---
title: Polar Phase in Liquid 3He
url: https://www.emergentmind.com/topics/polar-phase-in-liquid-3he
type: topic
---

# Polar Phase in Liquid 3He

The polar phase in liquid $^3$He is an unconventional superfluid state realized via anisotropic confinement, notably in “nematic” aerogels composed of nearly parallel strands. This phase exhibits a nodal p-wave order parameter with unique topological, symmetry, and dynamical attributes. The discovery and characterization of the polar phase have transformed understanding of anisotropy-induced phenomena in Fermi superfluids and extended the paradigm of the Anderson theorem to triplet-paired systems.

## 1. Order Parameter Structure and Symmetry Breaking

The order parameter of the polar phase is written in the spin–orbital basis as a $3\times3$ complex matrix $A_{\mu j}(\mathbf{r})$, with $\mu$ denoting spin and $j$ orbital components. For the uniform polar state:
\[
A_{\mu j}(\mathbf{r}) = \Delta\, \hat d_\mu\, \hat z_j,
\]
where $\Delta$ is the amplitude, $\hat d$ is a unit spin vector, and $\hat z$ is the global axis of anisotropy defined by the columnar aerogel strands [2101.10762, 1911.01193, 1901.05025]. The gap function on the Fermi surface is
\[
\Delta(\mathbf{k}) = \Delta_0\, k_z/k_F = \Delta_0 \cos\theta,
\]
implying a line node at the equator ($k_z=0$, $\theta = \pi/2$) [2101.10762].

The polar phase is an equal-spin-pairing (ESP) state with orbital angular momentum $\ell=1$, projection $\ell_z=0$. Symmetry breaking proceeds as follows:
- Full gauge U(1) symmetry is broken.
- Spin rotations reduce to SO(2) about $\hat d$; orbital rotations reduce to SO(2) about $\hat z$.
- A residual discrete $\mathbb{Z}_2$ symmetry remains: $(\varphi, \hat d) \sim (\varphi+\pi, -\hat d)$ [1508.02197, 1711.02915].

## 2. Stabilization in Nematic Aerogel and Anderson’s Theorem

Highly anisotropic aerogels, such as nafen or aligned mullite strands (“nematic” aerogels), stabilize the polar phase by lifting the degeneracy among $p$-wave pairing states through strong uniaxial scattering. The Ginzburg–Landau free-energy functional incorporates an explicit anisotropy term:
\[
F_\mathrm{ani} = N(0)\,A_{\mu j}^* \left[K_{jl} + n_{jl}(\mathbf{r})\right] A_{\mu l},
\]
where $K_{jl}$ is the global anisotropy tensor, and $n_{jl}(\mathbf{r})$ models spatially random local anisotropy [1901.05025].

A crucial property is the immunity—under ideal conditions—of the polar transition temperature $T_c$ to non-magnetic impurity scattering by columnar strands. This is a direct analog of Anderson’s theorem: specular, $k_z$-conserving scattering does not suppress $T_c$ for the $\ell_z=0$ channel. Self-energy and vertex corrections to the gap equation cancel exactly, so $T_c$ matches the bulk polar result [1803.03804, 2003.09652]. Magnetic (spin-flip) or diffuse scattering, however, reduces $T_c$ via standard pair-breaking effects.

## 3. Quasiparticle Spectrum, Topology, and Collective Modes

The Bogoliubov–de Gennes spectrum in the polar phase displays a Dirac line node: zero energy excitations occur when $|\mathbf{k}| = k_F$ and $k_z=0$. This nodal ring supports a nontrivial Berry phase for each spin sector and defines a topologically protected class distinct from chiral superfluids with Weyl point nodes [1710.07616, 1705.02231]. The low-energy Hamiltonian around the nodal line can be written (suppressing spin indices for clarity):
\[
H(\mathbf{p}) = v_F(p - k_F)\,\tau^3 + c_\perp\, (\hat z \cdot \mathbf{p})\, \tau^1,
\]
where $\tau^{1,3}$ are Pauli matrices in particle–hole space [1710.07616].

Collective modes include:
- Nambu–Goldstone bosons from broken gauge and spin symmetries,
- Gapped “Higgs” amplitude modes, whose masses obey a Nambu sum rule relating the mode gaps to the angular average of the squared gap on the Fermi surface [1705.02231].

Spin–orbit coupling splits the Fermi line of nodes into two Weyl points, but the nodal line is robust in the absence of such effects.

