Sound of Fermi arcs

TL;DR

This paper investigates the collective charge excitations of Fermi arcs in Weyl semimetals, revealing gapless, linearly dispersing plasmon modes influenced by surface state penetration and anisotropy, with implications for their electron liquid behavior.

Contribution

It introduces a theoretical analysis of Fermi arc plasmons in Weyl semimetals, highlighting their unique gapless and linear dispersion characteristics due to surface state penetration and anisotropic effects.

Findings

Fermi arc plasmons are gapless and linearly dispersing.

Surface state penetration influences plasmon gaplessness.

Fermi arc electron liquid exhibits mixed 1D and 2D characteristics.

Abstract

Using Green's function of semi-infinite Weyl semimetals we find that the collective charge excitations of Fermi arcs in undoped Weyl semi-metals are linearly dispersing gapless plasmon modes. The gaplessness comes from proper consideration of the deep penetration of surface states at the end of Fermi arcs into the interior of Weyl semimetal. The linear dispersion -- rather than square root dispersion of {\it pure 2D} electron systems with extended Fermi surface -- arises from the strong anisotropy introduced by the Fermi arc itself, due to which the continuum of surface particle-hole excitations in this system will have strong resemblence to the one-dimensional electron systems. This places the Fermi arc electron liquid in between the 1D and 2D electron liquids. Contamination with the particle-hole excitations of the bulk gives rise to the damping of the Fermi arc sounds.

Figures

Equations

$${\epsilon }_{z,z^{\prime }}(q,\omega )=1-V_{z,z^{\prime }}(q){\Pi }_{z,z^{\prime }}(q,\omega ),$$

$${\Pi }_{z,z^{\prime }}(\vec{q},\omega )=\sum _{\vec{k},\nu }{G}_{z,z^{\prime }}(\vec{k},\omega )G_{z^{\prime },z}(\vec{k}+\vec{q},\omega +\nu ).$$

$$\int \epsilon (\mathbf{r},{\mathbf{r}}^{\prime },\omega ){\varphi }_n({\mathbf{r}}^{\prime },\omega )d{\mathbf{r}}^{\prime }={\epsilon }_n(\omega ){\varphi }_n(\mathbf{r},\omega ),$$

$${\Pi }_{z,z^{\prime }}^{\uparrow \uparrow (\downarrow \downarrow )}=$$

$$\displaystyle+\cosh(q_{x}\zeta/2)\big{[}(4b^{2}\zeta^{2}+4+4b\zeta\mp iq_{x}q_{y}\zeta^{2})\sinh\zeta b-(4b\zeta+4b^{2}\zeta^{2})\cosh b\zeta\big{]}\bigg{\}},$$

$${\Pi }_{z,z^{\prime }}^{\uparrow \downarrow (\downarrow \uparrow )}=$$

$$\displaystyle+\cosh(q_{x}\zeta/2)\big{[}-(\zeta+b\zeta^{2})\sinh\zeta b+b\zeta^{2}\cosh\zeta b\big{]}(q_{y}\mp 2iq_{x})\bigg{\}},$$

$${\Pi }^{\alpha \beta }(q,\phi ,\omega )\to \alpha \beta {\Pi }^{\bar{\alpha }\bar{\beta }}(q,-\phi ,-\omega )$$

$$\epsilon =\left(\begin{array}{cc}1-V_0{\Pi }^{\uparrow \uparrow }& -V_0{\Pi }^{\uparrow \downarrow }\\ -V_0{\Pi }^{\downarrow \uparrow }& 1-V_0{\Pi }^{\downarrow \downarrow }\end{array}\right)$$

$$\hslash \omega =\hslash q{v}_F\sin \phi (1\pm {\gamma }_1)+{\Delta }_1$$

$$\int \epsilon (\mathbf{r},{\mathbf{r}}^{\prime },\omega ){\varphi }_n({\mathbf{r}}^{\prime },\omega )d{\mathbf{r}}^{\prime }={\epsilon }_n(\omega ){\varphi }_n(\mathbf{r},\omega ).$$

