Holographic model of superfluidity (0809.4870v3)

Abstract: We study a holographic model of a relativistic quantum system with a global U(1) symmetry, at non-zero temperature and density. When the temperature falls below a critical value, we find a second-order superfluid phase transition with mean-field critical exponents. In the symmetry-broken phase, we determine the speed of second sound as a function of temperature. As the velocity of the superfluid component relative to the normal component increases, the superfluid transition goes through a tricritical point and becomes first-order.

• The paper identifies a second-order superfluid phase transition with mean-field critical exponents occurring below a critical temperature.
• The paper employs an Euclidean path integral framework to analyze thermodynamic properties and construct phase diagrams with tricritical points.
• The paper derives hydrodynamic equations from an Einstein-Maxwell charged scalar model, elucidating conservation laws and second sound dynamics in superfluids.

Analysis of a Holographic Model of Superfluidity

This essay provides an analysis of the holographic model of superfluidity as presented by Herzog, Kovtun, and Son. The paper leverages gauge/gravity duality to model relativistic quantum systems exhibiting superfluidity. The primary focus of this paper is the behavior of a quantum system with a global $U(1)$ symmetry, explored through the lens of holography, particularly in systems with superfluid phase transitions.

Main Contributions of the Paper

The research explores several key areas of holographic superfluidity:

1. Superfluid Phase Transition: The authors identify a second-order superfluid phase transition occurring below a critical temperature. This transition evidences mean-field critical exponents, suggesting that the transition can be captured by mean-field theories.
2. Thermodynamic Analysis: A detailed exploration of the thermodynamic properties of the quantum system is performed. The authors use the Euclidean path integral framework to introduce chemical potentials and further analyze states with finite winding, eventually characterized by the thermodynamic pressure $P(T, \mu, \chi)$.
3. Hydrodynamics of Superfluids: The hydrodynamic equations derived are consistent with previously established formulations by Carter, Khalatnikov, and others. Through a Josephson equation and a set of hydrodynamic variables, the model integrates the conservation laws for energy, momentum, and particle number within the superfluid context.
4. Dual Gravity Description: The authors implement an Einstein-Maxwell theory with a charged scalar to explore the dual gravitational description. The paper shows that fluctuations in the boundary gauge fields can provide insights into the associated field theory's thermodynamics and dynamics.
5. Numerical Results: Significant numerical simulations support their theoretical findings, including the construction of phase diagrams exhibiting tricritical points where the nature of the phase transition shifts from second-order to first-order as a function of the superfluid velocity.

Key Numerical Results and Implications

• The critical velocity for the transition to a first-order phase is calculated, revealing insights into novel phenomena in relativistic superfluids, previously observed primarily in non-relativistic systems.
• The speed of second sound is evaluated, providing essential details on hydrodynamic excitations within the superfluid phase.

Implications and Future Directions

This work effectively uses holography to gain deeper insights into complex quantum systems, particularly those that contain strongly interacting elements like superfluids. The findings have potential applications in understanding other strongly interacting systems, such as the quark-gluon plasma.

For future studies, it would be promising to incorporate back-reaction effects within the gravitational model, potentially offering more precise predictions, particularly in the low-temperature regime. Furthermore, extracting dynamic properties such as correlation functions and exploring sound wave poles in density fluctuations may yield interesting insights into the behavior of relativistic quantum fluids.

The presented model not only furthers the theoretical understanding of holographic superfluidity but also paves the way for utilizing such approaches in broader contexts, potentially influencing experimental studies in ultra-cold atoms and materials exhibiting quantum phase transitions. Overall, this paper exemplifies how theoretical models based on holographic principles can resonate deeply with the characteristics observed in complex physical systems.