Surge

cond-mat and hep-th, the best of friends

Lately, as you may have noticed from my previous posts, I have become increasingly interested in learning condensed matter physics (side-by-side with some high energy physics). It’s incredible - something as simple as a linear chain of spins (say $\vert\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow…\rangle$) exhibits a wonderful range of phenomena, including duality, phase transitions, emergent phenomena, spinon waves, and can describe even paramagnets and ferromagnets! Indeed, the appeal of these statistical models to me lay not only their ability to illuminate features of QFT that are somewhat obscure in HEP, but also in their intrinsic wealth of features and applications. Here’s what I’ve been reading over the last week:

• John McGreevy’s “Where do QFTs come from?“. Most of my (and presumably many others’) first experience with QFT comes from high energy physics where its presented as a speedrun to the Standard Model. But these wonderfully engaging lecture notes explore a different (and very important!) perspective: microscopic phenomena, dualities, emergence and CFTs in the context of simple statistical and condensed matter systems. Wen’s Quantum Field Theory of Many-body Systems is another useful reference, employing more familiar field-theoretical languages and introducing topological order (the fractional quantum Hall effect is far more deep and interesting that I had originally thought!)

• To supplement the above, I am also reading Kitaev’s notes on Topological phases and quantum computation, which skew even further on the side of practicality (lattice models, toric code, anyons). Kitaev in particular is someone whose work I wish to get far more familiar with.

• Khomskii, Basic Aspects of the Quantum Theory of Solids. An excellent self-contained textbook on many-body quantum systems. I believe it takes a relatively modern approach too, focusing on order and excitations as its main themes in CMT. It’s quite refreshing to see, for instance, the interpretation of Feynman diagrams for electron-phonon-photon systems, as well as Fermi liquids.

• Jeff Harvey, Magnetic Monopoles, Duality, and Supersymmetry. I found this after it was cited by David Tong in his (rather dense, but well-referenced) Solitons notes. Extremely helpful, it covers topics right up to monopoles in N=2/4 supersymmetric gauge theories, fermion coupling and S-duality - I also found Jose Figueroa-O’Farrill’s Electromagnetic Duality for Children to be a good supplement.

• Some dedicated works on superconductivity - Carsten Timm’s Theory of Superconductivity is mathematical and modern, while I found Superconductivity, Superfluids and Condensates by Annett to be more conceptual. Incidentally, I’m going to be adding a post on the similarities between the Higgs mechanism, confinement and superconductors soon.

• Since I am a bit more comfortable with supersymmetric gauge theories now, I decided to explore a bit about Seiberg-Witten theory. I found an interesting introduction in What do Topologists want from Seiberg–Witten theory by Kevin Iga, explaining some of the history, motivation, and connection with Donaldson invariants. Arun Debray’s lecture notes on his webpage (which includes four-manifolds) also look really comprehensive and well-written - in fact, many of them approach physics from a mathematical POV, which is really nice to see.

Color charge is not the analogue of electric charge

Let’s say it loud and clear one more time for the people at the back: “Color charge is not the analogue of electric charge”.

This is one of those misconceptions which is spawned from annoying terminology: “color charge” as referred to in popular science accounts (red, green, blue) doesn’t refer to a charge in the conventional sense. There are a couple of places you may have noticed this: firstly, how come electric charge is observable while color charge isn’t? Secondly, why does electric charge appear in the gauge transformation of a particle coupled to the photon, while no colors appear in that for a particle couples to the gluon? The reason is that these two objects - electric charge and color charge - are completely different. Not many sources mention the difference between the two outright (and popular science accounts love to peddle this incorrect view), presumably leaving it to the reader to discern it for themselves.

$\mathrm U(1)$ representation theory

Let’s recall where electric charge $e$ comes from. The irreducible unitary representations of the group $\mathrm U(1)$ are

All such representations are of course one-dimensional. For a general representation, we have a matrix $K$ rather than a number $k$, which can be diagonalised to yield the charges of its irreducible components as its eigenvalues ${k_1, k_2, k_3, …}$ along the diagonal. Importantly, an elementary particle with charge $+2$ transforms differently than a particle composed of two $+1$’s: the former under $\mathbf 1_{+2}$, the latter under $\mathbf 1_{+1}\oplus\mathbf 1_{+1}$. To recap, to specify an irreducible representation of $\mathrm{U}(1)$, all one needs is an integer $k$. Since the representation is always one-dimensional, there is no explicit “internal” space for an electron: $\mathcal H\otimes\mathbb C^*\cong\mathcal H$.

