Topologically Protected Doubling of Tilted Dirac Fermions in Two Dimensions

TL;DR

This paper demonstrates that the doubling of tilted massless Dirac fermions in two-dimensional lattices is topologically protected by generalized chiral symmetry, even when the Dirac cones are tilted, extending the Nielsen-Ninomiya theorem.

Contribution

It introduces a topological protection mechanism for Dirac fermion doubling in tilted systems via generalized chiral symmetry, applicable to a broad class of lattice models.

Findings

Doubling of tilted Dirac fermions is topologically protected.

Generalized chiral symmetry can be transformed to conventional chiral symmetry.

Number of zero modes remains invariant under transformations.

Abstract

The doubling of massless Dirac fermions on two-dimensional lattices is theoretically studied. It has been shown that the doubling of massless Dirac fermions on a lattice with broken chiral symmetry is topologically protected even when the Dirac cone is tilted. This is due to the generalized chiral symmetry defined for lattice systems, where such models can be generated by a deformation of the chiral-symmetric lattice models. The present paper shows for two-band lattice models that this is a general way to produce systems with the generalized chiral symmetry in that such systems can always be transformed back to a lattice model with the conventional chiral symmetry. We specifically show that the number of zero modes is an invariant of the transformation, leading to the topological protection \`{a} la Nielsen-Ninomiya of the doubling of tilted and massless Dirac fermions in two dimensions.

Equations

$$\Gamma H\Gamma =-H.$$

$$H=v_F\left[(\mathbf{X}\cdot \mathbf{\sigma }){\pi }_x+(\mathbf{Y}\cdot \mathbf{\sigma }){\pi }_y\right],$$

$$H=\sum _{a\in A,b\in B}{t}_{ab}{c}_a^{†}{c}_b+h.c.,$$

$$H=\left(\begin{array}[]{cc}O&T\\ T^{\dagger}&O\end{array}\right),\quad\Gamma=\left(\begin{array}[]{cc}I&O\\ O&-I\end{array}\right).$$

$$H=v_F\left[(-X_0+\mathbf{X}\cdot \mathbf{\sigma }){\pi }_x+(-Y_0+\mathbf{Y}\cdot \mathbf{\sigma }){\pi }_y\right].$$

$${\gamma }^{†}H\gamma =-H$$

$$H_n(q)=T^{-1}{H}_c{T}^{-1}$$

$$T=\exp (q\mathbf{n}\cdot \mathbf{\sigma }/2)$$

$${\gamma }^{†}H\gamma =-H$$

$$\gamma=\left(\begin{array}[]{cc}a_{11}&a_{12}\\ a_{21}&a_{22}\end{array}\right)=w_{0}I_{2}+w_{x}\sigma_{x}+w_{y}\sigma_{y}+w_{z}\sigma_{z},$$

$${\gamma }^2$$

$$w_x^2+w_y^2+w_z^2=1.$$

$$w_i=w_R^i+i{w}_I^i,$$

$${\mathbf{w}}_R^2-{\mathbf{w}}_I^2=1$$

$${\mathbf{w}}_R\cdot {\mathbf{w}}_I=0,$$

$${\mathbf{w}}_R=(\cosh q){\mathbf{n}}_R,{\mathbf{w}}_I=(\sinh q){\mathbf{n}}_I,$$

$${\mathbf{n}}_R\cdot {\mathbf{n}}_I=0$$

$$\gamma =\left[(\cosh q){\mathbf{n}}_R+i(\sinh q){\mathbf{n}}_I\right]\cdot \mathbf{\sigma }.$$

$$\gamma$$

$$\mathbf{\tau }={\mathbf{n}}_R\times {\mathbf{n}}_I$$

$$({\mathbf{n}}_R\cdot \mathbf{\sigma })\exp (-q(\mathbf{\tau }\cdot \mathbf{\sigma }))$$

$$\gamma =T_{\mathbf{\tau }}(q)({\mathbf{n}}_{\mathbf{R}}\cdot \mathbf{\sigma })T_{\mathbf{\tau }}(q{)}^{-1}$$

$$T_{\mathbf{\tau }}(q)=\exp (q(\mathbf{\tau }\cdot \mathbf{\sigma })/2).$$

$$H_0=T_{\mathbf{\tau }}(q)H{T}_{\mathbf{\tau }}(q),$$

$$({\mathbf{n}}_R\cdot \mathbf{\sigma })H_0({\mathbf{n}}_R\cdot \mathbf{\sigma })=-H_0,$$

