Quasilinear quantum magnetoresistance in pressure-induced nonsymmorphic superconductor chromium arsenide

In conventional metals, modification of electron trajectories under magnetic field gives rise to a magnetoresistance that varies quadratically at low field, followed by a saturation at high field for closed orbits on the Fermi surface. Deviations from the conventional behaviour, for example, the observation of a linear magnetoresistance, or a non-saturating magnetoresistance, have been attributed to exotic electron scattering mechanisms. Recently, linear magnetoresistance has been observed in many Dirac materials, in which the electron–electron correlation is relatively weak. The strongly correlated helimagnet CrAs undergoes a quantum phase transition to a nonmagnetic superconductor under pressure. Here we observe, near the magnetic instability, a large and non-saturating quasilinear magnetoresistance from the upper critical field to 14 T at low temperatures. We show that the quasilinear magnetoresistance may arise from an intricate interplay between a nontrivial band crossing protected by nonsymmorphic crystal symmetry and strong magnetic fluctuations.

In Figure 3 of the main text, we show the calculated band structure of CrAs at 14.3 kbar and CrP at ambient pressure. We focus on the band structure near the Y point, where a small Fermi surface pocket is located. According to Figure 3, the dispersion is linear along Γ-Y direction and parabolic along Y-S ( k x ) direction, which is analogous to the scenario of the semi-Dirac point discussed in the literature [1,2], except that the parabolic branch with a negative mass does not exist in our case. The dispersion relation around the Y point near E F can be written as . (1) Note that (k x , k y , k z ) is a wavevector measured from Y. For the semiclassical approach, the areal quantization condition in a field µ 0 H ( k z ) has the form of S(ε) = 2π(n + γ)eµ 0 H/ , where γ ∈ [0, 1] is the phase factor which can not be determined by the semiclassical treatment. The area of an orbit at k z =0 is Thus, the Landau levels at k z = 0 are Therefore, the energy spacing between the n = 0 and n = 1 Landau levels is where According to Abrikosov's theory, when all the carriers occupy only the zeroth Landau Level (the 'extreme quantum limit'), the MR will linearly depend on the magnetic field [3]. To realize this situation, the Landau level splitting between the zeroth and the first Landau level must be larger than the Fermi energy of the system, i.e. ∆ 1 > E F −E c , at T = 0 K. At finite temperature, the effect of thermal fluctuation must be taken into account [4]. These considerations lead to equation (1) in the main text, i.e.
where (E F − E c ) is the energy difference between the Fermi energy and the crossing point. where The fitting parameter b immediately gives (E F − E c ), the results are summarized in Supplementary Table . The other fitting parameter a is related to the mass m, the slope of the dispersion c and f (γ). To estimate m and c, we need to consider another constraint based on the anisotropy of the Fermi surface on the k x − k y plane.
As mentioned in the main text, we can define an anisotropic factor α = k x F /k y F . Furthermore, we have the following From Supplementary Equation (9) and (10), we have the relation quoted in the main text Combine Supplementary Equation (8) and (11), we have Using the band structure displayed in Figure 3(b) of the main text, we adjust the Fermi energy to match the experimentally extracted (E F − E c ) from the fitting parameter b for a given pressure, while keeping the band structure intact (rigid band shift). This allows us to calculate m and c for p 1 =9.2 kbar, p 2 =13.1 kbar, and p 5 =18.0 kbar, with the results tabulated in Supplementary  ) is the energy separation between the crossing point and the Fermi energy, α is the anisotropic factor, c and m parametrise the linear and the parabolic energy-momentum dispersion relations, respectively. me is the bare electron mass.

Supplementary Note 2: H-T scaling
To get the H-T scaling plot as shown in Fig. 4 of the main text, we need to estimate ρ(0, 0). We use linear extrapolation of the base temperature ρ(µ 0 H) curve above µ 0 H * and get ρ(0, 0) ≈ 0.83 µΩ.cm, as shown in Supplementary  Figure 2 (a). For the other two pressure values, i.e., p 1 =9.2 kbar, and p 5 =18.0 kbar, we follow the same procedure. In Supplementary Figure 2   To contrast with the H-T scaling in the preceding section, we construct the Kohler plot for the same datasets. As shown in Supplementary Figure 3, a serious violation of Kohler's rule is observed for all three pressures. In fact, the new H-T scaling is not compatible with Kohler's rule [5]. Hence, we do not expect Kohler's rule to hold given the applicability of H-T scaling. In this section, we provide a detailed analysis based on the nonsymmorphic symmetry to prove the band crossing on the Y-S line. On the Y-S line which lies on the face of the first Brillouin zone, flipped wave vectors m y k = (k x , −k y , k z ) and m z k = (k x , k y , −k z ) are equivalent to k = (k x , k y , k z ), because k z = 0 and m y k = k+G with G being a reciprocal lattice vector. Thus, the k-group involves the glide symmetry G z and the mirror symmetry M y , which are symmetry elements of the space group P nma. The glide symmetry G z = {M z | x 2 } is a combined symmetry of mirror reflection M z and half translation along the x-axis. Thus, the nonsymmorphic symmetry is preverved on the Y-S line.
Because four Cr atoms are involved in the unit cell, the symmetry operators are represented by three sets of Pauli matrices, s i , σ i , and τ i (i = 0, 1, 2, 3). While s i acts on the spin space, σ i and τ i are operators in the sublattice space. Two inequivalent sites in zigzag chains are operated by σ i , and two inequivalent zigzag chains are operated by τ i (see Supplementary Figure 4).
The symmetry operators are represented bŷ whereÎT is inversion-time-reversal symmetry preserving all the wavevectors k, and K is the complex conjugate operator. In the following, we consider reduced operators on the Y-S line, Because the single electron part of the HamiltonianĤ 0 (k) commutes with the mirror reflection operatorM y (k) on the Y-S line, the Hamiltonian is block-diagonalized on the basis spanned by eigenstates ofM y (k). From the