Polarization, plasmon, and Debye screening in doped 3D ani-Weyl   semimetal

Jing-Rong Wang; Guo-Zhu Liu; Chang-Jin Zhang

arXiv:1706.03582·cond-mat.str-el·June 13, 2017

Polarization, plasmon, and Debye screening in doped 3D ani-Weyl semimetal

Jing-Rong Wang, Guo-Zhu Liu, Chang-Jin Zhang

PDF

Open Access

TL;DR

This paper analyzes the polarization, plasmon modes, and Debye screening in doped 3D anisotropic-Weyl semimetals, revealing unique density-dependent behaviors that can be experimentally tested.

Contribution

It provides a detailed calculation of polarization and plasmon modes in doped anisotropic-Weyl semimetals, highlighting their unique density dependencies and screening properties.

Findings

01

Plasmon frequency in the basal plane scales as n_e^{3/10}.

02

Plasmon frequency along the third direction scales as n_e^{1/2}.

03

Debye screening behavior is characterized at finite chemical potential and temperature.

Abstract

We compute the polarization function in a doped three-dimensional anisotropic-Weyl semimetal, in which the fermion energy dispersion is linear in two components of the momenta and quadratic in the third. Through detailed calculations, we find that the long wavelength plasmon mode depends on the fermion density $n_{e}$ in the form $Ω_{p}^{⊥} \propto n_{e}^{3/10}$ within the basal plane and behaves as $Ω_{p}^{z} \propto n_{e}^{1/2}$ along the third direction. This unique characteristic of the plasmon mode can be probed by various experimental techniques, such as electron energy-loss spectroscopy. The Debye screening at finite chemical potential and finite temperature is also analyzed based on the polarization function.

Tables1

Table 1. Table 1: Plasmon mode in various doped semimetals. Here, n e subscript 𝑛 𝑒 n_{e} is the fermion density, and the function n e ( μ ) subscript 𝑛 𝑒 𝜇 n_{e}(\mu) describes the relation between n e subscript 𝑛 𝑒 n_{e} and chemical potential μ 𝜇 \mu . The plasmon mode is characterized by Ω p subscript Ω 𝑝 \Omega_{p} .

Material	Fermion dispersion	$n_{e} (μ)$	$Ω_{p}$	Reference
2D Meta	$E = \pm A {\| 𝐤 \|}^{2}$	$n_{e} \propto μ$	$Ω_{p} \propto n_{e}^{\frac{1}{2}} {\| 𝐪 \|}^{\frac{1}{2}}$	GiulianiBook ; MaierBook ; Sarma09
doped 2D DSM	$E = \pm v \| 𝐤 \|$	$n_{e} \propto μ^{2}$	$Ω_{p} \propto n_{e}^{\frac{1}{4}} {\| 𝐪 \|}^{\frac{1}{2}}$	Wunsch06 ; Hwang07
doped 2D Semi-DSM	$E = \pm \sqrt{A^{2} k_{x}^{4} + v^{2} k_{y}^{2}}$	$n_{e} \propto μ^{\frac{3}{2}}$	$Ω_{p}^{x} \propto n_{e}^{\frac{2}{3}} {\| q_{x} \|}^{\frac{1}{2}}, Ω_{p}^{y} \propto n_{e}^{0} {\| q_{y} \|}^{\frac{1}{2}}$	Pyatkovskiy16
3D Meta	$E = \pm A {\| 𝐤 \|}^{2}$	$n_{e} \propto μ^{\frac{3}{2}}$	$Ω_{p} \propto n_{e}^{\frac{1}{2}}$	GiulianiBook ; MaierBook ; Sarma09
doped 3D DSM/WSM	$E = \pm v \| 𝐤 \|$	$n_{e} \propto μ^{3}$	$Ω_{p} \propto n_{e}^{\frac{1}{3}}$	Sarma09 ; Lv13 ; Panfilov14 ; Zhou15
doped 3D NLSM	$E = \pm \sqrt{A^{2} {(k_{⊥}^{2} - k_{F}^{2})}^{2} + v^{2} k_{z}^{2}}$	$n_{e} \propto μ^{2}$	$Ω_{p} \propto n_{e}^{\frac{1}{4}}$	YanZhongbo16 ; Rhim16
doped 3D double-WSM	$E = \pm \sqrt{A^{2} k_{⊥}^{4} + v^{2} k_{z}^{2}}$	$n_{e} \propto μ^{2}$	$Ω_{p}^{⊥} \propto n_{e}^{\frac{1}{2}}, Ω_{p}^{z} \propto n_{e}^{\frac{1}{4}}$	Ahn16
doped 3D triple-WSM	$E = \pm \sqrt{B^{2} k_{⊥}^{6} + v^{2} k_{z}^{2}}$	$n_{e} \propto μ^{\frac{5}{3}}$	$Ω_{p}^{⊥} \propto n_{e}^{\frac{3}{5}}, Ω_{p}^{z} \propto n_{e}^{\frac{1}{5}}$	Ahn16
doped 3D ani-WSM	$E = \pm \sqrt{v^{2} k_{⊥}^{2} + A^{2} k_{z}^{4}}$	$n_{e} \propto μ^{\frac{5}{2}}$	$Ω_{p}^{⊥} \propto n_{e}^{\frac{3}{10}}, Ω_{p}^{z} \propto n_{e}^{\frac{1}{2}}$

