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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 D´epartement de physique; Institut quantique; and Regroupement qu´eb´ecois sur les mat´eriauxde pointe; Universit´ede Sherbrooke; Sherbrooke; Qu´ebec; 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 asym- metric 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 !8 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, 0 1 We start from a Weyl semimetal model defined on a M −t 0 ::: −t B 0 C cubic lattice and add a particle-hole symmetric Hubbard B C B C interaction. In the zero-field case, it reads B −t M1 −t : : : 0 C H"" = B C ; (4a) B .. .. .. C ^ ^ X X B 0 . C H = H0 + U (^nr" − 1=2)(^nr# − 1=2) − µ n^r (1) B C r r @ A −t 0 ::: −t Mq−1 where U is the strength of the Hubbard interaction,n ^r" (^nr#) are occupation numbers for spin up (down), and µ is 0 1 the chemical potential. The non-interacting Hamiltonian A0 −it 0 : : : it B C H^ in second quantized form is25 B C 0 B C B it A1 −it : : : 0 C X H"# = B C ; (4b) H^ = C^ y [−(2t cos k + 2t cos k + 2t cos k )σ B .. .. .. C 0 k x y z z z B 0 . C k B C @ A + (2t sin ky)σy + (2t sin kx)σx]C^ 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 = Hy ; H = −H ; (5) written in spin-space, so the creation and annihilation #" "# ## "" operators are two-spinors, C^ = (^c ; c^ ). 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.
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