Interaction and Temperature Effects on the Magneto-Optical Conductivity Of

Interaction and temperature eﬀects on the magneto-optical conductivity of Weyl liquids

S. Acheche, R. Nourafkan, J. Padayasi, N. Martin, and A.-M. S. Tremblay Département de physique, Institut quantique, and Regroupement québécois sur les matériauxde pointe, Universitéde Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1 (Dated: July 29, 2020) Negative magnetoresistance is one of the manifestations of the chiral anomaly in Weyl semimetals. The magneto-optical conductivity also shows transitions between Landau levels that are not spaced as in an ordinary electron gas. How are such topological properties modified by interactions and temperature? We answer this question by studying a lattice model of Weyl semimetals with an on-site Hubbard interaction. Such an interacting Weyl semimetal, dubbed as Weyl liquid, may be realized in Mn3Sn. We solve that model with single-site dynamical mean-field theory. We find that in a Weyl liquid, quasiparticles can be characterized by a quasiparticle spectral weight Z, although their lifetime increases much more rapidly as frequency approaches zero than in an ordinary Fermi liquid. The negative magnetoresistance still exists, even though the slope of the linear dependence of the DC conductivity with respect to the magnetic field is decreased by the interaction. At elevated temperatures, a Weyl liquid crosses over to bad metallic behavior where the Drude peak becomes flat and featureless. We comment on the effect of a Zeeman term.

