Second Harmonic Generation Spectroscopy of the Surface Charge Density Wave in the Weyl Semimetal CoSi

Abstract: Using temperature-dependent rotational anisotropy second harmonic generation (RA-SHG), we identify a charge density wave (CDW) instability on the CoSi (001) face with an onset temperature at $90.0 \pm 0.8$ K. The SHG response tracks the order parameter amplitude, dominated by two nonlinear tensor elements, whose background-subtracted intensity evolves with temperature as a power law. The extracted critical exponent $β= 0.30 \pm 0.03$ is consistent with the 3D XY universality class, an unexpected result for a nominally two-dimensional surface layer. No corresponding anomaly is observed in the bulk response, establishing the transition as a purely surface-driven phase transition. We attribute this unexpected scaling to the coupling between surface Fermi-arc states and the bulk topology they are tied to, grounded in a previously observed intra-unit-cell phase relationship between surface sublayers that gives the order parameter intrinsic three-dimensional structure.

• The paper details the temperature-dependent second harmonic generation spectroscopy of CoSi, providing a critical exponent and revealing a surface charge density wave at 90.0 ± 0.8 K. This study identifies the transition's 3D XY universality class determined by the surface order parameter.
• The continuous symmetry transition is governed by a surface mechanism in Weyl semimetal CoSi resulting in surface CDW, evidenced by spectral filtering on the bulk confocal transitions. Contrasts with bulk topological constraints inferred by the dimensional requirements of the phase -- contrasting the dimensional constraints.
• The use of second harmonic generation spectroscopy in CoSi temperature fit confirms surface CDW surface transition while indicating the proximity to bulk multifold transitions.

Overview

This paper reports temperature-dependent rotational anisotropy second harmonic generation (RA-SHG) measurements on the (001) face of the chiral Weyl semimetal CoSi, identifying a continuous phase transition at $T_{CDW} = 90.0 \pm 0.8$ K that the authors attribute to the incommensurate surface charge density wave (CDW) previously observed by scanning tunneling microscopy (2608.17303). The central result is a critical exponent $\beta = 0.30 \pm 0.03$, consistent with the 3D XY universality class ($\beta_{3DXY} = 0.3485$), which is unexpected for an order parameter confined to a nominally two-dimensional surface layer where the Mermin–Wagner theorem forbids true long-range order with a continuous symmetry and where a Berezinskii–Kosterlitz–Thouless (BKT) transition would instead be anticipated.

Experimental approach and symmetry selection

CoSi crystallizes in the Sohncke space group 198 (B20 family) and hosts multifold band crossings at $\Gamma$ and R connected by large surface Fermi arcs. At near-normal incidence on the (001) surface, the incident field is purely in-plane, so the only probed tensor elements are the six in-plane components of $\chi^{(2)}_{ijk}$ allowed by the C$_1$ surface point group. Critically, the sole bulk-allowed element in this space group, $\chi^{(2)}_{xyz}$, requires an out-of-plane field component; since a dipole along the beam axis does not radiate along that axis, the bulk contribution is eliminated at normal incidence, cleanly isolating the surface response.

The measurements used a 1500 nm fundamental from an OPA pumped by a Ti:sapphire amplifier (5 kHz, ~35 fs), with four polarization geometries ($I_H$, $I_V$, $I_\parallel$, $\beta = 0.30 \pm 0.03$0) on polished (001) crystals oriented to within 1% by X-ray diffraction. Global fits at each temperature recover the six tensor elements with low cross-parameter correlation.

Identifying the transition

The angular shape of the RA-SHG traces is unchanged across the transition, as expected: an incommensurate CDW has no fixed phase relationship to the lattice and therefore does not lower the C$\beta = 0.30 \pm 0.03$1 symmetry, so the signature appears entirely as a change in magnitude of the susceptibility elements. Within a perturbative treatment, the opening of the CDW gap modifies both the ground-state population factor and the dipole matrix elements entering $\beta = 0.30 \pm 0.03$2, yielding a term linear in the order parameter $\beta = 0.30 \pm 0.03$3 and hence $\beta = 0.30 \pm 0.03$4. The authors subtract the high-temperature background at the tensor-element level—rather than at the intensity level, which would mix background and order-parameter terms—and reconstruct the intensity from the difference, fitting $\beta = 0.30 \pm 0.03$5.

