Detection of relativistic fermions in Weyl semimetal TaAs by magnetostriction measurements

Thus far, a detection of the Dirac or Weyl fermions in topological semimetals remains often elusive, since in these materials conventional charge carriers exist as well. Here, measuring a field-induced length change of the prototype Weyl semimetal TaAs at low temperatures, we find that its c-axis magnetostriction amounts to relatively large values whereas the a-axis magnetostriction exhibits strong variations with changing the orientation of the applied magnetic field. It is discovered that at magnetic fields above the ultra-quantum limit, the magnetostriction of TaAs contains a linear-in-field term, which, as we show, is a hallmark of the Weyl fermions in a material. Developing a theory for the magnetostriction of noncentrosymmetric topological semimetals and applying it to TaAs, we additionally find several parameters characterizing the interaction between the relativistic fermions and elastic degrees of freedom in this semimetal. Our study shows how dilatometry can be used to unveil Weyl fermions in candidate topological semimetals.

. Left: Photograph of the commercial dilatometer cell [1] mounted on a cold finger of a dilution refrigerator. This setup is equipped with a piezoelectric rotator for angle-dependent magnetostriction. Right: Back-reflection Laue photographs of the TaAs single crystal (sample 1). Figure (a) shows the diffractogram along the c axis, whereas (b) shows the diffractogram along the a axis.

Supplementary Note 1: Formula for the magnetostriction
We now derive the general formula for the magnetostriction [formula (1) in the main text], assuming, for definiteness, the tetragonal symmetry of a crystal (i.e., the symmetry of TaAs). This formula can be obtained by a minimization of the Ω potential for the deformed crystal placed in the external magnetic field H with respect to the strain tensor u ik , Ω = 1 2 C 11 (u 2 xx + u 2 yy ) + 1 2 C 33 u 2 zz + C 13 (u xx + u yy )u zz + C 12 u xx u yy where C mn are the elastic moduli of the crystal [2], B = µ 0 H is the magnetic induction in the sample, ∆Ω el (u kl , B) ≡ Ω el (u kl , B)−Ω el (0, B), and Ω el (u kl , B) = ∑ N i=1 Ω i (u kl , B) is the deformation-dependent part of the electron Ω potential for the crystal with N pockets of charge carriers. This part depends on the deformation, the magnetic induction B, and also on the temperature T and the chemical potential ζ (we do not indicate explicitly T and ζ here). The first six terms in Eq. (1) give the total elastic energy of a deformation. This energy is partly produced by ∆Ω el (u kl , 0), and hence the difference of the elastic terms and ∆Ω el (u kl , 0) is the elastic energy that is not associated with the electron pockets under study. The term ∆Ω el (u kl , B) describes the total change in the Ω potential of these pockets in the magnetic field under the deformation. As in Ref. [3][4][5][6], we assume that an elastic deformation shifts the electron bands as a whole and does not change their shape. This rigid-band approximation commonly is well justified for sufficiently small pockets (see below). In particular, this approximation seems to be valid for TaAs family of the semimetals, see Figs. 6e and 6f in Ref. [7]. In this rigid-band approximation, a shift ∆ε i of the ith electron energy band ε i (p) under the deformation coincides with the shift of its edge and is proportional to the strain tensor u kl , ∆ε i = ∑ k,l λ (i) kl u kl , where p is the electron quasimomentum, and the constants λ (i) kl are the so-called deformation potential of the ith band [8]. Therefore, in the minimization of Ω potential, one can use the following relation: where n i (B) = −∂Ω i (ζ, B)/∂ζ is the density of charge carriers of the pocket i in the magnetic field. The minimization gives the set of the equations in the tensor u kl , which defines the magnetostriction, i.e., the deformation of the crystal in the magnetic field, The same formula (2) describes (u xx + u yy )/2 but with the other Λ i , In Eqs. (3) and (4),λ (i) xx ≡ (λ a1 xx +λ a2 xx )/2 for all i belonging to the appropriate group of the equivalent pockets. It is significant that Λ c i and Λ ⊥ i have one and the same values inside each of these groups. On the other hand, the coefficients Λ i in formula (2), which also describes u xx and u yy , take on the two different values for the equivalent Fermi pockets: where a1 and a2 mark such pockets lying near the two different reflection planes. When the magnetic field is along the z axis, all n i (B) are one and the same for the equivalent Fermi surfaces, the contributions of the second terms in Λ a1 i and Λ a1 i to the sums over i cancel each other out, and we obtain from the above set of the equations that u xx = u yy , and u xy = u zx = u zy = 0. In this situation, the u zz is the elongation ∆L/L of the crystal along the direction [001], whereas u xx is the elongation in the direction [100]. Formulas (3) and (4) show that the constants Λ c i and Λ ⊥ i extracted from the measurements of u zz and u xx enable one to find the constants λ xx of the deformation potential if the the elastic moduli C mn are known. In particular, these moduli for TaAs are given in Ref. [9], and for this semimetal, formulas (3) and (4) yield the following equations: xx are expressed in eV. Note that formula (2) was used in a number of papers [4][5][6]. (In Refs. [6], corrections to Eq. (2) were also taken into account when the chemical potential is close to the edge of a Landau subband.) For definiteness, we shall discuss only u zz in the subsequent Supplementary Notes 2-7.
