Patterns of electromagnetic response in topological semimetals

Srinidhi T. Ramamurthy and Taylor L. Hughes

Phys. Rev. B 92, 085105 – Published 3 August 2015

Abstract

Topological semimetals are gapless states of matter which have robust and unique electromagnetic responses and surface states. In this paper, we consider semimetals which have pointlike Fermi surfaces in various spatial dimensions $D = 1, 2, 3$ which naturally occur in the transition between a weak topological insulator and a trivial insulating phase. These semimetals include those of Dirac and Weyl types. We construct these phases by layering strong topological insulator phases in one dimension lower. This perspective helps us understand their effective response field theory that is generally characterized by a 1-form $b$ which represents a source of Lorentz violation and can be read off from the location of the nodes in momentum space and the helicities/chiralities of the nodes. We derive effective response actions for the two-dimensional (2D) and 3D Dirac semimetals and extensively discuss the response of the Weyl semimetal. We also show how our work can be used to describe semimetals with Fermi surfaces with lower codimension as well as to describe the topological response of 3D topological crystalline insulators.

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Received 17 July 2014
Revised 28 May 2015

DOI:https://doi.org/10.1103/PhysRevB.92.085105

Authors & Affiliations

Srinidhi T. Ramamurthy and Taylor L. Hughes

Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Illinois 61801, USA

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Vol. 92, Iss. 8 — 15 August 2015

