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301. Relating field theories via stochastic quantization

Posted by Domenico Orlando

at March 11, 2009

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Domenico Orlando, a full-time member of IPMU (Japan), is known for his works on many aspects of string theory such as string theory on curved backgrounds, quantum crystals and exact solutions in string theory. Dmitry.

First of all I would like to thank Dmitry for inviting me to write about Stochastic Quantization in connection with a recent paper (arXiv:0903.0732) by Robbert Dijkgraaf, Susanne Reffert and myself.

In one sentence, Stochastic Quantization is a quantization scheme, introduced by Parisi and Wu, in which quantum mechanics is seen as the thermal equilibrium of a stochastic process with respect to an extra (fictious) time dimension. But let me start from the beginning.

The main idea. The main idea behind Stochastic Quantization is that using the analogy between the Euclidean path integral measure and the Boltzmann distribution for a statistical system in equilibrium one can write the Euclidean Green’s functions as limits of equal-time correlators for an appropriate stochastic process. To see how to do this in practice, let us consider as an example a scalar field theory in d dimensions, described by a field $\phi(x)$ with action $S_{\text{cl}}[\phi]$ . To construct the associated stochastic process we do the following:

introduce a new time dimension t, so that $\phi$ is promoted to a function of (d+1) variables $\phi (t,x)$ ;
impose a time evolution for $\phi(t,x)$ such that for $t \to \infty$ the system goes to equilibrium and “describes the same physics” as the quantum d-dimensional system. A simple way of doing this consists in imposing a Langevin equation
$\partial_t \phi(t,x)=– \frac{1}{2} \frac{\delta S_{\text{cl}}}{\delta \phi} + \eta (t,x)$

where $\eta(t,x)$ is a white Gaussian noise.

One can then show that for $t \to \infty$ the averages over the noise $\eta$ tend to the path integral averages for the d-dimensional system. That is,

$\lim_{t \to \infty} \langle \phi(t,x_1) \phi(t,x_2) \dots \rangle_\eta=\langle \phi(x_1) \phi(x_2) \dots \rangle$ .

Up to this point this is by now a well known construction. What we argue in our work is that a completely analogous approach can be followed to quantize a discrete system, such as a spin chain. In this case, starting from a system that can assume a set of configurations $\alpha$ with energy $H(\alpha )$ we add the time direction t and impose a master equation for the probability to be in the configuration $\alpha$ at time t:

$\partial_t P_{\alpha }(t)=\sum_{\beta \neq \alpha } W_{\alpha \beta} P_\beta (t) – W_{\beta \alpha } P_\alpha (t)$ ,

where the transition rates are written as

$W_{\alpha \beta}=e^{- \left( H(\alpha ) – H(\beta) \right)}$

if the system can pass from $\beta$ to $\alpha$ in a single elementary step (such as flipping a spin).

One can then prove that for $t \to \infty$ , the probability $P_\alpha (t)$ tends to the unique ground state of the matrix which corresponds – just like before – to the Boltzmann weight of the initial “classical” system

$\lim_{t \to \infty} P_\alpha (t)=P_\alpha^{(0)} \propto e^{-H(\alpha)}$ .

Now you might wonder: why should I use stochastic quantization? What is the advantage of this quantization procedure? There are different types of answers to this question:

from a purely technical point of view it turns out that stochastic quantization is particularly useful in the case of systems with a lot of symmetries since it does not require gauge fixing;
this approach is naturally suited for numerical simulations of large systems;
one can decide (and this is what we do in our paper) to take the extra time dimension seriously and study the (d+1)-dimensional systems.

Lagrangian and Hamiltonian description. In order to understand the dynamics in (d+1) dimensions better it is useful to translate the Langevin (or Master Equation) into a Lagrangian (resp. Hamiltonian) formalism:

choosing a Lagrangian approach one can see that it is natural to introduce fermionic partners of the field $\phi$ , collect them into a supermultiplet $\Phi$ and write the (d+1)-dimensional action in a manifestly supersymmetric:
$S_q[\Phi]=D \Phi {\bar D} \Phi + S_{\text{cl}}[\Phi]$ ,

where D is the supercovariant derivative and $S_{cl}[\Phi]$ now has taken the role of a superpotential. This points to a deep connection between Brownian motion and supersymmetry that is unfortunately beyond the scope of this post.
Choosing the Hamiltonian approach we found that in the discrete case, the system is described by a Schrodinger equation and the Hamiltonian is precisely the Laplacian on the state graph, i.e. the graph whose nodes are the classical configurations and whose lines join two configurations if they are connected by an elementary move (see the picture). This means that for $t \to \infty$ , the probability amplitudes are described by a harmonic function on the graph:
$H_{\text{q}} | \Psi_0 \rangle=\nabla^2 | \Psi_0 \rangle=0$ .

It is also a common property of all these models that the norm squared of the ground state is precisely the classical partition function in d dimensions:

$\langle \Psi_0 | \Psi_0 \rangle=Z_{\text{cl}}$ .

