Not all triangles are the same

Just the other day I wrote about a revelation I had about the self healing process, a hot topic in our lab these days. As often happens, the first impression is simplistic and not quite right, but eventually, we hopefully converge on the truth. However, my fallacy of yesterday gave me insights today, which I continue to pursue.

On another front, we have completed a new paper that we are submitting to Physical Review Letters, the highest impact physics journal. I always have reservations about sending a manuscript to a journal just because it is prestigious. What counts is the quality of the paper. On the other hand, if our work is as significant as we believe, then appearing in a top journal will give it more visibility.

I am excited by the science, and the possibility that we may have started a new branch of study. At the heart of our work are calculations of the optical nonlinearity of quantum wires. This in itself is totally new (to the best of our knowledge), but we are taking the elevator down a level to the realm of fundamental science. For those more practically minded, our work may also have some useful applications.

Science is often focused on a particular thing. While a researcher may be interested in solving the problem of global warming -- a grand problem, the actual work may involve studying the behavior of a particular kind of electrode dipped in a specific chemical. In fact, many groups around the world may be studying exactly the same thing, trying to work out a detail that could make a battery store 5% more energy. Such a leap would indeed be important.

Rather than focusing on details, our work is painting a big picture. I like ideas that have broad influence; views of the world from unique perspectives; and unexpected results on topics that have not crossed anyone's mind, but that resonate with all scientists as being really neat.

Our new work falls beyond the typical boundaries of what others are doing. We are interested in the abstract concept of how the shape (geometry) and topology of an object determine its optical properties. These ideas go beyond specific molecules or materials. To allow us to focus on the basics, we need to remove other complications. To that end, we study what is often called a toy model -- one that brings out the qualities of interest and suppresses the rest. In our case, we are considering structures made of connected wire segments that carry a sole electron.

Consider a continuous loop of wire in the shape of a triangle. If we deform this triangle into other triangles with differing edge lengths and angles, we find that the nonlinearity changes smoothly and not a hell of a lot. In fact, deform the triangle into a quadrangle and then into a quintangle, and nothing much new happens. Any closed loop, independent of the shape, is of the same topology. Thus, we might conclude that the geometry has little effect on the nonlinear response.

A bent wire that does not form a loop is of a different topology. So, consider the simple experiment of a triangle whose nonlinear-optical response is being measured. Now cut a vertex of the triangle so that two of the edges no longer touch. This is still a triangle but its topology has changed. Interestingly, the nonlinear response is found to be profoundly different with the snip of the wire cutters. Thus a change in topology for fixed geometry leads to a dramatic change of the nonlinear-optical response.

This work has applications in the design of better materials because it suggests that taking a molecule (modeled as a wire) and lopping off just a single bond could yield a dramatic improvement. Or, our work could inform nano-technologists on how to make better quantum wires.

We have only evaluated a small number of shapes, including loops made into triangles, quadrangles, quintangles, bent wires, split triangles, and star graphs. Star graphs, which are lines radiating from a central point, represent a topology that yields the larges hyperpolarizability.

To sample the space of all possible shapes, we let the computer randomly pick triangles, quadrangles, quitangles, star graphs, and whatever other shape we can squeeze in. Then we can see what is possible. With enough random tries -- we usually run our simulations over tens of thousands of configurations -- we can test the influence of any parameter, such as topology.

Below is a plot of the first (left) and second (right) hyperpolarizability, which tells us how strongly two and three photons interact with a molecule. Included are triangles (red), simple quadrilaterals (with no crossing edges - green), and all quadrilaterals (blue). Each point (and there are 10,000 here of each color), represents one configuration. A casual glance at the pattern reveals that geometrical effects do not make a big difference. To see the effects of topology, you'll have to read our paper on The Physics Archives.



I find this work really neat (and I hope the reviewers will agree) because we are sampling a very fundamental property of a molecule in terms of some very simple mathematical concepts that go back hundreds to thousands of years. The ancient Greeks heard the music of the spheres in planetary motion using a the metaphorical geometric ear. In our work, we can literally see the effects with light on our eyes when the system's structure changes so ever subtly. And, we get to enjoy a vision of the underlying process with the minds eye as portrayed in very pretty and colorful plots.

Heavy Matters

I am now sitting at my desk and preparing for class, leafing through an old heavy dust-covered book with the simple title Gravitation., by Misner, Thorne and Wheeler. As an undergraduate, I had great aspirations that included reading this 1200-page tome. As is the case with many of my books, it sat on my shelf for decades without notice.

It all started a few minutes ago, when I recalled that this book clearly explained differential forms, a topic that I plan to introduce in my class today. So, I climbed my library ladder and hauled it to my desk. Paging through the book, I marveled at its beauty, both in presentation style and illustrations. I still consider it a difficult read, but I am now better equipped to understand the physics.

This wonderful book not only took the authors and their helpers many person-years of effort, but it covers a range of topics that have been developed by the most brilliant minds of the past century. I am thankful to the efforts of all those who contributed to this field, and as a result, enriched my life.