Full text

I Introduction

The excitation spectrum of electronic systems in atomic length scales is composed of individual electrons, while beyond a certain length scale, their collective character becomes manifest Phillips (2012); Pines (1997); Bohm and Pines (1953). Among the salient collective excitations in electron liquids are plasmon which are longitudinal collective density oscillations Pines (1997). Based on general hydrodynamic arguments, plasmons of three dimensional (3D) systems are always gapped and given by $\omega_{\rm{p,3D}}^{2}\sim ne^{2}/\epsilon_{0}m$ Kittel et al. (1976); Phillips (2012). In two-dimensions, both quantum mechanical Giuliani and Vignale (2005) and hydrodynamic arguments Fetter (1973) establish that the long wave length limit of the plasmon dispersion must be $\propto\sqrt{q}$. This holds in both Dirac materials Hwang and Sarma (2009); Wunsch et al. (2006) and 2D materials with quadratic band dispersion Giuliani and Vignale (2005). Plasmons in a two-dimensional electron gas were first observed for a sheet of electrons on liquid helium Grimes and Adams (1976a, b) and then in inversion layers Allen et al. (1977), as well as in graphene Bostwick et al. (2007).

On the other hand, in one-dimension (1D) the fixed point of interacting gapless systems is the Luttinger liquid Voit (1995), rather than a Fermi liquid Baym and Pethick (2004). The low-energy sector of the spectrum of interacting fermion systems in 1D is exhausted by collective excitations only. In 1D, the collective charge density excitations appear in the form of linearly dispersing sound waves Sénéchal (2004); Gogolin et al. (2004); Rao and Sen (2001). The intuitive picture for collective charge and spin excitations in 1D is based on Fig. 1-a. At low energies and low momenta, the particle-hole (PH) continuum (PHC) is composed of a narrow band of free PH excitations all moving at the same group velocity, such that any amount of Coulomb forces binds them into coherent collective modes Giamarchi (2004). This is in contrast to the PHC of higher dimensional electron gases with extended Fermi surface in Fig. 1-b Giuliani and Vignale (2005) where due to rotational degree of freedom of the wave vector ${\boldsymbol{q}}$ there is no coherence in the group velocity of the particle-hole pairs. Therefore the electron liquids in dimensions higher than one, do not sustain plasmonic sound waves, except for a genuine non-equilibrium situation recently proposed for the bulk of 3D Weyl semimetals Song and Dai (2019). In this work we will show that the electron gas formed by Fermi arc states on the surface of Weyl semimetals (WSMs) supports a linearly dispersing gapless plasmon mode.

In general, restricting the material in half space gives rise to surface plasmons which were employed by Ritchie to explain the energy loss of fast electrons passing through thin films Ritchie and Marusak (1966). Since the penetration depth of electromagnetic waves in metals is rather short, the plasmons of metal surfaces are major players in technical applications of the collective oscillations Maier and Atwater (2005); Moore et al. (1965); Ozbay (2006); Atwater (2007); Heber (2009).

In recent year, Weyl semimetals have been added to the list of conducting materials Armitage et al. (2018); Rao (2016); Yan and Felser (2017) which are additionally endowed with topological indices protecting their gapless nature. The bulk electronic states in these systems is composed of linearly dispersing bands that touch each other at nodal points Lu et al. (2015); Huang et al. (2015). WSMs are characterized by very peculiar surface states known as Fermi arc Xu et al. (2016) which are localized on the surface if the Fermi wave vector is in the middle of the arc, but penetrate deeply into the bulk as the (Fermi) wave vector approaches the two ends of the Fermi arc Faraei et al. (2018).

Bulk plasmons and associated surface plasmon resonance (resulting from the projection of dielectric function on the surface) has been studied in WSMs Lv and Zhang (2013); Andolina et al. (2018). In these systems the gap in the 3D plasmons is given by $\sim e^{2}n^{2/3}v_{F}$. Assumption of specular reflection of incident electrons is equivalent to projection of the bulk dielectric function to the $z=0$ boundary surface. Using this approach, Lošić obtained gapped surface plasmon resonances for doped Weyl semi-metals Lošić (2018). Employing a simple dispersion of Fermi arcs, and neglecting the associated matrix elements, Lošić finds a plasmon mode dispersing as $\sqrt{q_{y}}=\sqrt{q\sin\theta}$ ($y$ direction is perpendicular to the Fermi arc). Apart from the angular dependence, the $\sqrt{q}$ behavior is expected for a pure 2D system with extended Fermi surface. Andolina and coworkers have studied the same problem Andolina et al. (2018). If only the contribution of arc states is taken into account, the plasmons for the Fermi arc states in their work becomes gapped.

