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FQ Hall effect: has Vafa superseded Laughlin?

A stringy description of a flagship condensed matter effect could be superior

Harvard's top string theorist Cumrun Vafa has proposed a new treatment of the fractional quantum Hall effect that is – if correct – more stringy and therefore potentially more unifying and meaningful than the descriptions used by condensed matter physicists, including the famous Laughlin wave function:

Fractional Quantum Hall Effect and M-Theory (cond-mat.mes-hall, arXiv)
Laughlin's theoretical contributions to the understanding of this effect are admired by string theorists and physicists who like "simply clever" ideas. But the classification of the effects and possibilities seemed to be a bit contrived and people could have thought that a more conceptual description requiring fewer parameters could exist.

Let me start at the very beginning of Hall-related physics, with the classical Hall effect. In the 19th century, they failed to learn quantum mechanics (the excuse was that it didn't exist yet) so they called it simply "the Hall effect".

At any rate, Edwin Hall took a conductor in 1879, a "wire" going in the \(z\) direction, and applied a transverse magnetic field (orthogonal to the wire) pointing in the \(x\) direction. What he was able to measure was a voltage not just in the \(z\) direction, as dictated by Ohm's law, but also in the third \(y\) direction (the cross product of the current and the magnetic field – a direction perpendicular both to the current as well as the magnetic field).

This Hall voltage was proportional to the magnetic field and its origin is easy to understand. The charge carriers that are parts of the electric current are subject to the Lorentz force \[

\vec F = q\cdot \vec v \times \vec B

\] and because the electrons in my conventions are pushed in the \(y\) direction by the Lorentz force, there will be a voltage in that direction, too.

That was simple. At some moment, quantum mechanics was born. Many quantities in quantum mechanics have a discrete spectrum. It turned out that given a fixed current, the Hall voltage has a discrete spectrum, too. It only looks continuous for "regular" magnetic fields that were used in the classical Hall effect. But if you apply really strong magnetic fields, you start to observe that the "Hall conductance" (which is a conductance only by the units; the current and the voltage in the ratio are going in different directions)\[

\sigma = \frac{I_{\rm channel}}{V_{\rm Hall}} = \nu \frac{e^2}{h}

\] only allows discrete values. \(e\) and \(h\) are the elementary charge and Planck's constant (perhaps surprisingly, \(h/e^2\) has units of ohms, it's called the von Klitzing constant \(25815.8\) ohms) but the truly funny thing is that the allowed values of \(\nu\) (pronounce: "nu"), the so-called filling factor, has to be a rather simple rational number. (The classical Hall effect is the usual classical limit of the quantum Hall effect prescribed by Bohr's 1920 "correspondence principle"; the integers specifying the eigenstate are so large that they look continuous.)

Experimentally, \(\nu\) is either an integer or a non-integral rational number. In the latter case, we may call \(\nu\) the "filling fraction" instead. The case of \(\nu\in\ZZ\), the "integer quantum Hall effect", is easy to explain. You must know the mathematics of "Landau levels". The Hamiltonian (=energy) of a free charged particle in the magnetic field is given by\[

\hat H = \frac{m|\hat{\vec v}|^2}{2} = \frac{1}{2m} \zav{ \hat{\vec p} - q\hat{\vec A}/c }^2

\] Note that in the presence of a magnetic field, it's still true that the kinetic energy is \(mv^2/2\). However, \(mv\) is no longer \(p\). Instead, they differ by \(qA/c\), the vector potential. This shift is needed for the \(U(1)\) gauge symmetry. If you locally change the phase of the charged particle's wave function by a gauge transformation, the kinetic energy or speed can't change, and that's why the kinetic energy has to subtract the vector potential which also changes under the gauge transformation.

At any rate, for a uniform magnetic field, \(\hat{\vec A}\) is a linear function of \(\hat{\vec r}\) and you may check that the Hamiltonian above is still a bilinear function in \(\hat{\vec x}\) and \(\hat{\vec p}\). For this reason, it's a Hamiltonian fully analogous to that of a quantum harmonic oscillator. It has an equally-spaced discrete spectrum, too (aside from some continuous momenta that decouple). The excitation (Landau) level ends up being correlated with the filling factor. The mathematics needed to explain it is as simple as the mathematics of a harmonic oscillator applied to each charge carrier separately.

However, experimentally, \(\nu\) may be a non-integer rational number, too. It's the truly nontrivial case of the fraction quantum Hall effect (FQHE).\[

\nu = \frac 13, \frac 25, \frac 37, \frac 23, \frac 35, \frac 15, \frac 29, \frac{3}{13}, \frac{5}{2}, \frac{12}{5}, \dots

\] How is it possible? The harmonic oscillators clearly don't allow levels "in between the normal ones". It seems almost obvious that the interactions between the electrons are actually critical and they conspire to produce a result that looks simple and theoretical condensed matter physics is full of cute phenomenological explanations for such a conspiracy – and various quasiparticles etc.

