297. Exact gravity dual of a gapless superconductor

Posted by George Koutsoumbas

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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8 Comments »

Comment by Instanton

2009-03-10 11:04:07

hi George

At some point you say in the paper that matter does not backreact on the metric, so, the limit M_P\to\infty is taken, isn’t it? If this is so, what is the meaning to even discuss BH entropy which is inversely proportional to G?

Instanton

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Comment by Dmitry

2009-03-10 14:08:58

Dear Instanton,

glad that you did not forget us and drop by time after time

Cheers,
Dmitry.

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Comment by g.koutsoumbas

2009-03-11 12:18:42

Dear Instanton,

Thank you for your comment. With respect to the lack of back reaction we have been refering to the work of Hartnol, Herzog and Horowitz, Phys. Rev.Lett. 101 031601.
Best regards,
George

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Comment by Dmitry

2009-03-10 13:59:37

Dear George,

thanks for the nice post! What kind of duality is it that you study – I noticed that you work with 4d gravity, so are you talking about AdS_4/CFT_3 ?

Cheers,
Dmitry.

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Comment by g.koutsoumbas

2009-03-11 12:20:27

Hi Dmitry,

Yes, you are right.

Best regards,
George

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Comment by Dmitry

2009-03-12 13:06:11

Hi George,

by the way, is the temperature in the system just the temperature of Hawking radiation? What does it mean then that “the scalar black hole absorbs energy from the heat bath”?

Cheers,
Dmitry.

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Comment by g.koutsoumbas

2009-03-13 10:55:10

Hi Dmitry!

The temperature is $\frac{f'(r_+)}{4\pi}$ . We have also tried to give some more colorful description of the phase transition.

297. Exact gravity dual of a gapless superconductor

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