In this paper we use our previously developed Green’s function for WMSs Faraei et al. (2018) which allows us to isolate the effects solely arising from the Fermi arcs states. Note that our Green’s function approach does not assume any effective Hamiltonian to describe the Fermi arc states. Fermi arc states in our approach are faithfully produced from appropriate boundary conditions Faraei et al. (2018). Then we find that the deep penetration of end-of-Fermi-arc states into the bulk, combined with the peculiar form of PHC – which bears certain resemblance to the 1D case as in Fig. 1-c – will compromise to produce a gapless branch of linearly dispersing Fermi arc plasmons. Let us briefly elaborate on these points: (i) As can be seen in Fig. 2, the penetration length of Fermi arc states into the interior of the WSM increases more and more by approaching the two ends of Fermi arcs. The method employed in Ref Andolina et al. (2018) uses a projection of the polarization function onto the surface, and avoids dealing with a determinantal equation in real-space coordinates $z,z^{\prime}$. Evading the determinant misses the long tail of Fermi arc states. We will show how systematic treatment of the penetration effects gives rise to gapless plasmon mode by progressively increasing the size of the determinant involved. (ii) The Fermi arc per se, creates enormous anisotropy in such a way that the phase space for the scattering of electrons will resemble those of one-dimensional systems as depicted in Fig. 1-c. This is in sharp contrast to the systems with extended closed Fermi surface where a rotational degree of freedom of the wave vector ${\boldsymbol{q}}$ of low-energy scattering processes gives rise to a PHC of type (b) rather than type (c) in Fig. 1. In this way the resulting plasmon mode will become a (linearly dispersing) sound wave. Note that such a Fermi arc sound wave is entirely different from the chiral zero sound (CZS) considered in Ref. Song and Dai (2019) in the following respects: (i) the most important difference is that to generate the CZS one needs to generate a non-equilibrium population around the right and left Weyl nodes. (ii) to realize the CZS the inter-valley relaxation rate must be much smaller than intra-valley time scales. (iii) realization of chiral zero sound requires a background B field to generate right/left movers, i.e. it is built on the chiral anomaly.

II Model and formalism

We consider a semi-infinite Weyl semimetal in $z<0$ part of the space bounded by $z=0$ surface as illustrated in Fig. 2. In our previous works Faraei et al. (2018); Faraei and Jafari (2019) we have used the Green’s function approach to formulate the Fermi arcs. We do not assume any particular Hamiltonian for the Fermi arc states. We rather obtain the Fermi arcs from appropriate boundary conditions Faraei et al. (2018). In this way, we ensure that all the necessary matrix elements, as well as the tail of Fermi arc states in the bulk (see Fig. 2) are properly taken into account. The deep penetration of the states near the ends of Fermi arc into the interior of WSM is one of the essential differences of the electron gas formed by the Fermi arc with respect to the standard 2D electron gases. Missing this effect will give rise to $\sqrt{q}$ behavior Lošić (2018) of the normal 2D electron gas systems.

When the translational symmetry along the $z$ axis is broken by presence of a surface at $z=0$, the dielectric function will become a matrix in spatial $z,z^{\prime}$ indices. The ”dielectric matrix” in the random phase approximation is given by,

[TABLE]

where $V_{z,z^{\prime}}(\vec{q})=2\pi e^{2}e^{-q|z-z^{\prime}|}/q$ is the Coulomb matrix element with $\vec{q}=(q_{x},q_{y})$ denoting the wave vector in the plane. Here $z$ and $z^{\prime}$ are the distance of the two electrons from the boundary. When they are at the same plane, $z=z^{\prime}$, this reduces to the Coulomb matrix element of the 2D systems Katsnelson (2012). $\Pi_{z,z^{\prime}}(\vec{q},\omega)$ is the polarizability tensor which can be written in terms of the Green’s function as,

[TABLE]

From this point we do not perform any further approximations such as projection of the polarization function into the surface Lošić (2018) or approximating the determinants such as ${\det}\leavevmode\nobreak\ \varepsilon_{z,z^{\prime}}=0$ by related trace operations Andolina et al. (2018). Instead, following Andersen and coworkers Andersen et al. (2012) the plasmon resonance in a generic situation with broken translation invariance are obtained numerically. For this purpose, one needs to numerically solve the eigenvalue equation,