I. INTRODUCTION nodes.11,12 This is a consequence of the parabolic density of states. It has been observed in the low-temperature, low-frequency optical spectroscopy of the known Weyl Weyl semimetals are three-dimensional (3D) analogs of 13 graphene with topologically protected band crossings and semimetal TaAs. Adding a magnetic field, not only interesting transport phenomena. Monopnictides TaAs, leads to the above-mentioned chiral-anomaly in the DC TaP, NbAs and NbP are prime examples1–3. In the pres- conductivity, it modifies the optical conductivity that ence of a uniform magnetic field, a Weyl node splits into now consists of a narrow low-frequency peak, whose DC degenerate Landau levels with a chiral zeroth level that value manifests the chiral nomaly, and of a series of asymmetric peaks from interband transitions superimposed on crosses the Fermi level and gives a non-zero DC conduc- 11,14 tivity. The degeneracy of the zeroth Landau level de- the linear background from the no-field case . The case of hybridized Weyl nodes has also been considered pends on the amplitude of the magnetic field. Hence, the 15 resistivity in the presence of parallel electric and mag- both theoretically and experimentally for NbP . The netic fields acquires a magnetic-field-dependent contribu- effect of long-range Coulomb interactions on interband magneto-optical absorption has been studied theoreti- tion, the so-called chiral anomaly contribution, which is 16 negative in contrast with the conventional metal.4 Nega- cally using GRPA . tive magnetoresistance has been experimentally observed Here, using dynamical mean-field theory17, we study, 5,6 7 in Weyl semimetals such as TaAs or Mn3Sn even based on Ref. 18, the effects of both local Hubbard-type though it is not always clear whether its origin is the interactions and temperature on the magneto-optical chiral anomaly.8–10 conductivity. We show how interactions and temperature The standard Drude contribution and the chiral- broaden the low-frequency peak, renormalize and redis- anomaly contribution compete with each other in gen- tribute optical spectral weight between the low-frequency eral, leading to a nonmonotonic dependence of the con- peak and the inter-band transitions between Landau lev- ductivity (or resistivity) on magnetic field. Whether the els and how interactions transfer optical weight to the chiral anomaly-related conductivity dominates or not de- high-frequency incoherent satellites. Theoretically, all pends on the ratio of different scattering times, such as this is related to the robustness of the quasiparticle pic- inter-Weyl point scattering time and transport scatter- ture, which we thoroughly analyze. We will refer to the ing time and the location of the chemical potential with interacting Weyl semimetal as a Weyl liquid19, by anal- respect to Weyl points. These quantities are impacted ogy with the Fermi liquid. arXiv:2003.14246v2 [cond-mat.str-el] 28 Jul 2020 by both temperature and electron-electron interaction, in We also investigate the temperature-induced transition 7 particular in correlated Weyl semimetals such as Mn3Sn , to bad-metal behavior. This occurs as follows in normal raising the question of their influence on the magnetore- metals. The half bandwidth of the low-frequency peak sistance of Weyl semimetals. in optical spectra, proportional to the scattering rate, The frequency dependence of the conductivity in Weyl grows with increasing temperature because of thermally semi-metals also has interesting features. At zero tem- induced scattering events grow. Upon approaching the perature and in the absence of a magnetic field, the op- so-called Mott-Ioffe-Regel20, the scattering rate becomes tical conductivity in the continuum limit and without of the order of the bandwidth, and the low-frequency interactions exhibits a linear frequency dependence with peak becomes flat and essentially featureless21–24. Quasi- vanishing DC limit when the chemical potential is at the particles and Fermi liquid behavior disappear. How this 2 physics manifests itself in a Weyl liquid is another subject spins and in the continuum limit, the Zeeman term sim- of this study. ply shifts the position of the Weyl nodes and does not After we introduce the model in Sec.II, we show the ef- influence the spectrum at all. So from now on, we do fect of interactions on single-particle properties in Sec.III not include a Zeeman term. We pick magnetic field val- and then of interactions and temperature on the con- ues commensurate with the original lattice, namely we ductivity in Sec.IV. AppendixA shows that the Zee- take eBa2/h = p/q with rational values of p/q28. In the man term has little influence on the magneto-optical con- rest of this paper, and with no loss of generality, we set ductivity, even though it shifts the Landau levels. Ap- p = 1. The applied magnetic field is in the z-direction pendixB contains a perturbative estimate of the imag- and we use the Landau gauge (i.e. a vector potential inary part of the self-energy, and shows that despite its A = (0, Bx, 0)) to preserve translation symmetry along ω8 dependence, the usual quasiparticle renormalization Z the y-direction. This leads to the following Harper ma- follows. AppendixC explains how to extract the weight trix: of an isolated Drude peak directly from imaginary fre- H↑↑ H↑↓ quency. HH = . (3) H↓↑ H↓↓ Each of the sub-matrices in the above equation is of di- II. MODEL AND METHODS mension q × q. They are defined as follows,   We start from a Weyl semimetal model defined on a M −t 0 ... −t  0  cubic lattice and add a particle-hole symmetric Hubbard     interaction. In the zero-field case, it reads  −t M1 −t . . . 0  H↑↑ =   , (4a)  ......  ˆ ˆ X X  0 . . . .  H = H0 + U (ˆnr↑ − 1/2)(ˆnr↓ − 1/2) − µ nˆr (1)   r r   −t 0 ... −t Mq−1 where U is the strength of the Hubbard interaction,n ˆr↑ (ˆnr↓) are occupation numbers for spin up (down), and µ is   the chemical potential. The non-interacting Hamiltonian A0 −it 0 . . . it   Hˆ in second quantized form is25   0    it A1 −it . . . 0  X H↑↓ =   , (4b) Hˆ = Cˆ † [−(2t cos k + 2t cos k + 2t cos k )σ  ......  0 k x y z z z  0 . . . .  k     + (2t sin ky)σy + (2t sin kx)σx]Cˆ k, (2) −it 0 . . . it Aq−1 where t and tz are independent parameters and σx, σy with and σ represent Pauli matrices. The Hamiltonian is z H = H† , H = −H , (5) written in spin-space, so the creation and annihilation ↓↑ ↑↓ ↓↓ ↑↑ operators are two-spinors, Cˆ = (ˆc , cˆ ). We also k k,↑ k,↓ and with the definitions Mn = −2t cos (ky + 2πnp/q) − take Boltzman’s constant k equal to unity and t = t = B z 2tz cos kz and An = 2it sin (ky + 2πnp/q). 1 for all calculations and use units where ¯h = 1, e = Equation3 describes a magnetic unit cell with q sites in 1. In addition, the lattice constant a equals unity. In the x direction. The full Hamiltonian includes a periodic this regime, t = tz, there are four Weyl nodes in the extension of Harper matrix along the x axis, the chemical first Brillouin zone of the non-interacting Hamiltonian potential and the Hubbard interaction. For the values of located at (0, π, ±π/2) and (π, 0, ±π/2). Time-reversal q that we choose, the Harper matrix is sufficiently large symmetry is broken, but this model has other symmetries and the corresponding reduced Brillouin zone along the detailed in Ref. 26. kx direction sufficiently small that dependencies on kx We introduce the orbital effects of a uniform magnetic can be neglected. These dependencies are associated with 27 field through the Peierls substitution . The magnetic the periodicity of the magnetic unit cell and they become Hamiltonian (in the tight-binding form) then includes a important in the large field limit where the magnetic unit site-dependent Peierls phase φnm(B) that changes the cell has only a few sites. Consequently, the sites inside the symmetries of the model. What about a Zeeman term? magnetic unit cell have identical local density of states When the Pauli matrices act on spin, there is a Zeeman (spin up + spin down) at half-filling. Therefore, we are term whose effect on the Hamiltonian has been discussed in the Landau regime defined in Ref. 29. in Ref. 26. In AppendixA we show that in the non- In this approximation, the free Hamiltonian that takes interacting case the Zeeman term influences the magneto- maximum advantage of translational invariance is optical spectrum only when lattice effects are important X † X and we show that the effects are very small. The Lan- Hˆ 0 = Cˆ HH Cˆ − µ nî (6) dau wave functions are a combination of up and down ky ,kz i 3 where the creation and destruction operators are defined bands (incoherent satellites around ω = ±10 on Fig.1a), ˆ in the basis: C = (ˆc1,ky ,kz ,↑,... cˆq,ky ,kz ,↓). the interaction tends to shrink the coherent quasiparti- We solve the interacting Hamiltonian non- cle bandwidth by the quasiparticle weight Z. However, perturbatively using the Dynamical Mean Field Theory the density of states at the Fermi level is not affected by (DMFT) framework17. The expected correction to the electronic correlations. This can be explained as follows. DMFT self-energy30 is an additive static momentum Physically, the spectral function is proportional to the dependent self-energy that would renormalize the energy quasiparticle weight Z, but the one-dimensional density dispersion and lead to corresponding vertex corrections. of states of the chiral level is proportional to one over the These static corrections should not be important when renormalized Fermi velocity ZvF so that the two factors long-wavelength spin fluctuations are absent. Ref. 29 of Z cancel each other. The density of states for chiral contains the derivation of the DMFT equations that Landau levels in the quasipaticle approximation is then include the orbital effects of a uniform magnetic field. In given by summary, the local self-energy depends on the magnetic field and the self-consistency equation itself is unaltered. Θ Z−1(ω/v ) + k − Θ Z−1(ω/v ) − k ρ(ω) ∝ F c F c , The effect of the magnetic field on the self-energy comes |vF | from the non interacting density of states of the Landau (8) 29,31 levels and the self-consistency equation. where vF stands for the Fermi velocity, kc is an energy We use an exact diagonalisation (ED) impurity solver cut-off and Θ is the step function. Equation8 shows for the impurity problem with a finite number of bath that the density of states is insensitive to the quasiparti- 32,33 sites, nb. Though still of considerable size, the nb = 5 cle spectral weight Z, but the range of frequencies where orbital Hamiltonian in this scheme can be diagonalized it applies is narrowed down. There is another way to ex- exactly to compute the local Green’s function at finite plain that the density of states at the Fermi level is inde- temperature. pendent of interactions: It is a general property of single- site DMFT at low enough temperature. Indeed, one can show, using Luttinger’s theorem for a momentum- III. MAGNETIC FIELD AND ELECTRONIC independent self-energy34, that the density of states at INTERACTION EFFECTS ON the Fermi level (ω = 0) is independent of interactions. SINGLE-PARTICLE PROPERTIES One has to assume that this remains valid for a range of energies close to the Fermi level. We first consider the effects of magnetic field and interactions on the density of states and then discuss the self-energy. This leads us to comment on the resilience B. Self-energy of quasiparticles in the presence of interactions in a Weyl semimetal. The scattering amplitude is related to the imaginary part of the self-energy, which depends on U. At fixed A. Density of states U, we investigate the effect of the magnetic field on the imaginary part of the self-energy at low temperature (corresponding to an inverse temperature β = 80) by com- For a non-interacting Weyl semimetal without mag- paring the case without a magnetic field, i.e. B = 0, netic field, described by Eq.2, the density of states at with the case where B = 2π/40, and with the case where low energy is characterized by ω2 behavior, with a van- B = 2π/16, i.e. in the quantum limit. The latter is char- ishing density of states at ω = 0. With a magnetic field, acterized by a clear separation between the zeroth and a finite field-dependent density of states appears at low the first non-zero Landau levels. energies as shown in Fig.1a. The non-interacting density Figure1b illustrates the effect of magnetic field on the of states in this figure is obtained from imaginary part of the self-energy for the first few Mat- Z 1 X dkydkz subara frequencies at β = 80, U = 12 and half-filling. ρ(ω) = δ( − ω) (7) N (2π)2 n,ky ,kz As in the case of the Hubbard model on the square lat- n tice26, the magnetic field does not have a large effect on where N is the total number of Landau levels, n is the the self-energy. For B = 2π/40, there are few differences

