# Tag Archives: Broken Symmetries

## 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.

## Transition Metal Dichalcogenide CDWs

There is an excellent review paper by K. Rossnagel on the origin of charge density waves (CDWs) in the transition metal dichalcogenide compounds (such as 2H-NbSe$_2$, 1T-TiSe$_2$, 2H-TaSe$_2$, etc.) . A lot of the work on these materials was undertaken in the 70s and 80s, but there has been a recent revival of interest because of the nature of superconductivity in a few of these compounds.

By “nature”, I mean that  the phase diagrams in these materials bear a striking resemblance to the phase diagram in the cuprates, except that the anti-ferromagnetism is replaced by a CDW phase. Shown below is the phase diagram for 1T-TiSe$_2$ under pressure and with copper intercalation (taken from this paper).

Strangely, with copper intercalation, the Hall resistance is negative, while  it is positive under pressure. This is interesting because like the cuprates, superconductivity can be brought about with either electrons or holes as majority carriers. A similar phase diagram is also observed for another TMD 1T-TaS$_2$ (see here for instance). 1T-TaS$_2$ has also been shown to exhibit Mott physics at low temperature in the parent compound.

It is suspected that the origin of the CDWs in 1T-TiSe$_2$ and 1T-TaS$_2$ are at least in part electronically driven (see Rossnagel’s review article and references therein). This makes the observation of the superconductivity in these compounds all the more interesting — as the superconductivity may also be primarily electronically driven. I have also blogged previously about another set of CDW materials (the rare earth tritellurides) that exhibit cuprate-like phase diagrams including an antiferromagnetic phase, and also about the interplay between CDWs and superconductivity in NbSe$2$.

It seems to me that there are some really quite fundamental open questions in the study of these compounds, which is in part why I keep re-visiting this topic myself.

## Transport Signatures in Charge Density Wave Systems

This post is inspired in part by Inna’s observation that a Josephson junction can act as a DC-AC converter. It turns out that CDWs can also act in a similar manner.

Sometimes I feel like quasi-1D charge density waves (CDWs) are like the lonely neglected child compared to superconductors, the popular, all-star athlete older sibling. Of course this is so because superconductors carry dissipationless current and exhibit perfect diamagnetism. However, quasi-1D CDWs can themselves exhibit pretty stunning transport signatures associated with the CDW condensate. Note that these spectacular properties are associated with incommensurate CDWs, as they break the translational symmetry of the crystal.

To make a comparison with superconductivity (even though no likes to be compared to their older sibling), here is a cartoon of the frequency-dependent conductivity (taken from G. Gruner’s Review of Modern Physics entitled Dynamics of Charge Density Waves):

Frequency-dependent conductivity for (a) a superconductor and (b) an incommensurate CDW

In the superconducting case, there is a delta function at zero frequency, indicative of dissipationless transport. For the CDW, there is also a collective charge transport mode, but in this case it is at finite energy (as it is pinned by impurities), and it is dissipative (indicated by the finite width).

This collective charge transport mode can be “depinned” and results in a nonlinear conductivity known as  a sliding CDW. This is evidenced below in the I-V characteristics. Below a threshold electric field/voltage, usual Ohmic characteristics are observed, associated with the “normal” non-condensed electrons. However, above the threshold electric field/voltage the collective mode is depinned and contributes to the I-V characteristics.

Non-linear IV characteristics indicative of collective charge transport in the CDW phase

Even more amazingly, once this CDW has been depinned, applying a DC field results in an AC response. Below is an image from a famous paper by Fleming and Grimes showing the Fourier transformed AC response with several harmonics. As the voltage is turned up, the fundamental frequency increases markedly (the voltage is highest in (a) and is decreased slowly until (e) where the CDW is no longer sliding).

AC response to a DC applied voltage in order of decreasing DC voltage in NbSe3. (a) V=5.81mV, (b) V=5.05mV, (c) 4.07mV, (d) V=3.40mV (e) V=0

The observed oscillation frequency is due to the collective mode getting depinned from its impurity site and then getting  weakly pinned successively by impurities, though this picture is debated. N.P. Ong, who did some great early work on CDW transport, has noted that the CDW “sings”. A nice cartoon of this idea is presented in the ball-and-egg-crate model shown below. One can imagine the successive “hits in the road” at periodic time intervals resulting in the AC response seen above.

Ball and egg crate model of CDW transport

Hopefully this post will help people appreciate more the shy younger sibling that is the charge density wave.

All images taken from G. Gruner RMP 60, 1129 (1988).

## Plasmons, the Coulomb Interaction and a Gap

In a famous 1962 paper entitled Plasmons, Gauge Invariance and Mass (pdf!), P.W. Anderson described the relationship between the gap in the plasmon spectrum and the idea of spontaneous symmetry breaking. It is an interesting historical note that Higgs cites Anderson’s paper in his landmark paper concerning the Higgs mechanism.

