# The Vortex Phase Diagram of Rotating Superfluid $^3$He-B

## TL;DR

This paper theoretically maps the vortex phase diagram of rotating superfluid $^3$He-B, revealing distinct vortex phases, phase transitions, and the effects of magnetic fields on vortex stability, advancing understanding of superfluid vortex structures.

## Contribution

First theoretical calculation of the pressure-temperature-field vortex phase diagram for rotating superfluid $^3$He-B, including vortex structures and stability analysis.

## Key findings

- Identified first-order phase transition line $T_v(p)$ separating vortex phases.
- Discovered the metastability and supercooling of the A-core vortex phase.
- Showed magnetic fields extend the stability region of the A-core phase.

## Abstract

We present the first theoretical calculation of the pressure-temperature-field phase diagram for the vortex phases of rotating superfluid $^3$He-B. Based on a strong-coupling extension of the Ginzburg-Landau theory that accounts for the relative stability of the bulk A and B phases of $^3$He at all pressures, we report calculations for the internal structure and free energies of distinct broken-symmetry vortices in rotating superfluid $^3$He-B. Theoretical results for the equilibrium vortex phase diagram in zero field and an external field of $H=284\,\mbox{G}$ parallel to the rotation axis, $\vec{H}\parallel\vec{\Omega}$, are reported, as well as the supercooling transition line, $T^{*}_ {v} (p,H)$. In zero field the vortex phases of $^3$He-B are separated by a first-order phase transition line $T_ {v} (p)$ that terminates on the bulk critical line $T_{c}(p)$ at a triple point. The low-pressure, low-temperature phase is characterized by an array of singly-quantized vortices that spontaneously breaks axial rotation symmetry, exhibits anisotropic vortex currents and an axial current anomaly (D-core phase). The high-pressure, high-temperature phase is characterized by vortices with both bulk A phase and $\beta$ phase in their cores (A-core phase). We show that this phase is metastable and supercools down to a minimum temperature, $T^{*}_ {v} (p,H)$, below which it is globally unstable to an array of D-core vortices. For $H\gtrsim 60\,\mbox{G}$ external magnetic fields aligned along the axis of rotation increase the region of stability of the A-core phase of rotating $^3$He-B, opening a window of stability down to low pressures. These results are compared with the experimentally reported phase transitions in rotating $^3$He-B.