$$H_0(\mathbf{k})=\mathbf{R}(\mathbf{k})\cdot \mathbf{\sigma }$$

$$\mathbf{R}(\mathbf{k})\cdot {\mathbf{n}}_R=0,$$

$$E(\mathbf{k})=\pm \sqrt{R_x(\mathbf{k}{)}^2+R_y(\mathbf{k}{)}^2+R_z(\mathbf{k}{)}^2},$$

$$\mathbf{R}({\mathbf{k}}_0)=0,$$

Full text

I Introduction

Since the discovery of grapheneNG ; Kim , Dirac electrons in two dimensions have attracted much attention in the last decade Neto ; Sarma ; Neto2 , which is now extended to Weyl and Dirac semimetals in three dimensions Wan ; Young1 ; Young2 ; TypeI ; TypeII ; TypeII-2 ; Weng ; DDSM ; AMV . In such extensive studies of Dirac fermions, massless Dirac fermions have been realized in a wide variety of two-dimensional systems such as an organic compound Tajima ; Kajitaetal ; KKSF , cold atoms on optical latticesColdAtom , and photonic crystals PC . There, we can note that massless Dirac cones can in general be tilted as in the organic metal $\alpha$- (BEDT-TTF) 2I3, where the conventional chiral symmetry is broken. The Dirac cones usually arise in pairs, which is dubbed doubled Dirac fermions. The doubling is usually recognized as a topological consequence of the chiral symmetry NN ; HFA ; Hatsugai1 ; Springer , so that one might think that the phenomenon would be degraded for tilted Dirac cones. However, the tilted Dirac fermions do seem to always appear in pairs in such materials Kajitaetal ; KKSF , which raises an interesting question of whether the doubling remains robust in systems without the conventional chiral symmetry.

To understand the topological stability of the tilted Dirac fermions, we examine the role of a generalized chiral symmetry introduced in Refs.KHMA ; HKA for the doubling of Dirac fermions. The generalized chiral symmetry is an extension of the chiral symmetry, which has been proposed to characterize the tilted Dirac fermions. The extended symmetry can be defined for lattice models as well as in continuum models KAH . While the previous study KAH is rather restricted to the deformation of lattice models that retains the extended symmetry, here we examine, from a more general point of view, what can be deduced when we require lattice models to preserve the extended symmetry. We shall find for two-band models that, if a lattice model respects the generalized chiral symmetry globally, the doubling of the tilted Dirac fermions is topologically protected. A key observation is that the system in such a case can be transformed from a lattice model having the conventional chiral symmetry without changing the number of fermion species. The present result shows that, even when the conventional chiral symmetry is broken, the generalized chiral symmetry protects the topological stability of the doubling of tilted Dirac fermions, which would be regarded as a natural extension of the Nielsen-Ninomiya’s theorem NN to tilted Dirac fermions in two dimensions.

II Generalized Chiral Symmetry

Let us first introduce an extension of the conventional chiral symmetry, which we call the generalized chiral symmetry. In general, a system is called chiral symmetric if the Hamiltonian $H$ anti-commutes with a chiral operator $\Gamma$ as $\{H,\Gamma\}=H\Gamma+\Gamma H=0$, where the chiral operator $\Gamma$ is a hermitian operator with $\Gamma^{2}=1$. We have thus

[TABLE]

When the conventional chiral symmetry is respected, the energy eigenstate $\psi_{E}$ with the eigenenergy $E$ ($H\psi_{E}=E\psi_{E}$) can be related to the eigenstate $\psi_{-E}$ having an eigenenergy $-E$ by $\psi_{-E}=\Gamma\psi_{E}$, since $H\psi_{-E}=H\Gamma\psi_{E}=-E\Gamma\psi_{E}=-E\psi_{-E}$. The conventional chiral symmetry therefore guarantees the particle-hole symmetry irrespective of the details of the Hamiltonian. The conventional chiral symmetry holds for vertical Dirac fermions, typically in graphene. The effective Dirac field Hamiltonian for a vertical Dirac fermion is generally given by

[TABLE]

where $\bm{\pi}=\bm{p}+e\bm{A}$ with $\bm{A}$ being the vector potential is the dynamical momentum, $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are Pauli matrices, $v_{F}$ the Fermi velocity, and $\bm{X}$ and $\bf{Y}$ are three-dimensional real vectors. It is then easy to verify that $\Gamma H\Gamma=-H$ is satisfied with $\Gamma=(\bm{X}\times\bm{Y})\cdot\bm{\sigma}/|\bm{X}\times\bm{Y}|$.

