Partially Mixed Helical Regime

• The partially mixed helical regime is characterized by incomplete helicity and competing order parameters that result in mixed quantum states.
• It exhibits distinct features such as helical mixing in quantum edge states, RG flow-controlled gaps, and hybrid modes in plasmas and fluids.
• Studying this regime advances applications in topological quantum computing, spintronics, and fluid dynamics through insights on novel transport and instability mechanisms.

The partially mixed helical regime is a concept emerging across condensed matter physics, quantum fluids, plasma physics, and classical and geophysical fluid mechanics. It denotes states or regimes in which helicity—either in spin, vorticity, or structural flow sense—is not maximally pure due to symmetry breaking, mixing, or coexistence of multiple order parameters or excitations. This regime is typified by the partial admixture of helical and non-helical characters, the interplay of competing orderings, or partial preservation of helical structure in the presence of perturbations and interactions. Given its broad relevance, this article synthesizes rigorous definitions, mathematical formalisms, and key phenomenology from recent developments in quantum materials, quantum Hall edge transport, superfluids, plasmas, and instability theory.

1. Quantum Helical Edge States: Symmetry Breaking and Helical Mixing

In quantum spin Hall and topological insulator edge channels, helical states arise from strong spin–orbit coupling with time-reversal symmetry (TRS) protection, leading to robust spin–momentum locking. When TRS is weakly broken—such as by a perpendicular Zeeman field—helicities of right- and left-movers become mixed and the energy spectrum is gapped, transitioning to a “partially mixed helical” (PMH) regime (Bakhshipour et al., 16 Jun 2024, Bakhshipour et al., 29 Aug 2025). The resulting low-energy theory is described by a bosonized Hamiltonian containing a sine-Gordon (cosine) term, $$\mathcal{H}_{\Delta_z} \propto (\Delta_z/\pi a_0) \cos(\sqrt{4\pi}\Phi + 2 k_F x)$$, promoting mixing of spin channels.

In this PMH regime:

• The Zeeman-induced gap is RG-relevant for strong electron–electron repulsion (Luttinger parameter $K < 2$), which enhances the helical mixing and leads to a strong gap phase.
• For attractive or weakly repulsive interactions ($K > 2$), the gap is RG-irrelevant and the helical state remains weakly mixed (weak gap regime).
• The RG flows satisfy: $\frac{d\mathcal{Y}}{dl} = (2-K(l))\mathcal{Y}(l)$ (gap coefficient $\mathcal{Y}$), and correlations in spin and charge density waves reflect the relevancy regimes.

Correlation function analysis demonstrates that, in the strong gap PMH regime, spin-density-wave (SDW) order dominates, especially in the direction perpendicular to the Zeeman field, while charge-density-wave (CDW) order is suppressed. The charge transport, computed using the Memory function approach, displays a non-uniform temperature dependence: a power-law at high temperatures ($\sigma \propto T^{3-2K}$) and exponential decay ($\sim e^{-\Delta_{\mathrm{eff}}/T}$) when the chemical potential lies within the Zeeman-induced gap. These features reflect the fundamental effects of helical mixing and strong correlations (Bakhshipour et al., 16 Jun 2024).

In systems with both Zeeman field and proximity-induced s-wave superconductivity (hybrid systems), the PMH regime occurs when the Zeeman gap is dominant but singlet pairing remains present; this modifies not only SDW and CDW correlations but also singlet and triplet pairing. Here, the RG equations include coupled flows for the Zeeman and superconducting sine-Gordon terms, with competition leading to logarithmic corrections in correlation functions, and interaction-dependent temperature and frequency scaling in spin transport properties (Bakhshipour et al., 29 Aug 2025). The key RG scaling relations are: $$\frac{d\mathcal{Y}_z}{dl} = (2-K)\mathcal{Y}_z,\qquad \frac{d\mathcal{Y}_s}{dl} = (2-1/K)\mathcal{Y}_s$$ where $\mathcal{Y}_z$ and $\mathcal{Y}_s$ denote Zeeman and singlet-pairing couplings, respectively.

