Jun 13, 2010

Free Lecture Notes #1

It's been a while since I wrote the last post. It was a very busy week. My contract in Aston finishes on September and, as life as a post-doc dictates, I have been looking for a job. Also, I have to prepare to some upcoming conferences, try to finish my projects and publish (or perish, of course), worry about paying bills, visas and many other things. Science is tough to do on days like these.

Therefore, to keep the ball rolling while I have not the time to write more elaborated things, I decided to start listing the lecture notes I have been accumulating from arXiv. That's a good trick to do if you do not have time. :) So, I will start with these first five:

  1. Lectures on holographic methods for condensed matter physics, Sean A. Hartnoll
  2. Lecture notes on the physics of cosmic microwave background anisotropiesAnthony Challinor & Hiranya Peiris
  3. Les Houches Lectures on Black HolesAndy Strominger
  4. Three lectures on Newton's lawsSergey S. Kokarev
  5. Gravity & Hydrodynamics: Lectures on the fluid-gravity correspondenceMukund Rangamani
In particular, the first one seems to be quite interesting. If you have comments about them, that would be nice as well. Have fun.

Jun 7, 2010

Spin Liquids

[Simulation of a quantum spin-liquid performed on a flat honeycomb structure - by The University of Stuttgart]

As I have been reading many posts, specially via Condensed Concepts, about spin liquids, I decided to learn a bit about that. So, following the new plan for this blog of using it to help me understand things, I am writing what I have found in the following arXiv article, which is a compilation of lectures given at the famous Les Houches Conference in France:
There are some minor typos, so just be sensible when reading. Nothing serious. It seems that although the definition of a spin liquid may make sense in the classical setup, it would not be realisable there, so the concept is usually only applied to quantum spin models. Anyway, let me describe what I understood from the above paper. If anyone have more interesting things to add or corrections to make, I would be very happy to hear them and learn more about the subject!

The term (quantum) spin liquid refers to the ground state of a spin model where no symmetry of the model Hamiltonian is broken.

Let me include at this point two paragraphs for the less technical audience that has been brave enough to continue reading up to here. Physicists may feel free to skip the next two paragraphs as I am going to explain the basic concepts for a broader audience. I will get back to a more technical description after them. First, a spin model is a mathematical model describing particles with spin (the quantum version of magnetic moment) usually, but not necessarily, on a lattice (a collection of points linked by lines). These models are defined through a so-called Hamiltonian function, which is just a formula that gives the energy of the model for each configuration of the spins in the lattice. The ground state is the minimum energy configuration, that should be favoured at zero temperature as all physical systems like to minimise the energy and there are no thermal fluctuations at T=0. You can think of the energy as a cost function that you always try to keep at a minimum. Finally, a symmetry of the Hamiltonian is some kind of modification that you do to the spins or any other variable in the Hamiltonian such that when you put this modified variable back into the formula, the Hamiltonian does not change.

The symmetry part requires some more explanation, I know. Consider an Euclidean vector with two coordinates  v=(x,y) and let us assume that in some system there is a Hamiltonian depending on it given by H=xy, i.e., the product of its two coordinates. If we multiply the vector v by -1, then each coordinate is multiplied by -1 and the Hamiltonian becomes H=(-x)(-y)=xy. It doesn't change. Therefore, the multiplication of v by -1 is a symmetry of the Hamiltonian. Of course physically meaningful symmetries are more interesting, although the one I have just gave you may be considered as a very special case of a more general local gauge symmetry, but we are not going to talk about that now. The important is the idea.

Consider now, for instance, the ground state of the (ferromagnetic) Heisenberg model. This is the classical example of symmetry breaking. The Hamiltonian is rotationally invariant as it is given by the scalar product of the spin vectors, but the ground state has magnetic order with all spins pointing in the same direction and obviously changes if they are rotated (although a rotation takes it to another ground state). The word liquid in spin liquid is an analogy with the transition from the liquid to the solid state. The liquid is homogeneous and looks the same everywhere, so it has a continuous translational symmetry of the liquid. On the other hand, a crystalline solid (let us not talk about glasses at this point...) breaks that symmetry in the sense that it is not invariant by a continuous translation, but by very specifically ones. The same works for rotations, but the basic idea is what matters. This can also be stated as the fact that the spins do not develop any long range order (LRO) at zero temperature, they are completely disordered even then.

That is the reason why these ground states can be realised only in the quantum setup. Classically, there are no fluctuations at zero temperature to destroy an ordered state. However, at the quantum level there exist quantum zero point fluctuations that can do job. They can disorder the spins from their ordered states guaranteeing that they will not break any symmetry.

An interesting characteristic of ground states with broken symmetries is that they are degenerate. By applying the symmetry operation that is itself broken by these states, you get another ground state. This kind of degeneracy should not happen in the spin liquids, but their ground state can still be degenerate, the degeneracy coming from another kind of order called topological order, about which I am certainly going to write a more detailed post in the future.

