Counterflow and coflow instabilities in miscible binary superfluids

TL;DR

This paper develops a unified thermodynamic criterion for the onset of counterflow and coflow instabilities in miscible binary superfluids, showing that instability arises when the static susceptibility (the Hessian of the thermal free energy) loses positive definiteness, i.e., when $det(\bar{\chi})$ diverges and changes sign or when $\chi_{++}\chi_{--}-\chi_{+-}^2=0$. The authors validate this criterion across three frameworks: (i) dissipative binary superfluid hydrodynamics, (ii) zero-temperature Gross-Pitaevskii theory, and (iii) holographic (gauge-gravity duality) models of strongly coupled, finite-temperature superfluids, demonstrating consistent onset conditions for instabilities in counterflow and coflow configurations. They reveal a universal scaling of critical velocities with inter-component coupling $\nu$, with three of four critical velocities following $\nu^{1/2}$, and show that nonlinear evolution leads to vortex formation and annihilation that restores stability. The results bridge weakly and strongly interacting regimes, elucidating how thermodynamic susceptibilities govern dynamical instabilities and their nonlinear fate, with potential implications for experimental and numerical studies of multicomponent quantum fluids. The work also highlights the power of holography in capturing dissipative, strongly coupled hydrodynamics beyond quasiparticle pictures.

Abstract

We explore instabilities in binary superfluids with a nonvanishing relative superflow, particularly focusing on counterflow and coflow instabilities. We extend recent results on the thermodynamic origin of finite superflow instabilities in single-component superfluids to binary systems and derive a criterion for the onset of instability through a hydrodynamic analysis, which applies to interacting many-body systems at finite temperature. We find that the onset of these instabilities is signaled by the determinant of the Hessian of the thermal free energy diverging and changing sign. We verify this hydrodynamic prediction in a holographic binary superfluid modeled with gauge/gravity duality, which naturally incorporates strong coupling, finite temperature, and dissipation. We also compare to results obtained using the Gross-Pitaevskii equation for weakly interacting Bose-Einstein condensates and find that the same criterion continues to apply at zero temperature, where it reduces to evaluating derivatives of the supercurrents with respect to the superfluid velocities. We observe that the critical velocities of these instabilities follow a general scaling law related to the interaction strength between superfluid components. Finally, the nonlinear stages of the instabilities are studied by full time evolution using gauge/gravity duality, where vortex annihilation leads to a decrease of superfluid velocity back to a value where the binary superfluid phase is stable.

Figures (10)

• Figure 1: The QNMs spectrum with respect to $k$ and $v_y$ for the counterflow case (top) and the coflow cases (bottom). The stationary configuration is dynamical unstable whenever $\mathrm{Im}\omega_k>0$. We have considered holographic miscible binary superfluids at $T/T_c=0.677$ and $\nu=-0.2$.
• Figure 2: The QNM spectrum for holographic miscible binary superfluids with $T/T_c=0.677$, $\nu=-0.1$ and $v_y=0$. The left panel shows real part of QNMs while the right panel shows imaginary part. The red lines are the superfluid sound modes. The blue and black lines correspond to the type $\mathrm{II}$ Goldstone boson when $\nu=0$. At nonzero $\nu$, we expect one branch of this mode to otain an imaginary gap, which is indeed what we observe.
• Figure 3: QNM spectrum at low momentum for the counterflow case. Top: QNM spectrum for superfluid velocity $v_{y1}=0.283$, so that $v_{c1}<v_{y1}<v_{c2}$. Bottom: QNM spectrum for superfluid velocity $v_{c2}<v_{y2}=0.314$. The red lines are the sound modes, the blue line is the diffusive mode and the black line is the gapped mode. The relevant parameter values are $T/T_c=0.677$, $\nu=-0.2$. The unstable mode for $v_{y1}$ is the sound mode, while for $v_{y2}$, the unstable mode is the gapped mode. For reference, the two critical velocities at $T/T_c=0.677$ and $\nu=-0.2$ are numerically found to be $v_{c1}=0.251$, $v_{c2}=0.308$.
• Figure 4: Top:Velocity $v_s$ of the sound mode for different superfluid velocities $v_y$ in the counterflow case. Bottom: Susceptibility $\chi_{++}$ and free energy density difference $\Delta f=f_{\mathrm{binary}}-f_{\mathrm{single}}$. The vertical dashed lines denote the critical velocities $v_{c1}$ and $v_{c2}$. Above $v_{c1}$, $\chi_{++}$ changes sign, the sound velocity becomes purely imaginary, and one sound mode becomes unstable. Above $v_{c2}$, $\Delta f$ becomes positive. When $\Delta f>0$, the binary superfluid phase is globally thermodynamically unstable, and the gapped mode becomes unstable with a positive imaginary part at $k=0$. Relevant parameters are $T/T_c=0.677$, $\nu=-0.2$.
• Figure 5: QNM spectrum for the coflow case. Top: QNM spectrum for a superfluid velocity $v'_{y1}=1.257$, such that $v'_{c1}<v'_{y1}<v'_{c2}$. Bottom: QNM spectrum for a superfluid velocity $v'_{y2}=2.199$, such that $v'_{y2}>v'_{c2}$. The red lines are the superfluid sound modes, the blue line is the diffusive mode and the black line is the gapped mode. Relevant parameters are $T/T_c=0.677$, $\nu=-0.2$. At $v'_{y1}$, only the diffusive mode is unstable, while at $v'_{y2}$, one of the sound mode also becomes unstable. For reference, the two critical velocities at $T/T_c=0.677$ and $\nu=-0.2$ are numerically found to be $v'_{c1}=0.691$, $v'_{c2}=2.136$.