A quantum critical point from flavours on a compact space

Veselin G. Filev

doi:10.1007/jhep08(2014)105

Abstract

We analyse a $2+1$ dimensional defect field theory on a two sphere in an external magnetic field. The theory is holographically dual to probe D5-branes in global AdS$_5\times S^5$ background. At any finite magnetic field only the confined phase of the theory is realised. There is a first order quantum phase transition, within the confined phase of theory, ending on a quantum critical point of a second order phase transition. We analyse the condensate and magnetisation of theory and construct its phase diagram. We study the critical exponents near the quantum critical point and find that the second derivatives of the free energy, with respect to the bare mass and the magnetic field, diverge with a critical exponent of $-2/3$. Next, we analyse the meson spectrum of the theory and identify a massless mode at the critical point signalling a diverging correlation length of the quantum fluctuations. We find that the derivative of the meson mass with respect to the bare mass also diverges with a critical exponent of $-2/3$. Finally, our studies of the magnetisation uncover a persistent diamagnetic response similar to that in mesoscopic systems, such as quantum dots and nano tubes.

Highlights

The existence of a quantum critical regime accessible at finite temperature is crucial for such applications, away from the critical point one needs more than just an effective field theory and the analysis often relies on physical arguments and approximations rather than rigorous calculations
We study the critical exponents near the quantum critical point and find that the second derivatives of the free energy, with respect to the bare mass and the magnetic field, diverge with a critical exponent of −2/3
We find that the derivative of the meson mass with respect to the bare mass diverges with a critical exponent of −2/3

Summary

The theory at zero magnetic field

To introduce fundamental flavours in the quenched approximation to the dual field theory we will consider a probe D5-brane extended along the radial coordinate r and wrapping two spheres in both the S5 and the AdS5 parts of the geometry. Generally the possible embeddings of the D5-brane split into two classes - embeddings which close at some radial distance above the origin of AdS5 by having the S2 inside the S5 part of the geometry shrink to zero size (Minkowski embeddings), and embeddings which reach all the way to the origin of AdS5 and close there, which we call “Ball” embeddings This construction has been investigated in ref. Note that to arrive at the last result one has to introduce appropriate counter terms [2] (see [19]) With this definition one obtains a constant non-zero value of the condensate at infinite bare mass, suggesting that an additional counter term is needed. Since we are mainly interested in the properties of the theory near the phase transition, we will take the same approach as in ref. [2] and use the present definition of the condensate (being proportional to −c)

General set up

Branes and magnetic monopoles

Fundamental condensate

Phase diagram

Π mqR2

Magnetisation

Critical behaviour

Meson spectrum

Spectrum along χ

Critical behaviour of the spectrum

Conclusion