## 4. Experimental Signatures and NMR Phenomenology

The polar phase is unambiguously identified via nuclear magnetic resonance (NMR). The primary NMR frequency shift $\Delta\omega$ is determined by the order-parameter amplitude and the relative orientation between magnetic field and aerogel axis. In ideal (uniform) polar states and for $\mu=0$ (field along strands):
\[
2\omega_L\Delta\omega = K_\mathrm{mf}\Omega_A^2,\quad K_\mathrm{mf} = 4/3,
\]
where $\Omega_A$ is the A-phase Leggett frequency [1911.01193, 1507.04275]. For fields perpendicular to the strands ($\mu=90^\circ$), $\Delta\omega=0$.

In realistic aerogels, spatial fluctuations of anisotropy generate longitudinal and transverse order-parameter fluctuations, suppressing both the average amplitude and the NMR shift, especially near $T_c$. The leading correction diverges as $\tau^{-1/2}$ with $\tau = T - T_c$ [1901.05025].

Table 1: Key NMR shifts in polar phase (ideal/real limit)

| Aerogel Quality  | $K(T)$ at $T \ll T_c$ | $\Delta\omega(\mu=90^\circ)$ | Fluctuations     |
|------------------|----------------------|-----------------------------|------------------|
| Ideal (nafen)    | 1.32–1.33            | 0                           | Negligible       |
| Disordered       | $<$1.3 (downturn)    | 0 or small negative         | Strong near $T_c$|

Half-quantum vortices (HQVs), accessible only in the polar phase, are detected as additional satellite peaks in NMR, with intensity and splitting functions of the HQV density and orientation [1508.02197, 1601.04190].

## 5. Quantum Criticality and the Polar–PdA Transition

At low but finite pressure $P_c(0)$, the polar phase undergoes a continuous quantum phase transition to the polar-distorted A (PdA) phase, where an additional imaginary order parameter component appears:
\[
A_{2j}(x) = \delta_{j3}\Delta + i\,\delta_{j1} e(x).
\]
The effective Gaussian action for low-energy fluctuations $e(q, \omega)$ is:
\[
S_\mathrm{eff}^{(2)}[e] =
N(0)\sum_{q, \omega} \left[ c_m(P) + \frac{|\Delta|\tau}{8} \left(\frac{\omega}{|\Delta|}\right)^2 \left(|\ln|\omega/\Delta|| + c_2\right) + \xi_0^2 c_{ij} q_i q_j \right]|e(q,\omega)|^2,
\]
with $c_m\propto P_c(0)-P$ and a nonanalytic $\omega^2\ln|\omega|$ term arising from the nodal line [2101.10762].

Dynamical scaling near criticality is characterized by a quantum critical exponent $z=1$, and the correlation length exponent is $\nu=1/2$. The compressibility diverges weakly at $P\rightarrow P_c^{-}$:
\[
\Delta\kappa \sim \sqrt{\ln[1/(P_c - P)]}.
\]
This $\sqrt{\ln}$ upturn (similarly, $\sqrt{\ln(T_0/T)}$ at $P = P_c$) is a characteristic signature of polar-phase quantum criticality.

## 6. Geometric and Surface Effects, Extensions

In periodic geometries and nano-fabricated arrays with strong pair-breaking boundaries, spatially inhomogeneous polar order can be stabilized. The phase diagram in the $(T, d, L)$ parameter space (where $d$ is post size and $L$ lattice spacing) defines a range in which only the polar phase exists, its amplitude modulated and pinned by lattice symmetry [1307.7308].

Surface preparation critically alters phase stability.  $^4$He preplating suppresses spin-flip (magnetic) scattering, maximizing anisotropy and favoring the polar phase, whereas a solid-like $^3$He surface layer enables exchange scattering, suppressing the polar phase and polar-distorted A window, as observed via both phase diagrams and spin-diffusion anisotropy [1804.01073].

## 7. Topological Excitations and Quantum Simulation Context

The polar phase supports unique topological defects:
- Half-quantum vortices, characterized by $\pi$ windings in both phase and spin vectors, observable via NMR satellites [1508.02197, 1601.04190].
- Domain walls (solitons) in spin–orbit textures.
- At PdA transition, Majorana zero modes can localize at HQV cores, with enhanced isolation due to strong vortex pinning [1508.02197].

The polar phase’s line-node topology enables simulation of emergent nontrivial geometry (“degenerate tetrad,” “anti-spacetime”), fractional QED actions, and drumhead surface states, connecting condensed matter and quantum field theory [1710.07616].

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The polar phase in $^3$He thus stands as a paradigmatic realization of a nodal p-wave superfluid stabilized by anisotropy and protected both by engineered disorder and symmetry, providing a testbed for topological quantum phenomena, collective mode quantum criticality, and analogies to high-energy physics and unconventional superconductivity. Key experimental diagnostics are based on precise NMR shifts, sound propagation, and the direct observation of topological defects and quantum critical characteristics [2101.10762, 1911.01193, 1901.05025, 1507.04275].

Source: https://www.emergentmind.com/topics/polar-phase-in-liquid-3he