Landau level index and n,ky ,kz the corresponding disper- with the self-energy without a magnetic field. However, sion energy obtained from the Harper Matrix. The finite for B = 2π/16 we can clearly see that the self-energies field-dependent density of states at low energy increases depart from each other at larger Matsubara frequencies. with increasing field. Apart from this, a magnetic field Those differences at intermediate frequencies come does not influence the main characteristics and the band- from the significant modification of the local density of width of the density of states of a non-interacting Weyl states due to the orbital effect of the magnetic field. It semimetal. is also interesting to note that despite these differences, Fig.1a also shows the density of states of the interact- the three self-energies have the same value at the first ing system. Apart from the lower and upper Hubbard Matsubara frequency. This indicates that the scattering 4

0.30 self-energy at low frequency for a single Weyl node (see U = 0 a) the derivation in AppendixB): 0.25 U = 12 2 8 00 U ω 0.20 Σ (ω) ∝ − 5 9 , (9) π vF ) ω ( 0.15 ρ where vF is the Fermi velocity. Nevertheless, one ex- pects that since this suggests the existence of quasipar- 0.10 ticles, this result should be valid for larger values of U. 0.05 Note that this unusual behavior is very different from Fermi liquid theory where Σ00(ω) is quadratic in fre- 0.00 quency. From the point of view of lifetime, the “Weyl 10 5 0 5 10 − − liquid” seems to lead to even more stable quasiparticles ω than Fermi liquids. But what about the quasiparticle spectral weight Z? B = 0 AppendixB shows that there is a finite value of Z and 0.050 calculations show that, as expected, it decreases (roughly − b) B = 2π/40 as −U 2) as U increases. The smallness of the imaginary B = 2π/16 0.075 part of the self-energy at low frequency offers a clue for − )

n the robustness of quasiparticle physics. This justiﬁes a iω

( 0.100 quasiparticle approach where the main eﬀect of the in-

00 −

Σ teractions is encoded in the quasiparticle weight with an 0.125 (extermely) small lifetime for the quasiparticles. How- − ever, the above derivation is only valid in the absence 0.150 of magnetic field since it relies on a vanishing density − of states at the Fermi level. When an external magnetic field is applied, a finite density of states appears and that 0.050 0.075 0.100 0.125 0.150 0.175 0.200 could change the physics. ωn

