Microscopic origin of the chirality-induced Hall anomaly in MnSi under pressure: texture identity, field thresholds, quantitative topological-charge mapping, and relation to the partial-order state

In MnSi under hydrostatic pressures of 6–12 kbar, just below the critical pressure $P_c\approx 14.6$ kbar where helimagnetic long-range order is suppressed, the Hall conductivity acquires an additional contribution $\sigma_{xy}^C$ confined to a narrow field window between $H_1\approx 0.1$ T and $H_2\approx 0.45$ T (below the spin-saturation field $H_s\approx 0.6$ T), with a step-like onset and disappearance, electron-like in sign, and a magnitude at 5 K roughly ten times the ordinary and anomalous Hall terms (about 1% of the zero-field longitudinal conductivity); the contribution first appears just below the pressure-reduced Curie temperature $T_C$ as a shallow feature, deepens with decreasing temperature, and vanishes both above $T_C$ and at the lowest temperatures (below about 2.5 K). This contribution is interpreted as arising from an emergent real-space magnetic field generated by chiral, noncoplanar spin textures, most plausibly fluctuations toward multi-q states (superpositions of several spiral propagation vectors) such as skyrmion lattices or bcc spin crystals. The open problem is to establish this interpretation microscopically: to identify the fluctuating spin textures in the actual pressure–temperature–field window, to explain the fields $H_1$ and $H_2$ (in particular why the signal vanishes below $H_s$ and so abruptly), to extract a controlled quantitative mapping between the Hall amplitude and the topological (skyrmion) charge density across pressure, temperature, and field, and to determine how these below-$P_c$ fluctuation textures connect to the partial-order (non-Fermi-liquid) regime above $P_c$.

Background

MnSi is a cubic chiral (B20-structure) itinerant helimagnet. The absence of inversion symmetry combined with weak spin–orbit coupling generates a Dzyaloshinskii–Moriya interaction, which twists the ferromagnetic order into a long-wavelength spiral with period $\ell\approx 18$ nm; a small magnetic field $H$ unwinds the spiral into a conical state whose cone angle shrinks to zero at the spin-saturation field $H_s$. In a small field–temperature pocket just below the Curie temperature $T_C$, MnSi instead hosts the A phase, shown by small-angle neutron scattering (SANS) to be a triple-q superposition of three helices, i.e. a triangular lattice of skyrmions — vortex-like, topologically nontrivial knots of the magnetization field.

The Hall conductivity of a metal with a noncollinear spin texture contains, besides the ordinary Lorentz-force term $\sigma_{xy}^N$ and the conventional anomalous Hall term $\sigma_{xy}^A$ (which in MnSi scales with the uniform magnetization), a third contribution generated by real-space spin chirality: an itinerant electron whose spin adiabatically follows the local magnetization $\mathbf{S}(\mathbf{r})$ accumulates a Berry phase on any closed path, equivalent to an emergent magnetic field $B_\phi$ proportional to the skyrmion density $\Phi_z=(8\pi)^{-1}\hat{n}\cdot(\partial_x\hat{n}\times\partial_y\hat{n})$, with $\hat{n}=\mathbf{S}/|\mathbf{S}|$. This 'topological' Hall contribution was measured in the ambient-pressure A phase of MnSi by Neubauer et al. (PRL 102, 186602) and identified as the emergent flux of the skyrmion lattice.

Under hydrostatic pressure, helimagnetic long-range order in MnSi is suppressed at a critical pressure $P_c\approx 14.6$ kbar, where the resistivity enters an extended non-Fermi-liquid regime (varying with temperature as a power markedly weaker than the quadratic law expected of a conventional Fermi liquid) and neutron scattering shows 'partial order' — diffuse magnetic intensity spread over a sphere in reciprocal space rather than the sharp Bragg spots of the helix. Lee, Kang, Onose, Tokura, and Ong (PRL 102, 186601, 2009) measured the Hall effect at $P=6$–$12$ kbar, well below $P_c$, and found the additional contribution $\sigma_{xy}^C$: large at 5 K (of order 1% of the zero-field longitudinal conductivity, roughly ten times the ordinary and anomalous Hall terms), electron-like in sign, confined to $H_1<H<H_2$ with a step-like (mesa) profile, first appearing just below the pressure-reduced $T_C$ as a shallow feature, deepening with decreasing temperature, and vanishing both above $T_C$ and at the lowest temperatures (below about 2.5 K), with its onset and disappearance coincident with kinks in the magnetoresistance. A static calculation by Binz and Vishwanath for the bcc1 spin crystal — a body-centered-cubic multi-q superposition of six spirals proposed as a candidate high-pressure state — reproduces the mesa-shaped field profile, including a cutoff below $H_s$, but not the magnitude. The interpretation of Lee et al. is that, for $P<P_c$, thermal fluctuations near $T_C$ populate chiral multi-q textures — superpositions of several equivalent spiral propagation vectors, such as skyrmion lattices or bcc spin crystals — that act as a precursor of the partial-order state above $P_c$.

