# Tag Archives: Order Parameters

## 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.

## Spin Fluctuations in the Cuprates

In the history of the cuprate superconductors, many predictions have been put forth, but there was one non-trivial prediction that has stood out among the rest. This is the prediction of the $d_{x^2-y^2}$ order parameter symmetry from spin-fluctuation models that were put forth before its experimental verification.

The idea is quite simple, and you can read more about it in this set of lecture notes by A.J. Leggett, where he lays out the concepts very well. I summarize the main points below.

In the cuprates, the Fermi surface is usually assumed to look like so, which has been determined by ARPES experiments:

Schematic of Fermi Surface as Determined by ARPES

One also knows that the antiferromagnetic phase in the parent compound looks like so:

(a) Real space representation of the antiferromagnetic parent phase. (b) Reciprocal space representation with Q representing the antiferromagnetic Bragg wavevector.

i)     Now, one can see that the points on the Fermi surface close to ($\pi$, 0) and (0, $\pi$) are the ones connected by the antiferromagnetic Bragg wavevector, Q. One would then predict a singlet pair wavefunction, as those points on the Fermi surface would be expected to exhibit the largest gap.

ii)     The other input is that scattering should not change the sign of the pair wave function, $F_\textbf{k} = orbital\times spin$, which comprises the orbital and spin components. Since the spin part is a singlet, (i.e. $1/\sqrt{2} (\uparrow_1\downarrow_2-\downarrow_1\uparrow_2)$), it will change sign when the pair interacts through a spin-fluctuation. Therefore, to keep $F_\textbf{k}$ invariant, the orbital part must also change sign under the scattering/interaction of wavevector Q.

The two criteria leave $d_{x^2-y^2}$ symmetry as the only option, and hence spin-fluctuation theories explicitly predict this symmetry.

Obviously, this does not mean that spin-fluctuation theories are correct, but it is worth noting that they have made a non-trivial prediction.

While this historical note is well-known to those have been studying high-temperature superconductivity since its discovery, those of us who were born around the same time as the discovery of the cuprates sometimes lose this kind of historical context.

Images are taken from the lecture notes by A.J. Leggett linked above.

## Simple, beautiful and decisive.

Condensed matter physics has seen its fair share of landmark experiments. In the field of high temperature superconductivity, there is one series that stands out among the rest, however. These are the Josephson Interference experiments conducted in the early 90s by van Harlingen and co-workers. These investigations were technically challenging, and some hard work no doubt went into the experimental design.

To understand the impact of the experiments, a little background and historical context is needed. At the time, the field of high temperature superconductivity was approximately seven years old and the symmetry of the superconducting order parameter was an open question. It was known that the order parameter had nodes, but whether the order parameter changed sign (i.e. was d-wave or extended s-wave) was unresolved.

The Josephson Interference experiments unambiguously settled this issue. The second iteration of the experiment (PDF link!), was particularly decisive. Essentially, the expected critical current as a function of flux would look like so for an s-wave and d-wave superconductor respectively:

The authors obtained the latter pattern and the order parameter symmetry problem was solved (the symmetry was d-wave). It is not often in condensed matter physics that experiments are this clean, unambiguous and illuminating. When they are, however, they deserve to be celebrated.