The peak SHG intensity grows by nearly a factor of two below $\beta = 0.30 \pm 0.03$6. Two tensor elements dominate: $\beta = 0.30 \pm 0.03$7 and $\beta = 0.30 \pm 0.03$8 exhibit clear power-law scaling, while the remaining four are roughly an order of magnitude smaller and show no scaling. Notably, this two-element dominance mirrors RhSi (2608.17303) but contrasts with PdGa, where all photogalvanic tensor elements are comparable—an asymmetry the authors correlate with the proximity of the multifold band crossings to the Fermi level (~0.5–1 eV above them in PdGa), suggesting a possible topological origin for the selectivity that remains unresolved.

Surface confinement

Two complementary measurements establish that the transition is surface-confined. First, off-normal incidence (~30°) on the (001) crystal introduces an $\beta = 0.30 \pm 0.03$9 component that probes the bulk directly; only weak temperature dependence is observed, in contrast to the nearly twofold enhancement at normal incidence. Second, RA-SHG on a separately polished (111) crystal—which isolates the bulk-allowed $\beta_{3DXY} = 0.3485$0 element—shows no anomaly between 60 and 110 K. Together these rule out a bulk origin for the ordering.

An independent, model-free corroboration of $\beta_{3DXY} = 0.3485$1 comes from the ratios among fitted tensor elements: below ~90 K they vary modestly, while above it the four weak components fluctuate by more than a factor of two. The authors interpret this as loss of long-range phase coherence, with SHG averaging over short-range-ordered patches of uncorrelated phase above the transition. The ratio $\beta_{3DXY} = 0.3485$2 remains constant to within a few percent through the transition, indicating both dominant elements track a single common factor, though the data cannot distinguish whether it is the population factor or the matrix elements.

Universality class and the dimensional puzzle

The complex order parameter of an incommensurate CDW places the transition in the XY (U(1)) family. The data show power-law growth across the full measured range with no crossover signatures of a BKT transition—inconsistent with BKT's infinite-order character, its lack of a conventional exponent, and far from the mean-field value $\beta_{3DXY} = 0.3485$3. The extracted $\beta_{3DXY} = 0.3485$4 is consistent with 3D XY.

This poses a direct contradiction with the Mermin–Wagner theorem, since STM work places the charge modulation in the topmost atomic layers. The authors' resolution rests on the intra-unit-cell structure of the order: STM topographs show a nearly $\beta_{3DXY} = 0.3485$5 phase shift between adjacent sublayers separated by only 80–131 pm, occurring specifically across odd, termination-changing step edges and absent across even ones. Because different terminations host inequivalent Fermi-arc states (as established in NbP by ARPES), this phase relationship evidences genuine coherence linking arc states rather than a structural coincidence. The order parameter thus extends coherently along [001], acquiring intrinsic three-dimensional character and sufficient phase stiffness to escape the BKT regime—a scenario analogous to interlayer-coupling-driven crossovers in IrTe$\beta_{3DXY} = 0.3485$6 and contrasting with NbSe$\beta_{3DXY} = 0.3485$7, where a genuine 2D BKT surface CDW regime was identified. More conceptually, the authors argue that Mermin–Wagner assumes dynamical self-containment, an assumption that fails for Fermi arcs, which have no independent existence apart from the bulk topology they bound.

Limitations and open questions

The microscopic mechanism coupling the surface sublayers—and tying that coupling to the bulk topological invariant—remains unidentified; the authors state this explicitly as an open question. The linear scaling between $\beta_{3DXY} = 0.3485$8 and the CDW order parameter is argued perturbatively but not independently verified. Tensor-element fits constrain each $\beta_{3DXY} = 0.3485$9 as real, though components may be complex; individual tensor elements yield erratic exponents sensitive to fitting window, which is why the analysis relies on the reconstructed intensity. The proposed connection between dominant tensor elements and Fermi-level proximity of multifold crossings is suggestive but unproven. Finally, whether analogous surface instabilities appear in other Weyl semimetals with extended arcs is untested.

Conclusion

RA-SHG establishes the CoSi (001) surface CDW as a continuous transition at $\Gamma$0 K with 3D XY criticality, confined to the surface as confirmed by the absence of anomalies in bulk-sensitive geometries and on the (111) face. The result demonstrates that correlated electronic order can emerge intrinsically on topological boundary states, with the apparent violation of Mermin–Wagner constraints resolved by the three-dimensional coherence of the arc-state order parameter rooted in bulk topology.