State Graph

Some examples. In our paper we consider a number of examples of pairs of theories in d and (d+1) dimensions related by Stochastic Quantization. Interestingly enough some of these pairs where already known but not recognized as being connected by this quantization scheme. More precisely we describe in some detail

zero-dimensional field theory and one-dimensional super quantum mechanics;
d-dimensional free bosons and (d+1)-dimensional super quantum Lifshitz model;
classical and quantum crystals (and dimers);
the XXZ model as growth of integer partitions;
gauged WZW model and Chern Simons (with some caveats).

I would like to end this post concentrating on one of these examples. Our initial motivation for this paper (as string theorists) comes from the observation made some time ago (see hep-th/0309208) that the partition function for the topological string A-model on $\mathbf{C}^3$ is the same as the partition function for the classical three-dimensional crystal melting. This is a stochastic system in which cubes are stacked in an empty corner of 3D space and a cube can be added to a configuration if three of its sides will touch either the wall or other cubes (see figure).

On the one hand, the Stochastic Quantization of this system is very interesting in itself: in particular in one dimension less (where one considers squares instead of cubes), we proved in a previous work (arXiv:0803.1927) that the quantized system is precisely the integrable XXZ spin chain with kink boundary conditions.
On the other hand we argue that the construction is also relevant in string theory since, according to the correspondence with topological strings, each cube configuration corresponds to a geometry. Then the quantum wavefunctions are naturally interpreted as sums over geometries and the Laplace equation for the ground state can be read in the spirit of the Wheeler-de Witt equation.

To summarize: Stochastic Quantization is a quantization scheme that relates a system in d dimensions to a stochastic process in (d+1). In our paper we argue that it can be also used for discrete systems and we describe some examples of theories in different dimensions that are related by this scheme. These examples are of interest for different fields, ranging from solid state physics to string theory.

Further Reading

The original paper (quite difficult to find): G. Parisi and Y.-S. Wu, Perturbation Theory Without Gauge Fixing, Sci. Sin. 24 (1981) 483
A collection of relevant papers: P.H. Damgaard and H. Huffel: “Stochastic quantization’‘ (on google book search)
A comprehensive review: P. H. Damgaard and H. Huffel, Stochastic quantization, Phys. Rep. 152 (1987), no. 5-6 227-398.
A more recent book: M. Namiki: “Stochastic quantization”

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300. Event horizon of Sgr A*

Posted by Dmitry

at March 11, 2009

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Dear friends,

I am really sorry for leaving you without a physics post (the video clearly cannot be counted ) – the reason was my faulty internet connection. The post that you will see below was almost ready but I wanted to make a short last check before it gets posted, and that’s when my faithful internet provider decided to cut me out. Anyway, all this just means that today you are going to have two nice posts instead of one.

By the way, I cannot help acknowledging the very fact that this post’s number is 300 (while 200th wasn’t that long time ago)… Why this observation did make me happier, another thing which made my day yesterday is the paper by Avery Broderick (CITAzen!), Abraham Loeb and Ramesh Narayan titled “The event horizon of Sagittarius A*“.

About two months ago we had a discussion of BH physics with Ervin Goldfain. At some point he argued that “as explained to me by Sabine Hossenfelder, astrophysical data alone cannot settle the issue of where the BH entropy is coming from.” What we are going to talk about in this post is the first robust conclusion based on astrophysical data alone that Sagittarius A*, a compact object in the center of our Galaxy, has all features intrinsic to black holes – for example, event horizon. As Brits say, if it walks like a duck and quacks like a duck, then it must be a duck.

So, as I said, Sagittarius (Sgr) A* is an extremely bright (in radio waves) compact object in the center of the Milky Way which is believed to a supermassive black hole. (By the way, the Milky Way center is hidden from us by the dust cloud, and in order to study it we have to use wavebands other than optical: IR, X-ray and radio). That is how it looks like:

Sagittarius A* - a supermassive black hole in the center of the Milky Way

or, on a bit larger scale,

Where is Sagittarius A* exactly?

Why do we (or at least many of us) believe that Sgr A* is a black hole? First of all, by studying motion of stars in the vicinity of the Sgr A*, we can determine its mass – and the latter turns out to be about 4 million solar masses. Second, very long base line interferometry (VLBI) instruments currently allow us to estimate the angular size of the very object with rather good precision. The latter is found to be around 27 microarcsecs, which corresponds the linear size of the object about 44 million kilometres (diameter – compare it to 150 million kilometres – distance between the Earth and the Sun).

If you squeeze 4 million Suns into the volume of 22 million kilometres cubed, the outcome of this operation is almost inevitably black hole – although there are other non-BH configurations with the same mass and characteristic linear scale possible, all of them are expected to collapse very rapidly. Indeed, the Schwarzchild radius for the configuration of this mass is

$r_g=\frac{2Gm}{c^2}=2.95 {\rm km}\cdot\frac{m}{M_{\rm Sun}}\sim{}12\cdot{}10^6{\rm km}$ .