Equations188

E = \pm v^{2} (k_{x}^{2} + k_{y}^{2}) + A^{2} k_{z}^{4} .

E = \pm v^{2} (k_{x}^{2} + k_{y}^{2}) + A^{2} k_{z}^{4} .

S_{f} = \int d τ d^{3} x ψ^{†} (\partial_{τ} + μ + H) ψ,

S_{f} = \int d τ d^{3} x ψ^{†} (\partial_{τ} + μ + H) ψ,

H = i v \partial_{x} σ_{x} + i v \partial_{y} σ_{y} + A \partial_{z}^{2} σ_{z}

H = i v \partial_{x} σ_{x} + i v \partial_{y} σ_{y} + A \partial_{z}^{2} σ_{z}

S_{C} = \int d τ d^{3} x d^{3} x^{'} (ψ^{†} ψ)_{τ, x} \frac{e ^{2}}{2 κ ∣ x - x ^{'} ∣} (ψ^{†} ψ)_{τ, x^{'}},

S_{C} = \int d τ d^{3} x d^{3} x^{'} (ψ^{†} ψ)_{τ, x} \frac{e ^{2}}{2 κ ∣ x - x ^{'} ∣} (ψ^{†} ψ)_{τ, x^{'}},

G_{0} (i ω_{n}, k) = \frac{1}{i ω _{n} + μ - ( v k _{x} σ _{x} + v k _{y} σ _{y} + A k _{z}^{2} σ _{z} )},

G_{0} (i ω_{n}, k) = \frac{1}{i ω _{n} + μ - ( v k _{x} σ _{x} + v k _{y} σ _{y} + A k _{z}^{2} σ _{z} )},

G_{0}^{ret} (ω, k) = \frac{1}{ω + μ - ( v k _{x} σ _{x} + v k _{y} σ _{y} + A k _{z}^{2} σ _{z} ) + i η} .

G_{0}^{ret} (ω, k) = \frac{1}{ω + μ - ( v k _{x} σ _{x} + v k _{y} σ _{y} + A k _{z}^{2} σ _{z} ) + i η} .

V_{0} (q) = \frac{4 π e ^{2}}{κ ∣ q ∣ ^{2}} .

V_{0} (q) = \frac{4 π e ^{2}}{κ ∣ q ∣ ^{2}} .

V^{ret} (Ω, q) = \frac{V _{0} ( q )}{ϵ _{r} ( Ω , q )},

V^{ret} (Ω, q) = \frac{V _{0} ( q )}{ϵ _{r} ( Ω , q )},

ϵ_{r} (Ω, q) = 1 + V_{0} (q) Π^{ret} (Ω, q),

ϵ_{r} (Ω, q) = 1 + V_{0} (q) Π^{ret} (Ω, q),

ϵ_{r} (Ω_{p}, q) = 1 + V_{0} (q) Π^{ret} (Ω_{p}, q) = 0.

ϵ_{r} (Ω_{p}, q) = 1 + V_{0} (q) Π^{ret} (Ω_{p}, q) = 0.