Skipping ahead a bit, a field in an irreducible $\mathrm U(1)$ representation $q$ transforms as $\psi(x)\to e^{iq\lambda(x)}\psi(x)$ under a gauge transformation, with the gauge field transforming as $A_\mu(x)\to A_\mu(x)-\partial_\mu\lambda(x)$. As usual, we can then define the gauge covariant derivative $D_\mu\equiv\partial_\mu-iq A_\mu$ to construct the gauge-invariant interacting Lagrangian

Generalising to $\mathrm SU(3)$

This is straightforward. Now let’s do the same thing for QCD: we have a quark field $\psi(x)$ (just one flavour for now, the generalisation is obvious). The transformation rules are now

where the $T^a$ form a basis for the $\mathfrak{su}(N)$ representation under consideration. The gauge field (gluon) necessarily transforms in the adjoint (dimension $N^2-1$) representation, but quarks could theoretically be in any representation (they are in the fundamental representation in the Standard Model). How do you characterise a representation in a basis-independent manner? By using the Casimir operators for $\mathrm{SU}(N)$ (most commonly, the quadratic operator $C_2\equiv\mathrm{Tr}(T_aT_a)$ - though this can be degenerate for higher representations, and so necessitates an additional finite number Casimirs to distinguish them). Alternatively, just draw a Young diagram - it’s cooler.

Let’s be standard and take the quarks in the fundamental. This is a 3-dimensional representation (but the Lie algebra has rank 8). That is, the color state (not charge!) of a quark is given by the column vector $(r, g, b)^\top$, i.e. $\vert q\rangle=r\vert r\rangle+g\vert g\rangle+b\vert b\rangle$. This is just a fancy way of saying that the overall Hilbert space of the quark must must involve the tensor product of the color space $\mathbb C^3$.

These are the colors - they label the internal state of the quark in $\mathbb C^3$. Note that any such configurations related by a $\mathrm{SU}(3)$ gauge transformation are identified, so saying that a quark is “red” is completely meaningless. Antiquarks similarly transform in the complex conjugate representation, known as the antifundamental $\mathbf{\bar 3}$, and to emphasise that these are two different representations (even though their vector spaces are canonically isomorphic), we denote its basis by ${\vert \bar r\rangle, \vert \bar g\rangle, \vert \bar b\rangle}$. For both quarks and antiquarks, the $T_a$ above are a set of 8 3x3 matrices, which form a basis for all 3x3 traceless matrices (the fundamental representation of the Lie algebra $\mathfrak{su}(N)$ ). Finally, the total Hilbert space is built as $\mathcal H_\text{tot}\cong\mathcal H\otimes\mathbb C^3_\text{color}$, which gives the fields an internal index $i\in{1,2,3}$.

Next, since the gluons transform in the adjoint, and $\mathbf{3\otimes\bar3=8\oplus1}$, we can use ${\vert r\rangle\vert \bar r\rangle, \vert r\rangle\vert \bar g\rangle, \vert r\rangle\vert \bar b\rangle, \vert g\rangle\vert \bar r\rangle, … }$, with the sole proviso that the singlet $\mathbf 1$ be eliminated by enforcing for instance $\sum_i\vert i\rangle\vert \bar i\rangle\overset!=0$. This time the $T_a$ are 8x8 matrices.

Conclusion

Thus it is the choice of representation, indicated by the choice of $T_a$ (not the dimension of these matrices - distinct representations can have the same dimension!), that is the exact analogue of the electric charge. One ought to say that quarks have “fundamental” charge, anti-quarks have “anti-fundamental” charge, gluons have “adjoint charge”, and baryons have “$\mathbf{10\oplus8\oplus8\oplus1}$” charge. Catchy. Bear in mind that this seemingly “simple” characterisation of QCD charges is deceptive: the way the representations interface during gluon-mediated interactions is far more complicated in non-abelian gauge theory than its abelian counterpart, which gives QCD its distinctive set of exotic phenomena.