When the magnetic field deviates from the z axis, the densities n i (B) for different equivalent pockets generally do not coincides, and in this case, u xx ̸ = u yy , and the offdiagonal u ik ̸ = 0. The latter means that the diagonalization of the tensor u kl gives its principal axes that differ from the axes x, y, z, and hence, the deformation of the crystal in such a tilted magnetic field is a superposition of its elongations (contractions) along the three directions different from the crystallographic axes. Such elongations change the shape of the sample. For example, if the sample at B = 0 is a rectangular parallelepiped with the axes x, y, z, it becomes an oblique parallelepiped. When B deviates from the z axis in the direction of the x axis, the cross sections of this parallelepiped by the planes y =const. are parallelograms, the angle of which differs from π/2 by a small amount of the order of u xz . If we define ∆L as the change in the distance between the parallel surfaces of the parallelepiped, the relative elongation ∆L/L is still given by u zz (or u xx ). Another consequence of the tilt of B is that the different values of the factors Λ a1 i and Λ a2 i for the equivalents pockets manifest themselves in the expression for u xx , whereas the factor Λ c i is still one and the same for these pockets. This fact may lead to essentially different dependences of u xx and u zz on the direction of B (Supplementary Note  8).
Finally, let us comment on the applicability of the rigid-band approximation, which has been used above. In general case, the shift ∆ε i (p) = ∑ k,l λ (i) kl (p)u kl of the ith electron energy band ε i (p) under the deformation depends on p due to the p dependence of λ (i) kl (p) [8]. The deformation potential λ (i) kl (p) essentially changes on the scale of the order of the size of the Brillouin zone, and so one can consider λ (i) kl as constant for small pockets of the Fermi surface. However, a caution is required when a band of the trivial electrons is separated from a valence band by a sufficiently small gap ∆. In this case, the effective electron masses of the band are of the order of ∆/V 2 where V is a typical interband matrix element of the velocity operator (V ∼ 10 5 ÷ 10 6 m/s). If the small gap ∆ substantially depends on a deformation, the rigidband approximation can fail for a small pocket of this band since the effective masses, and hence the shape of the band, noticeably change with the deformation. Another situation occurs for the Weyl nodes. These nodes are topologically protected, a small deformation cannot create a gap in the spectrum, and so one may expect the applicability of the rigid-band approximation to the case of the Weyl pockets.

Supplementary Note 2: Magnetostriction produced by Weyl electrons in noncentrosymmetric crystals
As was shown in Sec. 3.3 of Ref. [10], for a noncentrosymmetric crystal, a contribution of a Fermi pocket of Weyl quasiparticles to any thermodynamic quantity (including the magnetostriction) can be calculated as half of the appropriate contribution of the Dirac pocket with the same electron dispersion. Using formulas for the electron spectrum in the magnetic field in the vicinity of a Dirac point [10,11], the Ω potential and hence the density n(B) of the Weyl electrons can be found. In particular, we obtain at zero temperature, where n(ζ − ε d , B) is the density of the Weyl quasiparticles in the magnetic field H = B/µ 0 at a given chemical potential ζ; the density in absence of the magnetic field, n(ζ − ε d , 0), is assumed to be positive in the case of the electrons (ζ − ε d > 0) and negative for the holes (ζ − ε d < 0); ε d is the energy of the Weyl point; [u] means the integer part of u ≡ F/B; the frequency of the quantum oscillations also defines the boundary of the ultra-quantum region at which these oscillations disappear, and S max is the maximal cross sectional area of the Fermi surface of the Weyl quasiparticles at a given direction of the magnetic field. This area is proportional to (ζ − ε d ) 2 . Of course, formula (6) is applicable to the case of the Dirac electrons, too. We also point out the two useful representations of the sum in Eq. (6), where ζ(−1/2, u) is the Hurwitz zeta function [12], and {u} ≡ u − [u] is the fractional part of u. We emphasize that formula (6) is valid for the most general form of the dispersion laws ε c (p) and ε v (p) describing the quasiparticles in the vicinity of the Weyl point [10], a 2 , a 3 ) and b ii are the constant parameters, with a i specifying the so-called tilt of the spectrum. For reference, let us also write the expressions for n(ζ − ε d , 0) and S max in terms of these parameters, δ ij is the Kronecker symbol, and n = (n 1 , n 2 , n 3 ) is the unit vector along the magnetic field. Formulas (2) and (6) describe the magnetostriction u zz produced by the Weyl electrons at zero temperature T and in absence of the electron scattering by impurities. It was demonstrated by Shoenberg [13] that the effect of the impurities on the oscillating part of the magnetization can be taken into account by replacing the Fermi energy E F (i.e. ζ at T = 0) by E F + iΓ where the parameter Γ = πT D characterizes the electron scattering by the impurities, and T D is the so-call Dingle temperature. We shall use this replacement both for the oscillating and for the smooth parts of the magnetostriction. In other words, to take into account the scattering, the parameter u + 0i in formula (9) is replaced by u(1 + iγ) 2 where γ ≡ Γ/(ζ − ε d ) = πt D is the dimensionless characteristic of the scattering, and t D is the dimensionless Dingle temperature. As to the magnetostriction at T > 0, it can be obtained as follows: Consider now various limiting cases. Ultra-quantum regime (µ 0 H > F ). At high fields µ 0 H > F , the sum in equation (6) disappears since u = F/B < 1 in this case. Then, with formula (2), we obtain the following contribution of a Weyl pocket to the magnetostriction: Fig. 4. The magnetostriction ∆L/L = uzz (in units of Λn(ζ0, 0)) produced by the Weyl electrons at T = 0 versus the dimensionless chemical potential ζ/ζ0 for µ0H/F = 2 and µ0H/F = 1 (the solid black lines, the numbers near these lines indicate the values of µ0H/F ). The chemical potential ζ is measured from the ε d , and ζ0 is its initial value. The solid red line shows the smooth part of the magnetostriction,ūzz, enlarged in 10 times for µ0H/F = 1/7 and for the nonzero temperature T /ζ0 = 0.02. The dashed line depicts the dependence of the magnetostriction on the chemical potential ζ at T = 0, µ0H/F = 2 for the crystal with the electron spectrum (18) and δ = 1/2 (the chemical potential is now measured from ε0). Note that this dependence, which vanishes below ε0, qualitatively differs from the dependence for the Weyl electrons.
where the quantities Λ, n(ζ − ε d , 0), F introduced above and a ≡ −Λn(ζ − ε d , 0), b ≡ −0.75a/F refer to this pocket. It is evident that the Weyl pocket produces the elongation ∆L/L that increases linearly with B at the high magnetic fields.
Consider the dependence of u zz on the chemical potential ζ at a fix value of B ( Supplementary Fig. 4). Since where ζ 0 is the initial value of the chemical potential.
Supplementary figure 4 shows that the magnetostriction changes its sign when ζ crosses the energy of the Weyl point ε d . Weak magnetic fields (µ 0 H ≪ F ). At µ 0 H ≪ F , the ratio u in Eq. (6) is large, and the sum contains many terms. Using representation (8) and the expansion of the Hurwitz zeta function ζ(−1/2, u) at u ≫ 1 [12], we find from formulas (2) and (6) that the magnetostriction u zz produced by a Weyl pocket comprises the smooth and oscillating parts in the weak-field range, u zz =ū zz + u osc zz . The smooth partū zz has the form and the oscillating part u osc zz results from the term The oscillating behavior of u osc zz with changing u = F/B becomes evident from the formula [12], ) . (15) Interestingly, with changing the chemical potential, the coefficient c sharply changes it sign when ζ crosses the energy of the Weyl point ( Supplementary Fig. 4), where ζ 0 is the initial value of ζ. Note that c(ζ) diverges at ζ = ε d . Formulas (11), (13), (14), and (16) are written at T = 0. A nonzero temperature tends to suppress the oscillations of the magnetostriction, but it has effect onū zz only in the region |ζ − ε d | T . In this region, the divergence of c(ζ) near ε d is cut off ( Supplementary Fig. 4), and the Finally, as an example, we calculate the magnetostriction produced by the two groups of the Weyl electrons (W1 and W2) and by the holes, assuming that the contribution of these holes to the magnetostriction are described by the term c h B 2 where c h is a constant. Let experimental data on the magnetostriction in the ultraquantum regime for the W1 and W2 electrons can be approximated by the polynomial a + bB + c h B 2 . Then, if the frequencies F W 1 , F W 2 are known, the constants a W 1 and a W 2 are found from the equations that follows from formula (11), With known a W 1 , a W 2 , F W 1 , F W 2 , the magnetostriction at an arbitrary B is calculated, using formulas (2) and (6). The dimensionless Dingle temperatures are determined from the best fit of the calculated curve at T = 0 to the appropriate experimental data at a low temperature. In particular, we find for TaAs that In the case of F W 2 = 5 T, the calculated magnetostriction together with the experimental data are presented in Supplementary Fig. 5.  (2), (6) and (9) for the W1 and W2 electrons and assuming that c h B 2 is the contribution of the holes to the magnetostriction. We use the following values of the parameters:

Supplementary Note 3: Magnetostriction produced by electrons with parabolic spectrum