Figure 1. (color online) a) Local density of states for non- interacting (black line) and interacting (continuous red line) IV. CONDUCTIVITY OF INTERACTING Weyl semimetals in the presence of an external magnetic field WEYL SEMIMETALS (B = 2π/16). A Lorentzian broadening η = 0.01 has been used for both plots. Features that are sharper than 0.01 in In this section, we use our knowledge of single-particle frequency then, cannot be resolved. Note that the flat part properties to compute the optical conductivity and find of the spectrum near ω = 0 is insensitive to a wide range out how it is affected by magnetic field, temperature and of values of η since smoothing the density of states when it is already frequency independent does not change anything. finite bandwidth. b) Imaginary part of the self-energy at the three lowest Mat- Let us first consider the current-current correlation subara frequencies for the case without a magnetic field, i.e. function in Matsubara frequency. In linear response the- B = 0, the case where B = 2π/40, and finally in the quantum ory at zero momentum and neglecting vertex corrections, limit where B = 2π/16. The inverse temperature is β = 80. this correlation function Πzz(q → 0, iνn) along the z direction is36,37

π X times at the Fermi level for half-ﬁlling are essentially the Π (iν ) = − Tr a (k)G(k, iω ) zz n Nβ zz m same with or without magnetic ﬁeld. kωm − vz(k)G(k, iωm + iνn)vz(k)G(k, iωm) , C. Resilience of quaiparticles (10)

Although it is not completely apparent from the above where β = 1/(kBT ) is the inverse of temperature, iωn is results, quasiparticles are remarkably resilient in a Weyl the fermionic Matsubara frequency, Tr is the trace over semimetal. To show this, let us momentarily remove 2q × 2q matrices, a (k) = ∂2 H is the inverse effective zz kz H the magnetic field. At small interaction strength, the mass tensor, vz(k) = ∂kz HH the velocity matrix along self-energy can be calculated with the iterated perturba- the z direction, and G(k, iωm) is the interacting Matsub- 35 Dia tion theory (IPT) solver but without the DMFT self- ara Green function. Πzz is composed of two parts: Π , consistency. Then one can use the quadratic effective the diamagnetic part and ΠP ara, the paramagnetic part density of state of Weyl semimetals near the Fermi en- (respectively, first and second term in Eq. 10). Gauge in- ergy and obtain analytically the imaginary part of the variance imposes that these two parts cancel each other 5

4,40 perfectly at νn = 0. After analytic continuation, the real optical conductivity. Upon increasing U, the Drude 0 part of the retarded conductivity σzz is peak decreases in intensity and in weight because of the quasiparticle weight Z, even if the density of state at Π00 (ω) ω = 0 is not affected by electron-electron interactions. σ0 (ω) = zz , (11) zz ω The optical spectral weight of the interband contribution is also reduced by interaction and the optical weight 00 where Πzz(ω) is the imaginary part of Πzz(ω). One can is transferred to the incoherent satellite at high energy. also compute the real part of conductivity directly in real Furthermore, the optical gap between low-frequency peak 38 frequency from and the interband contribution is decreased by the interaction, as can be seen from Fig.2a. π X Z σ0 (ω) = dTr v (k)A(k, )v (k)A(k, ω + ) A quantitative study of the Drude peak in real fre- zz N z z k quency is cumbersome since it requires analytic continu- f() − f(ω + ) ation of numerical data. This could introduce artifacts, × , (12) ω especially in presence of interactions. Fortunately, at low temperatures the weight of the Drude peak can be where A is the spectral function matrix and f is the extracted from the Matsubara-frequency current-current P ara Fermi-Dirac distribution function. In the DC limit, the correlation function Πzz (iνn). As shown in Fig.2b, P ara difference of the Fermi-Dirac distribution functions in Πzz (iνn) is a smooth function of the Matsubara fre- Eq. 12 can be replaced by the derivative with respect quencies except at the lowest frequency, i.e., ν0 = 0. The to frequency at ω = 0. Equation 12 is valid only when sudden increase at this frequency indicates a non-zero P ara the trace is real, otherwise, additional terms coming from DC conductivity. Indeed, in an insulator, Πzz (iνn) the paramagnetic term of the current-current correlation smoothly reaches its zero frequency value as one can see function have to be taken into account to obtain the from B = 0 result. Furthermore, in the interacting Weyl real part of the conductivity. Moreover, since the sys- semimetals at low temperatures, the Drude peak is sepa- tem breaks time-reversal symmetry, the spectral weight rated from the rest of the optical spectrum by a clear gap must be computed from the anti-hermitian part of the (see Fig.2a). This allows us to show that the jump in P ara Green’s function matrix Πzz (iνn) at the lowest frequency scales with the Drude peak weight. The details of the derivation are presented 1 A(k, ω) = G†(k, ω) − G(k, ω) , (13) in AppendixC. 2iπ With the help of Eq. C5, we are able to follow the R D where the spectral weight is normalized as follows, fate of the Drude weight, WD = dωσ (ω), with and 0 D R dωA(k, ω) = 1. The real-frequency dependent Green’s without interaction. Here, we define σzz(ω) = σzz(ω) + res functions are obtained by analytic continuation using the σ (ω) because the Drude peak is separated with an en- Padéapproximant method39 with a Lorentzian broaden- ergy gap from the rest of the optical spectrum. As one ing eta = 0.01. can see from Fig.3a, at low temperatures the interaction dependence of the low-frequency peak weight normalized by the non-interacting one follows exactly the quasipar- A. Interaction effects in the low-temperature limit ticle weight for all values of U tested. The normalized Drude weight scales like the quasiparticle weight Z, even near the Mott transition41 (around U = 20), a clear sign Figure2 shows the magneto-optical conductivity of of the robustness of quasiparticle physics. the interacting Weyl semimetal for several interaction The shrinking of interband contributions with increas- strengths and B = 2π/16. It is an even function of fre- ing U is also a sign of quasiparticle physics. Indeed, the quency. In the presence of Hubbard interactions, the quasiparticle weight Z renormalizes the whole band, so magneto-optical conductivity has three well defined fea- the Landau levels become closer to each other. This leads tures: The Drude Peak near zero frequency, the inter- to excitations at lower frequency than in the absence of band transitions between Landau levels11 and incoher- interactions. Contrary to interband transitions, transi- ents peaks that are a consequence of Hubbard bands in tions between incoherent Hubbard bands and quasipar- the density of states. The interband transitions at lower ticle bands and between the lower and the upper Hub- frequency are quite similar to the results obtained from bard bands is due to the frequency dependence of the a non-interacting continuum model of Weyl nodes11, i.e. self-energy and cannot be explained by the quasiparticle a series of asymmetric peaks superimposed on the linear spectral weight alone. However, Z still governs a large background from the no-field case. In our case, the lat- part of the optical conductivity. tice introduces a natural cutoff and differences with the In Fig.3b, we present W = R dωσres(ω) normalized continuum model at higher energy. res by its value at U = 0. This quantity is easily computed The three parts of the optical conductivity are affected with the help of Eq. C5 and of differently by the Hubbard interaction. It is the chi- Z +∞ ral zero’th Landau level that leads to a finite density res ˜ P ara Wres = dωσ (ω) = Πzz (iνn = 0), (14) of states at the Fermi level and to a Drude peak in the −∞ 6

