PHYSICAL REVIEW B 97, 035403 (2018) Dynamic current-current susceptibility in three-dimensional Dirac and Weyl semimetals Anmol Thakur, Krishanu Sadhukhan, and Amit Agarwal * Department of Physics, Indian Institute of Technology, Kanpur 208016, India (Received 2 July 2017; published 3 January 2018) We study the linear response of doped three-dimensional Dirac and Weyl semimetals to vector potentials, by calculating the wave-vector- and frequency-dependent current-current response function analytically. The longitudinal part of the dynamic current-current response function is then used to study the plasmon dispersion and the optical conductivity. The transverse response in the static limit yields the orbital magnetic susceptibility. In a Weyl semimetal, along with the current-current response function, all these quantities are significantly impacted by the presence of parallel electric and magnetic fields (a finite E · B term) and can be used to experimentally explore the chiral anomaly. DOI: 10.1103/PhysRevB.97.035403 I. INTRODUCTION Dirac and Weyl semimetals are materials with linearly dispersing bands touching at discrete Dirac/Weyl points [1–4]. Graphene is one of the most prominent examples of a Dirac material in two dimensions (2D). In 3D materials, Dirac points appear due to accidental band crossings, and are robust against a gap opening, only if protected by some crystallographic symmetry [1,2]. The presence of time reversal and crystal inversion symmetry forces the Dirac point to be fourfold degenerate, with two degenerate pairs of linearly dispersing bands. Breaking of the time-reversal (or crystal inversion) symmetry splits the Dirac node into a pair of Weyl nodes of opposite chiralities displaced in momentum (or energy). 3D Weyl fermions have been realized in TaAs [5–10], NbP [11], Mo x W 1−x Te 2 [12], and photonic crystals [13]. 3D Dirac semimetals have been realized in Na 3 Bi [14–16], Cd 3 As 2 [17–21], and ZrTe 5 [22,23]. A peculiar phenomenon related to Weyl semimetals is the chiral anomaly in crystals: pumping of charges between the nodes of opposite chirality in the presence of parallel electric and magnetic fields (finite E · B term) [24]. This nonconserva- tion of the number of particles in a given Weyl node is a direct consequence of the lowest Landau level carrying only right or left movers (depending on the chirality of the Weyl node), as demonstrated explicitly in Ref. [24]. Alternately, it can also be obtained in a semiclassical transport framework as shown in Ref. [25], or from a field theoretic framework of Ref. [26]. There have been several proposals to detect the chiral anomaly: in collective density excitations or plasmons [27,28], transport experiments [25,29–32], optical conductivity [33], circular and linear dichroism [34,35], etc. In this paper, we study the response of a single node of Dirac and Weyl semimetals to static and dynamic vector fields by explicitly calculating the current-current response function [36–38]. For each node, we consider a rotationally invariant system in which the current-current correlation function can * amitag@iitk.ac.in be expressed as a combination of longitudinal (wave vector ‖ to the vector field) and transverse (wave vector ⊥ to the vector field) response. The longitudinal current-current re- sponse function determines the optical conductivity [39] of the system. It is also related to the density-density response via the current continuity equation and hence determines the dielectric properties and the spectrum of collective density excitations (plasmons) as well [27,40–42]. The transverse current-current response function determines the diamagnetic/orbital suscep- tibility [43]. We present analytical results for the wave-vector- and frequency-dependent longitudinal as well as transverse current-current response function for a single Dirac node, and then use it to explore the impact of chiral anomaly in Weyl semimetals. In particular, the impact of chiral anomaly (a finite E · B term) can be observed via its impact on the plasmon dispersion, optical conductivity, and the diamagnetic susceptibility. The paper is organized as follows. In Sec. II, we set up the calculation of the current-current response function for a single Dirac node. The results of the longitudinal response function are discussed in Sec. III, followed by the results for the transverse case in Sec. IV. In Sec. V, we study the response of Weyl semimetals in context of the chiral anomaly. Section VI explores the implications for anisotropic systems, and we summarize our results in Sec. VII. II. CURRENT-CURRENT RESPONSE FUNCTION OF A SINGLE DIRAC NODE The effective low-energy continuum Hamiltonian to de- scribe a single isotropic massless 3D Dirac (or Weyl) node is given by H = ¯ hv F (k x σ x + k y σ y + k z σ z ), (1) where σ i are the Pauli matrices denoting real spins, and v F is the Fermi velocity. The response of this system to an elec- tromagnetic vector potential with spatiotemporal variations, A(q,ω), is determined by the current-current response function  j k j l (q,ω). In general,  AB (q,ω) describes the response of the observable ˆ A coupled to a second observable ˆ B , and is defined 2469-9950/2018/97(3)/035403(13) 035403-1 ©2018 American Physical Society