Chiral topological insulating phases from three-dimensional nodal loop semimetals

TL;DR

This paper introduces a topological Z index for 3D chiral insulators with P*T symmetry, linking nodal loops to topological invariants and edge states, verified through a Bi2Te3 family material.

Contribution

It defines a new topological invariant for 3D nodal loop semimetals with chiral symmetry and P*T symmetry, connecting geometric properties to topological phases.

Findings

Defined a Z invariant as a winding number for nodal loops

Established correspondence between winding numbers and Dirac cone edge states

Validated the approach with a Bi2Te3 family topological insulator model

Abstract

We identify a topological Z index for three dimensional chiral insulators with P*T symmetry where two Hamiltonian terms define a nodal loop. Such systems may belong in the AIII or DIII symmetry class. The Z invariant is a winding number assigned to the nodal loop and has a correspondence to the geometric relation between the nodal loop and the zeroes of the gap terms. Dirac cone edge states under open boundary conditions are in correspondence with the winding numbers assigned to the nodal loops. We verify our method with the low-energy effective Hamiltonian of a three-dimensional material of topological insulators in the Bi$_2$Te$_3$ family.

Equations

$$H_0=(m-k_x^2-k_y^2){\sigma }_x+k_z{\sigma }_z,$$

$$H=H_0+h_{yx}{\sigma }_y{s}_x+h_{yy}{\sigma }_y{s}_y+h_{yz}{\sigma }_y{s}_z,$$

$$E_{\pm }=\pm \sqrt{h_{x0}^2+h_{z0}^2+h_{yx}^2+h_{yy}^2+h_{yz}^2}.$$

$${\nu }_{NL}={\oint }_{NL}\frac{h_{2}d{h}_1-h_{1}d{h}_2}{|{\mathbf{h}}_y{|}^2},$$

$$H$$

$$\mathbf{h}(k)$$

$$h_{x0}$$

$$h_{z0}$$

$${\nu }_{sum}={\nu }_{NL}^{+}+{\nu }_{NL}^{-}.$$

$$h_{yy}$$

$$h_{yz}$$

$$\mathbf{h}(k_x)=(-\cos k_x,0,0,0,-\sin k_x).$$

$${\mathbf{h}}^0=(\mu -\cos k_y,-\cos k_z,0,-\sin k_y,{\mu }_s-t_z\sin k_z),$$

$${\mathbf{h}}_{\parallel }^0$$

$${\mathbf{h}}_{\perp }^0$$

$$|{\mathbf{h}}_{\parallel }^0|=\sqrt{(\mu -\cos k_y{)}^2+({\mu }_s-t_z\sin k_z{)}^2}<1.$$

$$E_{\pm }=|{\mathbf{h}}_{\perp }^0|=\pm \sqrt{{\cos }^2{k}_z+{\sin }^2{k}_y}.$$

$$(\mu -1{)}^2<1-{\mu }_s^2,$$

$$({\mu }_s\mp t_z{)}^2<2\mu -{\mu }^2,$$

$$H$$

$$U_d$$

$$U_{od}$$

$$\displaystyle H(k)=\epsilon_{0}(k)I_{4\times 4}+\left(\begin{array}[]{cccc}M(k)&A_{1}k_{z}&0&A_{2}k_{-}\\ A_{1}k_{z}&-M(k)&A_{2}k_{-}&0\\ 0&A_{2}k_{+}&M(k)&-A_{1}k_{z}\\ A_{2}k_{+}&0&-A_{1}k_{z}&-M(k)\end{array}\right),$$

$$H(k)=M(k){\sigma }_z{s}_0+A_1{k}_z{\sigma }_x{s}_z+A_2{k}_x{\sigma }_x{s}_x+A_2{k}_y{\sigma }_x{s}_y,$$

$$H_1(k)$$

$$H_2(k)$$

Full text

Introduction.- Topological insulators (TIs) in three dimensions (3D) having time reversal symmetry can be characterized by $\mathbb{Z}_{2}$ numbers defined on some discrete momentaTopological1 ; Topological2 ; classes ; 3DTI_1 ; 3DTI_2 , which is equivalent to a quantized invariant expressed as an integral over the entire Brillouin Zone (BZ)3DTI_3 . There are then two types of TI’s, strong and weak, according to whether there is an odd or even number of Dirac cone surface states, respectively3DTI_2 . Besides the time reversal TIs, there is a class of chiral TIs which are described by a $\mathbb{Z}$-type topological invariantcTI . The geometrical representation of a topological invariant in some vector spaces provides an intuitive way to analyze the topological nature of many systemsGeometry1 ; Geometry2 ; Geometry3 ; Geometry4 ; Geometry5 .

