# Tag Archives: L. Landau

## Landau Theory and the Ginzburg Criterion

The Landau theory of second order phase transitions has probably been one of the most influential theories in all of condensed matter. It classifies phases by defining an order parameter — something that shows up only below the transition temperature, such as the magnetization in a paramagnetic to ferromagnetic phase transition. Landau theory has framed the way physicists think about equilibrium phases of matter, i.e. in terms of broken symmetries. Much current research is focused on transitions to phases of matter that possess a topological index, and a major research question is how to think about these phases which exist outside the Landau paradigm.

Despite its far-reaching influence, Landau theory actually doesn’t work quantitatively in most cases near a continuous phase transition. By this, I mean that it fails to predict the correct critical exponents. This is because Landau theory implicitly assumes that all the particles interact in some kind of average way and does not adequately take into account the fluctuations near a phase transition. Quite amazingly, Landau theory itself predicts that it is going to fail near a phase transition in many situations!

Let me give an example of its failure before discussing how it predicts its own demise. Landau theory predicts that the specific heat should exhibit a discontinuity like so at a phase transition:

However, if one examines the specific heat anomaly in liquid helium-4, for example, it looks more like a divergence as seen below:

So it clearly doesn’t predict the right critical exponent in that case. The Ginzburg criterion tells us how close to the transition temperature Landau theory will fail. The Ginzburg argument essentially goes like so: since Landau theory neglects fluctuations, we can see how accurate Landau theory is going to be by calculating the ratio of the fluctuations to the order parameter:

$E_R = |G(R)|/\eta^2$

where $E_R$ is the error in Landau theory, $|G(R)|$ quantifies the fluctuations and $\eta$ is the order parameter. Basically, if the error is small, i.e. $E_R << 1$, then Landau theory will work. However, if it approaches $\sim 1$, Landau theory begins to fail. One can actually calculate both the order parameter and the fluctuation region (quantified by the two-point correlation function) within Landau theory itself and therefore use Landau theory to calculate whether or not it will fail.

If one does carry out the calculation, one gets that Landau theory will work when:

$t^{(4-d)/2} >> k_B/\Delta C \xi(1)^d \equiv t_{L}^{(4-d)/2}$

where $t$ is the reduced temperature, $d$ is the dimension, $\xi(1)$ is the dimensionless mean-field correlation length at $T = 2T_C$ (extrapolated from Landau theory) and $\Delta C/k_B$ is the change in specific heat in units of $k_B$, which is usually one per degree of freedom. In words, the formula essentially counts the number of degrees of freedom in a volume defined by  $\xi(1)^d$. If the number of degrees of freedom is large, then Landau theory, which averages the interactions from many particles, works well.

So that was a little bit of a mouthful, but the important thing is that these quantities can be estimated quite well for many phases of matter. For instance, in liquid helium-4, the particle interactions are very short-ranged because the helium atom is closed-shell (this is what enables helium to remain a liquid all the way down to zero temperatures at ambient conditions in the first place). Therefore, we can assume that $\xi(1) \sim 1\textrm{\AA}$, and hence $t_L \sim 1$ and deviations from Landau theory can be easily observed in experiment close to the transition temperature.

Despite the qualitative similarities between superfluid helium-4 and superconductivity, a topic I have addressed in the past, Landau theory works much better for superconductors. We can also use the Ginzburg criterion in this case to calculate how close to the transition temperature one has to be in order to observe deviations from Landau theory. In fact, the question as to why Ginzburg-Landau theory works so well for BCS superconductors is what awakened me to these issues in the first place. Anyway, we assume that $\xi(1)$ is on the order of the Cooper pair size, which for BCS superconductors is on the order of $1000 \textrm{\AA}$. There are about $10^8$ particles in this volume and correspondingly, $t_L \sim 10^{-16}$ and Landau theory fails so close to the transition temperature that this region is inaccessible to experiment. Landau theory is therefore considered to work well in this case.

For high-Tc superconductors, the Cooper pair size is of order $10\textrm{\AA}$ and therefore deviations from Landau theory can be observed in experiment. The last thing to note about these formulas and approximations is that two parameters determine whether Landau theory works in practice: the number of dimensions and the range of interactions.

*Much of this post has been unabashedly pilfered from N. Goldenfeld’s book Lectures on Phase Transitions and the Renormalization Group, which I heartily recommend for further discussion of these topics.