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Figure 1
Schematic illustration of the motion of point nodes in the $k_{z} = 0$ plane of a cubic, 3D BZ as a parameter $m$ is adjusted. As $m$ increases, two Weyl nodes with opposite chirality (as represented by the color shading) are created in the 2D subspace (i.e., $k_{z} = 0$ ) of a full 3D BZ. As $m$ increases further, the nodes move throughout the BZ, meet at the boundary, and then finally annihilate to create a gapped phase with a weak topological invariant proportional to the reciprocal lattice vector separation $\vec{G} = 2 \vec{ν}$ of the Weyl nodes before annihilation. The far left BZ represents a trivial insulator, the far right represents a WTI, and the intermediate slices represent the Weyl semimetal phase.
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Figure 2
The current and charge density of the 1D (semi-)metal are plotted vs time for half filling and for nearest-neighbor hopping $α = 1$ . The current has a periodic response, as expected, with a period of 200 time slices for an electric field of strength $E = \frac{h}{e T (200 a)}$ for some time scale $T$ that is long. The charge density is given by $ρ = e \frac{k_{F}}{π} = e / 2 a$ , as expected, and shows no time-dependent behavior.
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Figure 3
(Top) Energy spectrum of the Hamiltonian $H_{1 D v}$ , where each curve represents a different value of $β .$ The solid blue line is $β = 0$ , the magenta dashed line is $β = 0.1$ , and the dash-dotted tan line is $β = 0.25 .$ All curves have $α = 1 .$ (Bottom) This is a magnified region of the top figure slightly below half-filling, which is the regime for our calculation. Exactly at half filling $β$ has no effect, and the stronger $β$ is, the more the Fermi-wave vectors and velocities are modified at a fixed $μ$ .
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Figure 4
The current of $H_{1 D v}$ is plotted vs next-nearest-neighbor hopping strength $β$ near half filling. $κ_{F} = π / 2 a - π / 100 a$ was chosen and the nearest-neighbor hopping $α = 1$ with periodic boundary conditions. The current increases linearly as a function of $β$ , as expected from Eq. (28). Note that we have let $a = ℏ = 1$ .
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Figure 5
Charge density in units of $e / a$ as a function of position for an inhomogeneous system with $N = 1000$ lattice sites, where each segment has $L_{s} = 500$ sites. The chemical potential is $μ = 0$ , and if $ε_{0}$ was tuned to zero the density would be $ρ = e / 2 a .$ For our choice of $ε_{0} = 0.5 t$ we have $b_{1 (ℓ)} = (π / a) 0.46, b_{1 (r)} = (π / a) 0.54 .$ Away from the interfaces the values match the calculation from the effective response action. Near the interfaces there are damped oscillations due to finite-size effects that are not captured by the analytic calculation. Note that the finite-size boundary effects have nothing to do with the spurious “interface” terms in Eq. (51).
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Figure 6
The energy spectrum for the Hamiltonian in Eq. (63) tuned into the 2D Dirac semimetal. The figure shows exact diagonalization of this model in a strip geometry ( $x$ direction with open boundaries and $y$ direction with periodic boundaries) with $\pm k_{y c} = \pm π / 3$ and $b_{y} = π / 3 .$ The flat band of states stretched between the Dirac nodes are edge modes.
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Figure 7
We have plotted the deviation of the charge density from the average for $L_{x} = L_{y} = 120$ at half-filling in a 2D Dirac semimetal with $b_{y} = π / 3$ (i.e., same parameters as in the previous figure). The average background charge per site is $Q_{0} = 120 e .$ We notice peaks at the boundaries of the system due to the charge carried by localized midgap modes. The charge density exponentially decays to the value of $Q_{0} = 120 e$ within a few lattice sites. The total charge at the boundary calculated from summing the boundary charge near the right edge is $Q_{b} = - 19.6 e$ , which matches the expected result, $Q_{b} = P_{1}^{x} L_{y} = - \frac{e}{6 a} 120 a = - 20 e .$ The deviation from $- 20$ is a finite-size effect and the result will converge to the analytic value as the system size increases.
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Figure 8
The boundary charge is plotted vs the mass parameter $m .$ The solid blue curve represents $\frac{e b_{y}}{2 π}$ , where $b_{y}$ is calculated from the solutions to $cos k_{y} = - m / t_{y}$ for a range of $m$ and with $t_{y} = - 1 .$ The open circles are the numerically calculated boundary charge (per layer) for a system with open boundaries in the $x$ direction ( $L_{x} = 120$ ) and periodic boundary conditions in the $y$ direction. They match except near $m = 1$ , where the cusplike analytic result is cut off in the numerics due to finite-size effects.
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Figure 9
The Energy spectrum is shown for the DSM with $b_{0} \neq 0$ and different masses turned on. (a) With $m_{A} \neq 0$ , we see that the edge modes split and do not cross as they move between the Dirac nodes. (b) With $m_{B} \neq 0$ , it looks like the edge mode dispersion of a Chern insulator and they cross at $k = 0$ .
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Figure 10
The bound current $J_{y}$ localized near a single edge vs $b_{0}$ is plotted for the model in Eq. (63) with $b_{y} = \frac{π}{3}, m_{A} = 10^{- 3}, L_{x, y} = 120$ , and periodic boundary conditions in the $y$ direction. The current matches the field theory prediction.
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Figure 11
Plots of (a) $J_{x}$ vs $y$ , which is the current on the nontopological edge, and (b) $J_{y}$ vs $x$ , which is the current on the topological edge. This is for the Dirac semimetal considered in previous figures but with a nonzero $b_{0}$ . For this system $b_{y} = \frac{π}{3}, γ = 0.1, m_{A} = 0.1$ , and $L_{x, y} = 96 .$ There are open boundary conditions in both directions. We note that $j_{y}$ is exponentially localized, whereas $j_{x}$ is less-sharply localized and oscillates as it decays into the bulk. The oscillation wavelength coincides with the wave-vector location of the Dirac nodes in $k$ space. With open boundary conditions, we must be careful to properly fill the edge states by using a nonzero inversion-breaking mass term $m_{A} .$ The currents with $Θ_{y} = \pm 1$ are plotted in black and red. The total current near the boundaries is identical in both cases and thus the magnetization does not depend on how the edge states traverse on the edge BZ. The slight difference between the current profiles in (b) is due to the fact that the wave functions of the occupied edge modes that determine those boundary currents are different in the two cases with $Θ_{y} = \pm 1$ ; however, the total current is the same.