Π (i Ω_{m}, q)

Π (i Ω_{m}, q)

G_{0} (i ω_{n}, k) = - \int_{- \infty}^{+ \infty} \frac{d ω _{1}}{π} \frac{Im [ G _{0}^{ret} ( ω _{1} , k ) ]}{i ω _{n} - ω _{1}},

G_{0} (i ω_{n}, k) = - \int_{- \infty}^{+ \infty} \frac{d ω _{1}}{π} \frac{Im [ G _{0}^{ret} ( ω _{1} , k ) ]}{i ω _{n} - ω _{1}},

Π (i Ω_{m}, q)

Π (i Ω_{m}, q)

\frac{1}{β} i ω_{n} \sum \frac{1}{i ω _{n} - ω _{1}} \frac{1}{i ω _{n} + i Ω _{m} - ω _{2}} = \frac{n _{F} ( ω _{1} ) - n _{F} ( ω _{2} )}{ω _{1} - ω _{2} + i Ω _{m}},

\frac{1}{β} i ω_{n} \sum \frac{1}{i ω _{n} - ω _{1}} \frac{1}{i ω _{n} + i Ω _{m} - ω _{2}} = \frac{n _{F} ( ω _{1} ) - n _{F} ( ω _{2} )}{ω _{1} - ω _{2} + i Ω _{m}},

Π (i Ω_{m}, q)

Π (i Ω_{m}, q)

Im [G_{0}^{ret} (ω, k)]

Im [G_{0}^{ret} (ω, k)]

E_{k} = v^{2} k_{⊥}^{2} + A^{2} k_{z}^{4} .

E_{k} = v^{2} k_{⊥}^{2} + A^{2} k_{z}^{4} .

Π (i Ω_{m}, q)

Π (i Ω_{m}, q)

F_{k, q}

F_{k, q}

\frac{1}{x + i Ω _{m}} \to \frac{1}{x + Ω + i η} = P \frac{1}{x + Ω} - iπ δ (x + Ω) .

\frac{1}{x + i Ω _{m}} \to \frac{1}{x + Ω + i η} = P \frac{1}{x + Ω} - iπ δ (x + Ω) .

Im [Π^{ret} (Ω, q)]

Im [Π^{ret} (Ω, q)]

Re [Π^{ret} (Ω, q)]

Re [Π^{ret} (Ω, q)]

[θ (- E_{k} - μ) - θ (- E_{k + q} - μ)] = 0.

[θ (- E_{k} - μ) - θ (- E_{k + q} - μ)] = 0.

Re Π_{++}^{ret} (Ω, q)

Re Π_{++}^{ret} (Ω, q)

Re Π_{+-}^{ret} (Ω, q)

Re Π_{-+}^{ret} (Ω, q)

C_{++}^{⊥}

C_{++}^{⊥}

C_{++}^{z}

C_{+-}^{⊥}

C_{+-}^{⊥}

C_{+-}^{z}

Re Π_{++}^{ret} (Ω, q) ≫ Re Π_{+-}^{ret} (Ω, q),

Re Π_{++}^{ret} (Ω, q) ≫ Re Π_{+-}^{ret} (Ω, q),

Re Π_{++}^{ret} (Ω, q) ≫ Re Π_{-+}^{ret} (Ω, q),

Re Π^{ret} (Ω, q)

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Taxonomy

TopicsTopological Materials and Phenomena · 2D Materials and Applications · Graphene research and applications

Full text

Polarization, plasmon, and Debye screening in doped 3D ani-Weyl semimetal

Jing-Rong Wang

Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory of the Chinese Academy of Science,Hefei 230031, Anhui, China

Guo-Zhu Liu

Corresponding author: [email protected]

Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China

Chang-Jin Zhang

Corresponding author: [email protected]

Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory of the Chinese Academy of Science,Hefei 230031, Anhui, China

Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, P. R. China

Abstract

We compute the polarization function in a doped three-dimensional anisotropic-Weyl semimetal, in which the fermion energy dispersion is linear in two components of the momenta and quadratic in the third. Through detailed calculations, we find that the long wavelength plasmon mode depends on the fermion density $n_{e}$ in the form $\Omega_{p}^{\bot}\propto n_{e}^{3/10}$ within the basal plane and behaves as $\Omega_{p}^{z}\propto n_{e}^{1/2}$ along the third direction. This unique characteristic of the plasmon mode can be probed by various experimental techniques, such as electron energy-loss spectroscopy. The Debye screening at finite chemical potential and finite temperature is also analyzed based on the polarization function.