[TABLE]

and then to find zero eigenvalues $\varepsilon_{n}(\omega)=0$. By scanning range of $\omega$ values, one can find the energy scales $\omega_{p}$ at which the eigenvalues of the dielectric matrix vanish, $\varepsilon_{n}(\omega_{p})=0$. This will correspond to plasmon resonances. Various branches are labeled by $n$. We are interested in the lowest energy branch. Furthermore at a resonance the loss function $\imaginary(\varepsilon_{n}^{-1})$ will develop a peak which corresponds to the characteristic energy losses suffered by fast charged particles traversing the material Winther (2015). To take the long penetration depth of Fermi arc states into account, one needs to discretize the portion of space near the surface as depicted in Fig. 2. The separation between the Weyl nodes is $2|\vec{b}|$, where $b$ can be used as a unit of momentum, which will also set the unit of length as $\hbar/b$.

To calculate the $\varepsilon({\boldsymbol{r}},{\boldsymbol{r}}^{\prime},\omega)$, one basically needs the polarization function $\Pi({\boldsymbol{r}},{\boldsymbol{r}}^{\prime},\omega)$ as they are related by Eq. (8). In the geometry of Fig. 2 by translational symmetry in $(x,y)$ plane, the polarization will reduce to $\Pi(z,z^{\prime},{\boldsymbol{q}},\omega)$ where ${\boldsymbol{q}}=(q_{x},q_{y})$ is the wave vector corresponding to the surface. To calculate the polarization we can use the Green’s functions obtained for the Weyl Hamiltonian by applying a proper boundary condition Faraei et al. (2018). As we are interested in bare Fermi arc plasmons, we use the particular surface Green’s functions that are naturally separated from the bulk contribution. Without loss the generality, we rotate the coordinate system, such that the Fermi arc is oriented on the $k_{x}$ axis. In this situation the Green’s function will become much easier to work with Faraei and Jafari (2019). Convolution of the Green’s functions (which are $2\times 2$ matrices in their spin indices) as in Eq.(2) gives,

[TABLE]

where $2\vec{b}=(2b,0,0)$ is the vector that connects the Weyl nods in the bulk of the WSM and $\zeta=2(z+z^{\prime})$ with $z$ and $z^{\prime}$ measured from the surface. The dispersion of Fermi arc states involves single-particles moving along positive $y$ direction on the top surface. Therefore PH excitations with positive energy are restricted to $q_{y}>0$ on the top surface. The $q_{y}<0$ PH excitations correspondingly take place in the bottom surface. A nice property of the above polarization matrix elements is that under mirror reflection with respect to Fermi arc, $\varphi\to-\varphi$ or $q_{y}\to-q_{y}$ (which is equivalent to going from top surface to bottom surface) we have

[TABLE]

where $\alpha,\beta$ can take $\uparrow\equiv+1$ or $\downarrow\equiv-1$ values.

To solve for $\det\varepsilon=0$, we need to discretize the $z$ direction in Eq. (4) and form the $\Pi^{\alpha,\beta}_{z,z^{\prime}}$ matrix. Discretization by a mesh of size $N$, we will be dealing with $2N\times 2N$ matrices, where the extra $2$ comes from the above matrix structure in the spin space. The numerical details are given in the supplement. To get a feeling about the Fermi arc plasmon dispersion, let us start with the $N=1$ approximation which can be analytically handled. In this case, we just need to consider one layer at $z=0$. Eq (8) for one layer gives,

[TABLE]

where $V_{0}=V_{z=0,z^{\prime}=0}(q)=2\pi e^{2}/q$ is pure 2D Coulomb interaction matrix element. In ${\boldsymbol{q}}\to 0$ limit, the determinant of the above dielectric matrix has the following solution which after restoring various constants gives the following $q$-linear plasmon dispersion,

