Relativistic superfluid profiles near critical surfaces

TL;DR

The paper addresses the breakdown of Landau's two-fluid hydrodynamics in regions with rapid condensate variation and develops a relativistic Gross–Pitaevskii framework to analyze stationary superflows near phase boundaries. It derives an exact relativistic Landau critical velocity, demonstrates that classic Newtonian profiles such as vortex cores and boundary layers persist, and shows how Tolman temperature gradients in accelerating or rotating frames produce smooth superfluid–normal crossovers instead of sharp transitions. The work provides a consistent, causal relativistic description of superfluid equilibria, with implications for neutron-star physics and other extreme environments. Overall, it establishes a robust tool for studying relativistic superfluids near critical surfaces and near walls, where gradient effects are essential.

Abstract

Landau's two-fluid model of superfluidity ceases to apply in regions where the condensate amplitude exhibits rapid spatial variation, such as vortex cores or in the vicinity of container walls. A recently proposed relativistic Gross-Pitaevskii-type framework treats the condensate as an independent scalar degree of freedom, enabling a controlled analysis of such regimes. We use it to construct stationary superflows close to the superfluid-normal phase boundary, and examine their stability. We obtain an exact expression for Landau's critical velocity and show that the standard Newtonian profiles (such as the near-vortex condensate depletion or the boundary-layer decay) persist unmodified in the relativistic setting. We further analyse a genuinely relativistic configuration in which an accelerated superfluid develops a phase boundary induced by Tolman temperature gradients.

Figures (5)

• Figure 1: Phase diagram of the equilibrium condensate amplitude $\bar{\sigma}(T,v)$ at fixed $\mu$ for a fluid governed by the equation of state \ref{['EoS']}. The superfluid region (where $\bar{\sigma}\neq 0$) is shown in blue, while the normal phase is indicated in red. We will show in section \ref{['LandauIsBack']} that there is an intermediate magenta band $v_c/\sqrt{3}\, {\le}\, v\, {\le}\, v_c$, with $v_c\,{=}\,\sqrt{m_0(T_c{-}T)/\mu^2}$, in which the naive maximization of $P(\sigma)$ predicts a nonzero condensate, yet the corresponding configurations are unstable to fluctuations. This instability implies that superfluidity actually breaks down at $v_c/\sqrt{3}$, which therefore represents Landau's critical velocity in this model.
• Figure 2: Equilibrium profile of the condensate amplitude $\bar{\sigma}(x)$ near a hard wall occupying the region $x\leq 0$, as given by \ref{['tanhaccone']}. The condensate vanishes at the boundary and relaxes to its homogeneous value $\bar{\sigma}$ [cf. \ref{['nereee']}] over a characteristic distance of order the healing length $L = 1/\sqrt{\lambda \bar{\sigma}^2}$. This expression is formally identical to the standard nonrelativistic result.
• Figure 3: Solution $f(\xi)$ of \ref{['galactic']} with $\ell=50$, and boundary conditions $f(100)=f(120)=0$. The dashed line is the function $\sqrt{1-\ell^2/\xi^2}$, corresponding to the uniform prediction \ref{['nereee']}, which is the value of the condensate magnitude employed in Landau's two-fluid model.
• Figure 4: Solutions of \ref{['visually']} for the condensate profile of a superfluid in equilibrium between two uniformly accelerating walls, located at $\varrho_{1}=10\,L$ and $\varrho_{2}=40\,L$. The four panels correspond to $\xi_\star = 5$ (upper left), $10$ (upper right), $20$ (lower left), and $30$ (lower right). The dashed curve represents $\sqrt{1-\xi_\star/\xi}$, i.e. the uniform prediction \ref{['nereee']}, which is the condensate magnitude entering Landau's two--fluid model.
• Figure 5: Solutions of \ref{['biggone']} describing the condensate amplitude for a superfluid in equilibrium within a rotating cylindrical container. The four panels correspond respectively to the parameter sets $\{r_{\text{Bucket}}/L,\theta,\nu,\omega\}=\{10,1/12,0.2,0.1\}$ (upper left), $\{10,1/11,0.6,1\}$ (upper right), $\{20,1/21,0.1,0.1\}$ (lower left), and $\{20,1/21,0.5,0.5\}$ (lower right). The dashed curve represents the uniform prediction \ref{['nereee']}, which is the condensate magnitude entering Landau's two--fluid model.