This scale is approx. 2 times smaller than the observed scale 22 million km followed from the observations, but we also need to properly take relativistic effects into account – if Sgr A* is indeed a black hole, spacetime around it is strongly curved leading to the gravitational lensing effect (so that its angular size observed from the distance of 26000 ly is quite a bit larger than the actual one). When lensing is accounted for, the only conclusion that can be made is that Sgr A* is indeed a supermassive black hole.

Now, if metastasis of crackpotism have already poisoned your brain, you may be left unhappy since GR was quietly implemented in our considerations (in the estimation of the gravitational lensing effect) What if the underlying theory of gravity is not GR, can we still make a conclusion that Sgr A* is a black hole? (as we remember, existence of BHs is a general phenomenon not intrinsic to just GR) The paper by Broderick et al. shows that the answer to this question is positive.

Their proof that Sgr A* possesses a horizon is based on the following three simple assumptions:

1. Emission from Sgr A* is powered by accretion. This is now generally accepted and indeed sounds quite reasonable even for arrogant non-astrophysicist like myself. Since the population of stars (and matter density in general) is extremely high in the region near the center of the Galaxy, there is no surprise that part of this matter will be captured by the gravitational well of a dense object with a mass of 4 million Suns. Typically, in systems like neutron stars, BH candidates and simply binaries (where one of the components is heavier than the other and pulls matter from it) infalling matter forms a disk around gravitationally attracting body called accretion disk. It can speed up to relativistic velocities and rapidly loose energy due to radiative friction.

Accretion disk for close binary
The energy released in the process can be very high (like in AGNs, and that is what allows us to detect such systems albeit their angular size being extremely small.

2. Sgr A* lives for a sufficiently long time for the gravitationally attracted matter to achieve a steady state regime of infalling. This is again reasonable since the age of Sgr A* is estimated to be 10 Gyrs, while the correlation time in kinetics of infalling matter is of the order $GM/c^3\sim 20{\rm sec}$ .

3. Surface emission from compact object has the blackbody spectrum (in this respect, note that no thermal component is observed in the spectrum of Sgr A*). What surface emission we are talking about? Suppose that Sgr A* is not a black hole (that is, it does not have an event horizon). Instead, say, it is a very large star. In this case, all the gravitational binding energy of the infalling matter should be eventually emitted back into space. Partially, this energy is emitted in the process of accretion due to the radiative friction as we discussed above but some amount of matter certainly reaches the surface of the star and feeds thermonuclear reaction. In this case, the remaining energy should be emitted by the star itself. That is the part of the flux from Sgr A* we are looking for in the data – surface emission.

At the technical level, the authors want to estimate so called radiative efficiency for Sgr A*. What is it? Suppose a massive matter particle falls on Sgr A*, and its gravitational binding energy is reemitted back into space. Let us denote the overall outgoing flux at infinity as $L_\infty$ . Not all the energy is emitted in the form of radiation, but only a certain (small) fraction

$L_{\rm obs}=\eta_r{}L_\infty$ .

The flux $L_{\rm obs}$ is exactly what we observe by our instruments, and the coefficient $\eta_r$ is called the radiative efficiency. Another part of the gravitational binding energy can be released in the form of outgoing kinetic flows similar, say, to jets perpendicular to the plane of the accretion disk on the picture above; for that part of the energy we can write $L_{\rm out}=\eta_k{}L_\infty$ ).

We want to estimate $\eta_r+\eta_{\rm out}$ becuase we know the values of radiative efficiency for many systems such as active galactic nuclei (where $\eta_k+\eta_r$ is of the order 10%), etc. Clearly, it is very hard to get any information about the overall flux $L_\infty$ from observations, but what ultimately allows us to estimate radiative efficiency for Sgr A* are the accurate recent measurements of the flux $L_{\rm obs}$ from Sgr A* together with its apparent radius/angular size (vua VLBI).

Indeed, we can write for the surface flux (it is blackbody, see the cond. 3 above)

$L_{surf}=4\pi\sigma\R_a^2T_\infty^4$ ,

so the temperature of bb radiation from Sgr A* (as seen at infinity) will be given by

$T_\infty=\left(\frac{1-\eta_r-\eta_k}{\eta_k}\frac{L_{\rm obs}}{4\pi\sigma{}R_a^2}\right)^{1/4}$

The surface flux is

$F=\pi\left(\frac{R_a}{D}\right)^2B_\nu(T_\infty)$ ,

where $B_\nu{}(T_\infty)$ is the blackbody spectrum. Therefore, we can estimate the maximum possible temperature at the surface of Sgr A* as

$T_\infty<T_{\rm max}=\frac{h\nu}{k\log\left(1+\frac{2\pi{}h\nu^3R_a^2}{c^2F_{\rm obs}D^2}\right)}$

and derive a limit upon $L_{\rm surf}$ directly as

$\frac{L_{\rm surf}}{L_{\rm obs}}<\frac{L_{\rm surf, max}}{L_{\rm obs}}=\frac{\sigma{}R_a^4}{D^2F_{\rm obs}}T_{\rm max}^4$ .