I Introduction

Studying the intriguing properties of various semimetals have been one of the core subjects in condensed matter physics for more than one decade Vafek14 ; Wehling14 ; Wan11 ; Burkov16 ; Yan17 ; Hasan17 ; Burkov17 ; Armitage17 ; Weng16 ; FangChen16 , starting from the successful fabrication of monolayer graphene CastroNeto09 ; Kotov12 . The surface state of three-dimensional (3D) topological insulator bears strong similarity to graphene, and both of these two systems are classified as two-dimensional (2D) Dirac semimetal (DSM) Hasan10 ; Qi11 . It is currently clear that there are a variety of semimetals materials, including 3D DSM Vafek14 ; Wehling14 , 3D Weyl semimetal (WSM) Wan11 ; Burkov16 ; Yan17 ; Hasan17 ; Burkov17 ; Armitage17 , and nodal line semimetal (NLSM) Weng16 ; FangChen16 . It is interesting that some of such semimetals provide a nice platform to realize the interesting concepts of chiral anomaly by probing the negative magnetoresistance HuangChenGFGroup15 ; ZhangChengLong16 . In addition, there exit 2D semi-DSM Hasegawa06 ; Dietl08 ; Montambaux09B ; Pardo09 ; Banerjee09 ; Banerjee12 ; Dora13 ; Pyatkovskiy16 ; Isobe16 ; Cho16 ; WangLiuZhang17 , 3D double-WSM XuFang11 ; Fang12 ; Yang14A ; Huang16 ; Lai15 ; Jian15 ; WangLiuZhang16 ; BeraRoy16 ; Sbierski17 ; Roy17 ; Ahn16 ; Ahn17 ; Park17 , 3D triple-WSM Yang14A ; WangLiuZhang16 ; Roy17 ; Ahn16 ; Ahn17 ; Park17 ; LiuZunger17 ; Zhang16 ; WangLiuZhang17B , and 3D anisotropic-WSM (ani-WSM) Yang14A ; Yang13 ; Yang14B ; Moon14 .

In this paper, we pay attention to 3D ani-WSM where the Weyl fermion spectrum displays linear dependence on two components of the momenta and quadratic dependence on the third Yang14A ; Yang13 ; Yang14B ; Moon14 , namely

[TABLE]

Such 3D ani-WSM might be produced at the quantum critical point (QCP) between a normal insulator and a WSM, or at the QCP between a normal insulator and a topological insulator in 3D noncentrosymmetric systems Yang13 . For 3D ani-WSM, the chirality of the band-touching point is zero Yang13 . First principle calculations suggested that the topological quantum phase transition (QPT) from normal insulator to topological insulator in a noncentrosymmetric system can be tuned by applying pressure to BiTeI Bahramy12 . This theoretical prediction was subsequently confirmed by experiments carried out by means of $x$ -ray powder diffraction and infrared spectroscopy techniques Xi13 .

It is also proposed Yang14A that a semimetallic state in which the fermion excitations have the dispersion of Eq. (1) can emerge at the QCP between normal insulator and topological 3D DSM, or the QCP between 3D DSM and weak topological insulator or topological crystalline insulator. The analysis made by Yuan et al. Yuan17 showed that the fermion dispersion given by Eq. (1) may be realized in ZrTe5 at the QCP between insulator phase and DSM phase. Remarkably, recent quantum oscillation measurements provided important clue for the existence of such type of fermionic excitations in ZrTe5 under pressure ZhangJingLei17 .

In an intrinsic semimetal, the chemical potential is exactly zero, and the density of states (DOS) vanishes at the Fermi level. As a result, the dynamically screened Coulomb interaction between the fermion excitations is still long-ranged. Extensive renormalization group (RG) Shankar94 analysis revealed that the long-range Coulomb interaction in various intrinsic semimetals might be marginally irrelevant Kotov12 ; Isobe16 ; Cho16 ; WangLiuZhang17 ; Goswami11 ; Lai15 ; Jian15 ; Zhang16 ; WangLiuZhang16 , relevant Herbut14 ; JanssenSeries , or irrelevant Yang14B ; Huh16 , determined by the specific fermion dispersion and the spatial dimension of the system. In the special case of 3D ani-WSM, the Coulomb interaction is found to be irrelevant Yang14B and thus does not significantly modify the low-energy behaviors of free fermions, which is consistent with the previous work of Abrikosov Abrikosov72 .