It seems that there was no observation up to date of a spin liquid phase in a real system, although many simmulations on almost realistic models seem to observe it. For instance, the article Exotic Quantum Spin-Liquid Simulated: A Starting Point for Superconductivity?, from where I took the picture for this post, describe one of them. As I said, Ross McKenzie from the blog Condensed Concepts have been writing a lot about that recently, so let me just list some of his posts
Just to summarise things: Spin liquids are ground states that break no symmetry from the original Hamiltonian, they have no long range order an can only appear in quantum systems because these have fluctuations even at zero temperature and these quantum fluctuations can destroy the order. Although from the experimental side these states have not been realised so far, it seems that theoretically they are reasonably understood, although through the article in the beginning of this post it seems to me that this understanding is only at the mean field level. The detailed description of these states, even theoretically, is still lacking and seems to be an interesting topic of research.

As a finishing note, Misguich at the end of the paper gives an interesting connection of spin liquids with Kitaev's toric code (which the reader may already know from previous posts). The ground state of this topological quantum code is a spin liquid. Although the ground state has a degeneracy, this degeneracy is topological and has nothing to do with the breaking of a symmetry by the ground state, so we are still fine. This analogy is only a final observation in the article, but given that the search for an experimental realisation of a topological code is also a hot topic, this gives another path through which spin liquids may be observable in real systems, although in this case they would be engineered instead of natural.

    Jun 4, 2010

    JoP: Condensed Matter - Highlights 2009

    SEM micrograph of a strongly crumpled graphene sheet on a Si wafer - Condensed Matter Physics Group, The University of Manchester

    The Journal of Physics: Condensed Matter published here a list of 35 papers published on it in 2009 that were chosen "on the basis of a range of criteria including referee endorsements, citations and download levels, and simple broad appeal". These papers will be free to read until 31-Dec-2010. 

    An interesting thing to do is to compare the subject of the selected works. Graphene is the theme of 8 articles, about 23% of the total. Then comes superconductivity with 4 articles, with about 11%, multiferroics with 3, about  9%, and then comes all other subjects with 2 or 1 articles.

    The above list gives supporting evidence to the statement that graphene has been the biggest star in the latest year(s). There are many reasons and probably funding is one of them. We can argue that graphene has not only many interesting properties but also great potential for technological applications, which can be said also about superconductors but their time of impact seems to be gone at least by now. 

    Jun 3, 2010

    StatPhys and ECCs #3: From bits to spins

    [O(3) Spin Model - Credit: Paul Coddington, University of Adelaide]

    This was meant to be the final post in the series but while writing it I realised that I had too much to say and as such, I decided to split this into more parts (I still don't know how many...). If you feel lost while reading, try to remind the basic concepts from the previous posts:
    • Statistical Physics of Error-Correcting Codes #1:
    • Statistical Physics of Error-Correcting Codes #2: Statistical Physics Overview
    Also, take a look at the references therein. Now, at the end of post #2 on this subject I said that when working with statistical physics we are always interested in what happens in the thermodynamic limit, by which we understand the limit when the number of units N in our system is very large. This is not always true actually. There is a lot of interest in what is called finite size effects in statphys, which are the effects that appear when we consider N not being large. Although the difference between large and not large may seem not very well defined, mathematically it can be translated as taking the limit of N being infinite.

    Taking the system size to infinity may seem radical and even unnatural, but a bit of practice with calculations is enough for everyone to see that one mol of atoms is for all practical purposes (fapp, according to John Bell's definition) infinite. Even smaller numbers, like one-billion or one-million may be close enough to infinite to render any deviation completely negligible in most practical situations. Now think about the last film you downloaded (I know you did it, but I will not tell anyone). It was probably of the order of 1G, right? This means that it has about 10 billion bits. This may not be as big as a mol, but still is big enough to allow us to consider this system already in the thermodynamic limit fapp.

    Consider then some very large file. To be consistent with post #1 let us call it a message t. In the same way as the electron spin, bits can assume only two values. The former can assume +1 and -1, while the latter can assume 0 and 1. For calculational convenience, it is usual to work with the former representation (specially because statistical physicists were used to do that way before they started to look at bits). We can transform between these two representations in two ways. The first is the linear relation

    where $\sigma\in\{\pm1\}$ is the spin variable while $x\in\{0,1\}$ is the binary variable. You can see that 0 is mapped to -1 and 1 to +1 as would seem more natural. However, there is a second way to map these variables which seems less natural, but actually is much more beautiful, namely

    And where is the beauty?, you may ask as it is not only non-linear but also maps 0 to +1 and 1 to -1, which is kind of weird. In fact, the above mapping is a homomorphism between the Galois field of order 2 and the square roots of unity!