The latter condition can be rewritten in terms of radiative efficiency as

$\eta_r+\eta_k>\frac{1}{1+L_{\rm surf,max}}/L_{\rm acc}$

Turning to the results of observations (see the Fig. below), we find that in order for an emitting surface of Sgr A* to exist, it should be expected that $\eta_r+\eta_k$ > 99.6%, that is,

if matter falls onto Sgr A*, somehow 99.6% of the liberated gravitational binding energy must be radiated, powering either the observed luminosity or kinetic outflows. Otherwise the emission of the reminder upon settling onto the surface would have been detected.

Radiative efficiency of Sgr A*

Radiative efficiency like 99.6% is way larger than 10% observed for active galactic nuclei, so ultimately we either have to come out with absolutely new revolutionary mechanism of accretion/gravitational binding energy release or say that emitting surface is simply absent, that is, Sgr A* is a black hole.

Let me note that Sgr A* is a pretty much single black hole candidate in this sense, since the angular sizes of other candidates are very far from being determined at this moment, and knowing angular size of the object is crucially important for the Broderick-Loeb-Narayan argument.

Instead of making a conclusion (I am sure this story is very far from the end!), let me briefly mention another argument why the theory being black holes and Sgr A* in particular is most probably GR.

The flux outgoing from Sgr A* exhibits flares in mm, sub-mm, IR and X-ray wavebands (see for example this article in the Oct 2001 issue of Scientific American). These flares are supposed to take place when large chunks of infalling matter reach the black hole and the gravitational binding energy gets released. As it turns out, rise times for flares and variability timescales are compatible to the periods of innermost stable orbits around black holes described by General Relativity. This, however, is another story…

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299. Video of the day: Why gravity is weak

Posted by Dmitry

at March 10, 2009

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It seems I did not post those for quite a long time… Another promo video about string theory – enjoy

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298. Do all spherical viruses have icosahedral symmetry?

Posted by Antonio Garrido

at March 9, 2009

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This is a guest post by Eric Lewin Altshuler (UMDNJ, Newark) and Antonio Perez-Garrido (U. of Cartagena). Dmitry.

More than half a century ago Crick and Watson (Nature, 177, 473-475 (1956)) had the ingenious insight that viral capsids must be made of multiple units of the same small number of proteins, lest the viral genome be orders of magnitude too large – if coding for each of the hundreds or thousands of capsid proteins separately – to fit inside its capsid. Caspar and Klug (Cold Spring Harbor Symposia On Quantitative Biology 27, 1 (1962)) made a significant advance in appreciating that the structure of a number of viral capsids had icosahedral symmetry. They described capsids by a number T=a^2 + b^2 + ab (, non-negative integers) having N=10T + 2 subunits arranged into an icosadeltahedral lattice. However, determination of the structure of these large capsids is a tour de force of experimentation, and until recently capsid structure was actually determined by fitting relatively low resolution data to an assumption of an icosadeltahedral capsid (see Strauss, “Viruses and Human Disease”, p. 34, Academic Press, San Diego (2002) and refs. therein.) Recently, though high resolution studies have confirmed icosadeltahedral configurations for 1, 3 (Cardone et al., Nature 457, 694-698 (2009)) and 7 (Jiang et al., Nature 439 612-616 (2006) and Gertsman et al., Nature doi:10.1038/nature07686) viruses. So do all spherical viruses have icosahedral symmetry?

The subunits of the viral capsids themselves can often have many components so theory or modeling might seem most difficult. However, because electrostatic interactions between subunits are likely important, it turns out that a more than century old model problem, we have found (arXiv:0902.3566) may shed important light on spherical viral capsid structure and evolution.

Over one hundred years ago J. J.Thomson (Philos. Mag. 7, 237-265 (1904)) asked the question of the minimum energy configuration of unit point charges on (the surface of) a unit conducting sphere. Much theoretical, numerical and experimental work since then has made considerable progress on Thomson’s problem yielding interesting and nonobvious results: For N=4 charges the global minimum energy configuration is the geometrically symmetric configuration of a tetrahedron. However, for N=8 the minimum energy configuration is not a cube, but rather an anticube – four charges arranged in a square parallel to the equatorial plane in both the Northern and Southern hemispheres, but with the squares rotated by 45 degrees with respect to each other. This configuration has a lower energy than a cube as the rotation of the squares lowers the energy between nearest neighbor charges between the two squares. The case of N=8 also illustrates a general phenomenon in Thomson’s problem whereby the most symmetric configuration is not necessarily the configuration of minimum energy.

For $12 \le N\le 100$ (and likely for the most part up to N=200 ) attack of Thomson’s problem by multiple numerical and theoretical approaches and methods has likely found the minimum energy configurations. In most cases there are exactly twelve charges with five nearest neighbors – pentamers and the rest of the charges with six nearest neighbors – hexamers. Euler’s theorem for convex polygons – the number of vertices plus faces equals the number of edges plus 2 ( V + F=E + 2 ) – has the result for points on a sphere that there must be at least twelve pentamers with the rest of the charges being hexamers or pentamer/septamer pairs. Only for N of the form 10(a^2 + b^2 + ab) + 2 (where $a \le b\le 0$ ) is it possible for the twelve pentamers and the entire configuration to have icosahedral symmetry.