In realistic semimetal materials, the chemical potential usually takes a finite value, and its value can be adjusted by changing the gate voltage. For semimetals defined at a finite chemical potential, there are undamped collective modes, namely plasmon. The properties of plasmon is directly related to the energy dispersion of fermion excitations, and can be experimentally investigated by various techniques, such as the electron energy-loss spectroscopy GiulianiBook ; MaierBook . In recent years, there appeared certain amount of theoretic studies for the plasmon mode in several sorts of doped semimetals, including 2D DSM Wunsch06 ; Hwang07 ; Pyatkovskiy09 , 2D semi-DSM Banerjee12 ; Pyatkovskiy16 , 3D DSM/WSM Sarma09 ; Lv13 ; Panfilov14 ; Zhou15 , 3D NLSM YanZhongbo16 ; Rhim16 , 3D multi-WSMs Ahn16 , and some other novel fermion systems Stauber14 ; Chang14 . However, a systematic study for the plasmon in the case of doped 3D ani-WSM is still lacking.

In this paper, we calculate the polarization function in the context of doped 3D ani-WSM, and then analyze the asymptotic behavior of plasmon in the long wavelength limit. It is found that the plasmon within the basal plane exhibits the behavior $\Omega_{p}^{\bot}\propto n_{e}^{\frac{3}{10}}$ , and the one along the third direction is given by $\Omega_{p}^{z}\propto n_{e}^{\frac{1}{2}}$ , where $n_{e}$ is the density of fermions. These behaviors could be probed by experiments, which is expected to offer important evidence for the existence of fermions with the dispersion given by Eq. (1). On the basis of the polarization, the Debye screening at finite chemical potential and finite temperature is also analyzed.

The rest of the paper is organized as follows. The model considered by us is presented in Sec. II. In Sec. III, the polarization function is calculated and presented in the form a general expression. The plasmon in doped 3D ani-WSM is analyzed in Sec. IV. In Sec. V, we analyze the behaviors of the Debye screening at finite chemical potential and finite temperature. We summarize our results in Sec. VI, and present in the Appendices the calculational details for the derivation of the polarization function in the long wavelength limit, the integration over the azimuthal angle, and the analysis of Debye screening.

II Model and Feynman rules

For a doped 3D ani-WSM, the free fermion action is given by

[TABLE]

where $\psi$ represents the spinor field and $\mu$ is the chemical potential. The Hamiltonian density takes the from

[TABLE]

with $\sigma_{x,y,z}$ being the Pauli matrices. Here, $v$ and $A$ are two model parameters. The Coulomb interaction is described by

[TABLE]

where $e$ is the electric charge and $\kappa$ the dielectric constant.

The fermion propagator is written in the Matsubara formalism:

[TABLE]

where $\omega_{n}=(2n+1)\pi T$ . Making analytical continuation $i\omega_{n}\rightarrow\omega+i\eta$ leads to the retarded fermion propagator:

[TABLE]

The bare Coulomb interaction can be written as

[TABLE]

After including the dynamical screening caused by the collective particle-hole excitations, we write down the dressed retarded Coulomb interaction

[TABLE]

which is obtained by using the random-phase approximation (RPA). The dielectric function $\epsilon_{r}(\Omega,\mathbf{q})$ is defined as

[TABLE]

where $\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})$ is the retarded polarization function. The collective plasmon mode is determined by the condition

[TABLE]

In the following, we will calculate the polarization function and then analyze the asymptotic behavior of the plasmon mode in the long wavelength limit.

III General expression of polarization function

To the leading order, the polarization function reads

[TABLE]

where $\Omega_{m}=2m\pi T$ and $\beta=\frac{1}{T}$ . Utilizing the standard spectral representation

[TABLE]

we obtain

[TABLE]

Summing up all frequencies leads to

[TABLE]

which then yields

[TABLE]

The imaginary part of the retarded fermion propagator is then given by

[TABLE]

where the energy

[TABLE]

Substituting Eq. (16) into Eq. (15), we find that

[TABLE]

where

[TABLE]

Here, $n_{F}(E)=\frac{1}{e^{E/T}+1}$ is the Fermi-Dirac distribution function. We then perform the following analytic continuation:

[TABLE]

Now the imaginary and real parts of the retarded polarization function become

[TABLE]

and

[TABLE]

The derivation for the polarization function in the long wavelength limit is shown in Appendix A, and the calculation details for the imaginary and real parts of the retarded polarization function are presented in Appendix C. Since the polarization function is invariant under the transformation $\mu\rightarrow-\mu$ , reflecting the particle-hole symmetry, we will choose $\mu>0$ in the subsequent calculations.

IV Plasmon mode

In this section, we analyze the long wavelength plasmon in doped 3D ani-WSM at zero temperature.