0.8 1.0 1.0 U=4 U=8 0.6 a) 0.8 0.8 U=12 a) =0 U=16 0.6 0.6 D,U 0 zz 0.4 σ Z /W D 0.4 0.4 0.2 W 0.2 0.2 0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 0.0 0.0 ω 1.0 0.5 B = 0 0.8 B = 2π/40

=0 b) b) B = 2π/16

) 0.4 0.6 res,U n iν /W ( 0 zz res 0.4 Π 0.3 W 0.2 B =2⇡/16 B =2⇡/40 0.2 0.0 0 1 2 3 4 5 0 5 10 15 20 νn U 1.0 Figure 2. (color online) a) Magneto-optical conductivity of the interacting Weyl semimetal in an external magnetic field 0.8 for several interaction strengths and β = 80. b) Magneto- =0 c) optical polarization as a function of bosonic Matsubara fre- 0.6 quency for B = 2π/16, B = 2π/40 and B = 0, (i.e. without res,U external magnetic field in the latter case). DC conductivity /W scales with the jump at the lowest frequency. The figure shows res 0.4 W that DC conductivity (resistivity) is increasing (decreasing) upon increasing field, i.e., reducing q. 0.2 B =2⇡/16 B =2⇡/40 0.0 1.0 0.8 0.6 0.4 0.2 0.0 ˜ P ara where Πzz denotes the paramagnetic part of the pZ current-current correlation function without the jump at the lowest frequency. It can be obtained from an extrap- Figure 3. (color online) a) Interacting Drude-peak weight P ara normalized by the non-interacting one (left vertical axis) and olation to νn = 0 of a polynomial fit of Πzz (iνn 6= 0). quasi-particle weight for B = 2π/16 (right vertical axis) both In the weak to intermediate range of interaction, Wres depends only weakly on the interaction strength. For as a function of interaction strength. Both quantities show identical interaction dependence. b) Spectral weight of the interaction strengths larger than the bare band-width, Drude peak normalized by the interband spectral weight as a Wres decreases at a faster rate and eventually saturates function of U, calculated using Eq. 14. c) Normalized W√res to a finite value when the system undergoes a phase tran- as a function of square root of the quasiparticle weight Z sition to a Mott insulator. Hence the variation of WD or that suggests Z1/2 dependence. For all panels, β = 80. of the effective mass with interaction is more pronounced than the variation of Wres. Figure3c shows the normalized Wres √as a function of magnetoresistance phenomenon, which is a consequence square root of the quasiparticle weight Z. The point of the quantum limit where only the chiral Landau lev- at Z = 0 is in the insulator. The linearity of the curve els contribute to the Drude peak4. As one can see from and the value of the slope, equal to 1/2 over the whole Fig.4, the linear dependence of the conductivity is not range, except for the transition from metal to insulator, impacted by the interaction but the slope decreases upon indicates that the ratio is directly proportional to the increasing U. At higher U, the system undergoes a phase square root of Z. We have not found a simple argument transition to a Mott phase with zero DC conductivity. for this result. Electron-electron interactions do not destroy the negative Finally, consider the magnetoresistance of a Weyl liq- magnetoresistance, they only renormalize it, as expected uid. For the values of q tested in this paper (q ∈ from the quasiparticle picture. This statement finds its [16, 100]), the Drude weight increases linearly with the experimental proof since Weyl physics has been observed magnetic field (see figure4). This is the famous negative in correlated materials.7 7