On the other hand, a transition point between topologically different insulating phases can be viewed as a semimetal phase with nontrivial topology in its gap closing pointstransition1 ; transition2 . Topological semimetals (TSMs)TSM1 ; TSM2 ; TSM3 ; TSM4 ; TSM5 ; TSM6 ; TSM7 have a Fermi surface (FS) with reduced dimension. While a 3D normal metal has a two-dimensional (2D) FS, a TSM has a one-dimensional (1D) or zero-dimensional (0D) FS at half-filling. 3D systems with 0D FS are known as the WeylTSM1 or DiracTSM2 semimetals. In these systems, the two bands touch linearly at discrete gap closing points in the BZ, and hold topologically protected edge states under open boundary conditions (OPC), such as for instance, the Fermi arcs. More recently, 3D nodal line semimetalsTNL0 ; TNL1 ; TNL2 ; TNL3 ; TNL_Fang1 ; TNL_Fang2 have attracted growing attention. In such systems, the linear band touching points form one or several 1D lines in the BZ.

One of the most interesting cases is when the nodal lines form closed, nodal loops (NLs). A NL can be classified in either of two classes, according to whether it carries a $\mathbb{Z}_{2}$ monopole charge or not. The one without a monopole charge can shrink into a point and disappear, and is topologically trivial in this sense. NLs are protected by the combination of inversion and time reversal symmetries, $P*T$, for spinless systems, while additional symmetries are required to protect NLs in 3D systems with spin-orbital couplingTNL_Fang1 ; TNL_Fang2 . On the other hand, NL semimetals have also been studied in 2DTNL_2D ; TNL_Linhu . In this case, although the NL itself does not carry topological charge, the addition of some chiral gap terms can make the system become topological and insulating, where the topological invariant is given by a winding number defined along the NLTNL_Linhu . An interesting question to address is, what effect can gap terms have on a 3D NL semimetal?

In this paper, we study a spin-1/2 system with $P*T$ symmetry, and show that anticommuting mass gap terms can drive a 3D NL semimetal into a chiral TI, which can be characterized by a integer winding number defined along each NL. This winding number is determined by the geometric relation between the NL and the zeroes of the gap terms. Although the gap terms may be initially considered small, our results only depend on their zeroes, so that it holds valid for any finite terms that gap out the NLs. The system’s surface states may hold an odd or even number of Dirac cones, and their existence has a correspondence to the NL winding number. In this sense, the NL can serves as an indicator of the topological properties of a 3D insulator. In order to show the utility of our theory, we apply it to a 3D material of the topological Bi2Te3 family and give a brief discussion.

Minimal model.- We begin our discussion with a simplest two band model for $P*T$ symmetry-protected NL semimetals:

[TABLE]

with $\sigma$ the Pauli matrices acting on an orbital space. For spinless system, the $P*T$ symmetry is simply given by the complex conjugation and a unitary matrix, $P$, such that the Hamiltonian satisfies $PH_{0}^{*}(k)P^{-1}=H_{0}(k)$. In this case $P=1$ and the $P*T$ symmetry ensures the absence of the second Pauli matrix. The nodes of $H_{0}$ yield a 1D solution, a NL with $k_{z}=0$ and $k_{x}^{2}+k_{y}^{2}=m$. Introducing a $\sigma_{y}$ term, $H=H_{0}+h_{y}\sigma_{y}$, not only breaks $P*T$ symmetry but $h_{y}$ also serves as an effective mass term, which can be either $k$-independent or related with $k$. In the former case, it opens a stable gap in the BZ, which drives the system into a trivial insulator. If $h_{y}$ is a function of $k$, the nodes of $H$ may be pairs of points, and the system is a Weyl semimetal.