[TABLE]

where $\gamma_{1}=\alpha_{\rm WSM}\sqrt{\sin^{2}\varphi+4\cos^{2}\varphi}/(2\pi^{2})$ is the relative difference in the slope of two modes and is determined by the ”fine structure” constant $\alpha_{\rm WSM}=e^{2}/(\hbar v_{F})$ which is of the order of unity for Fermi velocities that are hundreds of times less than the speed of light. The gap for $N=1$ approximation is given by $\Delta_{1}=\frac{2e^{2}b}{\pi^{3}}\sin\varphi$. But the two-dimensional electron systems are not expected to have gapped plasmons Giuliani and Vignale (2005); Katsnelson (2012). In fact anticipating that $\gamma,\Delta$ will depend on the mesh size $N$, we have introduced subscript $"1"$ in $\gamma_{1}$ and $\Delta_{1}$. To investigate this systematically, in Fig. 3 we study the effect of taking progressively deeper layers around the surface into account. As can be seen, by increasing $N$ both $\gamma_{N}$ (left panel) and $\Delta_{N}$ (right panel) approach zero.

This figure clearly shows that the gap $\Delta_{N=1}$ and the slope difference $\gamma_{N=1}$ are artifacts of considering a mesh of size $N=1$ and quickly vanish by taking progressively larger mesh sizes into account. As pointed out, this form of finite size effect in surface states arises because the Fermi arc states at the two ends of the Fermi arc deeply penetrate into the bulk as in Fig. 2. The conclusion is that the numerical evaluation of the determinant of $\varepsilon$ for reasonably large $N$ can not be avoided. Evading this determinant by projecting it to the surface – which might be valid for typical non-Weyl electronic systems – introduces sever errors in the dispersion of Fermi arc plasmons Lošić (2018); Andolina et al. (2018).

Although the $N=1$ approximation to determinant of $\varepsilon$ does not give correct gap and slope, but the linear nature of the plasmon dispersion and its angular dependence survive the larger $N$ limit. To establish the linear $q$ dependence and $\sin\varphi$ angular dependence, in Fig. 4 we have plotted the dispersion of the (doubly degenerate) lowest plasmon mode for large enough $N=50$ to ensure that $\gamma_{N}$ and $\Delta_{N}$ are already zero. The direction of the wave vector ${\boldsymbol{q}}$ is fixed by $\varphi=\pi/6$ where \varphi=\arctan\big(q_{y}/q_{x}\big{missing}) is the polar angle of ${\boldsymbol{q}}$. Note that in this figure both $\omega$ and ${\boldsymbol{q}}$ are plotted in their natural units. As can be seen the linear dispersion of the $N=1$ approximation, surprisingly robust at much larger $N$. Extrapolating the dispersion to $q\to 0$ limit clearly confirms that for this value of mesh size, the plasmon mode is gapless. Also the fact that in Fig. 3 the relative slope difference $\gamma_{N}$ is already zero for $N=50$ indicates that this lowest mode is doubly degenerate. The double degeneracy corresponds to $\uparrow$ and $\downarrow$ spin directions.

Let us now investigate the angular dependence of the linear plasmon mode of the Fermi arcs. The slope of the lowest Fermi arc plasmon mode in Fig. 4 within the machine precision is given by $\sin(\pi/6)=0.5$. To explore this further, in Fig. 5 we have plotted the lowest mode plasmon frequency as a function of the polar angle $\varphi$ for a fixed $|{\boldsymbol{q}}|=0.01b$. The magnitude of wave vector has been chosen small enough to ensure that it already lies in the linear dispersion regime. The dashed line denotes the numerical data, while the solid line represents the $\sin\varphi$ profile. It is surprising to see that the $\sin\varphi$ angular dependence also robust even in the limit of larger mesh sizes. Therefore properly handling the penetration of Fermi arc states into the interior of WSM gives rise to a branch of linear plasmon mode which is doubly (spin) degenerate and disperses as $\omega=v_{F}q\sin\varphi=v_{F}q_{y}$.

Finally in Fig. 6 we have plotted the eigen-functions of Eq. (3) that correspond to zeros of the dielectric function $\varepsilon_{n}(\omega)$. The eigen-function corresponds to lowest mode evaluated for parameters $N=150$ and $db=75$ and $q=0.1b$. For such a large value of $N$ the slope difference $\gamma_{N}$ is already zero, and this electrostatic profile is doubly degenerate corresponding to $\uparrow$ and $\downarrow$ spin densities. The actual density of electrons according to Poisson equation will be the second derivative of this curve. The mode is clearly bound to the surface of the WSM.