0 T =1.0 T =0.25 U = 12 a) 0.06 U = 0 T =0.67 T =0.2 T =0.5 T =0.13 1 T =0.4 T =0.1 T =0.33 0.04 0 DC

σ 2 ) n ! 0.02 ( 00

⌃ 3

0.00 4 0.00 0.01 0.02 0.03 0.04 0.05 0.06 1/q B ∼ 5 Figure 4. DC conductivity of an interacting Weyl semimetal at β = 80 as a function of the external magnetic field. Conductivity is linearly increasing upon increasing the field 0 2 4 6 8 10 strength. The slope depends on the interaction strengh and !n decreases when U increases. The dashed parts are linear ex- 0.12 trapolation based on the first three values of the DC conduc- B =0 0.11 T ⇤ tivity. B =2⇡/40 b) 0.10 B =2⇡/16 i # ˆ n

" 0.09

B. Temperature eﬀects ˆ n h 0.08

1.00 0.07 0.5 T = 1.0

) 0.75 T = 0.5 0.06 0.4 ∞ T = 0.3333 B =0 0.50 )/W( T = 0.25 1.50 )

ω B =2⇡/40 0.3 T = 0.025 0 0 zz W( 0.25 / !

σ B =2⇡/16 )

T = 0.0125 0 1.25

! c) 0.2 ( 0.00 1 0 1 10− 10 10 00 ω ⌃ 1.00 0.1 1 (

/ 0.75 1 0.0 0.0 0.5 1.0 1.5 2.0 0.50 ω 0.0 0.2 0.4 0.6 0.8 1.0 Figure 5. (color online) Magneto-optical conductivity of an T interacting Weyl semimetal for several temperatures at B = 2π/40 and U = 12. The inset shows the magneto-optical Figure 6. (color online) a) Imaginary part of the self-energy spectral weight W (ω) as a function of cutoff frequency (cf. as a function of Matsubara frequencies for different tempera- Eq. 15). Three distinct plateaux corresponding to the three tures. U is fixed to 12 and the system is at half-filling. The features discussed inIVA can be seen in the inset continuous lines are computed using DMFT. The dashed lines are a zero frequency extrapolation of the self-energies based Bad metal behavior is a consequence of interac- on a fourth-order polynomial fit of the first five values of the 21,22,42 self-energy. At low temperature, the self-energy extrapolates tions. In a Fermi liquid (FL) the crossover between to zero like in ordinary metals. However, at higher tempera- a coherent metal and a bad metal can be identified from ture, it extrapolates to a value far from being zero at zero fre- the frequency dependence of the optical conductivity at quency. b) Double occupancy as a function of temperature for low frequencies22. At low-T , the Drude peak of a FL de- U = 12 and for several magnetic fields. T ∗ can be defined by cays as 1/ω2. The crossover out of the FL regime leads the inflection point around 0.45, marked by a vertical dashed to a broader low-frequency peak whose frequency depen- line. c) T ∗ coincides with the minimum in the approximation 2 −1 dence is no longer 1/ω . At the transition to the bad (1 − ImΣ(ω0)/ω0) for the single-particle spectral weight Z. −1 metal regime the Drude and interband features merge.22 Values of (1 − ImΣ(ω0)/ω0) larger than unity are not phys- Similar behavior can be seen in the optical conductivity ical and in fact quasiparticles disappear in the regime where of a Weyl semimetal. At low-T , the Drude peak is sepa- there is an increase after reaching a minimum. rated from the interband contribution and it decays very quickly with frequency. At higher temperatures though, both Drude peak and interband contributions broaden and merge together as one can see from Fig.5 for large in- 8 teraction, here U = 12 equal to the non-interating band- V. CONCLUSION width. In fact, the merging of the Drude peak with inter- In summary, our study reveals that an interacting Weyl band transitions is just one of the manifestations of semimetal is extremely robust to short-range interac- what happens on much larger energy scales. Indeed, the tions. The density of states at the Fermi level coming crossover affects the optical conductivity on all frequency from the magnetic-field-induced chiral level is not mod- scales. To illustrate this, we calculated the optical spec- ified by interactions. Interactions are manifest mostly tral weight integrated up to a cutoff ω through a quasiparticle renormalization Z, as in a Fermi Z ω liquid, but with a frequency-dependent self-energy that 0 W (ω) = dν σzz(ν), (15) vanishes much faster with frequency as one approaches 0 the Fermi level. The slope of the negative magnetore- and we plotted it as an inset in Fig.5. At low tem- sistance dependence on the field is reduced by Z. At peratures, three frequency ranges can be identified: the elevated temperatures, Weyl liquids exhibit a crossover Drude weight for ω ∈ [0, 1] followed by the intraband con- to a bad metal phase at a crossover temperature that is tributions and finally the Hubbard band contributions at essentially magnetic-field independent. higher energies. This three-part structure is unchanged until the crossover between metal and bad metal occurs at T ∗. Above T ∗, the Drude and finite frequency fea- ACKNOWLEDGMENTS tures merge. That temperature affects dynamical properties over scales much larger than thermal energies is This work has been supported by the Natural Sciences characteristic of strongly correlated systems. and Engineering Research Council of Canada (NSERC) In bad metals, the quasi-particles become ill-defined. under Grant No. RGPIN-2014-04584, by the Research This can be seen from the quasi-particles scattering rate, Chair in the Theory of Quantum Materials, by the given by the imaginary part of the self-energy on the real Canada First Research Excellence Fund and by the Cana- axis. At low enough temperature, it can be approximated dian Institute for Advanced Research. Simulations were by the imaginary part of the Matsubara self-energy using performed on computers provided by the Canadian Foun- ´ Σ00(ω )−1 dation for Innovation, the Ministèrede l’Education des Z ≈ 1 − 0 . (16) Loisirs et du Sport (Québec), Calcul Québec, and Com- ω0 pute Canada. In Fig.6a, the continuous lines represent the value of the self-energy obtained from DMFT for U = 12 and at different temperatures. The dashed lines represent a zero Appendix A: Effect of the Zeeman term on the frequency extrapolation based on a fourth order poly- magneto-optical conductivity nomial fit of the first five values of the self-energy. It extrapolates to zero frequency at low temperature, con- In this Appendix, we show that even unrealistically sistent with existence of the low-energy well-defined ex- large values of the Zeeman term h do not appreciably citations, but at higher temperature this behavior is no change the magneto-optical spectrum, even though the longer observed. Landau levels can be strongly modified. Fundamentally, The crossover temperature can be identified clearly this comes from selection rules and correlated changes of from static quantities as well. Consider double occu- the eigenenergies in the valence and conduction bands. pancy, shown in Fig.6b. The crossover temperature is We start with the weak magnetic-field case B = 2π/40 around the inflection point of these curves at T ∗ ≈ 0.45, and then consider in more detail a stronger field, B = consistent with the optical conductivity. Moreover, we 2π/16 where lattice effects are more apparent and modifi- also find that the crossover temperature in the presence cations of the magneto-optical conductivity more notice- of a magnetic field is very close to the crossover temper- able. We close by studying the case where the Zeeman ature obtained without magnetic field. This is because, term is so strong that there is a topological transition as shown for example in Ref. 22, the bandwidth and U from four to two Weyl nodes. Even though we restrict are clearly the two energies that control T ∗. The energies ourselves to the non-interacting case, it suffices to keep in associated with the magnetic fields that we consider are mind the quasiparticle picture, Hubbard bands and Mott small compared with the bandwidth and with U, and are transition to guess the qualitative effects of interactions. thus not very relevant. It has been shown in Ref. 26 that, for our model, the The crossover can also be seen from the temperature Weyl nodes remain at µ = 0 even when h differs from −1 dependence of (1 − ImΣ(ω0)/ω0) , plotted at the lower zero. The Landau levels for B = 2π/40 and h = 0 appear panel of the Fig.6c. At low- T , this quantity gives the in Fig.7. The corresponding magneto-optical conductiv- quasi-particle Z. As one can see, it reaches a minimum ity for three values of the Zeeman term, h = 0, 0.5, 1, is around T ∗ ≈ 0.45 and becomes even larger than unity at in Fig.8. We already know from the density of states in high enough temperature. This behavior is not physical the non-interacting case, Fig.1a, that when h = 0, lat- and coincides with the disappearance of quasi-particles. tice effects become important only around ω ∼ ±2. This 9 explains why the spectrum is essentially independent of 0.40 h = 0 h for frequency less than about ω = 4. Indeed, adding 0.35 a Zeeman term to the Hamiltonian in Eq. (5) of Ref. 11 h = 0.5 h = 1 shows that, in the continuum model, the Zeeman term 0.30 only shifts the Weyl nodes in the kz direction without changing the energy spectrum. 0.25