For comparison, consider the magnetostriction u zz produced by the "trivial" electrons with the parabolic dispersion, where ε 0 is the edge of the electron band, and m x , m y , m z are the effective electron masses. In the magnetic field H these electrons have the well-known energy spectrum, where n is an integer (n ≥ 0); p ∥ is the quasimomentum along the magnetic field; m ∥ and m * are the longitudinal effective mass and the cyclotron mass, respectively, and the electron magnetic moment composed of its spin and orbital parts is written as δ(e /m * ), with δ being a constant. Since due to the time reversal symmetry, the contribution of any charge-carrier pocket to the Ω potential for a noncetrosymmetric crystal can be considered as a half contribution of the same pocket with the states that are doubly degenerate in spin, one should imply that the parameter δ in the Zeeman term δ(e B/m * ) of Eq. (18) takes the values +δ and −δ (in this situation, δ can be represented in terms of the g factor, δ = gm * /4m). With this spectrum, we obtain at zero temperature, (19) where n(ζ − ε 0 , B) is the density of the quasiparticles in the magnetic field H = B/µ 0 at a given chemical potential ζ; the frequency F in the ratio u ≡ F/B is still defined by formula (7). However, for the trivial electrons, the maximal cross sectional area S max of their Fermi surface is proportional to (ζ − ε 0 ). Ultra-quantum regime. At δ < 1/2 and µ 0 H > B uq = F/(0.5 − δ), the sum in equation (19) disappears. Then, with formula (2), we obtain the constant contribution of the trivial-electron pocket to the magnetostriction in this ultra-quantum limit, Interestingly, at δ = 1/2, this limiting value of u zz cannot be reached since at any H, the lowest Landau subband remains occupied by the electrons. In this situation Eq. (19) reduces to formula (6) for the Weyl electrons. When δ > 1/2, the behavior of the magnetostriction in the ultra-quantum limit cardinally changes. It follows from Eq. (19) that for 1/2 < δ < 3/2 at µ 0 H > F/(1.5−δ) and for 3/2 < δ < 5/2 at µ 0 H > F/(2.5−δ), respectively. In both the cases, n(ζ − ε 0 , B) ∝ B 3/2 at B ≫ F . However, if a limited range of the high magnetic fields is considered, formulas (21) lead to the curves u zz (B), the shapes of which are close to the straight line described by Eq. (11) (cf. Fig. 1 in the main text). Nevertheless, the slopes of these curves λ ≡ du zz /dB depend on δ and differ from the slope characteristic of the Weyl quasiparticles. Indeed, the slope for the Weyl fermions, The special case δ = 1/2, for which the formula for the magnetostriction exactly coincides with the expression for the Weyl electron, separates the two essentially different behaviors of the magnetostriction in the ultraquantum limit.
Weak magnetic fields (µ 0 H ≪ F ). At µ 0 H ≪ F , the ratio u in Eq. (19) is large, and the sum contains many terms. Similarly to the case of the Weyl electrons, we find from formulas (2) and (19) that the magnetostriction u zz produced by the trivial electrons comprises the smooth and oscillating parts in the weak-field range, u zz =ū zz + u osc zz . These smooth (ū zz ) and oscillating (u osc zz ) parts have the following forms: Interestingly, at δ → 0 when the Zeeman energy is negligible as compared to the cyclotron spacing between the Landau subbands, we find that . This value is of the opposite sign as compared to c for the Weyl electrons. On the other hand, the phase of the oscillating part (23) at δ → 0 is shifted by π with respect to the phase of the oscillations described by Eq. (14). This shift is caused by the difference in the Berry phases for these two cases [10,14]. On the other hand, at δ = 1/2 the constant c exactly coincides with that given by Eq. (13), and equation (23) for the oscillating part reduces to Eq. (14). Thus, we find that the B-dependence of u zz for the trivial electrons with δ ̸ = 1/2 essentially differs from the appropriate dependence for the Weyl quasiparticles (see also Fig. 1 in the main text). This result enables one to distinguish between the Weyl and trivial electrons with measurements of the magnetic-field dependences of the magnetostriction. However, if δ = 1/2, the formulas for the magnetostriction exactly coincide in these two cases.
According to Ref. [15,16], the quantity δ is close to 1/2 for a band in a centrosymmetric crystal if this band is separated from another by a small gap that is less than the strength of the spin-orbit interaction. These bands are doubly degenerate in spin for such crystals. Strictly speaking, the spectrum (18) (with ±δ) is applicable to this situation only if the chemical potential measured from the edge of the band, ζ − ε 0 , is noticeably less than the gap ∆ in the spectrum. However, it turns out that even if ζ − ε 0 ∆, the B dependence of the magnetostriction for the gapped spectrum coincides with its B dependence for Dirac electrons. In this context, it is worth noting that the spectrum of bismuth is sufficiently well described by a two-band model, the parameter δ is close to 1/2 [16], and so the field dependence of its magnetostriction [5,17] is reminiscent of the appropriate dependence for the Dirac electrons. However, the situation essentially changes for the noncentrosymmetric Weyl semimetals considered here. In particular, the holes in TaAs are located near the nodal ring that would occur in this semimetal in neglect of the spin-orbit interaction [18]. The spin-orbit coupling lifts the degeneracy of the bands, and there are four nearby bands in the region of the Brillouin zone where the holes exist. In this situation, there is no reason to expect that δ ≈ 1/2 for them.