III Fermi arc as a unique electron gas

Typically for an extended Fermi surface in 2D, one expects a $\sqrt{q}$ plasmon dispersion. But the linear plasmon $q\sin\varphi=q_{y}$ arising from the Fermi arc states (of undoped WSMs) is in sharp contrast to the usual 2D plasmons. Even more surprising is that this behavior does not receive any correction by numerically computing the plasmons for larger $N$. So there must be a reason for this behavior. To elaborate on this, let us start by noting that apart from diploar $\sin\varphi$ angular dependence (which is a 2D feature), the linear sound mode is reminiscent of the 1D bosonic modes. Indeed due to strong anisotropy of the Fermi arc, the PHC of Fermi arc states in Fig. 1-(c) bears more resemblance to the PHC of 1D systems in panel (a), rather than the PHC of 2D systems in panel (b).

If the dispersion $\varepsilon(k_{y})=k_{y}$ of the Fermi arc was a 1D dispersion, then a straightforward bosonization arguments Gogolin et al. (2004) would give the charge bosonic collective mode with $\omega(q_{y})\propto q_{y}$ dispersion. The dispersion of Fermi arc states is quite similar, $\varepsilon^{\rm arc}({\boldsymbol{k}})=k_{y}$, albeit with the importance difference that $k_{y}$ is now a component of a 2D vector ${\boldsymbol{k}}=(k_{x},k_{y})$. The restricted phase space for the bosonic mode formation in Fig. 1-(c) within the random phase approximation gives the bosonic mode $\omega({\boldsymbol{q}})\sim q_{y}=|{\boldsymbol{q}}|\sin\varphi$. The surprising fit of the numerical data obtained from RPA dielectric matrix suggest that the anisotropy of the Fermi arc states is likely to admit interesting forms of bosonization which may allow one to go far beyond the above simple RPA treatment. From this point of view, the electron gas formed by the Fermi arc states is actually something between 1D and 2D.

The peculiar 1D-like continuum, the Fermi arc plasmons will protect them from Landau damping into the surface PH excitations. However, contamination with the bulk PHC of excitations which is expected to be stronger for those Fermi arc states closer to the ends of Fermi arc can lead to the damping of Fermi arc plasmon sounds.

IV Summary and discussion

Within the random phase approximation, we studied the collective charge dynamics of the Fermi arc states of undoped Weyl semimetals. Fermi arc states near the projection of Weyl nodes have infinite localization lengths. As such, their tail extends well inside the bulk (see Fig. 2). The proper treatment of these tails by considering large enough grids makes the collective excitations gapless and the $\uparrow$ and $\downarrow$ modes degenerate. Failing to account for this long tail, will generate gapped plasmons Lv and Zhang (2013); Andolina et al. (2018) from pure Fermi arc states. The peculiar dispersion of Fermi arc states places them somewhere in between the 1D and 2D electron liquids (Fig. 1). As such, unlike the typical 2D electron liquids possessing extended Fermi surface (and hence a finite matter density) for which the plasmon disperses as $\sqrt{q}$, for Fermi arcs we find a plasmon branch dispersing like $q\sin\varphi$ where $\varphi$ is the polar angle of the wave-vector ${\boldsymbol{q}}$. The linear dispersion is more like a 1D feature which arises from the limited phases space for particle-hole excitations in Fig. 1 (c). This peculiar PH continuum endows the Fermi arc states with a sound branch arising from charge oscillations of the electrons. The p-wave profile (i.e. $\sin\varphi$ dependence) is the essential difference of the present sound from similar sounds (bosons) in 1D electronic systems.

Unlike the recent proposal for the chiral zero sound of bulk electronic degrees of freedom in Weyl semimetals Song and Dai (2019), the present sound is due to Fermi arcs states and moreover, does not require non-equilibrium population of the two Weyl nodes. Mixing of the Fermi arc states with bulk states gives rise to contamination of the bulk PHC that opens up a damping channel for the Fermi arc sounds Andolina et al. (2018).

V acknowledgments

T.F. appreciates the financial support from Iran National Science Foundation (INSF) under post doctoral project no. 96015597. S. A. J. appreciates research deputy of Sharif University of Technology, grant no. G960214. Z.F. was supported by a post doctoral fellowship from the Iran Science Elites Federation (ISEF).