6 ρ 0.20

4 0.15

2 0.10 ) z

k 0.05

( 0 n 0.00 2 − 6 4 2 0 2 4 6 − − − ω 4 − Figure 9. Local density of states for B = 2π/16 and three 6 − values of the Zeeman term, h = 0, 0.5, 1. 3 2 1 0 1 2 3 − − − kz 6 Figure 7. Landau levels for B = 2π/40, h = 0 4

0.4 2 ) h = 0 z k

( 0 n

h = 0.5 0.3 h = 1 2 − 4

zz −

σ 0.2 6 − 3 2 1 0 1 2 3 − − − 0.1 kz Figure 10. Landau levels for B = 2π/16, and h = 0, 0.5, 1 with the colors corresponding to the previous figure. 0.0 0 2 4 6 8 10 ω than for lower B fields. Compared with B = 2π/40, the Figure 8. Magneto-optical spectrum for B = 2π/40, h = 0, 0.5, 1. It is only at the highest energies that small changes weight of the interband transitions has decreased com- can be detected. In the low-energy limit, up to ω ∼ 4, one pared with the Drude peak and deviations from the con- can understand the lack to h dependence from the continuum tinuum model appear at lower frequency. model. Finally, consider the extreme case near h = 2 where there is a topological transition from four to two Weyl Fig.9 shows the density of states for larger magnetic nodes. The results for the magneto-optical conductivity field, B = 2π/16, and for three values of the Zeeman are in Fig. 12. While there is clearly a difference between field h = 0, 0.5, 1. The Landau levels are shown with the h = 0 and h = 1.9, the change across the transition, from corresponding colors in Fig. 10. The difference between h = 1.9 to h = 2.1, is barely noticeable. the three cases is striking. Note also that the magnetic field is so strong that lattice effects, as measured by the deviation from ω2 dependence of the density of states, appear at lower frequency for the largest Zeeman term. Nevertheless, the magneto-optical spectrum, shown in Appendix B: Perturbation theory of the Self-energy in the DMFT framework Fig. 11 is not very dependent on h. Landau levels in the conduction and valence band vary in synchrony and selection rules enforce this near h independence of the In order to obtain Eq.9, we use second order pertur- spectrum, even though lattice effects are more noticeable bation theory for the self-energy. The latter is given by 10