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Figure 12
With a similar setup to the previous figure, we use a density plot for the current vs $x, y$ position for the 2D Dirac semimetal with $b_{y} = \frac{π}{3}, γ = m_{A} = 0.1, L_{x, y} = 96$ , and we have open boundary conditions in both directions. We calculated the current density in the $x$ direction and summed it with the current density in the $y$ direction to produce this pseudocolor plot. We see that the currents are spatially localized at the edges, strongly for the one moving along the edges parallel to the $y$ axis and less strongly and oscillatory for the one moving along the edges parallel to the $x$ direction. The total magnitude of the current in the neighborhood of each edge is the same, and the current circulates around the boundaries of the sample.
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Figure 13
In (a) and (b) we show two arbitrary pairings of the four Dirac nodes with opposite (red and blue) helicity in the 2D square-lattice BZ. In (c) and (d) we show cases when $Θ_{i}$ for $i = x, y$ both take their nontrivial values such that the edge states/Dirac-node pairing pass through the BZ boundary. Below and to the right of each 2D BZ we show the projection onto the respective edge BZs. In cases where there are overlapping edge states and a $Z_{2}$ cancellation we show the resulting modified effective helicities in a second projected edge BZ subfigure. Finally, we show a diagram for each edge-state projection showing the calculated boundary charge resolved vs $k_{edge}$ when a uniform half-filled background charge has been subtracted and in units of $e .$ The black curves show the results after the cancellation of overlapping edge states and the black + red curves show the result if the overlapping edge-state regions all contribute. In (c) and (d) there are cases where there are red curves below and above the axes. These are contributions coming from each overlapping edge state which cancel when added together.
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Figure 14
Rules for helicity modifications and line removal for use with the determination of the boundary charge for Dirac semimetals with four nodes. The signs inside the enclosed circles represent the helicities, and the oriented arrows refer to the nodal pairing in the text.
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Figure 15
We illustrate the dispersion of the edge states of the model $H = sin k_{x} σ^{x} + (m_{B} sin k_{y} - m_{A}) σ^{y} + (1 - m - cos k_{x} - cos k_{y}) σ^{z}$ in the limits $m_{A} = 0, 0.05, 0.10, 0.15, m_{B} = 0.2$ with various dashed lines. The Dirac nodes are located at $\pm \frac{π}{2}$ . The crossing point has shifted to a nonzero $k_{y}$ once we turn on an inversion-breaking mass and moves towards one of the Dirac points as $m_{A}$ is increased further. This leads to a nonzero polarization which is decided by the ratio of $m_{A}$ and $m_{B}$ , while the sign of the polarization is decided by $m_{A}$ .
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Figure 16
The zeroth Landau level of the WSM in a uniform magnetic field is plotted vs $k_{z}$ before (in black) and after (in red) switching on a $γ$ , which gives us $b_{0} = (γ / ℏ) sin 2 π / 3 = 0.17$ . The blue line is shown to indicate $E = 0$ . The model parameters have $b_{z} = 2 π / 3, m = 1 / 2$ , and $L_{x} = L_{y} = L_{z} = 60$ with the magnetic flux per unit cell given by $ϕ = 2 π / 60$ . $b_{0}$ was then switched on to plot the curve in red. We see that the Landau level is simply shifted in momentum space and is akin to turning on an external electric field in the 1D model.
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Figure 17
The zeroth Landau level is plotted vs $k_{z}$ before (in black) and after (in red) switching on a $b_{0}$ using the NNN velocity term. The blue line is shown to indicate $E = 0$ . The model had $b_{z} = 2 π / 3, m = 1 / 2$ , and $L_{x} = L_{y} = L_{z} = 60$ , with $ϕ = 2 π / 60$ . We then switch on a term to change the velocity of the two Weyl nodes with $t_{NNN} = 0.2$ . The shift we expect is then given by $2 t_{NNN} m \approx 0.2$ , as seen in the figure. In effect, near $E = 0$ the zeroth Landau level is shifted.
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Figure 18
The current is plotted vs $b_{0}$ for the two-band model of the WSM. The current is linear and the slopes match almost exactly. This plot is generated for $L_{x} = 30$ and the flux per plaquette is $ϕ = - 2 π / L_{x}$ . We use $L_{y} = 30, L_{z} = 30$ , and $b_{z} = \frac{π}{2}$ to generate this plot.
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Figure 19
The charge density is plotted vs position in the $x$ direction with open boundary conditions. The system is composed of a WSM with $b_{z, L} = π / 5$ for $0 < x < L_{x} / 2$ and $b_{z, R} = π / 3$ for $L_{x} / 2 < x < L_{x}$ . The total number of sites in the $x$ direction was $L_{x} = 30$ with magnetic flux per unit cell in the $x - y$ plane $ϕ = - 2 π / 30$ . Also, $L_{z} = 30$ and $L_{y} = 30$ . The bulk charge density is given by $N_{x} = - L_{z} L_{y} b_{z} B_{z} / 4 π^{2} = - 3, - 5$ , as predicted by the action.
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Figure 20
Setup to generate an EM response in a 3D Dirac semimetal. To get a nonzero response, there must be two adiabatic parameters $θ$ and $ϕ .$ The parameter $θ$ represents an interpolation between a 3D Dirac semimetal with, for example, $b_{z} \neq 0$ to a trivial insulator with $b_{z} = 0 .$ The parameter $ϕ$ represents a magnetization domain wall on the $x z$ surface plane. There will be a branch of low-energy fermion modes trapped on the domain wall which can bind charge or can carry current if $b_{0} \neq 0$ .
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Figure 21
The localized charge on a magnetic domain wall on the surface of a 3D DSM resolved vs $k_{z}$ , i.e., the direction in which the Dirac nodes are separated in momentum space. We note that there is a half charge bound at the domain wall only for each state satisfying $| k_{z} | < {cos}^{- 1} m$ . In the plot, we have used $m = 0.5$ , which means $b_{z} = \frac{π}{3}$ .
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Figure 22
The total current localized at the magnetic domain wall is plotted vs $b_{0}$ for the 3D DSM. The expected value of the total current localized on the domain wall is $\frac{e b_{0}}{2 π}$ . The system size is a cube of $L = 30$ lattice sites in every direction with $b_{z} = \frac{π}{2}$ . We used open boundary conditions in both the $x, y$ directions and periodic boundary conditions in the $z$ direction. The red dots are the theoretical result and the black line is the numerical result. The deviation arises due to the importance of lattice effects at larger values of $b_{0}$ .
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