The “colors” merely label the internal state of a fundamental quark (or anticolors for an antiquark) - the blame of this misunderstanding lies in the fact that $\mathbf 3$, $\bar{\mathbf3}$ and $\mathbf 8$ are the only representations used in QCD, which means the terms “red, green, blue” get reused everywhere as if they were true charges. If quarks were to transform under $\mathbf{15}$ then none of this confusion would arise.

An Introduction to QCD

This post mostly serves to describe a few preliminary concepts in quantum chromodynamics as an introduction to a few more in-depth QCD posts that I have planned. Later on in this series I’ll aim to illustrate analogues between a few phenomena from gauge theory in the context of condensed matter and other low energy phenomena.

Introduction

Electrical charge is a reasonably simple concept to understand: electrons have a charge of $-e$, positrons (anti-electrons) and protons have a charge of $+e$, and the charge of a composite object is simply equal to the algebraic sum of the charges of its constituent particles. Photons, constituting electromagnetic radiation, can interact with electrically charged particles but are themselves uncharged. The theory with quantum fields corresponding to only fundamental, electrically charged particles (electrons, positrons, muons, antimuons, etc.) and photons is called quantum electrodynamics, and has been very successful indeed since its inception in the 1930s.

The photon is not, however, the only massless gauge boson (read: “force mediator”) in the Standard Model. It is assisted by the gluon, which mediates the strong force between quarks. Due to the magical properties of the theory describing them at low energies, the strong force is not a simple $1/r^2$ law. Instead, at short distances it is far stronger than the electromagnetic force, but its strength begins to drop very rapidly with increasing distance. The theory of quarks and gluons and how they interact is called quantum chromodynamics (QCD).

[image]

Consider ordinary electric attraction, albeit described with quantum diction. The charge on a particle determines its coupling with the photon, and so determines the strength of the electric force with other electrically charged particles. Analogously in QCD, the nature and strength of the interaction between two quarks is determined by their charge - but this charge is not a simple, additive number! Instead, it’s a more complicated object that generalises the idea of charge being a real number.

Hadrons

One peculiar feature of QCD, though nobody has managed to prove this rigorously from first principles ($1 million awaits you if you do), is that a solitary quark cannot be observed. The internal state of a quark, determining the nature of its interaction with other quarks, cannot be probed directly (at least at ordinary energies)! You may well wonder why electric charge is observable then, given the similarity in their underlying description, and the crux of the answer lies in the fact that, contrary to popular belief, electric charge is not the analogue of color charge. I go into this in another blog post here.

We can have composite particles made of multiple quarks/antiquarks of course - provided that they are colorless (“color singlets”). These are known as hadrons. The simplest example is a bound state of a quark and an anti-quark, known as a meson. For instance, the $\pi^+$ meson comprises the quarks $u\bar d$ - an up quark and a down anti-quark. If the quark has a color of red, and the antiquark has a color of antired, then the resulting combination is effectively colorless, and is allowed to exist.

This is in contrast to a combination of two quarks, which can never be colorless. That said, three quarks can form a colorless combination, known as a baryon. The ubiquitous proton and neutron are examples of this, having composition $uud$ and $udd$ respectively. Further combinations of certain quarks and antiquarks - tetraquarks, pentaquarks, and so on - are theoretically colourless, but are too heavy to see at ordinary energies. The LHC does manage to get a glimpse of them though: see here.

Confinement

All right, but nature can be tricked, you say. You can grab hold of a meson and cunningly begin to separate the quark and the antiquark, thus freeing them, can’t you? Nope, nature wins again. In the confining phase of QCD (when only color singlets are observable), there is a characteristic linear energy cost of separating a quark and an antiquark, i.e. $V(r)=\sigma r$, for some constant sigma. We say that the quark and antiquark are connected by a flux tube. Note that this flux tube is not like an ordinary spring, which stores an energy proportional to $r^2$ rather than $r$.