    Let me explain this better. Isolated bits are usually summed using a sum mod 2 operation, also known as exclusive OR or simply XOR. The operation is defined by the following table at the right and many times it is expressed using the symbol $\oplus$ to differentiate it from the normal + sign. That's the notation I am going to use. By also defining the product of two bits to be the usual product of two numbers, the bits form a mathematical structure called a finite field or a Galois field and its order is the number of elements, in this case 2. But under this addition mod 2, this so-called binary field is also a group. The two square roots of 1 also form a group under the usual multiplication, and the non-linear mapping defined above maps one group to the other preserving the group operation. This can be easily seen by writing the mapping in a more formal way as $\sigma(x) = (-1)^x$ and observing that

    \[\sigma(x_1\oplus x_2) = (-1)^{x_1\oplus x_2} = (-1)^{x_1}(-1)^{x_2}=\sigma(x_1)\sigma(x_2).\]
    Homeomorphisms are always beautiful. This one in special can be even generalised to higher orders. When the Galois field has a prime order p, the addition operation of the field can be taken to be the addition mod p (be careful, as it doesn't work for a general order p!) therefore representing the elements by the integers from 0 to p-1. The mapping can then be generalised to

    \[\sigma(x)=\exp\left(\frac{2\pi i}{p} x\right).\]
    I will leave to you to check that this is indeed a homeomorphism. This formula can be interpreted as a mapping from the Galois field to spins pointing in some direction, as the exponentials are nothing more than the p-th complex roots of unit. I guess that all this beautiful mathematical structure is enough to justify using the non-linear mapping instead of the linear one, and that is what I am going to do from now on.

    Keeping the potential for generalisation in the back of our minds for now, let's come back to the case of bits, the binary field. There are three classic ensembles in statistical physics. The word ensemble refers to a particular situation of study where some microscopic variables are allowed to vary around a mean value and others are kept fixed. When we fix the number of particles (or units, I will use it particles many times as I am a physicist and some habits are difficult to lose) and the average value of the energy, we call this the canonical ensemble, and this will imply that the temperature of the system will be kept constant. There are many ways to show (my favorite being the maximum entropy that you can find in Jaynes book) that when the system is in equilibrium the canonical ensemble is describedby the Gibbs distribution

    \[P(\sigma)=\frac{e^{-\beta H(\sigma)}}{Z},\]
    where $H$ is the Hamiltonian (energy, cost function etc) of the system. $Z$ is the partition function and is one of the stars of the show. Actually, from the partition function we can derive all important properties of the system.

    Then, our first task will be to find out what is the analogue of the Hamiltonian in our case. Let's first understand what is that we want when analysing error-correcting codes. The interesting thing is to understand how much noise the code can stand before we cannot perfectly recover the message. We will not accept errors in the recovered message because if we do, the message will degrade each time we decode it. Imagine that in your HD. If you access to much a document, you loose it. Just to give the right credits here, the first person who noticed the mapping between codes and spin systems was Nicola Sourlas back in 1989:
    However, we will deal with a slightly different model from his. Our basic idea will be to use Bayes' rule to infer the original codeword from the corrupted one. Once we have the codeword, the original message is automatically retrieved as the correspondence is 1-to-1. Let us call the dynamical variable representing the possible codewords by $\tau$ and the received codeword, already corrupted by noise, by $r$. Then, according to Bayes' rule we have

    The first term in the numerator is the so-called likelihood of $\tau$ and represents the action of the noise in the codeword. Therefore, this term contains the noise model of the channel. The second term is a prior distribution over $\tau$, where we include any information we have that may help in the decoding. Finally, $Z$ is just the normalisation factor, although you probably already noticed that I am using the same notation as for the partition function because that is exactly what it will be.

    In our case of parity-check codes, we know that the codeword is in the kernel of the parity-check matrix. That's the only prior information we have about the codeword and we will symbolise it by the indicator function $\chi(A,\tau)$ which will be 1 when $\tau$ is in the kernel of $A$ and zero otherwise. There are many ways to include this term in the calculation, but the most physical one is to see it as an interaction term between each coordinate of $\tau$. Then, we can write the probability of $\tau$ as

    \[P(\tau)=\frac{e^{-\beta H}}{Z},\]

    \[H=-\ln P(r|\tau)-\ln \chi(A,\tau).\]
    This finally has the form of a Gibbs probability. The inverse temperature $\beta$ was introduced for convenience and can be taken to be 1 to coincide with the original problem. The first term of the Hamiltonian usually factorises in the spins, with each component of $r$ acting in a different component of $\tau$, which confers them the role of local fields.

    There are many subtleties in what I wrote above, but the general idea is that. Bayes' rule allows us to make the connection with the statistical physics of the problem.  In the next post I will enter into the details of this formulation and show how the methods of statistical physics allow us to extract what we want from the problem.