The dedication says it best:

We dedicate this book to our fellow citizens who, for love of truth, take from their own wants by taxes and gifts, and now and then send forth one of themselves as dedicated servant, to forward the search into the mysteries and marvelous simplicities of this strange and beautiful Universe, our home.

Differential Geometry

I have been enjoying the last couple of days preparing for my Statistical Mechanics class. My new angle is to throw in a bit of differential geometry to expose students to its beauty as well as to give them a new and powerful tool to solve problems.

This morning, I was going over my notes on canonical transformations - a topic that I also covered in Classical Mechanics last semester. One concept that I still find a tad uncomfortable is in the manipulations of the quantities q, the coordinate, and q dot, its associated velocity. Central to the calculus of variations is the assumption that these two variables are independent. I can logically understand how this is the case, but my gut protests.

One of my colleagues, who shares an interest in the beauty of differential geometry, gave me the book Applied Differential Geometry, by William L. Burke. I noticed recently the most magnificent dedication on its inner cover, which reads, "To all those who, like me, have wondered how in the hell you can change q dot without changing q."

I can't wait to dive into this wonderful book; but, it will have to wait until I meet a looming proposal deadline and have prepared/delivered an invited talk in a few days. I must get back to preparing for my class, which meets in 45 minutes. Mornings before class are one of my favorite times...

Another semester and another new class - general relativity and thermodynamics

When I was an undergraduate at the University of Pennsylvania, I took a class in general relativity (GR). It was taught by the eccentric Professor Jeffrey Cohen, not to be confused with the equally eccentric Professor Michael Cohen.

Michael Cohen had instilled in me a deep appreciation for truly understanding physics. Just as Michael Cohen felt that he would never attain the depth of understanding commanded by his adviser, THE Richard Feynman , I too feel that I will never approach the physical intuition of Michael Cohen. It is fortunate that the singularities that we call great physicists are born with abilities far superior to their contemporaries.

After more than three decades have passed, I recall little from my undergraduate flirtation with GR. However, some of the mathematical formalism of differential forms has taunted me for much of my career. I recall Jeffrey Cohen mentioning a paper on the topic of the properties of a black hole in some complex geometry that took forty pages of derivations in an article that appeared in The Physical Review. Using the trickery of differential forms, he was able to solve the problem in just a few steps.

The trick was to formulate the problem in a coordinate independent way, then to project the results into the coordinate system that reflected the symmetry of the problem. In contrast, the Physical Review paper used the inelegant brute-force approach of picking the coordinate system up front, and then by necessity painstakingly plodding through all the messy mathematics.

Given the complexity of the problems that we work on as a matter of daily routine in our research, I am always looking for simplifying tools. In teaching various classes, my intention is to sneak in a little bit of differential forms to wet the appetites of my acolytes and to teach my old brain some new tricks. Furthermore, the geometric interpretation of the mathematics adds a deeper layer of understanding.

In the upcoming spring semester, I am teaching graduate statistical mechanics for my first time. As usual, preparing for a new course if filled with grand excitement. You can imagine my elation when I realized that a homework assignment in the textbook could be done with ease using differential geometry. Since then, it has been difficult for me to think about anything else.

The problem is a simple one that normally requires a bit of math. The student is to show that the 6N-dimensional volume element in phase space for an N-particle system is invariant under a canonical transformation. To put this into simple English, the problem seeks to show that a transformation of coordinates does not change the nature of the results. Be reformulating the problem so that the volume element is represented as a wedge product of what are called one-forms, the volume element is shown to be the same when the so-called Poisson bracket yields unity -- the requirement of a canonical transformation. Thus, the problem is solved without the need for messy mathematics.

This realization makes me feel like a kid at Christmas. Ironically, tonight is Christmas Eve, the focal point of my family's celebration. My father has made what may be his last trip to Pullman from Philadelphia. He is 94 and still lives on his own, drives a car, and prepares meals for senior citizens at the Ukrainian Cultural Center in Fox Chase, Pennsylvania. Though still vigorous, his body betrays the telltale signs of wear and tear due to old age. Both of my children are home for the holidays, and all the fragrance from the traditional Ukrainian foods simmering on the stove and in the oven permeate the house. As I write this post, my wife is busily making last-minute preparations.

It is fortunate for me that my family values my passion for physics, and allows me to occasionally be a recluse. Just a few minutes ago, my wife called out a query about my whereabouts. I simply answered, "I am excited about something." Though she undoubtedly had some mundane duty for me to perform, she immediately signaled her understanding of my state of mind, and left me alone. I am truly fortunate to be living with someone who shares in my passions.

The intensity and meaningfulness of spirituality that I derive from physics far exceeds all others, including the times in my distant past when I had embraced religion. As my family turns in for the night, I continue to sit at my desk, full of excitement in my new-found understanding, and looking forward to sharing this understanding with my family and my students. It is a truly privileged life that allows me to rekindles the child-like wonder of Christmas on a daily basis.