As $F(r) = -\frac{\mathrm d}{\mathrm dr}V(r)$, the force between two quarks is independent of distance. But as you increase the separation $r$, the energy stored in this “rubber band” increases linearly, eventually becoming high enough to excite a new quark-antiquark pair out of the QCD vacuum! It’s sort of like trying to cut a bar magnet into its separate poles.

If you’d like to learn more, I invite you to read a series on Matt Strassler’s website about particle physics.

A Tale of $(\mathcal N =)$ 2 Supersymmetries

I didn’t know much supersymmetry a couple of weeks ago - only as far as supermultiplets and superfields. In particular, I didn’t know how to construct supersymmetric gauge theories and the like, so that was my starting point. Also, I’ve found the new exotic territory (that I describe in this post) extremely appealing: I love the way that SUSY gauge theories can be lifted to string theory, the way that statistical field theoretic notions like phase transitions and superconductors are found in high-energy QCD.

• High-energy particle physics models - It might sound odd, but this is one of the areas adjoint to raw supersymmetry that I’m not as interested in currently! Nonetheless the immediate extensions of the Standard Model and gauge/gravity-mediated SUSY breaking are incredibly important tools which were appealing to me.

• Supersymmetric solitons - David Tong’s TASI notes, and the Shifman compendium are the go-to sources on these. When they’re powered (that is, constrained) by SUSY, solitons become really powerful, while still maintaining their elegant geometrical nature

• Supersymmetric moduli/pseudomoduli spaces - especially that of supersymmetric quantum chromodynamics, which displays a wide array of intriguing phenomena (gaugino condensation, phase transitions, etc.)

• Even phases of regular ol’ gauge theories are super interesting (so many cool analogies with the ‘t Hooft model and superconductors!), but the one thing that SQCD really has going for it is Seiberg duality, which is simply fantastic (and, not to mention, completely unexpected!). Matt Strassler has some great lectures on the full-fledged duality cascade (passing D3 branes on conifolds (!) along the way)

• Supersymmetric sigma models (particularly for $\mathcal N = 2$, you get to see hyperKähler and special Kähler manifolds in the wild) - I find these incredibly elegant: you get to encode the low-energy spectrum of an extremely complex supersymmetric gauge theory geometrically as a manifold with a lot of structure on it!

• Non-renormalization theorems, holomorphy, instantons - very pragmatic tools, it goes without saying.

• Cosmology - I don’t know much beyond the basic equations of $\Lambda$CDM, so I’m going to be reading about some more advanced things like inflation, early-universe matter generation and cosmological perturbation theory

• Statistical field theory - the most underrated topic in physics in my opinion. As I mention in my quantum field theory guide, you should learn graduate-level statistical field theory before starting quantum field theory (I didn’t, and I wish I did). The analogies between the two are vast, but the intuition that statistical field theory provides is far greater, since it’s “closer to Earth” (and Ken Wilson). I was learning about domain walls, RG flows, non-trivial fixed points, sigma models, and a smörgåsbord of dualities. But that’s not all! There’s universal criticality, minimal models, WZW models (I’m learning from the Yellow book of course) that can manifest before your eyes in a condensed matter lab.

• Superfluidity and superconductivity: Finally getting to bring my basic solid-state QM knowledge into a more advanced context! Plus it’s nice seeing a quantum field theory that’s outside of HEP. Superconductivity is the correct classical analogy of the Higgs mechanism, interestingly enough.

• Holographic descriptions of quark-gluon plasma: The ultimate “real-life” application of string theory! You use the AdS/CFT correspondence to link SYM (and even SQCD, if you’re feeling brave) to a 5D black hole and its perturbations. The classical description of QGP is also a nice foray into condensed matter physics. On that note, something that I want to learn about is holographic descriptions of hadron and glueball spectra.

• Sphalerons - introducing the instantons’ wilder cousin. They trigger B and L violation in the Standard Model, nonperturbatively.

I might conclude with the fact that I no longer do much string theory - not because of some misplaced disillusionment, but because supersymmetry on its own is just way too cool. I still find the S-matrix properties of string theory extremely alluring, but perhaps I’m going to learn about something like “practical” AdS/CFT, non-commutative geometry or “lifting” QFTs to string theory rather than, say, superstrings.