Polar Phase in Liquid 3He

• The polar phase in liquid 3He is an unconventional p-wave superfluid defined by a nodal order parameter and stabilized by anisotropic nematic aerogels.
• It exhibits distinctive topological characteristics, including a Dirac line node and associated collective modes like Nambu–Goldstone and Higgs excitations.
• Experimental NMR diagnostics and quantum critical behavior validate its unique symmetry breaking, phase transitions, and robustness against non-magnetic impurity scattering.

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 (Nakashima et al., 2021, Dmitriev et al., 2019, Fomin, 2019). 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 ($k_z=0$, $\theta = \pi/2$) (Nakashima et al., 2021).

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: $(\varphi, \hat d) \sim (\varphi+\pi, -\hat d)$ (Autti et al., 2015, Autti et al., 2017).

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 (Fomin, 2019).

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 (Fomin, 2018, Fomin, 2020). 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 (Nissinen et al., 2017, Zubkov, 2017). 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 (Nissinen et al., 2017).

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 (Zubkov, 2017).

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 (Dmitriev et al., 2019, Dmitriev et al., 2015). 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 $\tau = T - T_c$ (Fomin, 2019).

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 (Autti et al., 2015, Zavjalov, 2016).

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 (Nakashima et al., 2021).

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 (Wiman et al., 2013).

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 (Mineev, 2018).

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 (Autti et al., 2015, Zavjalov, 2016).
• 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 (Autti et al., 2015).

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 (Nissinen et al., 2017).

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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 (Nakashima et al., 2021, Dmitriev et al., 2019, Fomin, 2019, Dmitriev et al., 2015).