In condensed matter physics – with many interacting charged particles – it's possible for the electrons to seemingly get fractional (FQHE); for the charge and spin to separate much like when your soul escapes from your body (spinon, chargons), and so on. These phenomena may be viewed as clever phenomenological ideas designed to describe some confusing observations by experimenters. However, closely related and sometimes the same phenomena appear in string theory when string theorists are explaining various transitions and dualities in their much more mathematically well-defined theory of fundamental physics (I am talking about fractional D-branes, Myers effect, and tons of other things).

The importance of thinking in terms of quasiparticles for string theory – and the post-duality-revolution string theory's ability to blur the difference between fundamental and "composite" particles – is another reason to say that string theory is actually much closer to fields like condensed matter physics – disciplines where the theorists and experimenters interact on a daily basis – than "older" parts of the fundamental high-energy physics.

At any rate, I've mentioned the Landau-level-based explanation of the integer quantum Hall effect. Because we're dealing with harmonic oscillators of a sort, there are some Gaussian-based wave functions for the electrons. Robert Laughlin decided to find a dirty yet clever explanation for the fractional quantum Hall effect, too. His wave function contains the "Landau" Gaussian factor as well, aside from a polynomial prefactor:\[

\psi(z_i,\zeta_a) &= \prod_{i,a} (z_i-\zeta_a)\prod_{i\lt j}(z_i-z_j)^{\frac{1}{\nu}}\times\\
&\times \exp\left(-B\sum_i |z_i|^2\right)

\] Here, \(z_i\) and \(\zeta_a\) are complex positions of electrons and quasi-holes, respectively. This basic wave function explained the FQHE with the filling fraction \(\nu = 1/m\) and Laughlin could have shared the Nobel prize. Note that for \(\nu=1/m\), the exponent \(1/\nu\) is actually integer which makes the wave function single-valued.

We're dealing with wave functions in 1 complex dimension i.e. 2 real dimensions so it looks like the setup is similar to the research of conformal field theories in 2 real dimensions (or 1 complex dimension: we want the Euclidean spacetime signature), the kind of mathematics that is omnipresent in the research of perturbative string theory (and its compactifications on string-scale manifolds, e.g. in Gepner models). Indeed, the classification of similar wave functions and dynamical behaviors has been mapped to RCFTs (rational conformal field theories), basically things like the "minimal models".

Also, the polynomial prefactors may remind you of the prefactors (especially the Vandermonde determinant) that convert the integration over matrices in old matrix models to the "fermionic statistics of the eigenvalues".

Cumrun Vafa enters the scene

Cumrun is the father of F-theory (in the same sense in which Witten is the mother of M-theory). He's written lots of impressive papers about topological string theory; and cooperated with Strominger in the first microscopic calculation of the Bekenstein-Hawking (black hole) entropy using D-branes in string theory. Also, he's behind the swampland paradigm (including our Weak Gravity Conjecture) etc.

In this new condensed-matter paper, Cumrun has shown evidence that a more unified description of the FQHE may exist, one that may be called the "holographic description". First of all, he has employed his knowledge of 2D CFTs to describe the dynamics of the FQHE by the minimal models. The minimal models that are needed are representations of either the Virasoro algebra that we already know in bosonic string theory; or the super-Virasoro algebra that only becomes essential in the case of the supersymmetric string theory.

If that part of Vafa's paper is right, it's already amusing because I believe that Robert Laughlin, as a hater of reductionism and a critic of string theory, must dislike supersymmetry as well. If super-Virasoro minimal models are needed for the conceptually unified description of the effect he is famous for, that's pretty juicy. If I roughly get it, Cumrun may de facto unify several classes of "variations of FQHE" into a broader family with one parameter only.

Laughlin with a king in 1998. Should Cumrun have been there instead?

But Cumrun goes further, to three dimensions. 2D CFTs are generally dual to quantum gravity in 3 dimensions. Why it shouldn't be true in this case? The question is what is the right 3D theory of gravity. He identifies the Chern-Simons theory with the \(SL(2,\CC)\) gauge group as a viable candidate.

Note that Chern-Simons theory, while a theory with a gauge field and no dynamical metric, seems at least approximately equivalent to gravity in 3 dimensions. One may perform a field redefinition – that was sometimes called a 3D toy model of Ashtekar's "new variables" field redefinition in 4D. In 3 dimensions, things are less inconsistent because pure 3D gravity has no local excitations. The Ricci tensor and the Einstein tensor have 6 independent components each; but so does the Riemann tensor. So Einstein's equations in the vacuum – Ricci-flatness – are actually equivalent to the complete (Riemann) flatness. No gravitational waves can propagate in the 3D vacuum.

