. The concept of symmetry is a good guiding principle in trying to seek similar structures in different phenomena. But once the concept of symmetry is invoked, one must be aware of all the mathematical aspects it brings along because of its articulation in terms of group theory. From quarks to new topological phases, symmetry arguments are extremely powerful.

In this chapter, we will be studying such parallel structures that arise in black hole spacetimes and quan- tum Hall effect through symmetries, specifically in the context of thermality. On one hand, in the quantum Hall effect one starts with a simple consideration of electrons in a magnetic field in two dimensions. This gives rise to Landau levels, non-commutativity in the lowest Landau levels, quantized Hall conductance, edge localisation, topological band structure, fractionalisation, anyons and many more fascinating phenomena[18]. This problem is one of the simplest instances of introducing gauge fields into quantum mechanics result- ing in non-trivial effects such as the Berry’s phase. On the other hand, black holes are one of the most intriguing astrophysical objects. They are one of the simplest macroscopic objects in nature in that they are described by their mass, angular momentum and charge[10]. Their ‘construction’, at least classically, is purely geometrical and yet they exhibit a plethora of features such as the singularities, one-way membrane, quasinormal modes and wormholes[11] . Further, on introducing quantum mechanics, there is the emergence of Hawking radiation[15] followed with the issue of its thermal interpretation and the information paradox.

These phenomena continue to baffle us to this day and is an ongoing field of intense research. A black hole is the key phenomenological entity in nature that forms the ground for interplay between quantum mechanics and gravity. In fact, describing black hole thermality would be a test bed for theories of quantum gravity. In this chapter, we will see that certain bare essential structures that appear in the description of thermality in the presence of black hole horizons, also appear in the context of quantum Hall effect and the common model appearing in both contexts is a simple quantum mechanical Hamiltonian of the inverted Harmonic oscillator. This model is realised through applying a saddle potential to a quantum Hall system. The saddle potential has been studied and well known in the field of quantum Hall effect and used to model systems with point contacts[190, 191]. Quantum point contacts are an integral part of many experimental set ups for conductance measurements. These are very important for studying shot noise and Anyon interferometry. Therefore, the parallels constructed between the black hole phenomena and quantum Hall effect are easily accessible to experiments. While it might be ambitious to expect such experimentally accessible parallels to help in investigations on quantum gravity, the process of investigating these parallels is in itself a very fruitful process. It helps us re-appreciate certain aspects of fundamental quantum mechanics. Casting the well known physics of quantum Hall effect in the light of structures parallel to black hole physics, could lead to better understanding of the quantum Hall phenomena.

This chapter is organised as follows: In the next section, an overview is given about the kind of parallels we will be drawing between the two settings. This will be followed by a section where the basics of black hole physics is reviewed. Black hole/Rindler thermality is derived using both path integrals and mode exapnsion and the derivation of black hole quasinormal modes is presented. Then, the basics of quantum Hall physics and lowest Landau levels is presented. Finally, the exact parallels of black hole thermality and quasinormal modes are shown in lowest Landau level in the presence of an applied potential