An Integral from the SSH Model

A while ago, I was solving the Su-Schrieffer-Heeger (SSH) model for polyacetylene and came across an integral which I immediately thought was pretty cool. Here is the integral along with the answer:

Just looking at the integral, it is difficult to see why no matter what the value of , the integral will always give +1 or -1, which only depends on the sign of . This means that if or if , you would get the same result, in this case +1, as the answer to the integral! I’ll leave it to you to figure out why this is the case. (Hint: you can use contour integration, but you don’t have to.)

It turns out that the result actually has some interesting topological implications for the SSH model, as there are fractional statistics associated with the domain wall solitons. I guess it’s not so surprising that an integral that possesses topological properties would show up in a physical system with topological characteristics! But I thought the integral was pretty amusing anyhow, so I thought I’d share it.

Aside: For those who are interested in how I arrived at this integral in the SSH model, here are some of my notes. (Sorry if there are any errors and please let me know!) Also, the idea of solitons in the SSH model actually bears a strong qualitative resemblance to the excellent zipper analogy that Brian Skinner used on his blog.

Reflecting on General Ideas

In condensed matter physics, it is easy to get lost in the details of one’s day-to-day work. It is important to sometimes take the time to reflect upon what you’ve done and learned and think about what it all means. In this spirit, below is a list of some of the most important ideas related to condensed matter physics that I picked up during my time as an undergraduate and graduate student. This is of course personal, and I hope that in time I will add to the list.

1. Relationship between measurements and correlation functions
2. Relationship between equilibrium fluctuations and non-equilibrium dissipative channels (i.e. the fluctuation-dissipation theorem)
3. Principle of entropy maximization/free-energy minimization for matter in equilibrium
4. Concept of the quasi-particle and screening
5. Concept of Berry phase and the corresponding topological and geometrical consequences
6. Broken symmetry, the Landau paradigm of phase classification and the idea of an order parameter
7. Sum rules and the corresponding constraints placed on both microscopic theories and experimental spectra
8. Bose-Einstein and Cooper Pair condensation and their spectacular properties
9. Logical independence of physical theories on the theory of everything
10. Effects of long-range vs. short-range interactions on macroscopic properties of solids
11. Role of dimensionality in observing qualitatively different physical properties and phases of matter

The first two items on the list are well-explained in Forster’s Hydrodynamics, Fluctuations, Broken Symmetry and Correlation Functions without the use of Green’s functions and other advanced theoretical techniques. Although not yet a condensed matter phenomenon, Bell’s theorem and non-locality rank among the most startling consequences of quantum mechanics that I learned in graduate school. I suspect that its influence will be observed in a condensed matter setting in due time.

Please feel free to share your own ideas or concepts you would add to the list.

General Aspects of Topology in Quantum Mechanics

Condensed matter physics has, in the past ten years or so, made a left turn towards studying topological properties of materials. Following the discovery of the Quantum Hall Effect (QHE) in 1980, it took about 25 years to experimentally discover that similar phenomenology could occur in bulk samples in the absence of a magnetic field in topological insulators. In the current issue of Nature Physics, there are three papers demonstrating the existence of a Weyl semimetal in TaAs and NbAs. These states of matter bear a striking similarity to quantum mechanical effects such as the Aharonov-Bohm effect and the Dirac monopole problem.

So what do all of these things have in common? Well, I vaguely addressed this issue in a previous post concerning Berry phases, but I want to elaborate a little more here. First it should be understood that all of these problems take place on some sort of manifold. For instance, the Aharonov-Bohm effect takes place in a plane, the Dirac monopole problem on a 3D sphere and the problems in solid-state physics largely on a torus due to periodic boundary conditions.

Now, what makes all of these problems exhibit a robust topological quantization of some sort is that the Berry connection in these problems cannot adequately be described by a single function over the entire manifold. If one were to attempt to write down a function for the Berry connection, there would necessarily exist a singularity somewhere on the manifold. But because the Berry connection is not an observable, one can just write down two (or more) different functions on different parts (or “neighborhoods”) of the manifold. The price one has to pay is that one has to “patch” the functions together at the boundary of the neighborhoods. Therefore, the existence of the topological quantization in most of the problems described above arise because of a singularity in the Berry connection somewhere on the manifold that cannot be gotten rid of with a gauge transformation.

For instance, for the Aharonov-Bohm effect, the outside of the solenoid and the inside of the solenoid must be described by different functions, or else the “outside function” would be singular at the center of the solenoid.  Qualitatively, one can think of the manifold as a plane with a hole punched in the middle of it. In the case of the Dirac monopole, the magnetic monopole itself is the position of the singularity and there is a hole punched in 3-dimensional space.

