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arXiv:2401.04275

Revisiting the Landau criterion: a hydrodynamic and holographic approach to superfluid instabilities

Filippo Sottovia

Published Jan 8, 2024• 0 citations
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Abstract

In this thesis we investigate the instabilities of superfluids at finite superflow by means of a hydrodynamical approach. We find that at a finite value of the background superfluid velocity a hydrodynamic collective mode crosses to the upper half complex frequency plane, thereby signalling a dynamical instability. At the same time, however, this instability is also thermodynamic, as its onset is controlled by one of the second derivatives of the free energy changing sign. We carry out our analysis in two main setups: the "probe limit", where the fluctuations of the temperature and the normal fluid's velocity are frozen, and a complete approach, which includes them. In both cases we test our results with the help of gauge-gravity duality, finding good agreement between the hydrodynamic modes of the boundary theory and the quasinormal modes of the gravity theory. Our criterion for the onset of the instability, which is formulated in a model-independent way, applies to interacting systems irrespective of the strength of interactions, does not rely on boost invariance and does not assume any specific quantum statistics. As a final check, we also show that it yields the Landau critical velocity for Galilean superfluids with Bose-Einstein quasiparticles.

DOI
10.48550/arxiv.2401.04275

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In this thesis we investigate the instabilities of superfluids at finite superflow by means of a hydrodynamical approach. We find that at a finite value of the background superfluid velocity a hydrodynamic collective mode crosses to the upper half complex frequency plane, thereby signalling a dynamical instability. At the same time, however, this instability is also thermodynamic, as its onset is controlled by one of the second derivatives of the free energy changing sign. We carry out our analysis in two main setups: the "probe limit", where the fluctuations of the temperature and the normal fluid's velocity are frozen, and a complete approach, which includes them. In both cases we test our results with the help of gauge-gravity duality, finding good agreement between the hydrodynamic modes of the boundary theory and the quasinormal modes of the gravity theory. Our criterion for the onset of the instability, which is formulated in a model-independent way, applies to interacting systems irrespective of the strength of interactions, does not rely on boost invariance and does not assume any specific quantum statistics. As a final check, we also show that it yields the Landau critical velocity for Galilean superfluids with Bose-Einstein quasiparticles.
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In the collisional region at finite temperatures, the collective modes of superfluids are described by the Landau two-fluid hydrodynamic equations. This region can now be probed over the entire BCS--Bose-Einstein-condensate crossover in trapped Fermi superfluids with a Feshbach resonance, including the unitarity region. Building on the approach initiated by Zaremba, Nikuni, and Griffin in 1999 for trapped atomic Bose gases, we present a variational formulation of two-fluid hydrodynamic collective modes based on the work of Zilsel in 1950 developed for superfluid helium. Assuming a simple variational Ansatz for the superfluid and normal fluid velocities, the frequencies of the hydrodynamic modes are given by solutions of coupled algebraic equations, with constants only involving spatial integrals over various equilibrium thermodynamic derivatives. This variational approach is both simpler and more physical than a direct attempt to solve the Landau two-fluid differential equations. Our two-fluid results are shown to reduce to those of Pitaevskii and Stringari for a pure superfluid at T=0T=0T=0.
In the collisional region at finite temperatures, the collective modes of superfluids are described by the Landau two-fluid hydrodynamic equations reviewed in Chapter 14. In the case of trapped Bose gases, these are coupled differential equations with position-dependent coefficients associated with the local thermodynamic functions. Building on the approach initiated by Zaremba et al. (1999) for trapped atomic Bose gases, in this chapter we develop an alternative variational formulation of two-fluid hydrodynamics. This is based on the work of Zilsel (1950), originally developed to deal with superfluid He. Assuming a simple variational ansatz for the superfluid and normal fluid velocity fields, this approach reduces the problem of finding the hydrodynamic collective mode frequencies to solving coupled algebraic equations for a few variational parameters. These equations contain constants that involve spatial integrals over various equilibrium thermodynamic derivatives. Such a variational approach is both simpler and more physical than a direct attempt to solve the Landau two-fluid equations numerically. This chapter is mainly based on Taylor and Griffin (2005), Taylor (2008) and Zilsel (1950). In it, we discuss the normal modes of the non-dissipative Landau two-fluid equations for a trapped superfluid. In Section 16.3, we illustrate this formalism by deriving expressions for the frequencies of the dipole and breathing modes of a trapped Bose superfluid. In Chapters 17 and 18, we discuss an extended version of the two-fluid equations that includes hydrodynamic damping. The hydrodynamic damping of the collective modes is calculated in Chapter 19 using a generalized version of the variational approach developed in this chapter. Zilsel's variational formulation Since two-fluid hydrodynamics only describes a system in local equilibrium, all thermodynamic quantities are functions of position and time. Even in static equilibrium, in the presence of a trapping potential, most thermodynamic quantities will be position dependent.
In this chapter, we review the famous Landau theory of superfluidity at finite temperatures. This theory is based on coupled hydrodynamic equations for the superfluid and normal fluid components. Landau's two-fluid description is only valid when collisions among the thermal excitations making up the normal fluid are strong enough to produce local hydrodynamic equilibrium. These two-fluid equations were originally developed for liquid. He but are thought to be generic in form, describing the collision-dominated hydrodynamic region of all Bose superfluids. In this chapter, we will consider the solutions of the two-fluid equations mainly for a uniform superfluid. We discuss the existence of second sound (involving the out-of-phase motion of the superfluid and normal fluid components) as a characteristic feature of a Bose superfluid at finite temperatures. This chapter gives background material needed for Chapters 15–19. In Chapter 15, we will show that, in the appropriate limit, the Landau two-fluid equations can be derived from the ZNG coupled equations given in Chapter 3 for a trapped dilute Bose-condensed gas. In Chapters 17–19, we extend this discussion and derive the Landau–Khalatnikov two-fluid equations, which include hydrodynamic damping associated with various transport coefficients. Useful reviews of the two-fluid equations in the context of dilute spatially uniform Bose-condensed gases are given by Pethick and Smith (2008, Chapter 10) and Pitaevskii and Stringari (2003, Chapter 6). History of two-fluid equations The original discovery of superfluidity in liquid He was dramatically announced with the publication of the famous back-to-back papers of Kapitza (1938) in Moscow and Allen and Misener (1938) in Cambridge. These and subsequent experiments in the next few years showed that superfluid He could exhibit very bizarre hydrodynamic behaviour compared to classical liquids.
We study the collective excitation and stability of superfluid Fermi gases flowing with a constant velocity in three-dimensional free space. In particular, we investigate a possible gapless superfluid state induced by the superflow using the mean-field theory and the generalized random-phase approximation (GRPA). For weak attractive interactions, we find that the mean-field superfluid order parameter can take a nonzero value even after the superflow velocity exceeds the threshold for the onset of Bogoliubov quasiparticle excitations. Since the Cooper pairs are only partially broken by the quasiparticle excitations, a gapless superfluid state can be formed over a certain range of superflow velocity above the pair-breaking onset. In addition to the usual quasiparticle-pair continuum and the Anderson-Bogoliubov collective mode, the GRPA excitation spectrum of the gapless superfluid state has a quasiparticle-quasihole continuum and a second collective mode. We find that the long-wavelength excitations of the second collective mode eventually cause dynamical instability of the system when the superflow velocity increases. However, the gapless superfluid state still remains stable in a narrow but finite range of superflow velocity.