Now, for N = 12, 32, 72, 122, 132, 192, 212, 272 and 282 (arXiv:0902.3566, Table 1, Figure 1) the icosahedrally symmetric configuration (an icosadeltahedral configuration) is the best known energyminimum(and the presumed global energy minimum configuration). But for N = 42, 92, 162, 252 (and then numbers larger than 282) the icosadeltahedral configuration is not the global energyminimum. Instead configurationswith exactly twelve pentamers, but with the pentamers arranged in $D_{5h}$ , D_2 , D_3 and C_2 symmetries are the global energy minima for N = 42, 92, 162 and 252 respectively (arXiv:0902.3566, Table I, Figure 2). These symmetries of global energy minima hold not only for the 1/r Coulomb potential, but for other representative electrostatic potentials as well (arXiv:0902.3566, Table II).

Why aren’t icosadeltahedral configurations the global minima for N = 42, 92 and 162 ((2, 0), (3, 0) and 4, 0))? For these N (see Figure 1) one sees that the vertices of the pentamers (known as disclinations in the language of elasticity and continuum mechanics) are lined up point to point. If the pentamers could be arranged with a different energy the cost of the disclinations being lined up and relatively closer to each other could be removed, as long as the new configuration itself does not impose an energy cost even higher. Conversely, for N = 32, 72 ( N = 12 is a uniquely special case) one sees (Figure 1) that the pentamers are rotated with respect to each other to reduce strain of aligned disclinations. In general it seems that in seeking global energy minima for N<200 Nature uses the general strategies of moving and rotating pentamers. In some cases either because of pure geometrical constraints or just to minimize the energy, occasionally a pentamer/heptamer defect pair (dislocation defect in the language of elasticity) is needed to achieve a global energy minimum. As N grows larger – $\ge 500$ or so – important papers by Dodgson and Moore (Dodgson J., Phys. A 29, 2499 – 2508 (1996) and Dodgson and Moore, Phys. Rev. B 55, 3816 – 3831 (1997)) showed that the energy strain of the pentamers which is necessitated by the topology of a sphere but which distort the pure hexagonal lattice that would be the energy minimum on a flat sheet, is such that to lower the energy pentamer/heptamer defect pairs are needed between all of the pentamers.

Icosadeltahedral configurations

Fig. 1. Icosadeltahedral configurations.

Further, as grows the number of local energy minima, M(N) (1 $2\le{}N{}\le{}112$ ) was found to grow exponentially (Erber and Hockney, Phys. Rev. Lett. 74, 1482 (1995)):

$M(N)\approx 0.382\times \exp \left( 0.0497N\right)$ .

Thus, if Nature is using an energy minimization strategy to find the configuration of alignment of molecules in a viral capsid it would seem that as N grows it will become increasingly difficult if not impossible to find the global energy minimum configuration. Furthermore, the number and relative depth and breadth of good local minima could also be a constraint on kinetic strategies that Nature may use to find the ultimate configuration. In arXiv:0902.3566, Table I we show the number of times we found the various local minima in runs where we started the charges from 5000 random configurations and then used standard conjugate gradient methods to go to a local minimum. We see that for N = 12 , 32 and 72 the minimum energy configuration and overwhelmingly the most common is in fact the icosadeltahedral configuration. For N = 42 , 92 and 162 however, the icosadeltahedral configuration is not only not the minimum energy configuration, it is not reached from random configurations ever. Similarly, we found that for others interaction potentials again the icosahedral configurations for N = 42 , 92 and 162 are not global minima and virtually never occur in the simulations (arXiv:0902.3566, Table II).

Icosadeltahedral configuration, global minima and overlap

Fig. 2. Icosadeltahedral configuration b) Global minima, c) Icosadeltahedral and global minima overlap.

Models of icosahedron

Fig. 3. Models of an icosahedron (left) and one with D_5h symmetry (right) that resembles the global minimum for N=42 .

For N = 42 ( T4 ) icosahedral and $D_{5h}$ symmetries look quite similar (Figure 2 and Figure 3). If we only take into account the balls’ positions in Fig. 3 we can pass from the left model to the right one just by rotating an hemisphere by an angle of $2 \Pi /10$ . Based on these electrostatic model potentials we would predict that T 1 , T 3 and T 7 virus would have icosadeltahedral configurations as recently found experimentally. T 13 viruses may have an icosadeltahedral configuration. We believe that high resolution studies of T 4 viruses will show a $D_{5h}$ configuration in particular and not an icosadeltahedral configuration. We predict that T 9 , T 16 and T 25 viruses will not be found to have an icosadeltahedral configuration though we do not have a clear prediction for the structure of these viruses. Conversely, if a T 4 , T 9 , T 16 or T 25 virus were found to have an icosadeltahedral configuration given the essentially vanishing possibility of this from energetic considerations or statistical considerations based on electrostatic potentials, it would indicate a mechanical rule of assembly to be discovered that is of expontentially good precision. It is still also a mystery why nature seems to so prominently use capsids with numbers of subunits to the exclusion of other numbers of subunits. The energetics and statistics are so favorable for N = 12, 32 and 72 ( T 1 , T 3 , T 7 ) that protein subunits consistent with these configurationsmust have emerged. The same factors would suggest a $D_{5h}$ configuration for N = 42. A key question could be understanding the evolution of T 16 capsids. In general, geometry and topology seem to be important constraints that need to be considered in viral evolution or even possible treatments for viral.

