Summary

Quantum phase transitions in holographic superfluids

We start with applying the methods of the AdS/CFT correspondence on two different gravity setups which are related to phenomena of strongly coupled systems inthermal equilibriumsuch as quantum phase transitions and quantum critical points. We use the bottom-up approach in section 3.3 and the top-down approach in section 3.4 for the constructing of holographic superfluids at finite baryon and isospin chemical potential. We study the quantum phase transitions and look for a possible quantum critical point in our setups.

3.1 Introduction and motivation

Gauge/gravity duality, a generalized and extended version of the AdS/CFT cor-respondence [10–12] as discussed in section 2.2, maps a more general quantum field theory to a given background with the same global symmetries. It provides a novel method for studying strongly correlated systems at finite temperature and densities. In recent years, remarkable progress has been made towards application of gauge/gravity duality for describing strongly coupled condensed matter physics at low temperatures, see e. g. [19–22]. At low temperatures condensed matter shows many interesting properties including superconduc-tivity and superfluidity. Some of the first applications of gauge/gravity duality towards the holographic description of superfluids and superconductors are described in [15, 17] and of (non-) Fermi liquids in [112]. Of special interest are the studies of quantum critical theories which arise at continuous phase transitions at zero temperature [24]. A phase transition at zero temperature is not driven by thermal fluctuations, but rather by quantum fluctuations. If the quantum phase transition is continuous, i. e. at least of second order, there is a quantum critical point. This quantum critical point influences the phase diagram also at non-zero temperature, see fig. 3.1 for a schematic picture of

the phase diagram near the quantum critical point. In this influenced region, the quantum critical region, the system may be described by a critical the-ory even at finite temperature [45–47]. There are speculations that quantum phase transitions might be important in describing highTc-superconductors like cuprates, non-Fermi liquids or superconducting-insulator transitions in thin metallic films, see e. g. [19, 20] and references therein. Up to date, there is still a lack of a complete and satisfactory theoretical description for high Tc-superconductors, ultra-cold Fermi gases and non-Fermi liquids, and it is believed that a better understanding of the critical region may shed light on this branch of physics.

g T

g

0

Insulator Superfluid

Quantum critical T

FIGURE3.1: Typical phase diagram near a quantum critical point. The quantum critical point at zero temperature is triggered at some critical parametersg =gcwhich might be the chemical potential, the magnetic field, the chemical composition, etc. The superconducting and normal conducting phases are separated by a quantum critical region which may be described by a scale-invariant theory at finite temperature. The dashed lines denote the cross-overs, while the solid line denotes a possible Kosterlitz-Thouless phase transition at the temperatureTKT>0. This figure is taken from [45].

Using the gauge/gravity duality it is possible to construct physical systems which show a phase transition from a normal to a superconducting phase, see e.g. the reviews [19–22]. Studying these systems, many properties of superconductors and superfluids can be recovered such as infinite direct current (DC) conductivities, mass gap for single-particle excitation and remnant of the Meisser-Ochsenfeld effect, see e. g. [15–18]. In general, the dual gravity solution for a superconducting state is a chargedAdSblack hole which develops hair. For black holes in asymptotically flat space, there is a no-hair theorem, see [113] for a review, which postulates that all black hole solutions of the Einstein-Maxwell equations in general relativity can be characterized by three classical parameters: mass, angular momentum and electric charge. This situation is changed if the embedding space-time is asymptoticallyAdS. AdS space-times are vacuum solutions of Einstein’s field equations with negative

cosmological constant. The crucial difference to flat space-times is that an AdSspace-time acts like a confining box such that charged particles cannot escape to infinity. This property allows AdS black holes to develop hair made of condensing charged particles [21, 114]. Depending on whether it is a scalar or vector hair, the dual field theory will be a s-wave or p-wave theory of superconductivity. Such a theory may describe superconductors or superfluids. In general, a superconducting condensate breaks anU(1)symmetry spontaneously. If the underlying broken symmetry is a local symmetry, the system describes a superconductor, while for the case of a global symmetry the system describes a superfluid.