And the gravitational sources in 3D only create "deficit angles". The space around them is basically a cone, something that you may create by cutting a wedge from a flat sheet of paper with the scissors and by gluing. Not much is happening "locally" in the 3D spacetime of quantum gravity which is also why the room for inconsistencies is reduced.

The gauge group \(SL(2,\CC)\) basically corresponds to a negative value of the cosmological constant. In this sense, the 3D gravitating spacetime may be an anti de Sitter space and Cumrun's proposed duality is a low-dimensional example of AdS/CFT. This is a somewhat more vacuous statement because there are no local bulk excitations in Chern-Simons theory – you can't determine the precise geometry – so I think that the assignment "AdS" is only topological in character. Moreover, the big strength of Chern-Simons and topological field theories is that you may put them on spaces of diverse topologies so even the weaker topological claim about the AdS space can't be considered an absolute constraint for the theory.

Also, Chern-Simons theory may actually deviate from a consistent theory of quantum gravity once you go beyond the effective field-theory description – e.g. to black hole microstates. But maybe you should go beyond Chern-Simons theory in that case. Cumrun proposes that the black holes that should exist in the 3D theory of quantum gravity should be identified with Laughlin's quasi-holes. If true, it's funny. Couldn't have the people checked that the two kinds of holes may actually be physically equivalent in the given context?

At any rate, if the 3D theory has boundaries, there is FQHE-like dynamics on the boundary and he may make some predictions about this dynamics. In particular, he claims that some excitations exist and obey exotic statistics. In 4 dimensions and higher, we can have bosons and fermions. In 2+1 dimensions, the trip of one particle around another is topologically different from no trip at all (the world lines may get braided which is why knot theory exists in 3D), so there may be the generic interpolations between the bosons and fermions, the anyons (with a phase). And you may also think about non-Abelian statistics and Cumrun actually claims that it has to be true if his model is correct.

Many non-string theorists have played with topological field (not string) theory and related things and you could think that Vafa's paper is just a "paper by a string theorist", not a "paper using string theory per se". (Strominger semi-jokingly said that he defined string theory as anything studied by his friends. I am sort of annoyed by that semi-joke because that's how you would describe an ill-defined business ruled by nepotism which string theory is certainly not.) But you would be wrong. At the end, Vafa constructs his full candidate description of the FQHE in terms of a compactification of M-theory. Well, he picks the dynamics of M5-branes in M-theory, the \((2,0)\) superconformal field theory in \(d=6\). Many of us have played with this exotic beast in many papers.

These six-dimensional theories with a self-dual three-form field strength at low energies are classified by the ADE classification. Cumrun compactifies such theories on the genus \(g\) Riemann surfaces \(\Sigma\) and claims that the possibilities correspond different forms of the FQHE. Lots of very particular technical constructions in the research of string/M-theory are actually used. Many facts are known about the compactifications which is why Cumrun can make numerous predictions for the actual lab experiments.

I can't reliably verify that Vafa's claims are right. It looks OK according to the resolution I have but to be sure about the final verdict, one has to be familiar with lots of details about the known theoretical and experimental knowledge of the FQHE as well, not to mention theoretical knowledge about the compactifications of the \((2,0)\) theory etc., and I am not fully familiar with everything that is needed.

However, I am certain that if the paper is right and the observed FQHE behavior may be mapped to compactifications of an important limit of string/M-theory, then condensed matter theoretical physicists at good schools should be trained in string/M-theory. A string course – perhaps a special "Strings for CMT" optimized course – should be mandatory. Subir Sachdev-like AdS/viscosity duality has been important for quite some time but in some sense, this kind of Vafa's description of FQHE – and perhaps related compactifications that describe other quantized, classifiable behaviors in condensed matter physics – could make string/M-theory even more fundamental for a sensible understanding of experiments that condensed matter physicists actually love to study in their labs.

It seems that we may be getting closer to the "full knowledge" of all similar interesting emergent behaviors exactly because the ideas we encounter start to repeat themselves – and sometimes in contexts so seemingly distant as Laughlin's labs and calculations of (for Laughlin esoteric) compactifications of limits of M-theory.


Aside from a nicety, Cumrun added in an e-mail:
I will only add that the experiments can settle this one perhaps even in a year, as I am told: Anomalies in neutral upstream currents that have already been observed, if confirmed for \(\nu=n/(2n+1)\) filling fraction, will be against the current paradigm and in line with my model.
Good luck to science and Cumrun in particular. ;-)

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