We extend this model by including the spin degree of freedom and write the Hamiltonian with $P*T$ symmetry as

[TABLE]

where the Pauli matrices $s_{j}$ act in spin space. $P*T$ symmetry now reads $Ps_{y}H_{0}^{*}(k)s_{y}P^{-1}=H_{0}(k)$, with $P=1$ and satisfies $(P*T)^{2}=-1$, Here we use the labels $h_{ij}$ to represent the term of $\sigma_{i}$ and $s_{j}$, with $i=0$ or $j=0$ for the identity matrix in the corresponding subspace. These five terms form an anticommuting set of Dirac matrices, but we note that there are also other equivalent choicesArfken . The spectrum of (1)-(2) is simply given by

[TABLE]

The effective mass gapping out the NL is now $|\mathbf{h}_{y}|$, where $\mathbf{h}_{y}=(h_{yx},h_{yy},h_{yz})$. Requiring $\mathbf{h}_{y}=0$ may give a solution of points (0D), lines (1D) or surfaces (2D), depending on the number of non-zero $k$-dependent terms it has.

If $|\mathbf{h}_{y}|$ contains two non-zero $k$-dependent terms, $\mathbf{h}_{y}=0$ shall give one or several 1D lines. Thus the system is generally an insulator, as the gap closing condition requires the crossing of the NL and these 1D lines, which is accidental. Such a four-component Dirac Hamiltonian describes a chiral topological insulatorcTI , as the model satisfies $SH(k)S^{-1}=-H(k)$, with the chiral operator $S$ given by the absent fifth Dirac matrix. In the absence of time reversal symmetry, the system belongs to the AIII class and can be characterized by a Z invariantclasses . We define a winding number of $\mathbf{h}_{y}$ along the NLTNL_Linhu ,

[TABLE]

can be shown to be equivalent to a Berry phase of the occupied Bloch bands at half fillingBerry . Here, $h_{1}$ and $h_{2}$ denote the two $k$-dependent terms of $\mathbf{h}_{y}$. If we consider the $k_{x}-k_{y}$ plane that contains the NL, the intersection of the 1D lines and the plane produces a series of singularities in the plane, and the winding number (4) of the NL is simply the summation of the windings around the singularities within the NL, as shown in Fig.1. This winding number may take on any integer value, as it is only associated with the number of lines going through the NL. We also note that in the presence of time reversal symmetry, this model would fall into the CII class, which is described by a $Z_{2}$ topological index insteadclasses .

Finally, if all the three terms of $\mathbf{h}_{y}$ are depended on $k$, $h_{y}=0$ shall give one or several 0D points. In this case, the intersection of these points and the plane of the NL is also accidental. From the symmetry classification point of view, the presence of the fifth Dirac matrix breaks the chiral symmetry, and the model falls into the A class, which is non-topological in 3D. In other words, we could smoothly move a singularity out of the NL without closing the gap.

Winding numbers and geometry of the loops for a lattice model.- In order to reveal the topological properties described by the NL winding number, we next consider a lattice model described by an anticommuting set of Dirac matrices $\mathbf{\Gamma}=(\sigma_{x}s_{0},\sigma_{z}s_{0},\sigma_{y}s_{x},\sigma_{y}s_{y},\sigma_{y}s_{z})$, as

[TABLE]

with

[TABLE]

which form two NLs in $k_{x}-k_{y}$ plane for $k_{z}=\pi/2$ and $-\pi/2$ when $0<|\mu/t_{\parallel}|<2$. The position and shape of the NLs are only associated with the ratio of $\mu$ and $t_{\parallel}$, hence we can choose $t_{\parallel}=t_{\perp}=1$ for the sake of simplicity. In the following we only consider the case with positive $\mu$, with the center of the NLs given by $k_{x}=k_{y}=0$. For negative $\mu$, the center of the NLs is at $k_{x}=k_{y}=\pi$, and a similar discussion applies. The two NLs of $k_{z}=\pm\pi/2$ give two independent winding numbers, $\nu_{NL}^{\pm}$, respectively, and we define the total winding number of the system as