As is clear from Supplementary Note 1, if not only ε 0 but also the gap in the spectrum ∆ depend on a deformation (and ζ − ε 0 is not-too-small as compared to this gap), the rigid-band approximation can fail. In this case, formula (2) will contain an additional term, and in the weak-field range, this term is proportional to B 2 .

Supplementary Note 4: Magnetostriction and magnetization
Let us compare the magnetostriction with the magnetization, which is considered as another thermodynamic probe of the Weyl electrons [19][20][21]. The magnetization M characterizes the change of the electron energy in the magnetic field, and this change occurs for all electrons, including those lying far below the Fermi level. In particular, a band completely filled by electrons can give a noticeable contribution χ 0 H to the magnetization where χ 0 is a constant. It is significant that the absolute value of this constant increases when a top of the filled band approaches the lowest boundary of unoccupied electron states at the same point of the Brillouin zone. This increase is due to virtual interband transitions of electrons in the magnetic field. On the other hand, the magnetostriction probes the energy of the interaction between electrons and the elastic deformations of the crystal, and within the rigid-band approximation, it is determined by the change of the charge-carrier density in the magnetic field. This density remains unchanged for the filled band, and so this band does not contribute to the magnetostriction.
In the weak magnetic fields, the magnetization of trivial electrons is proportional to (ζ − ε 0 ) 1/2 B ∝ n 1/3 B [8] and increases with increasing n. In contrast, the coefficient c in Eq. (22), which characterizes the change of the electron density in the magnetic field, decreases with n, As to a Weyl point, it was found (see review [10] and the reference therein) that the appropriate magnetization looks like M ∝ ln(|E F − ε d |)B ∝ ln(n 1/3 )B. In other words, the smaller Weyl pocket, the larger contribution of the Weyl point to the M , and this increase is reminiscent of that given by Eq. (16) for the coefficient c: However, these enhancements of the magnetization and of the magnetostriction have different origin. The magnetostriction of the Weyl electrons grows to higher values due to the decrease of their pocket, whereas in the case of the magnetization, its logarithmic enhancement is caused by the lower filled band since the Fermi level, i.e., the boundary of the unoccupied electron states, tends to the top of this band ε d when the Weyl pocket shrinks.
In the ultra-quantum limit, the magnetization of the Weyl electrons is proportional to B ln(CB/F )−6F where C is a constant [10,11,22], see also [19,20]. Except for the logarithmic factor, this formula looks like Eq. (11). However, this factor has the same origin as the factor ln |E F − ε d | in the weak-field expression, and it can be obtained by the replacement of the difference |E F − ε d | by the Landau-level spacing, which becomes larger than this difference in the ultra-quantum limit. As to the magnetization of the trivial electrons in this limit, it tends to zero [20] if the parameter δ < 1/2 and is proportional to B 3/2 in the opposite case of δ > 1/2. (Indeed, for δ > 1/2, simple estimates give: n ∝ B 3/2 , the B-dependent contribution to the Ω potential, δΩ ∝ (δ − 1/2)(e B/m * )n, and M = −∂δΩ/∂B ∝ B 3/2 .) By and large we may conclude that although the results for the magnetostriction and the magnetization are quite similar, they stem from the different contributions to the thermodynamic potential and have different physical origin.

Supplementary Note 5: Magnetostriction of TaAs
In Supplementary Notes 2 and 3 the B-dependences of the magnetostriction are analyzed under the assumption that a variation of the magnetic field does not change the chemical potential ζ of electrons. This situation does can occur when a crystal contains a large charge-carrier group that maintains the constancy of the chemical potential. However, in TaAs there are eight equivalent pockets of the W1 electrons, sixteen equivalent pockets of the W2 electrons, and eight equivalent pockets of the holes, with all the pockets being relatively small. In this case it is necessary to take into account the magnetic-field dependence of the chemical potential in analyzing the magnetostriction. This dependence ζ(B) is found from the conservation condition of the total density of the charge carriers, where the summation is over all the pockets, ζ 0 is the initial value of ζ at H = 0, and the charge-carrier density in the magnetic field for the ith pocket, n i (ζ, B), is described by formula (6) (2) is rewritten as follows: It is clear from Eqs. (24) and (25) that if all the constants Λ i were equal, the magnetostriction u zz would be zero. Consider a special case when the magnetic field is considerably less than the fields F i /µ 0 for all the groups of the charge carriers. In this situation, equation (24) in the chemical potential ζ is approximately solvable since the difference n i (ζ, B) − n i (ζ 0 , 0) can be rewritten as follows: where ν i ≡ ∂n i (ζ 0 , 0)/∂ζ 0 , and we have taken into account that ∂n i (ζ, B)/∂ζ ≈ ν i at the weak magnetic fields µ 0 H ≪ F i . Then, equations (24) and (25) give where ν ≡ ∑ i ν i . Therefore, in this weak-field range the B-dependence of the chemical potential can be taken into account by the renormalization of the constants Λ i . For stronger magnetic fields, it is necessary to solve equation (24) numerically.