There is an excellent discussion on both these problems in Sakurai’s quantum mechanics textbook. I particularly like the approach he takes to the Dirac monopole problem, which he adapted from Wu and Yang’s elegant solution. The explanation of the QHE using similar ideas was developed in this great (but unfortunately quite mathematical) paper by Kohmoto (pdf!). I realize that this post only sketches the main point (with perhaps too much haste), but I hope that it will be illuminating to some.

Update: I have written a guest post for Brian Skinner’s blog Gravity and Levity where I discuss the topics here in a little more detail. You can read the post here if you’re interested.

Modern Theory of Polarization

It is quite curious that the simple concept of polarization in a solid was not understood until the early to mid-90s. The solution to the problem actually came from the computational physics community because of their inability to calculate accurately the polarization in solids. Prior to the solution, there were papers seriously discussing whether polarization was a bulk phenomenon or whether it was a property of a crystal surface. The solution to the problem is commonly associated with these papers by Resta (pdf!) and King-Smith and Vanderbilt (pdf!) .

There are a few startling realizations that arose from the modern theory of polarization:

1. Polarization is not a well-defined quantity in that it is multi-valued.
2. Only polarization difference has any physical meaning.
3. Experiments only measure differences in polarization.
4. Polarization is deeply rooted in the concept of the Berry phase.
5. (not startling) Polarization is a bulk phenomenon.

There is a fantastic pedagogical introduction to the subject by Spaldin entitled A beginner’s guide to the modern theory of polarization.

As just a little preview of the Spaldin paper, let me outline the issues as she does. The problem with defining polarization as the dipole moment per unit cell can easily be understood using the picture of the one-dimensional chain below:

Using the box on the left, one would calculate the dipole moment per unit length as:

whereas the box on the right gives:

.

That these don’t match has to do precisely with the fact that polarization in a solid is multi-valued, as alluded to earlier. Now consider the following one-dimensional lattice with a distortion:

The polarization now for the distorted lattice (second row in the image above) for the left and right boxes are respectively:

Left Box:

Right Box:

We can therefore see that for both boxes, the change in polarization is , which is a single-valued and experimentally well-defined quantity.

While this illustration was classical, when one includes the wavefunction of the electrons, one is forced to consider the Berry phase of the Bloch electrons. While I have known about this result for some time now, I still find it quite surprising that the simple concept of polarization in a solid has any relationship to the Berry phase at all. I strongly recommend Spaldin’s eminently readable article as an excellent introduction to the subject.

It’s Not Just a (Berry) Phase

Just a few years after Berry’s breakthrough paper (pdf!) on the phase now named after him, it was recognized by Zak (pdf!) that this effect could play a role in a solid state setting. Zak realized that in the Bloch Hamiltonian, the crystal momentum, k, could be treated as a “parameter” similar to how other “parameters” had been treated in Berry’s original work. More concretely:

While phenomena that are now considered prototypical Berry phase effects were understood before the Berry paper, it was Berry who unified many concepts under a general framework. For instance, the Aharonov-Bohm effect, the Dirac monopole problem and the Integer Quantum Hall Effect were all understood separately, but the Berry phase concept presented a huge leap forward in consolidating these seemingly disparate phenomena.

It should be mentioned that the three cases mentioned above are all instances where the Berry phase is topological and results in a robust quantization through a topological invariant (known as the Chern number). However, the Berry phase does not have to be topological in nature and can have “geometrical” consequences. Many of these “geometrical” consequences are discussed in a great review article by Xiao, Chang and Niu.

One particular “geometrical” consequence stands out: the semiclassical equations of motion for electrons in a solid, as detailed in textbooks such as Ashcroft and Mermin, are incomplete! The application of an electric field in crystals with either broken time-reversal symmetry or broken inversion symmetry may, under certain circumstances, exhibit a transverse velocity associated with a non-zero Berry curvature (i.e. an application of an electric field in the x-direction will result in electrons travelling in the y-direction!). This experimental paper by NP Ong’s group demonstrates this effect (pdf!), and this paper by Karplus and Luttinger (paywall!) theoretically explains the transverse velocity in ferromagnets, again before Berry’s seminal paper.

I’ll also mention briefly that the seemingly mundane phenomenon of electrical polarization in solids is another case that cannot be explained without Berry-esque concepts, but I’ll leave that topic for another day.

It truly is stunning how many effects can be accounted for under the umbrella of Berry/Zak ideas. I’ll even restrain myself from mentioning topological insulators and Chern insulators (paywall) / the quantum anomalous hall effect (paywall).