While there are many different formulations, including Anderson’s, of why the plasmon is gapped at zero momentum in a 3D solid, they all rely on one crucial element: the long-range nature of the Coulomb interaction (i.e. the electrons are charged particles). Of these formulations, I prefer one “cartoon-y” explanation which captures the essential physics well.

Before continuing, let me stress that it is quite unusual for a fluid medium (such as the electrons in a metal) to possess no zero frequency excitations at long wavelengths. For instance,  the dispersion relation for surface gravity waves on water (pdf!) is:

$\omega^2(k)=gk \tanh kh$.

Now, in 3D and in the long-wavelength limit, the plasmon sets up opposite charges on the surfaces of the solid as pictured below:

The long-wavelength plasmon therefore sets up the same electric field as in a capacitor. The electric field for a capacitor is $\textbf{E} = \frac{\sigma\hat{x}}{\epsilon_0}$. This expression is surprisingly independent of the distance separating the surfaces of the solid. Therefore, it takes a finite amount of energy to set up this electric field, even in the limit of infinite distance. This finite energy results in the gapping of the plasmon.

This cartoon can be extended further to 2D and 1D solids. In the 2D case, the electric field for the 1D “lines of charge” bounding the solid falls off like $\textbf{E}\sim\frac{1}{\textbf{r}}$. Therefore, in the infinite distance limit, it takes no energy to create this electric field and the plasmon is not gapped at $\textbf{q}=0$. Similarly, for the 1D case, the electric field from the points bounding the solid falls of as $\frac{1}{\textbf{r}^2}$, and the plasmon is again gapless.

This reasoning can be applied further to the phenomenon known as LO-TO splitting in a polar solid. Here, the longitudinal optical phonon (LO) and the transverse optical phonon (TO) branches are non-degenerate down to the very lowest (but non-zero!) momenta. Group theory predicts these modes to be degenerate at $\textbf{q}=0$ for the zincblende crystal structure of typical semiconducting compounds. Below is the phonon dispersion for GaAs demonstrating this phenomenon:

Again, the splitting occurs due to the long-ranged nature of the Coulomb interaction. In this case, however, it is the polar ionic degree of freedom that sets up the electric field as opposed to the electronic degrees of freedom. Using the same reasoning as above, one would predict that the LO-TO splitting would disappear in the 2D limit, and a quick check in the literature suggests this to be the case as reported in this paper about mono-layer Boron Nitride.

I very much appreciate toy models such as this that give one enough physical intuition to be able to predict the outcome of an experiment. It has its (very obvious!) limitations, but is valuable nonetheless.

## How do we define states of matter?

Historically, many people seemed to lean towards defining a phase of matter by its (broken) symmetries. For instance, a ferromagnet has broken rotational symmetry and time-reversal symmetry, a solid has broken translational and rotational symmetry, etc. In light of the discoveries of the Quantum Hall Effect and topological insulators, it seems like this symmetry classification does not encompass all states of matter.

The symmetry classification is largely a theoretical construct, however. I would think that one defines a state of matter by particular experimental properties that it exhibits. For example, one could define a superconductor by requiring it to exhibit the following:

1. Zero Resistivity
2. Meissner Effect
3. Zero Peltier Coefficient

Put another way, to verify that one has discovered a superconductor, these three criteria must be satisfied.

Let us take another example: a simple metal. The criterion that must be satisfied for this case is the existence of a Fermi surface. This can be measured by quantum oscillation measurements, angle-resolved photoemission, or a few other probes.

Yet another example: a 2D topological insulator. What one must observe is:

1. The Fermi energy intersects an odd number of topologically protected edge states in half the edge Brillouin zone (which was shown by transport in this classic paper)
2. The existence of a spin-polarization associated with the edge states

While these three examples were chosen because they were simple, I have remaining doubts. Are these observations necessary and sufficient to define these states of matter? Are there cases where one can better define a state of matter theoretically?

For instance, a theorist may define a 2D topological insulator by the existence of a non-trivial topological number, which seems like a perfectly valid criterion to me. This topological number cannot be experimentally observed in a very direct way (to my knowledge) and has to be inferred from the edge states, band structure, etc.

The reason I started thinking about this is because I did not find the definition of a charge density wave in this widely-cited paper by Johannes and Mazin appropriate. It states:

[A charge density wave is a] Peierls-like instabilit[y] that occur[s] due to a divergency in the real part of the electronic susceptibility, so that the electronic subsystem would be unstable per se, even if the ions were clamped at their high symmetry positions.

This definition bothers me in particular because it defines a charge density wave by its cause (i.e. Peierls-like instability due to a divergence in the real part of the electronic susceptibility).  The main qualm I have is that one should not define a state of matter by its origin or cause. This is like trying define a superconductor by the mechanism that causes its existence (i.e. phonon-mediated electron-electron interaction for superconductors, which would exclude unconventional superconductors from its definition). This is obviously problematic. Therefore, shouldn’t one define a charge density wave by its experimentally measured properties?

So I come back to the original question: how does one define a state of matter?