In the case of TaAs, it is convenient to rewrite general formula (25) with the use of Eq. (24) as follows: where the summation is carried out over the W1 and W2 pockets only; the constants Λ W 1 , Λ W 2 , and Λ h refer to the W1, W2 electrons, and to the holes, respectively. This formula is valid at any strength of the magnetic fields. In the weak magnetic fields, µ 0 H ≪ F i , formulas (24) and (27) are equivalent to Eqs. (26).
Magnetic field parallel to the c axis. When the magnetic field is parallel to the c axis, all the pockets in the W1 electron group or in the W2 group or in the hole group give identical contributions to the magnetostriction. Let F W 1 , F W 2 , and F h denote the frequencies F i for the W1 and W2 electrons pockets and for the holes, respectively. According to Ref. [18], at this direction of B, one has F W 1 ∼ 7 T, F W 2 ∼ 5 T, and the field F h ∼ 19 T for the holes is larger than the maximal field 16 T in our experiments. Since F h is large, we shall consider the range µ 0 H ≤ 16 T as the low-field region for the holes and shall describe their contribution to Eq. (24) as follows: where ν h = ∂n h (ζ, 0)/∂ζ is density of states for the holes in zero magnetic field, whereas the function β(ζ) defines the variation of the hole density for the low magnetic fields, (The oscillation contribution to n h (ζ, B) is neglected.) If the holes can be described by the parabolic spectrum (17), (18), one arrives at [compare with Eq. (22)], where ε 0 is the edge of the hole band, and we have taken into account that n h (ζ, 0) ∝ −(ε 0 − ζ) 3/2 and n h (ζ, 0)/F 2 h (ζ) ∝ −(ε 0 − ζ) −1/2 for the parabolic spectrum. However, we emphasize that formula (28) is applicable to the holes even though their spectrum is not described by the parabolic model, and their Fermi surface noticeably differs from an ellipsoid. The only requirement of the applicability is the condition of the low-field limit for them (i.e., B < F h ).
Let us introduce the dimensionless deviation of the chemical potential from its initial value at where n W 1 , n W 2 are the densities of the electrons at zero magnetic field (i.e., at , and the frequencies F W 1 , F W 2 are taken at ζ = ζ 0 . The densities n i satisfies the relation n W 1 + n W 2 − |n h | = n imp where n h is the hole density at H = 0, and the charge carrier density n imp is caused by impurities. This n imp specifies the doping in the sample. With the doping, 1 + (n W 2 /n W 1 ) − (|n h |/n W 1 ) ̸ = 0. In the case of the parabolic spectrum for the holes, we obtain from Eqs. (29), Interestingly, if the magnetic field is tilted away from the c axis, the quantities F h , δ, and henceβ 0 , will change, but the ratioβ 1 /β 0 will remain unchanged for the parabolic spectrum. Of course, the parametersν h ,β 0 , andβ 1 may essentially differ from these estimates if the dispersion of the holes noticeably deviates from the parabolic law. According to expression (27), the general formula for the magnetostriction ∆L/L = u zz of TaAs at zero temperature takes the form: where

Supplementary Note 6: Estimates of the parameters characterizing Fermi-surface pockets in TaAs
Parameters characterizing Weyl Fermi pockets can be estimated, using the data for the quantum-oscillation frequencies and for the appropriate cyclotron masses [23]. In particular, if the frequency F produced by a Weyl pocket and the appropriate cyclotron mass m * have been measured at least for one direction of the magnetic field, the formula enables one to find the position of the chemical potential ζ relative to the energy ε d of the Weyl point. Here S max is the area of the extremal cross section perpendicular to the magnetic field for the Weyl pocket. The density n W of the Weyl charge carriers can be expressed in terms of directly-measurable frequencies of the quantum oscillations, where V is the volume of a Weyl pocket in the Brillouin zone, N W is the numbers of the equivalent Weyl pockets, and F i are the frequencies associated with the principal directions of the Fermi-surface ellipsoid. In other words, F 1 and F 3 are the maximal and minimal frequencies produced by the pocket when the magnetic field rotates in various planes, and F 2 corresponds to the direction of H perpendicular to the directions at which F 1 and F 2 occur. The cross sectional areas S (i) max correspond to the frequencies F i , and these cross sections are mutually orthogonal.