2. Collective Edge and Bulk Modes: Partially Mixed Helical Plasmons

The partially mixed helical regime also appears in edge collective modes of quantum Hall systems. For 2D electron systems at not-too-low temperatures ($\hbar\omega_c \gg k_B T \gg \hbar v_g/2\ell_0$), the “helical edge magnetoplasmon” emerges as a hybrid mode with weak damping even under strong edge dissipation (Silva et al., 2010). This regime is defined by a dimensionless dissipation parameter $$\eta_T = \xi/(k_x \ell_T) \gtrsim \ln[1/(k_x \ell_T)] \gg 1$$, where $\xi$ is the transverse-to-Hall conductivity ratio and $\ell_T = (k_B T\ell_0^2)/(\hbar v_g)$.

The spatial structure and robust propagation of this mode originate from a partial mixing of monopole and higher multipole (e.g., quadrupole) contributions, expressed via an expansion: $$\rho^{(s)}(\omega, k_x, y) = \tilde{R}_0(Y) e^{-Y} \sum_{n=0}^\infty \rho_n^{(s)}(\omega, k_x) L_n(Y)$$ Laguerre polynomials $L_n(Y)$ capture the mix, and as the edge magnetoplasmon propagates, the phase modulates this multipole content. The mode remains weakly damped due to strong electron–electron Coulomb coupling that renormalizes its dispersion, while other (pure) edge modes are overdamped in this regime. The Coulomb renormalization and modifications by external screening (e.g., gates or air interfaces) further typify the “partial mixing” and resilience of the helical edge mode over others (Silva et al., 2010).

3. Partially Mixed Helical Regimes in Quantum Fluids and Superfluids

In multicomponent quantum fluids—such as phase-separated two-component Bose–Einstein condensates (BECs)—partially mixed helical regimes arise during inter-component shear flow instabilities (Hayashi et al., 2013). Two regimes are distinguished:

• Helical vortex sheet: Wide cylindrical interface, with vorticity localized and instabilities dominated by ripple (“ripplon”) modes, described by effective interface theory and capillary-like dispersion.
• Core-flow vortex: Narrow interface or full overlap of components near the vortex core, instability dominated by vortex-characteristic (Kelvin, varicose, splitting) modes.

The crossover (partially mixed) regime, in which the interface thickness and radius are comparable, features competition and mixing of ripplon and vortex modes. The phase diagram is governed by parameters such as the intercomponent interaction $\gamma$, circulation quantum number $L$, equilibrium interface radius $r_0$, and stratification parameter $\nu$. The effective Lagrangian and resultant dispersion relations for interface perturbations (e.g., equation for ripple mode frequency $\omega_{\ell,q}$) encapsulate the intricate balance leading to partial mixing.

4. Helicity Mixing in Magnetized, Turbulent, and Layered Systems

Plasmas and Turbulent Flows

In partially magnetized beam-driven plasmas, the transition between quasi-neutral and non-neutral regimes as pressure varies leads to a mixed helical response: spiral-arm (2D) structures in the quasi-neutral (higher pressure) case arising from a lower-hybrid instability, and fully 3D helical-rotating structures in the non-neutral (low pressure) case due to diocotron-type Kelvin–Helmholtz instabilities (Chen et al., 2 Jan 2025). The helical structures are characterized by a measurable rotation frequency (e.g., $\omega = 2\pi |\lambda|/(B R^2)$) and their emergence modifies cross-field transport, with consequences for plasma confinement and mixing.

In statistically stationary, externally forced turbulent flows, the “partially mixed helical regime” designates a state with maximal helicity at all scales due to multi-scale, helical injection (Biferale et al., 2018). Here, energy and helicity transfers are both subdominant to external injection across the inertial range, so the mirror symmetry breaking (helicity imbalance) persists. Isotropic helicoids—parity-breaking probe particles—respond to this helical field by exhibiting preferential sampling, clustering, and effective light/heavy particle behavior depending on their intrinsic chirality coupling parameter ($C_0$), particle size, and inertia. Analytical and DNS studies reveal that such preferential sampling and clustering encode the persistent helical character of the flow at all scales.

