Rabi model exactly solved

Rabi model looks very simple: it describes a two-level atom coupled to a monochromatic beam of light through dipole interactions. However, an exact solution is not found until recently. A paper [the preprint of which is here] in PRL reports such a solution. Exactly solvable models are always interesting, because they can offer insights in many areas that may seem irrelevant at first glance.

Braak’s unexpected full analytical solutions of the quantum Rabi model are, however, worthy of celebration [2]. In mathematics and physics there are many, not necessarily compatible, criteria for a model to be both integrable and solvable. Examples include Frobenius’ condition for integrability in differential systems and Liouville’s condition for integrability in dynamical systems [6]. In the realm of quantum physics, it has been assumed that the existence of invariant subspaces associated with conserved quantities, other than energy, might be a necessary condition, as is the case in the Jaynes-Cummings model. The quantum Rabi model possesses naturally an additional discrete symmetry, the parity, which was assumed for some time to be insufficient for yielding a solvable model. Braak has proved that this is not the case and has presented exact analytical solutions of the quantum Rabi model for all parameter regimes. This is a remarkable achievement that adds the model to the short list of integrable quantum systems. Furthermore, Braak is able to take advantage of this key result to propose an operational criterion of integrability, inspired by the case of the hydrogen atom.

Integrability is, following Braak, equivalent to the existence of quantum numbers that classify eigenstates uniquely. It does not presuppose the existence of a family of commuting operators. Surprisingly, he has been able to apply this novel integrability criterion to a more elaborate case, the generalized quantum Rabi model, where a term that allows tunneling between the two atomic states is added. For the latter, he was able to prove that the model is not integrable, because it has an additional symmetry that is broken. However, the model is exactly solvable, and Braak presents the solutions. These are important results advancing the mathematical aspects of the quantum Rabi model in terms of integrability and solvability. Moreover, we should not overlook that the quantum Rabi model is a key physical model describing the interaction of quantum light and matter. [http://physics.aps.org/viewpoint-for/10.1103/PhysRevLett.107.100401]

Another simple and universal role in high Tc ?

These authors presented a very simple rule that seems validated by their analysis of experimental data [J. Phys.: Condens. Matter 23 (2011) 295701 (17pp)]. In this rule, the Tc of optimal compounds is essentially set by two length scales and the electron charge, i.e., Tc~e^2/l\times l'. What is striking is that, this rue was argued to cover a wide range of materials, including cuprates, pnictides and ruthenates. They proposed a paring mechanism via Compton scattering: e.g., the holes in the conducting layer is scattered by the electrons in the charge reservoir layer. Instead of forming excitons, superfluid forms. The following is a brief sojourn over this work [http://iopscience.iop.org/0953-8984/labtalk-article/46706]:

High-T_C superconductors have layered crystal structures, where T_C depends on bond lengths, ionic valences, and Coulomb coupling between electronic bands in adjacent, spatially separated layers. Analysis of 31 high-T_C materials—cuprates, ruthenates, rutheno-cuprates, iron pnictides and organics—has revealed that the optimal transition temperature T_CO is given by the universal expression k_B^-1e²Λ / ℓζ. Here, ℓ is the spacing between interacting charges within the layers, ζ is the distance between interacting layers, Λ is a universal constant, equal to about twice the reduced electron Compton wavelength, k_B is Boltzmann's constant and e is the elementary charge. Non-optimum compounds in which sample degradation is evident typically exhibit T_C below T_CO. Figure 1 shows T_CO versus (ση/A)^1/2/ζ—a theoretical expression determining 1 / ℓζ, where σ is the charge fraction, η is the layer number count and A is the formulaic area. The diagonal black line represents the theoretical T_CO. Coloured data points falling within ± 1.4 K of the line constitute validation of the theory.

Figure 1. Experimental test of theory.

Figure 2. Proposed pairing interaction.

The elemental building block of high-T_C superconductors comprises two adjacent and spatially separated charge layers. The factor e² / ℓζ, determining T_CO arises from Coulomb forces between them. Remarkably an explicit dependence on phonons, plasmons, magnetism, spins, band structure, effective masses, Fermi-surface topologies and pairing-state symmetries in high-T_C materials is absent. The magnitude of Λ suggests a universal role of Compton scattering in high-T_C superconductivity, as illustrated in figure 2 that considers pairing of carriers (h) mediated by electronic excitation (e) via virtual photons (ν). Several other important predictions are given. A conducting charge sheet is non-superconducting without a second mediating charge layer next to it, and a charge structure representing a room-temperature superconductor yet to be discovered is presented.

A grand unification

In science, there is one thing that always makes me curious and awe. Roughly, I'd like to call it 'grand unification', which means a simple concept that connects various seemingly unrelated and independent phenomena which occur in distant disciplines. Such unification corroborates the conviction that, the universe has a common underlying mathematical structure. Here I talk of one more example of this.

This example is about fast symmetry breaking phase transitions (FSPT), in contrast with the usual cases where one talks of phase transitions near equilibrium states which may be described within hydrodynamic jargon. Up to my knowledge, no general effective theory for the moment has been established for FSPT. Despite this, it is possible to find out on some quantities very generic constraints which derive from the basic structure (such as symmetry and causality) of the first principles theory. Suppose a physical system is in equilibrium. Now one changes an experimental knob (e.g., pressure). This system shall then evolve away from its initial equilibrium state according to the corresponding dynamic theory, and shall eventually reach another new equilibrium state. However, what this final state might be should depend on the evolution and how the knobs are changed. An interesting thing is that, during this evolution some topological defects (which are textures that break the overall symmetry) shall form. Generally, it is expected that, as constrained by causality, which means no physical signal can travel faster than a typical velocity special to the system, two textures separated by a typical distance, say x, could not develop significant correlations during the evolution. Assume that a texture be characterized by 'direction vector'. Then, the causality indicates that, two distant textures should choose their directions independently in statistical sense. Therefore, all direction vectors may be found with certain textures because of the symmetry respected by the first principles model. Now, an interesting question is, how does x scale with the phase transition rate ?

In 1976, Kibble made an estimation [1], which is based on a cosmological model. It deals with cosmic strings, which are inherently stable topological objects that are expected to survive to today. Experimental verification of his idea is however quite difficult, since it is about the whole universe, which is so vast. A breakthrough came by Zurek [2], who tried Kibble's idea on Helium, which undergoes super-fluid transition at much low temperatures. And in such system, topological defects, which are known as fluxons and antifluxons by their winding numbers, could form as the transition is passed. Therefore, the helium system poses a remarkable realization of the big universe in this respect. To obtain the relation between x and transition rate, Zurek employed the dynamic Landau's theory, which was supposed to govern the temporary evolution. It turns out that, the relation follows a simple power law. Verification of this law came afterward. The latest one was done on superconductor [3]. For a good review see Ref.[4].

It is exciting to see that, the thing speculated about cosmos has its image on earth. I'd like to pose another question concerning FSPT, how does x scale with the dimension of the system ?

It may be worth pointing out that, as was emphasized by Anderson in his 'more is different' address, to deal with emergent phenomena, it is not enough to know just the first principles model (e.g., BCS model), which are usually not much useful in obtaining intuition and understanding. It is more useful to come up with an effective model (e.g., Landau's model), which concerns the quantities under imminent interest instead of those in the first principles model.

[1]J.Phys.A, 9:1387(1976)
[2]Nature, 317:505(1985)
[3]PRB, 80:180501(R), 2009
[4]Phys.Today, 60:47(2007)