[TABLE]

Without loss of generality, here we choose $h_{yx}=0$ to preserve a chiral symmetry with the operator $S=\sigma_{y}s_{x}$. The other two gap terms of $\mathbf{h}_{y}$ are functions of $k$, and $\mathbf{h}_{y}=0$ gives 1D lines in the BZ. We consider the following form of $\mathbf{h}_{y}$ that breaks time reversal symmetry:

[TABLE]

and the system falls into the AIII class. $|\mathbf{h}_{y}|=0$ gives some 1D lines in the $k_{x}-k_{z}$ plane with $k_{y}=0$ or $\pi$, which may or may not be enclosed by the NLs. We would also like to point out that although the gap terms need be small for the system to preserve a NL like structure, the topological properties are not related to the exact value of these terms, but only to the ratios between them. For the sake of simplicity, we choose $t_{y}=t_{x}=1$ and positive $\mu_{s}$ hereafter.

We first consider a simple case with $t_{z}=0$. In this case, the 1D solution of $\mathbf{h}_{y}=0$ gives four lines perpendicular to $k_{x}-k_{y}$ plane, two with $k_{y}=0$ and two with $k_{y}=\pi$. For positive $\mu$, the pair of lines with $k_{y}=\pi$ are always outside the loop. By tuning $\mu_{s}$ and $\mu$, the NLs may enclose $2$, $1$ or [math] lines with $k_{y}=0$, as shown by Fig.2(a)-(c). However, the windings of these lines in the $k_{x}-k_{y}$ plane have opposite values (as in Fig.1), and the NL enclosing either [math] or $2$ lines will result in $\nu_{NL}=0$. On the other hand, as the system preserves a reflection symmetry along $z$ direction, each line will be enclosed by either two or none of the NLs, hence the total winding number, $\nu_{sum}$, in this case is always even.

In the presence of a nonzero $t_{z}$, the reflection symmetry is broken, and the lines of $|\mathbf{h}_{y}|=0$ will change shape with $t_{z}$ and eventually form a closed ring, as shown in Fig.2(d)-(f). In this case, an enclosed line will cross one of the NLs at some point, resulting in a topological phase transition. After this transition, the system has an odd winding number, $\nu_{sum}=1$, as only one of the NLs encloses a singularity.

edgestates and phase diagram.- The topological properties of a 3D topological insulator can be represented by the number of Dirac cones in the edge states under OBC. Next, we apply the method in Ref.edge_state to study the edge states in our model. The existence of edge states and their eigenenergies, under OBC, are associated with the bulk topology of the system, which can be seen by the trajectory of $\mathbf{h}(k)$ in the 5-component vector space formed by the Dirac matrices $\mathbf{\Gamma}$. Here we choose a surface plane perpendicular to the $x$ direction by fixing $k_{y}$ and $k_{z}$, and study the corresponding edge states as an example. Edge states in the other two directions can also be studied in this way, and give similar results are obtained. The Hamiltonian terms associated with $k_{x}$ give an elliptical trajectory of $\mathbf{h}$ in the 1-5 plane in $\mathbf{\Gamma}$ space,

[TABLE]

The remaining Hamiltonian terms,

[TABLE]

can be viewed as the vector from the origin of the vector space to the center of the ellipse $\mathbf{h}(k_{x})$. The parallel and perpendicular components of $\mathbf{h}^{0}$ to the 1-5 plane are given by

[TABLE]

The existence of edge states depends on whether the ellipse $\mathbf{h}(k_{x})$ encloses the point $\mathbf{h}^{0}_{\parallel}$, and this condition reads

[TABLE]

Provided that Eq.(13) holds, the edge state energies are given by

[TABLE]

Candidate Dirac cones at $(k_{y},k_{z})=(0,\pm\frac{\pi}{2})$ must satisfy the inequality (13). When $t_{z}=0$, Eq.(13) becomes

[TABLE]

for both $k_{z}=\pm\frac{\pi}{2}$. In other words, it gives either no Dirac cone, or a pair of Dirac cones at $(k_{y},k_{z})=(0,\pm\frac{\pi}{2})$. For nonzero $t_{z}$, there may exist [math], $1$ or $2$ Dirac cones depending on the parameters, as the condition (13) becomes

[TABLE]

for $k_{z}=\pm\frac{\pi}{2}$, respectively. In Fig.2 we display phase diagrams showing the number of Dirac cone edge states. These results are also in consistence with the winding numbers of the NLs, as shown in the figure.