Layered and Stratified Flows

The partially mixed helical regime also manifests in stratified Taylor–Couette flow (Leclercq et al., 2016) and mixed-boundary Rayleigh–Bénard convection (Ostilla-Monico et al., 2020). In Taylor–Couette systems, interactions between helical modes and their nonlinear mixing produce robust coherent patterns (“mixed ribbons”, “mixed cross-spirals”) and both dynamic (transient) and static (persistent staircase) layering—behavior sensitive to the Schmidt number due to the differing rates of momentum and density diffusion.

Rayleigh–Bénard convection with 50% conducting/adiabatic striped boundary conditions exhibits a transition between two regimes as the stripe wavelength crosses the cell height: in the large-stripe regime, bulk flows reorganize to develop strong, directional (horizontal) circulations with enhanced velocity fluctuations normal to the stripes—breaking the mirror symmetry and inducing large-scale helical-like winds, a template for partial mixing in geophysical and astrophysical flows (Ostilla-Monico et al., 2020).

5. Topological and Spin Chains: Mixed Helical States via Dissipation

Dissipative stabilization in quantum spin chains yields steady states that are nontrivial mixtures of helical orderings, even in the absence of a protecting gap. In the boundary-driven XXZ spin-1/2 chain, at special anisotropy values, the system’s steady state is a rank-2 mixture of two helical states of opposite winding numbers (wavenumbers), forming a partially mixed helical regime (Essink et al., 2019). Increasing the anisotropy away from these fine-tuned values results in higher-rank mixed states; at zero anisotropy, the steady state becomes fully mixed. These findings point to dissipative mechanisms as a route to engineer partially mixed helical and topological orders in open quantum systems.

6. Phenomenology and Broader Implications

The partially mixed helical regime is a unifying framework for:

• Systems with incomplete helicity due to competing perturbations (magnetic fields, superconducting pairing, disorder, or interaction-induced mixing).
• Mixed topological states where symmetry or order parameters change discretely via intermediate mixed states—either in quantum spin chains or in spatially extended topological phases.
• Fluid and plasma systems where mixed or partial helicity triggers novel transport, mixing, and instability characteristics, with practical consequences for plasma devices, geophysical flows, and turbulence diagnostics.

Mathematically, the regime is described by a combination or admixture of eigenstates, correlation functions, or collective modes, controlled by parameters encoding the balance of mixing, interaction, and symmetry breaking. Signatures include anisotropic response functions, nontrivial RG flows, non-universal scaling exponents, and distinctive transport phenomena (nonuniversal conductivity, altered scaling, and robust but incomplete quantization). In all cases, the regime is associated with emergent structures—be they vortex lattices, helical plasmon modes, or patterned turbulent patches—which reveal the interplay of symmetry, topology, and correlation.

7. Representative Mathematical Structures

Regime/System	Key Parameter(s)	Defining Feature(s) / Equation(s)
1D Helical Edge (TI, Zeeman)	$K, \Delta_z$	Sine–Gordon gap: $\mathcal{H}_{\Delta_z} \sim \cos(\sqrt{4\pi} \Phi)$
Magnetoplasmon (QHE edge)	$\eta_T, k_x$	Multipole expansion: $$\rho^{(s)} = \tilde{R}_0(Y) e^{-Y} \sum_{n=0}^\infty \rho_n^{(s)} L_n(Y)$$
Spin Chain (XXZ + Lindblad)	$\Delta, \Gamma$	Rank-2 steady state: $$R_0 = b|s\rangle\langle s| + (1-b)|a\rangle\langle a|$$
Turbulence (helical forcing)	$C_0, a, S, \mathrm{St}$	Coupled motion: $\dot{v}, \dot{\omega} = \ldots$ matrix with helicoidality
Plasma (beam-generated)	$p_{\mathrm{th}}, \omega$	Rotation frequency: $\omega = 2\pi |\lambda|/ (B R^2)$

8. Conclusion

The partially mixed helical regime encompasses a wide array of phenomena unified by the partial breaking, mixing, or competition of helical, topological, or parity-breaking order. Across quantum electronic materials, topological edge states, spin chains, quantum fluids, plasma, and turbulent flows, partially mixed helical regimes dictate the system’s spectral, transport, and structural characteristics. Their emergent behaviors and tunability have critical implications for topological quantum computing, spintronics, plasma confinement, and control of turbulent transport, situating them at the frontier of modern condensed matter and fluid dynamics research.