In the case of the parabolic dispersion of charge carriers, the formula that is similar to Eq. (34) looks as follows: and contains the additional factor 1/2 as compared to Eq. (34). Using the experimental data for the cross-sectional areas and the cyclotron masses (Table I in Ref. [18]) and the calculated value F W 2 ∼ 5 T for H parallel to the c axis [18], we obtain with formulas (34)-(36) that n W 1 ≈ 2.5 × 10 18 cm −3 , ζ 0 − ε d,W 1 ≈ 28.4 ± 3.5 meV [23], and n W 2 /n W 1 ∼ 0.15, ζ 0 − ε d,W 2 ≈ 11.9 ± 1 meV. Since the oscillations in the magnetostriction of our sample and the quantum oscillations observed by Arnold et al. [18] for the W1 electrons have practically the same frequency at H parallel to the c axis, we conclude that the Fermisurface parameters estimated above are appropriate for our samples, too. For this reason, in the calculation with formulas of Supplementary Note 5, we set Interestingly, in the case of the parabolic spectrum for the holes, we similarly can find ε 0 − ζ 0 using Eq. (37) and values of F and m * measured for the hole at two directions of H [18]. It turns that ε 0 − ζ 0 lies in the interval 12.6-20.6 meV if the data for H ∥ [100] are used, and ε 0 − ζ 0 is in the range 7.9-16.2 meV if we use the data for H ∥ [110]. Therefore, it is reasonable to assume that 12.6 meV < ε 0 − ζ 0 < 16.2 meV. With this ε 0 − ζ 0 and the above ζ 0 − ε d,W 1 , we arrive at the estimate, If the dispersion of the holes deviates from the parabolic dependence, the estimate of ε 0 −ζ 0 and the expression for β 1 /β 0 can noticeably change. Below we do not exclude the possibility of this deviation since the hole pockets of the Fermi surface in TaAs are associated with the bandcontact lines that would occur in absence of the spin-orbit interaction [18].

Supplementary Note 7: Calculation of the magnetostriction along the [001] direction and an analysis of the obtained parameters
Calculations. We calculate the magnetostriction with Eqs. (30), (32) at given frequencies F W 1 = 7.2 T, F W 2 = 5, and at the fixed values of the ratios (ζ 0 − ε d,W 1 )/(ζ 0 − ε d,W 2 ) and n W 2 /n W 1 , see Eq. (38). The values of the other parameters are chosen so that the calculated magnetostriction best matches the experimental data (set 1 in Supplementary Table I). The result is presented in Supplementary Fig. 6. Note that although the value F W 2 = 5 T was calculated in Ref. [18], the frequency F W 2 was not measured anywhere, and so it can, in principle, differ from this value. We find that if F W 2 decreases, the quality of such fits of the theoretical curve to the experimental data gradually improves, and the best fit is reached at F W 2 ≈ 1.35 T; see set 2 in Supplementary Table I and Supplementary Fig. 7. In this context, we shall analyze the two sets of the parameters below. The first set agrees with the value of F W 2 from Ref. [18], while the second one provides the best fit of the theoretical curve to the experimental data on the magnetostriction.
We also compare the magnetostriction measured at the temperature T = 4.2 K with the magnetostriction calculated at a finite dimensionless temperature t = T /(ζ 0 −  Table I (set 1).
Supplementary Table I Table I. The theoretical curve agrees with the experimental data at t ≈ 0.015. (The same t ≈ 0.015 is also obtained for the second set.) This value of t leads to the independent estimate of ζ 0 − ε W 1 : ζ 0 − ε W 1 ≈ 280 K ≈ 24 meV, which is only a little less than the value 28.4 ± 3.5 meV obtained from the data of Arnold et al. [18] in Supplementary Note 6.
We may now estimate the Dingle temperature T D,W 1 = (ζ 0 − ε W 1 )γ W 1 /π for the W1 electrons. The data of Supplementary Table I and ζ 0 − ε W 1 ≈ 280 K give T D,W 1 ≈ 2.2 K. This value of T D,W 1 is comparable with T D,W 1 ≈ 3.2 K obtained in Ref. [18]. As to T D,W 2 , we can only tentatively estimate its value T D,W 2 =  Table  I).
Note also that a reduction of T D,W 2 to 2.5 K only marginally affects the agreement between the experimental and calculated data for the first set of the parameters and has no effect at all on the agreement for the second set when F W 2 is small.
The results of these calculations for both the sets of the parameters are presented in Supplementary Table II. A comparison of the obtained constants Λ c i shows that in describing the magnetostriction, the use of the simplified approach, for which the B dependence of ζ is neglected, is reasonably well justified. (A caution should be given only to the case when one the parameters a W i is unusually small.) The same conclusion follows either from a comparison of Supplementary Figs. 5 and 6 or from the calculation of the dependence ζ(B) in the weak magnetic fields. Formula (30) for such fields yieldsδζ ≈ zB 2 where the coefficient z is determined from the equation, With the data of Supplementary Table I, we find that this coefficient is equal to z 1 ≈ 10 −5 for the first set and to z 2 ≈ −7.6 × 10 −5 for the second one. On the other hand, in the weak-field range, formula (32) for the .
Comparing, e.g., the terms 3δζ and B 2 /(16F 2 W 1 ), one obtains that the first term is smaller than the second one by the factor 40 for set 1 and by the factor 5 for set 2. In other words, in the first approximation, one really can neglect the dependence ζ(B).