As $T\rightarrow 0$ , the function $n_{F}(E)$ becomes the step function $\theta(E)$ . In the long wavelength regime with $\max(vq_{\bot},Aq_{z}^{2})\ll\Omega\ll\mu$ , it is easy to find that $\mathrm{Im}\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})=0$ , which implies the existence of undamped plasmon. According to the values of $\alpha$ and $\alpha^{\prime}$ , we can see that the real part of the polarization function is divided into four parts, namely $\mathrm{Re}\Pi_{++}^{\mathrm{ret}}(\Omega,\mathbf{q})$ , $\mathrm{Re}\Pi_{+-}^{\mathrm{ret}}(\Omega,\mathbf{q})$ , $\mathrm{Re}\Pi_{-+}^{\mathrm{ret}}(\Omega,\mathbf{q})$ , and $\mathrm{Re}\Pi_{--}^{\mathrm{ret}}(\Omega,\mathbf{q})$ . In the regime $\max(vq_{\bot},Aq_{z}^{2})\ll\Omega\ll\mu$ , it is easy to verify that $\mathrm{Re}\Pi_{--}^{\mathrm{ret}}(\Omega,\mathbf{q})=0$ due to the relation

[TABLE]

As shown in Appendix A, in the regime $\max(vq_{\bot},Aq_{z}^{2})\ll\Omega\ll\mu$ , we have

[TABLE]

where

[TABLE]

and

[TABLE]

Noticing that

[TABLE]

which is valid in the regime $\max(vq_{\bot},Aq_{z}^{2})\ll\Omega\ll\mu$ , we finally get

[TABLE]

As mentioned in Sec. II, the plasmon mode is determined by Eq. (10), which gives rise to

[TABLE]

From this equation, we can get the plasmon mode

[TABLE]

where $C_{0}=\sqrt{4\pi e^{2}\kappa}$ and $\phi$ is the angle between $\mathbf{q}$ and $z$ -axis. The plasmon mode within the basal plane and the one along the third direction can be respectively written as

[TABLE]

According to calculations presented in Appendix B, the relation between carrier density $n_{e}$ and chemical potential $\mu$ is

[TABLE]

It is now straightforward to obtain

[TABLE]

which is a unique characteristic of doped 3D ani-WSM, and can be probed experimentally.

We now would like to compare the plasmon mode in doped 3D ani-WSM with other analogous semimetals. The behaviors of plasmon mode obtained in the context of various doped semimetals as well as traditional metals are summarized in Table 1. It is clear that the plasmon in 3D fermion systems is always gapped, but the one in 2D fermion systems is gapless. The difference arises form the fact that the bare Coulomb interaction has distinct momentum dependence in 2D and 3D. The bare Coulomb interaction is given by Eq. (7) in 3D, and has the form

[TABLE]

in 2D. For a fixed spatial dimension, the behavior of the plasmon mode is closely related to the energy dispersion of the fermionic excitations. More concretely, in the isotropic case, the plasmon is also isotropic and its power takes different value in different semimetals. In semimetals with anisotropic fermion dispersion, the plasmon mode is also anisotropic.

V Debye screening

Based on the results obtained in Appendix D, we find that

[TABLE]

in the limit of $\Omega=0$ and $|\mathbf{q}|=0$ , which represents the Debye screening induced by finite chemical potential and finite temperature. At $T>0$ and $\mu=0$ , we have

[TABLE]

whereas at $T=0$ and $\mu>0$ we find that

[TABLE]

As a consequence of the Debye screening, the dressed Coulomb interaction becomes short-ranged.

VI Summary

In summary, we calculate the polarization function in the context of a doped 3D ani-WSM, and analyze the behavior of the long wavelength plasmon. We find that the plasmon within the basal plane depends on the fermion density in the form $\Omega_{p}^{\bot}=n_{e}^{\frac{3}{10}}$ , and that the plasmon along the third direction behaves as $\Omega_{p}^{z}\propto n_{e}^{\frac{1}{2}}$ . These behaviors can be experimentally detected and would provide a useful methods to verify the existence of 3D ani-Weyl fermions. We expect that such plasmon mode could be confirmed in BiTeI, ZrTe5, and other semimetals that host fermion excitations with dispersion Eq.(1).