that is even smaller than the ω2 of a Fermi liquid. Nev- h = 0 0.8 ertheless, there is a quasiparticle spectral weight that is h = 0.5 smaller than unity, as in Fermi-liquid theory. To show h = 1 this, start from the Kramers-Kronig relations imaginary 0.6 parts of self-energy

zz Z 00 σ d˜ω Σ (˜ω) Σ0(ω) = P , (B3) 0.4 π ω˜ − ω where P stands for Cauchy’s principal value. We can 0.2 solve this Eq. B3 by using the identity

Z d˜ω Σ00(˜ω) − Σ00(ω) Z d˜ω 1 0.0 Σ0(ω) = P + Σ00(ω)P . 0 2 4 6 8 10 π ω˜ − ω π ω˜ − ω ω (B4) Noting that Figure 11. Magneto-optical conductivity for B = 2π/16 and three values of the Zeeman term, h = 0, 0.5, 1. Even though ω˜8 − ω8 = (˜ω − ω)(˜ω + ω)(˜ω2 + ω2)(˜ω4 + ω4) (B5) the Landau levels are very different for different values of h, the magneto-optical conductivity is not very sensitive to h. and using a particle-hole symmetric model, we find

2 7 5 3 0 U 2D ω 2D ω 0.8 h = 0 Σ (ω) = − 6 9 + 40320π vF 7 5 h = 1.9 3 5 2D ω 7 8 ω − D h = 2.1 + + 2Dω − ω ln( ) (B6) 0.6 3 ω + D

zz with D the bandwidth of our toy model. Equation B6 σ 0.4 has a term linear in frequency that dominates at low frequency, other terms being of higher order in (ω/D)2. This behavior aﬀects the quasiparticle weight the same 0.2 way as in Fermi liquid theory since

0 −1 ∂Σ (ω) 0.0 Z = 1 − . (B7) 0 2 4 6 8 10 ∂ω ω=0 ω This proof can clearly be generalized to models without Figure 12. Magneto-optical conductivity for B = 2π/16 and particle-hole symmetry and for Σ00(ω) ∼ −ωn with n an three values of the Zeeman term, h = 0, 1.9, 2.1. There is a arbitrary integer because (˜ω − ω) is always a factor of topological transition from four to two Weyl nodes at h = 2, (˜ωn − ωn). but nevertheless, the spectrum dos not change much across the transition. Appendix C: Derivation of the Drude weight the formula: Here we show how the Drude Weight can be extracted Z 0 Z ω+1 from the imaginary-frequency results for Πzz(iνn). Tak- 00 2 Σω(ω) = −πU d1 d2ρ(1)ρ(2)ρ(ω+1−2), ing advantage of the fact that there is a gap in the optical −ω 0 (B1) conductivity, we can write it as a sum of Drude and of where ρ is the non-interacting local density of state of other contributions: the impurity. This is justiﬁed in the limit U → 0. Using σ0 = σD + σres. (C1) the density of states of the low-energy Hamiltonian of a zz zz zz Weyl semimetal Using gauge invariance, the f-sum rule43 can be written as follows 2 ρ() = (B2) Z Z Z 2π2v3 0 D res F dωσzz(ω) = dωσzz(ω) + dωσzz (ω) leads to formula9, which shows that the imaginary part Z 00 Πzz(ω) 8 = dω = πΠpara(iν = 0), (C2) of the self-energy scales like ω . ω zz n It seems that the zero-temperature Weyl semi-metal para is a sort of ”super Fermi liquid” with an imaginary part where Πzz is the paramagnetic contribution. 11

In Weyl semimetals subject to an external magnetic optical conductivity, so the spectral representation field, a selection rule on the conductivity along the di- Z 00 no Drude rection of the magnetic field applies, as can be deduuced ˜ para dω Πzz (ω) Πzz (iνn) = (C4) from Ref. 11. Indeed, with n being the index of the π ω − iνn th n Landau, the only possible transitions are −|n|→ |n|. tells us that at temperatures smaller than the gap, Then the contribution to the conductivity of the zeroth Π˜ para(iν ) can be extrapolated to ν = 0 from a poly- Landau level appears only very near zero frequency. This zz n n nomial fit of ΠP ara(iν 6= 0). Once Π˜ P ara is computed, selection rule allows us to do a gedanken experiment and zz n zz the Drude weight is easily obtained using Eq. C2 imagine that the zeroth Landau level is not present any- more. Then the f-sum rule becomes Z D P ara ˜ P ara dωσzz(ω) = π Πzz (iνn = 0) − Πzz (iνn = 0) . Z res ˜ para dωσ (ω) = πΠzz (iνn = 0), (C3) (C5) Given the remarkable robustness of quasiparticle physics ˜ para with Πzz (iνn = 0) the paramagnetic part of the corre- in Weyl semimetal at low temperature, we can use this lation function in the absence of the Drude contribution. result even for large values of U, as long as there is a gap In the absence of this contribution, there is a gap in the between the Drude peak and the interband transitions.

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