In order to visualize the edge states, next we choose open boundary condition in $x$ direction and rewrite the Hamiltonian as a tight-binding between planes

[TABLE]

with $\hat{c}_{n}$ is a vector of annihilation operators $\hat{c}_{n,\sigma,s}$ on plane $n$, $\hat{c}_{n}=(\hat{c}_{n,+,+},\hat{c}_{n,+,-},\hat{c}_{n,-,+},\hat{c}_{n,-,-})_{n}$, and

[TABLE]

We numerically diagonalize this Hamiltonian and show the four closest doubly degenete energy bands above and below zero energy in Fig.4, with OBC along $x$ from (a) to (c), and periodic boundary condition from (d) to (f) as comparison. Panels (a) and (b) are for the topologically nontrivial phases with $\nu_{NL}^{+}=\nu_{NL}^{-}=1$ and $\nu_{NL}^{+}=1,~{}\nu_{NL}^{-}=0$, and the spectra show edge states with two or one Dirac cone, respectively. Panel (c) is for the topologically trivial phase with $\nu_{NL}^{+}=\nu_{NL}^{-}=0$, where there is no edge state connecting the conduction and valence bands.

*A real material example.-*Finally, we apply our method to 3D topological insulators of the Bi2Te3 familyreal . These materials possess time reversal symmetry and are characterized by a $Z_{2}$ number. Nevertheless, their topological nature is determined by the physics near the time-reversal-invariant point $\Gamma(0,0,0)$ in the Brillouin zone, around which the low-energy effective Hamiltonian also satisfies a chiral symmetry. This Hamiltonian is given by

[TABLE]

with $k_{\pm}=k_{x}\pm ik_{y}$, $\epsilon_{0}(k)=C+D_{1}k_{z}^{2}+D_{2}|k_{+}|^{2}$ and $M(k)=M-B_{1}k_{z}^{2}-B_{2}|k_{+}|^{2}$. Using Dirac matrices, this Hamiltonian can be written as

[TABLE]

where we left out the identity matrix as it only changes the shape of the energy bands, not the topology of the system. The chiral operator is given by the absent fifth Dirac matrix, $S=\sigma_{y}s_{0}$.

$P*T$ symmetry is here implemented by $P=\sigma_{z}$. However, we note that particle-hole symmetry also exists in this case, which reads $CH^{*}(-k)C^{-1}=-H(k)$, with $C=\sigma_{y}s_{y}$, and satisfies $CC^{*}=1$. Thus the model Eq.(25) falls into the DIII class, which is also characterized by a $Z$ invariant in 3Dclasses . Similar to our previous discussion, we write $H=H_{1}+H_{2}$, with

[TABLE]

Then $H_{1}$ has a NL in $k_{x}-k_{y}$ plane, and the nodes of $H_{2}$ produce a single line enclosed by the loop. This gives a NL winding number $\nu_{NL}=1$, which corresponds to the topological properties of the $\Gamma$ point.

Summary.- In summary, we have studied Hamiltonians with $P*T$ symmetry where two terms define a NL which is gapped out by the other terms. In the presence of chiral symmetry, these gap terms can drive the system into a chiral TI, which can be described by a winding number defined along the NL. This winding number is associated with the geometric relation between the NL and the zeroes of the gap terms. We investigated a lattice model in detail, which has two NLs in the BZ, each of them with a winding number of $1$ or [math] due to the gap terms. This winding number corresponds to the emergence of a Dirac cone for the surface states under OBC. Finally, we applied our method to the 3D topological insulators of the Bi2Te3 family, and showed the connection between their topological nature and the NL winding number.

Acknowledgments.- Financial support from FCT through grant UID/CTM/04540/2013 is acknowledged. S. C. is supported by NSFC under Grants No. 11425419, No. 11374354 and No. 11174360.