In summary, we have shown that theory and simulations for a broad and representative range of electrostatic potentials for interaction of capsid subunits predict an icosaheltahedrally symmetric configuration for T 1 , T 3 , and T 7 viruses (and maybe for T 13 viruses). However, the prediction for T 4 , T 9 , T 16 and T 25 viruses Is for a capside of lower symmetry, in particular $D_{5h}$ for viruses. If, conversely, icosadeltahedral structures are found in high resolution studies, then any model of capsid assembly would have to account for such a structure in the face of energy and vast statistical disadvantages.

The other mystery – potentially linked? – is to uncover the principle of why Nature has chosen to make capsids only with numbers of subunits, whereas geometry and topology do not exclude other N . N = 12 has a relatively lower energy per subunit than other N near 12, so this one seems to have arose first. In terms of understanding the evolution of viruses one would guess that T 3 viruses somehow arose as a variation of T 1 viruses.

The next challenge is to understand how T 4 viruses evolved. This understanding of viral capsid structure and evolution is not purely a theoretical exercise, as such understanding could lead to the development of new therapeutic agents.

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297. Exact gravity dual of a gapless superconductor

Posted by George Koutsoumbas

at March 8, 2009

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This is a guest post by George Koutsoumbas from the National Technical University of Athens. Dmitry.

I would like to thank Dmitry for the invitation to write a blog entry on my recent work with E.Papantonopoulos and G. Siopsis entitled “Exact Gravity Dual of a Gapless Superconductor”, arXiv:0902.0733 [hep-th].

The AdS/CFT correspondence has become a powerful tool in studying strongly coupled phenomena in quantum field theory using results from a weak coupled gravity background. According to this correspondence principle, a string theory on asymptotically AdS spacetimes can be related to a conformal field theory on the boundary. In recent years, apart from string theory, this holographic correspondence, following a more phenomenological approach, has also been applied to nuclear physics in order to describe certain aspects such as heavy ion collisions at RHIC and to certain condensed matter systems. Phenomena such as the Hall effect and Nernst effect have dual gravitational descriptions.

Recently the AdS/CFT correspondence has also been applied to superconductivity in Phys. Rev. Lett. 101, 031601 (2008) [arXiv:0803.3295 [hep-th]]. The gravity dual of a superconductor consists of a system with a black hole and a charged scalar field, in which the black hole admits scalar hair at temperature smaller than a critical temperature, while there is no scalar hair at larger temperatures. A condensate of the charged scalar field is formed through its coupling to a Maxwell field of the background. Neither field was backreacting on the metric. This decoupled Abelian-Higgs sector can be obtained from an Einstein-Maxwell-scalar theory through a scaling limit in which the product of the charge of the black hole and the charge of the scalar field is held fixed while the latter is taken to infinity. Considering fluctuations of the vector potential, the frequency dependent conductivity was calculated, and it was shown that it develops a gap determined by the condensate. Away from the large charge limit, the backreaction of the scalar field to the spacetime metric has to be taken into consideration. It was found that all the essential characteristics of the dual superconductor were persisting. Moreover, even for very small charge the superconductivity was maintained.

We proposed in [arXiv:0902.0733 [hep-th]] a model of a gravity dual of a superconductor in which the charged scalar field is provided by the scalar hair of an exact charged black hole solution. It has been shown that, below a critical temperature, this black hole solution undergoes a spontaneous dressing up with the scalar hair, while above that critical temperature the dressed black hole decays into the bare black hole. We will show that even for small values of the charge a condensate is formed, while an electromagnetic perturbation of the background electromagnetic field determines the conductivity and the superfluid density of the boundary theory. There is evidence that these black hole solutions can be obtained from eleven-dimensional supergravity theory.