## Fractional quasiparticles and reality

As a condensed matter physicist, one of the central themes that one must become accustomed to is the idea of a quasiparticle. These quasiparticles are not particles as nature made them per se, but only exist inside matter. (Yes, nature made matter too, and therefore quasiparticles as well, but come on — you know what I mean!)

Probably the first formulation of a quasiparticle was in Einstein’s theory of specific heat in a solid at low temperature. He postulated that the sound vibrations in a solid, much like photons from a blackbody, obeyed the Planck distribution, implying some sort of particulate nature to sound. This introduction was quite indirect, and the first really explicit formulation of quasiparticles was presented by Landau in his theory of He4. Here, he proposed that most physical observables could be described in terms of “phonons” and “rotons“, quantized sound vibrations at low and high momenta respectively.

In solid state physics, one of the most common quasiparticles is the hole; in the study of magnetism it is the magnon, in semiconductor physics, the exciton is ubiquitous and there are many other examples as well. So let me ask a seemingly benign question: are these quasiparticles real (i.e. are they real particles)?

In my experience in the condensed matter community, I suspect that most would answer in the affirmative, and if not, at least claim that the particles observed in condensed matter are just as real as any particle observed in particle physics.

Part of the reason I bring this issue up is because of concerns raised soon following the discovery of the fractional quantum Hall effect (FQHE). When the theory of the FQHE was formulated by Laughlin, it was thought that his formulation of quasiparticles of charge e/3 may have been a mere oddity in the mathematical description of the FQHE. Do these particles carrying e/3 current actually exist or is this just a convenient mathematical description?

In two papers that appeared almost concurrently, linked here and here, it was shown using quantum shot noise experiments that these e/3 particles did indeed exist. Briefly, quantum shot noise arises because of the discrete nature of particles and enables one to measure the charge of a current-carrying particle to a pretty good degree of accuracy. In comparing their results to the models of particles carrying charge e versus particles carrying charge e/3, the data shows no contest. Here is a plot below showing this result quite emphatically:

One may then pose the question: is there a true distinction between what really “exists out there” versus a theory that conveniently describes and predicts nature? Is the physicist’s job complete once the equations have been written down (i.e should he/she not care about questions like “are these fractional charges real”)?

These are tough questions to answer, and are largely personal, but I lean towards answering ‘yes’ to the former and ‘no’ to the latter. I would contend that the quantum shot noise experiments outlined above wouldn’t have even been conducted if the questions posed above were not serious considerations. While asking if something is real may not always be answerable, when it is, it usually results in a deepened understanding.

This discussion reminds me of an (8-year old!) YouTube video of David who, following oral surgery to remove a tooth, still feels the affects of anesthesia :

## Paradigm Shifts and “The Scourge of Bibliometrics”

Yesterday, I attended an insightful talk by A.J. Leggett at the APS March Meeting entitled Reflection on the Past, Present and Future of Condensed Matter Physics. The talk was interesting in two regards. Firstly, he referred to specific points in the history of condensed matter physics that resulted in (Kuhn-type) paradigm shifts in our thinking of condensed matter. Of course these paradigm shifts were not as violent as special relativity or quantum mechanics, so he deemed them “velvet” paradigm shifts.

This list, which he acknowledged was personal, consisted of:

1. Landau’s theory of the Fermi liquid
2. BCS theory
3. Renormalization group
4. Fractional quantum hall effect

Notable absentees from this list were superfluidity in 3He, the integer quanutm hall effect, the discovery of cuprate superconductivity and topological insulators. He argued that these latter advances did not result in major conceptual upheavals.

He went on to elaborate the reasons for these velvet revolutions, which I enumerate to correspond to the list above:

1. Abandonment of microscopic theory, in particular with the use of Landau parameters; trying to relate experimental properties to one another with the input of experiment
2. Use of an effective low-energy Hamiltonian to describe phase of matter
3. Concept of universality and scaling
4. Discovery of quasiparticles with fractional charge

It is informative to think about condensed matter physics in this way, as it demonstrates the conceptual advances that we almost take for granted in today’s work.