ACKNOWLEDGEMENTS

We would acknowledge the support by the Ministry of Science and Technology of China under Grant Nos. 2016YFA0300404 and 2017YFA0403600, and the support by the National Natural Science Foundation of China under Grants 11574285, 11504379, 11674327, and U1532267. J.R.W. is also supported by the Natural Science Foundation of Anhui Province under Grant 1608085MA19.

Appendix A $\mathrm{Re}\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})$ in the limit $\max(vq_{\bot},Aq_{z}^{2})\ll\Omega\ll\mu$

A.1 $\mathrm{Re}\Pi_{++}^{\mathrm{ret}}(\Omega,\mathbf{q})$

In the limit $\max(vq_{\bot},Aq_{z}^{2})\ll\Omega$ , we have

[TABLE]

Since $E_{\mathbf{k}}=\sqrt{v^{2}\left(k_{x}^{2}+v_{y}^{2}\right)+A^{2}k_{z}^{4}}$ , it is clear that

[TABLE]

Now $\mathrm{Re}\Pi_{++}^{\mathrm{ret}}(\Omega,\mathbf{q})$ can be further written as

[TABLE]

We then employ the transformations

[TABLE]

which are equivalent to

[TABLE]

It is easy to verify that

[TABLE]

Utilizing the transformations Eqs. (49) and (53), one can get

[TABLE]

where

[TABLE]

A.2 $\mathrm{Re}\Pi_{+-}^{\mathrm{ret}}(\Omega,\mathbf{q})$

In the limit $\max(vq_{\bot},Aq_{z}^{2})\ll\Omega$ , $\mathrm{Re}\Pi_{+-}^{\mathrm{ret}}(\Omega,\mathbf{q})$ is approximately given by

[TABLE]

Substituting Eq. (46) into Eq. (57), one gets

[TABLE]

Performing the integration of azimuth angle, we obtain

[TABLE]

Using the transformations Eqs. (49) and (53), and performing the integration of $\delta$ , $\mathrm{Re}\Pi_{+-}^{\mathrm{ret}}(\Omega,\mathbf{q})$ can be further written as

[TABLE]

In the regime $\Omega\ll\mu$ , it can be approximated by

[TABLE]

where

[TABLE]

A.3 $\mathrm{Re}\Pi_{-+}^{\mathrm{ret}}(\Omega,\mathbf{q})$

In the limit $\max(vq_{\bot},Aq_{z}^{2})\ll\Omega$ , $\mathrm{Re}\Pi_{-+}^{\mathrm{ret}}(\Omega,\mathbf{q})$ takes the form

[TABLE]

Substituting Eq. (46) into Eq. (64), and carrying out the integration of azimuth angle, we can get

[TABLE]

Utilizing the transformations Eqs. (49) and (53), and performing the integration of $\delta$ , we can get

[TABLE]

In the regime $\Omega\ll\mu$ , it can be approximately written as

[TABLE]

where

[TABLE]

Comparing Eqs. (61)-(63) with Eqs. (67)-(69), we know that

[TABLE]

in the regime $\max(vq_{\bot},Aq_{z}^{2})\ll\Omega\ll\mu$ .

Appendix B The density of fermion

The density of fermion is given by

[TABLE]

where $E_{\mathbf{k}}=\sqrt{v^{2}k_{\bot}^{2}+A^{2}k_{z}^{4}}$ . Employing the transformations Eqs. (49) and (53), the density becomes

[TABLE]

Appendix C Further Calculation of polarization function

C.1 $\mathrm{Im}\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})$

Eq. (21) in the main body of the paper can be further written as

[TABLE]

It can be found that

[TABLE]

Therefore, we only need to calculate $I_{1}$ , $I_{3}$ , and $I_{7}$ , which are given by

[TABLE]

where

[TABLE]

Suppose that

[TABLE]

After tedious evaluation, we get

[TABLE]

where

[TABLE]

Here, $\varphi_{1}^{a}$ and $\varphi_{2}^{b}$ are the two angles satisfying $F_{1}(\varphi_{1,2}^{a})=0$ in the range $(0,2\pi)$ . We then let

[TABLE]

We similarly obtain

[TABLE]

where $\varphi_{1}^{b}$ and $\varphi_{2}^{b}$ satisfy $F_{2}(\varphi_{1,2}^{b})=0$ in the range $(0,2\pi)$ . After substituting Eq. (83) into Eqs. (77) and (79), and also substituting Eq. (87) into Eq. (78), and then carrying out the integration over $\varphi$ , we can get the simplified expressions of $I_{1}$ , $I_{2}$ , and $I_{3}$ . Making use of the relations Eqs. (74-(76), we eventually have