To obtain a black hole with scalar hair, we start with the four-dimensional action

$I=I_{gr} +I_{\mathrm{matter}}$

consisting of the Einstein-Hilbert action with a negative cosmological constant $\Lambda=– \frac{3}{l^2}$ , $I_{gr}=\frac{1}{16\pi G} \int{}d^4x\sqrt{-g}\left[ R+ \frac{6}{l^2} \right]$ , where is Newton’s constant, is the Ricci scalar, is the AdS radius and a charged scalar together with a Maxwell field

$I_{\mathrm{matter}}=\int{}d^4x\sqrt{-g}\left[ \frac{1}{2}g^{\mu\nu} D_\mu\phi (D_\nu\phi)^{*}\frac{1}{12}R\phi^*\phi-\frac{2\pi G}{3}(\phi^*\phi)^{2}\right]-$
$-\frac{1}{4 }\int d^{4}x\sqrt{-g}F^{\mu \nu}F_{\mu \nu }$ ,

where $D_\mu \phi \equiv\partial_\mu\phi + iqA_\mu \phi.$

The presence of a negative cosmological constant allows the existence of black holes with topology $\mathbb{R}^{2}\times\Sigma$ , where $\Sigma$ is a two-dimensional manifold of constant negative curvature. These black holes are known as topological black holes. The simplest solution reads

$ds^{2}=-f_{TBH}(\rho)dt^{2}+\frac{1}{f_{TBH}(\rho)}d\rho^{2}+\rho^{2}d\sigma^{2}$ ,
$f_{TBH}(\rho)=\rho^{2}-1-\frac{\rho_0}{\rho},$

with $\phi=0$ , $A_\mu=0$ and we have set the AdS radius l=1 . $\rho_0$ is a constant which is proportional to the mass and it is bounded from below ( $\rho_0\geq-\frac{2}{3\sqrt{3}}$ ). $d\sigma^{2}$ is the line element of the two-dimensional manifold $\Sigma$ , which is locally isomorphic to the hyperbolic manifold $H^{2}$ and of the form $\Sigma=H^{2}/\Gamma$ , $\Gamma\subset{}O(2,1)}$ , where $\Gamma$ is a freely acting discrete subgroup (i.e., without fixed points) of isometries. This space becomes a compact space of constant negative curvature with genus $\gen\ge{}2$ by identifying, according to the connection rules of the discrete subgroup $\Gamma$ , the opposite edges of a $4\gen$ -sided polygon whose sides are geodesics and is centered at the origin $\theta=\varphi=0$ of the pseudosphere. An octagon is the simplest such polygon, yielding a compact surface of genus $\gen=2$ under these identifications. Thus, the two-dimensional manifold $\Sigma$ is a compact Riemann 2-surface of genus $\mathrm{g}\geq2$ .

A static black hole solution with topology $\mathbb{R}^{2}\times \Sigma$ and scalar hair (MTZ black hole), is also allowed and given by

$d\hat{s}^{2}=-f_{MTZ} (r)dt^{2}+ \frac{dr^{2}}{f_{MTZ} (r)} +r^{2}d\sigma^{2} \quad, \quadf_{MTZ}=r^{2}-\biggl(1+\frac{r_0}{r}\biggr)^{2}~,$

with

$\phi(r)=\Psi(r)\equiv{}-\sqrt{\frac{3}{4\pi G}}\frac{r_0}{r+r_0}$ , $A_\mu=0$ , $\Psi(r)=\frac{r_0}{r+r_0}.$

It can be shown that

$\Delta F=F_{TBH}-F_{MTZ}=-\frac{\sigma}{8\piG}(T-T_{0})^{3}\pi^{3}+\dots,$

indicating a phase transition between MTZ and TBH at the critical temperature $T_0=\frac{1}{2 \pi}$ . It is easily seen that $r_+\le \rho_+,$ and the inequality is saturated for $r_+=\rho_+=1.$ Thermodynamically we can understand this phase transition as follows. We find that $S_{TBH}>S_{MTZ}$ and $M_{TBH}>M_{MTZ}$ for the relevant ranges of the horizons r_+ or $\rho_+.$ If $r_{+}>1$ ( $T>T_{0}$ ), both black holes have positive mass. As T>T_0 implies $F_{TBH} \le F_{MTZ},$ the MTZ black hole dressed with the scalar field will decay into the bare black hole. In the decay process, the scalar black hole absorbs energy from the thermal bath, increasing its horizon radius (from r_+ to $\rho_+>r_+$ ) and consequently its entropy. Therefore, in a sense the scalar field is absorbed by the black hole. If $r_{+}<1$ black holes have negative mass, but now $F_{TBH}> F_{MTZ},$ which means that the MTZ configuration with nonzero scalar field is favorable. As a consequence, below the critical temperature, the bare black hole undergoes a spontaneous “dressing up” with the scalar field. In the process, the mass and entropy of the black hole decrease and the differences in energy and entropy are transferred to the heat bath.

We will now discuss an exact gravity dual of a superconductor. For $A_{\mu}=0$ we have two different gravity backgrounds. If $\phi\neq 0$ then a condensate is formed, the field equations have as a solution the MTZ black hole and the scalar field is given by $\phi=-\Psi$ . If $\phi=0$ then no condensate is formed and the field equations have as a solution the TBH black hole.

Note that the mechanism of condensation of the scalar field here is different than the mechanism of condensation of the dual superconductor in the case of a black hole of flat horizon. There the scalar field condensed because a kind of an abelian Higgs mechanism was in operation. In our case, the condensation of the scalar field has a geometrical origin and is due entirely to its coupling to gravity. It may be shown that

$C_n \approx \frac{\pi\sigma}{3\sqrt 3 G}\ T \ \ , \ \ \ \ C_s \approx \frac{\pi\sigma}{2G}\ T \ ,$

therefore both heat capacities vanish linearly with temperature as $T\to 0$ , indicating that we have a gapless superconductor.