Assuming the parabolic spectrum of the holes and inserting the data of Supplementary Table I for set 1 into formulas (31), we obtain thatβ 1 /β 0 ≈ 0.94 and |n h |/n W 1 ≈ 0.32. The ratioβ 1 /β 0 agrees with the estimate derived from the data of Ref. [18] (Supplementary Note 6). The obtained value of |n h |/n W 1 means that the doping in our sample, n W 1 + n W 2 − |n h |, is of the order of 0.83n W 1 ≈ 2 × 10 18 cm −3 . As to the value ofβ 0 presented in Supplementary Table I for set 1, it can be reproduced with the first formula of (31) at |n h |/n W 1 = 0.32 and F h = 19 T if we assume that due to the spin-orbit interaction, the magnetic moment of the holes is sufficiently large (the parameter δ amounts to 2.2).
A similar analysis for the second set of the parameters in Supplementary Table I leads to the unrealistic value of ε 0 − ζ 0 ≈ 63 meV which essentially exceeds the estimate ε 0 −ζ 0 ≈ 12.6÷16.2 meV inferred from the results of Arnold et al. [18] under the assumption of the parabolic spectrum for the holes (Supplementary Note 6). Moreover, formulas (31) with set 2 give |n h |/n W 1 ≈ 2.05, the value of which disagrees with the electron doping of the specimen that was used in Ref. [18] and that is similar to ours. These discrepancies indicate that the second set of the parameters may be admissible only if a significant deviation of the dispersion of the holes from the parabolic law occurs and hence if formulas (31) and the estimate for ε 0 − ζ 0 in Supplementary Note 6 are not applicable to this charge carriers.  Consider the magnetostriction ∆L/L along the a axis for the magnetic field still aligned with the direction [001]. In this case, in describing the magnetostriction, only the values of A W 1 and A W 2 can change due to a change of the constants Λ W 1 , Λ W 2 , and Λ h . All the other parameters determining this ∆L/L should remain the same as in the case of the c-axis magnetostriction. Therefore, a comparison of theoretical and experimental results for the a-axis magnetostriction may enable one to choose between the two sets of the parameters presented in Supplementary  Table I. The magnetostriction ∆L/L measured along the a axis at B parallel to [001] is shown in Supplementary  Fig. 9. In this figure, we also show the magnetostriction calculated with both the sets of the parameters in Supplementary Table I The presented fit of the theoretical curve to the experimental data argues in favor of set 2. However, we emphasize that the orientation of the sample in respect to the magnetic field direction cannot be established in our experiments with a high accuracy, and hence the experimental curve presented in Supplementary Fig. 9 does not necessarily corresponds to the case of B ∥ [001]. Due to the high sensitivity of the a-axis magnetostriction to the angle between B and the c axis (see Fig. 6 in the main text), its true B dependence at B ∥ [001] may essentially differ from the experimental curve presented in Supplementary Fig. 9, and so a more precise orientation of the sample is required in order to reliably exclude the possibility of set 1.
Let us now discuss the unusual high sensitivity of the a-axis magnetostriction to small deviation of B from the c axis (Fig. 6 in the main text). When the magnetic field is tilted away from the direction [001] in the plane (010), electron pockets in the W1, W2 groups, as well as the hole pockets, produce different contributions to the magnetostriction, and one should take into account that with increasing the tilt, the parameters F W 1 , F W 2 , F h not only change in magnitude but also the number of these parameters increases (the charge-carrier pockets in any group have different values of these parameters). This makes the strict theoretical analysis of the magnetostriction very complicated. However, the low sensitivity of the c-axis magnetostriction to the deviation angle (Fig. 7 in the main text) indicates that the total differences of the densities n(B) − n(0) for each of the groups (W1, W2, h) change only slightly with the angle. As is explained in Supplementary Note 1, the a-axis magnetostriction contains an additional term that is proportional to n a1 (B) − n a2 (B) where n a1 and n a2 are the charge-carrier densities of the equivalent pockets lying near the reflection planes (010) and (100), respectively. Consider the situation when equivalent pockets have the shape of elongated ellipsoids, the longest axes of which are practically perpendicular to the c axis. This model seems to be applicable to the Fermi surface of the holes in TaAs. When the magnetic field is tilted in the (010) plane, the appropriate extremal cross-sectional areas of the ellipsoids lying near this plane abruptly decreases, whereas the appropriate areas for the ellipsoids located near the (100) plane remain practically unchanged. This means that in the weak magnetic fields, the difference |n a1 (B) − n a2 (B)| steeply increases with the tilt angle. which do indicate the large variation of c h with the angle Θ between B and the a axis. This sharp variation may be due to the above-mentioned strong Θ dependence of |n a1 (B) − n a2 (B)| for the holes. We cannot also exclude that the Fermi-surface pockets for the W2 electrons have a shape other than spherical, and so they can contribute to the angular dependence of the a-axis magnetostriction as well. (Although this assumption does not agree with the numerical calculation of the Fermi surface for the W2 electrons [18], however, subtle details of this small surface can hardly be obtained in the calculations.) On the other hand, the W1 pockets have the shape close to ellipsoids elongated in the [001] direction, and so they give a gradually varying contribution to this angular dependence.