[TABLE]

where

[TABLE]

C.2 $\mathrm{Re}\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})$

After further calculations, we write Eq. (22) of the main body of the paper in the form

[TABLE]

There exists a number of identities:

[TABLE]

It is thus only necessary to calculate $J_{1}$ and $J_{3}$ , given by

[TABLE]

where

[TABLE]

It is convenient to define $Z=e^{i\varphi}$ , which implies that

[TABLE]

Now $M_{1}$ can be expressed as

[TABLE]

where

[TABLE]

Using the residue theorem, we obtain

[TABLE]

Substituting Eq. (106) into Eqs. (100) and (101), it is straightforward to get the expressions of $J_{1}$ and $J_{3}$ . With help of (98) and (99), we finally find that

[TABLE]

In the derivation, a constant term that is independent of $\Omega$ , $\mathbf{q}$ , $\mu$ , and $T$ has been dropped.

Appendix D Debye Screening

At $\Omega=0$ and $|\mathbf{q}|=0$ , we have

[TABLE]

Adopting the transformations shown in Eqs. (49) and (53) gives rise to

[TABLE]

Bibliography69

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1(1) O. Vafek and A. Vishwanath, Annu. Rev. Condens. Matter Phys. 5 , 83 (2014).
2(2) T. O. Wehling, A. M. Black-Schaffer, and A. V. Balatsky, Adv. Phys. 63 , 1 (2014).
3(3) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83 , 205101 (2011).
4(4) A. A. Burkov, Nat. Mater. 15 , 1145 (2016).
5(5) B. Yan and C. Felser, Annu. Rev. Condens. Matter Phys. 8 , 337 (2017).
6(6) M. Z. Hasan, S.-Y. Xu, I. Belopolski, and S.-M. Huang, Annu. Rev. Condens. Matter Phys. 8 , 289 (2017).
7(7) A. A. Burkov, ar Xiv:1704.06660 v 1.
8(8) N. P. Armitage, E. J. Mele, and A. Vishwanath, ar Xiv:1705.01111 v 1.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Taxonomy

Polarization, plasmon, and Debye screening in doped 3D ani-Weyl semimetal

Abstract

I Introduction

II Model and Feynman rules

III General expression of polarization function

IV Plasmon mode

V Debye screening

VI Summary

ACKNOWLEDGEMENTS

Appendix A ReΠret(Ω,q)\mathrm{Re}\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})ReΠret(Ω,q) in the limit max⁡(vq⊥,Aqz2)≪Ω≪μ\max(vq_{\bot},Aq_{z}^{2})\ll\Omega\ll\mumax(vq⊥​,Aqz2​)≪Ω≪μ

A.1 ReΠ++ret(Ω,q)\mathrm{Re}\Pi_{++}^{\mathrm{ret}}(\Omega,\mathbf{q})ReΠ++ret​(Ω,q)

A.2 ReΠ+−ret(Ω,q)\mathrm{Re}\Pi_{+-}^{\mathrm{ret}}(\Omega,\mathbf{q})ReΠ+−ret​(Ω,q)

A.3 ReΠ−+ret(Ω,q)\mathrm{Re}\Pi_{-+}^{\mathrm{ret}}(\Omega,\mathbf{q})ReΠ−+ret​(Ω,q)

Appendix B The density of fermion

Appendix C Further Calculation of polarization function

C.1 ImΠret(Ω,q)\mathrm{Im}\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})ImΠret(Ω,q)

C.2 ReΠret(Ω,q)\mathrm{Re}\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})ReΠret(Ω,q)

Appendix D Debye Screening

Appendix A $\mathrm{Re}\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})$ in the limit $\max(vq_{\bot},Aq_{z}^{2})\ll\Omega\ll\mu$

A.1 $\mathrm{Re}\Pi_{++}^{\mathrm{ret}}(\Omega,\mathbf{q})$

A.2 $\mathrm{Re}\Pi_{+-}^{\mathrm{ret}}(\Omega,\mathbf{q})$

A.3 $\mathrm{Re}\Pi_{-+}^{\mathrm{ret}}(\Omega,\mathbf{q})$

C.1 $\mathrm{Im}\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})$

C.2 $\mathrm{Re}\Pi^{\mathrm{ret}}(\Omega,\mathbf{q})$