In this chapter we focus on the construction of holographic superfluids and study their phase diagrams. Of great interests are the quantum phase transitions and the possible existence of a quantum critical point in such systems. So far most of studied systems showing the transition to a holographic superfluid have only been considered with one control parameter, usually the ratio of the temperature to a chemical potential. In such systems the phase transition is at a finite temperature and thus these systems have no quantum phase transition, see e. g. [17, 48]. In this chapter we construct gravity systems which resemble a p-wave superfluid with continuous phase transitions at zero temperature and thus possess quantum critical points. That can be done by introducing a further chemical potential as a second control parameter. Varying this an additional parameter the phase transition temperature can be tuned to zero, so that studying quantum phase transition is possible. Studying the order of the phase transition at zero temperature by comparing the free energy of the normal and superconducting phase, it is possible to determine whether the system possesses a quantum critical point or not.

The main motivation for studying quantum phase transitions at finite baryon and isospin chemical potential using holographic methods is the following.

First, studying physics at quantum criticality using gauge/gravity duality has obtained very promising results related to transport phenomena in strongly cor-related systems, see again [19–22] and [47]. Moreover, gauge/ gravity duality seems to be the only known theory so far which is able to provide analytical results which are used to phenomenologically describe physical processes in this regime. Thus the recent developments encourage further studies using holographic methods to understand physical phenomena at quantum criticality.

The second main motivation for the work presented in this chapter is that there are studies about quantum phase transitions at finite baryon and isospin chemi-cal potential from QCD [49, 50], which can be used to compare with our results obtained from gravity models. For a clearer understanding what we could learn from our results combined with those from the works mentioned above, let us now briefly describe some for this chapter relevant results of [49, 50] in the next paragraph.

Usually, systems with two chemical potentials are called imbalanced

mix-0 0.2 0.4 0.6 0.8 1 0

0.05 0.1 0.15 0.2

µ_B(GeV)

T(GeV)

TCP

2nd

1st /=0

/&0

µ_I=0.2GeV

FIGURE3.2: Phase diagrams of real world systems: (a) Imbalanced Fermi mixture in the canonical ensemble [49]: The spin polarization is the thermodynamic conjugated variable to the ratio of the chemical potentials favoring the different spins. (b) QCD at finite baryon and isospin chemical potential [50]. In both phase diagrams we observe a superfluid phase at small temperature and small ratio of two chemical potentials. In addition in both diagrams the phase transition is second order for large temperature and becomes first order at low temperatures. Both diagrams show a first order quantum phase transition, thus in those systems there is no quantum critical point. The figures are taken from [49] and [50].

tures since two kinds of particles are present in imbalanced numbers. Examples are imbalanced Fermi mixtures where fermions with spin up and spin down are imbalanced [49], and QCD at finite baryon and isospin chemical potential where for instance up and down quarks are imbalanced [50] (see also [115]).

Interestingly the phase diagrams of both these systems are very similar (see figure 3.2). In both systems there is a superfluid state at low temperatures and at certain ratios of the two chemical potentials. In addition also the order of the phase transition agrees in both examples: At low temperatures (also at zero temperature) the transition is first order while at higher temperatures the tran-sition becomes second order. Thus, we want to address the question whether there is an universal structure which relates these two different systems? The answer we find in this chapter is no and will be explicated in subsection 3.4.4.

In the next section we will give a brief review how to construct supercon-ductors and superfluids using holographic methods relying on [15, 17], and study quantum phase transitions in holographic superfluidity at finite baryon density and isospin density in an imbalanced mixture. We construct the phase diagrams via two different gravity models and compare these results among each other and with the results obtained from QCD [49, 50]. The materials presented in sections 3.3 and 3.4 are my own results which are obtained in collaboration with Johanna Erdmenger, Viviane Grass and Patrick Kerner [3].