The second aspect of his talk that resonated strongly with the audience was what he dubbed “the scourge of bibliometrics”. He told the tale of his own formative years as a physicist. He published one single-page paper for his PhD work. Furthermore, once appointed as a lecturer at the University of Sussex, his job was to be a lecturer and teach from Monday thru Friday. If he did this job well, it was considered a job well-done. If research was something he wanted to partake in as a side-project, he was encouraged to do so. He discussed how this atmosphere allowed him to develop as a physicist, without the requirement of publishing papers for career advancement.

Furthermore, he claimed, because of the current focus on metrics, burgeoning young scientists are now encouraged to seek out problems that they can solve in a time frame of two to three years. He saw this as a terrible trend. While it is often necessary to complete short-term projects, it is also important to think about problems that one may be able to solve in, say, twenty years, or maybe even never. He claimed that this is what is meant by doing real science — jumping into the unknown. In fact, he asserted that if he were to give any advice to graduate students, postdocs and young faculty in the audience, it would be to try to spend about 20% of one’s time committed to some of these long-term problems.

This raises a number of questions in my mind. It is well-acknowledged within the community and even the blogosphere that the focus on publication number and short term-ism within the condensed matter physics community is detrimental. Both Ross McKenzie and Doug Natelson have expressed such sentiment numerous times on their blogs as well. From speaking to almost every physicist I know, this is a consensus opinion. The natural question to ask then is: if this is the consensus opinion, why is the modern climate as such?

It seems to me like part of this comes from the competition for funding among different research groups and funding agencies needing a way to discriminate between them. This leads to the widespread use of metrics, such as h-indices and publication number, to decide whether or not to allocate funding to a particular group. This doesn’t seem to be the only reason, however. Increasingly, young scientists are judged for hire by their publication output and the journals in which they publish.

Luckily, the situation is not all bad. Because so many people openly discuss this issue, I have noticed that the there is a certain amount of push-back from individual scientists. On my recent postdoc interviews, the principal investigators were most interested in what I was going to bring to the table rather than peruse through my publication list. I appreciated this immensely, as I had spent a large part of my graduate years pursuing instrumentation development. Nonetheless, I still felt a great deal of pressure to publish papers towards the end of graduate school, and it is this feeling of pressure that needs to be alleviated.

Strangely, I often find myself in the situation working despite the forces that be, rather than being encouraged to do so. I highly doubt that I am the only one with this feeling.

## Excitonic Insulator

The state of matter dubbed the excitonic insulator was first qualitatively discussed by Mott, Keldysh and Kopaev, and others and then expanded upon more systematically by Jerome, Rice and Kohn.

The excitonic insulating state can be considered from two normal states (pictured below). Either the system must be a small-gap semiconductor or a small indirect overlap semimetal. In fact, Mott had first considered the semimetallic limit, while Kohn and others had considered the semiconducting limit.

Intuitively, one can consider the following two heuristic arguments from the different limits, as presented in the article by Rossnagel, which was cited in the previous post:

1. Semiconducting limit: If one can somehow reduce the band gap energy, $E_G$, then at some point, the binding energy to form an exciton, $E_B$, will exceed $E_G$, and the system will unstable to the spontaneous formation excitons.
2. Semimetallic limit: In this case, one considers screening effects. If one decreases the band overlap, a characteristic energy, $E_1$, will be reached such that particle-hole pairs will be insufficiently screened, leading to a localization of the charge carriers.

Therefore, in the regime of $E_1$$E_G$ <$E_B$, the excitonic insulator state is expected. Properties of the excitonic insulator state are presented pedagogically in a Les Houches lecture by Kohn in this book, which is very difficult to find!

In a solid state context, it has been difficult to establish whether the excitonic insulator state has been realized because a lattice distortion is expected to accompany the transition to the excitonic insulator ground state. Therefore, it is difficult to isolate the driving mechanism behind the transition (this difficulty will be familiar to those who study high T-c superconductivity!).

There are a few materials suspected to possess excitonic insulator ground states in a solid state setting: 1T-TiSe$_2$, Ta$_2$NiSe$_5$ and TmSe$_{0.45}$Te$_{0.55}$. In my personal opinion, the case for 1T-TiSe$_2$ is probably the strongest purely because there have been far more experiments on this material than the other candidate materials.

Though this state of matter was considered almost 50 years ago, it still remains relevant today. As Landau once said,

Unfortunately, everything that is new is not interesting, and everything which is interesting, is not new.