What the transport data alone cannot do is: distinguish which multi-q texture fluctuates, pin down why the response switches on and off precisely at $H_1$ and $H_2<H_s$, convert the measured $\sigma_{xy}^C$ into a skyrmion or chirality density, or establish whether the fluctuation regime below $P_c$ and the partial-order regime above $P_c$ are one continuous evolution or merely neighboring states. Neighboring results frame these gaps: the ambient-pressure fluctuation regime above $T_C$ is known to host chiral skyrmion-liquid-like correlations (Pappas et al.) and fluctuating skyrmion textures undergoing weak crystallization (Kindervater et al.); under pressure, SANS shows that the helical propagation vector reorients from $\langle 111\rangle$ to $\langle 100\rangle$ at intermediate pressures and that above $P_c$ field-induced skyrmion lattices and conical spirals still form (Bannenberg et al.); the topological Hall signal persists, with unchanged sign and magnitude, into the non-Fermi-liquid regime above $P_c$ (Ritz et al., Nature 2013); and the suppression of order at $P_c$ appears to arise from collapse of the exchange interaction rather than of the moment (Dalmas de Réotier et al.).

Problem Statement

Establish the microscopic origin of the chirality-induced Hall anomaly $\sigma_{xy}^C$ observed in MnSi under hydrostatic pressures of 6–12 kbar, below the critical pressure $P_c\approx 14.6$ kbar, with a single coherent objective: convert the transport observation into an identified, quantitatively understood magnetic state. Specifically, for the field window $H_1\approx 0.1\,\mathrm{T}<H<H_2\approx 0.45\,\mathrm{T}$ below the spin-saturation field $H_s\approx 0.6\,\mathrm{T}$ and temperatures between roughly 5 K and the pressure-reduced $T_C$ over which the step-like contribution is observed:

1. Identify the spin textures that generate $\sigma_{xy}^C$ — determine whether the response arises from fluctuations into specific multi-q states (skyrmion lattices, bcc spin crystals, or other noncoplanar chiral textures), established independently of the Hall signal itself.

2. Explain microscopically the mechanism that fixes the onset and cutoff fields $H_1$ and $H_2$, in particular why the anomaly vanishes abruptly at $H_2$ below the nominal spin-saturation field $H_s$, and why the onset and disappearance are so sharp and coincident with magnetoresistance anomalies.

3. Extract the quantitative relation between the measured magnitude of $\sigma_{xy}^C$ and the underlying topological (skyrmion) charge or chirality density, as a controlled mapping valid under varying pressure, temperature, and field — including which corrections to the simple emergent-field picture (incomplete spin polarization, non-adiabaticity, band-structure and scattering effects) control the amplitude.

4. Determine the relation between these below-$P_c$ fluctuation-induced textures and the partial-order, non-Fermi-liquid regime above $P_c$: whether they form one continuous evolution of the same fluctuating chiral textures across the pressure-driven suppression of helimagnetic order, or are distinct states merely adjacent in parameter space.

These four facets are physically linked aspects of one question — what, microscopically, does the pressure-induced Hall anomaly of MnSi measure: the texture identification (1) is what fixes the topological density entering the mapping (3), the threshold mechanism (2) is the field-scale counterpart of the same texture physics, and the pressure evolution (4) is what makes the anomaly a probe of the partial-order state rather than an isolated curiosity. Ambient-pressure analogues (the static A-phase skyrmion lattice and the fluctuating skyrmion textures above $T_C$ at ambient pressure) are useful comparisons but do not by themselves answer the under-pressure question.

References

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