To these exact gravity backgrounds we shall apply an electromagnetic perturbation. In the case without condensate the wave equation for perturbing the vector potential reads

$f_{TBH}(f_{TBH}A')'+\left(\omega^{2}\right)A=0~,$

where is an appropriately defined component of the vector potential and we considered the lowest angular eigenvalue.

The solution of the equation behaves asymptotically as

$A=A^{(0)} + \frac{A^{(1)}}{r} + \dots$

and it may be shown that the conductivity reads $\sigma(\omega)=\frac{A^{(1)}}{i\omega{}A^{(0)}}=1.$ This solution holds if the temperature is above the critical temperature and it tells us that the boundary conducting theory is in the normal phase, as expected.

If the temperature is below the critical temperature the vacuum TBH acquires hair, and a condensate forms. In this case, the corresponding wave equation for the vector potential reads

$f_{MTZ}\left(f_{MTZ}A'\right)'+\left(\omega^{2} -q^{2}\Psi^{2}f_{MTZ}\right)A=0.$

This equation can not be solved analytically in general, but we can solve this equation for weak coupling $q^{2}$ using perturbation theory. Also, a numerical analysis of it has been done. The behaviour we observed of the boundary conducting theory can be found in materials with paramagnetic impurities and to unconventional superconductors like the chiral p-wave superconductor.

Let us now discuss the numerical solution of the wave equation in the interval $r\in [r_+, \infty)$ and compare it with the analytical results obtained above using perturbation theory. By curve fitting the solution of the wave equation, we calculated the coefficients $A^{(0)}$ and $A^{(1)}$ referred to above and deduced the conductivity $\sigma$ . In the limit $\omega\to 0$ , the conductivity yields the densities of the superfluid and normal components.

On the basis of the analytic results we expect that, at low temperature, the normal fluid density can be expanded as $\ln{}n_n=\gamma \ln T + \dots$ whereas near the critical temperature, the superfluid density is expanded as $n_s=\alpha (T-T_0)^2+ \dots$ We therefore fit the data accordingly. The table contains numerical values for $\alpha$ and $\gamma$ obtained through the fit and compares them with their analytical counterparts. It is clear that the agreement between numerical and analytical results is quite satisfactory for the superfluid density, while serious discrepancies appear for the normal density.

$q/\sqrt G$	$\gamma_{\mathrm{num}}$	$\gamma_{\mathrm{an}}$	$\alpha_{\mathrm{numerical}}$	$\alpha_{\mathrm{an}}$
0.1	0.0020	0.0024	0.0225	0.024
0.5	0.053	0.0597	0.552	0.589
1.0	0.187	0.239	2.196	2.356
2.0	0.684	0.955	8.678	9.425
3.0	1.325	2.15	20.35	21.21
5.0	2.522	5.97	52.90	58.90

Numerical vs. analytical results for the normal and superfluid densities for various values of the charge.

We analyzed the $\omega$ dependence of the transport coefficients for various values of the temperature. It turns out that at low temperatures there result rather small values for the real part of the conductivity, while for larger temperatures this real part tends to the value 1, which is the outcome for the topological black hole.

The figures contain the dependences of the normal and superfluid densities on the temperature (Figure 1), the charge (Figure 2) and the frequency (Figure 3) (click on the figure to get a version with larger resolution):

Fig. 1. The logarithm of the normal fluid density as a function of the logarithm of the temperature (left) and the superconducting fluid density as a function of (T-T₀)² (right) for q/G^1/2 = 5.0. The solid lines represent the fits ln n_n = 2.45 ln T + 5.3 (left) and n_s=52.9 (T-T₀)² (right).

Fig. 2. Numerical and analytical results for the normal (left) and superfluid (right) densities vs q². Numerical data are fitted by 0.176 q² – 0.0030 q⁴ (left) and 2.29 q² – 0.007 q⁴ (right).

Fig. 3. The real (left) and the imaginary (right) part of the conductivity versus ?/T for q/G^1/2=5 and T=0.0032, 0.032, 0.064$ The lowest curve corresponds to the lowest temperature for the real part. The uppermost curve corresponds to the lowest temperature for the imaginary part.

Conclusion

We presented a model of an exact gravity dual of a gapless superconductor, in which a condensate forms as a result of the coupling of a charged scalar field to gravity. The charged scalar field responsible for the condensation is a solution of the field equations and below a critical temperature dresses up a vacuum black hole of a constant negative curvature horizon (TBH) with scalar hair. Perturbing the background Maxwell field and using the AdS/CFT correspondence, we determined the conductivity of the boundary theory and analysed the behaviour of the normal and superconducting fluid densities using both analytical and numerical techniques.

AdS/CFT, Condensed Matter, Journal club, Master Postdoc, Quantum field theory, String theory

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