The conventional chiral symmetry can also be defined for lattice models with a bipartite structure, such as the honeycomb lattice in two dimensions. For a bipartite lattice having transfer integrals $t_{ab}$ only between A and B sub-lattices, the Hamiltonian can be expressed as

[TABLE]

where $c_{a}(c_{b})$ denotes an annihilation operator of an electron on an A(B) sub-lattice site. We can then define a chiral operator as $\Gamma=\exp(i\pi\sum_{b\in B}c_{b}^{\dagger}c_{b})$ which anti-commutes with the Hamiltonian. If we express the Hamiltonian and the chiral operator in a matrix form using a basis ($a_{1},\ldots,b_{1},\ldots$) with $\{a_{i}\}(\{b_{i}\})$ the basis on the A(B) sub-lattice in the $i$-th unit cell, we have

[TABLE]

Here $T$ stands for the hopping matrix with $t_{ab}$ between A and B sub-lattices as its elements, and $I$ is the identity matrix. It is then straightforward to see that the equation $H\Gamma+\Gamma H=0$ holds, which implies the Hamiltonian $H$ is chiral-symmetric irrespective of the details of the matrix elements $t_{ab}$. The chiral symmetry for lattice models has been essential to the topological protection of the doubling of fermions in two and also in four dimensions NN ; Hatsugai1 .

For a tilted Dirac fermion, the effective Hamiltonian has additional terms $-v_{F}(X_{0}\pi_{x}+Y_{0}\pi_{y})$ as

[TABLE]

We have shown that this Hamiltonian satisfies a relation,

[TABLE]

with $\gamma=T\Gamma T^{-1}$ and $T=\exp(q\bm{\tau}\cdot\bm{\sigma}/2)$ as long as $|\bm{X}\times\bm{Y}|^{2}-|\bm{\eta}|^{2}>0$ with $\bm{\eta}\equiv Y_{0}\bm{X}-X_{0}\bm{Y}$ HKA . Here $\bm{\tau}$ is a unit vector parallel to $(\bm{X}\times\bm{Y})\times\bm{\eta}$, and a real parameter $q$ is determined by $\tanh q=|\bm{\eta}|/|\bm{X}\times\bm{Y}|$. This extended symmetry, which we call the generalized chiral symmetry, has been shown to protect the zero-mode Landau levels of tilted Dirac fermions in two dimensions KHMA . Note that the operator $\gamma$, which we call the generalized chiral operator, is not hermitian, although we still have $\gamma^{2}=1$. The requirement, $|\bm{X}\times\bm{Y}|^{2}-|\bm{\eta}|^{2}>0$, for the generalized chiral symmetry is nothing but the geometrical condition that the cross section of the tilted dispersion with a constant energy plane is an ellipse (Fig. 1(a)). When the dispersion is so much tilted that the cross section becomes an open hyperbola (Fig. 1(b)), as in the case of the type-II Dirac and Weyl fermions TypeII ; TypeII-2 , the requirement is no longer satisfied, and the present generalized chiral symmetry does not apply.

The generalized chiral symmetry can be defined for lattice models as well. We have shown that a series of lattice models respecting the generalized chiral symmetry can be systematically generated by a simple algebraic deformation of a conventionally chiral-symmetric model such as the honeycomb lattice model or the $\pi$-flux model on the square lattice KAH . Namely, if we have a chiral symmetric lattice model with the bipartite structure and denote the Hamiltonian in the momentum space as $H_{c}$, the algebraic deformation,

[TABLE]

with

[TABLE]

creates a series of deformed Hamiltonian $H_{n}(q)$. Here $\bm{n}$ is an arbitrary three dimensional real vector with $\bm{n}^{2}=1$ and $q$ a real number. Since the Hamiltonian $H_{c}$ is chiral symmetric by definition, there exists an operator $\Gamma_{c}$ satisfying $\Gamma_{c}H_{c}\Gamma_{c}=-H_{c}$ with $\Gamma_{c}^{2}=1$. It is then straightforward to see that we can define a lattice version of the generalized chiral operator $\gamma$ as $\gamma=T\Gamma_{c}T^{-1}$ so that $\gamma^{\dagger}H_{n}(q)\gamma=-H$ is satisfied. Hence this provides a systematic way for generating lattice models with generalized chiral symmetry from those with conventional chiral symmetry KAH . We can note that, when the original chiral-symmetric model has massless Dirac fermions, the deformed lattice models respecting the generalized chiral symmetry have still massless but tilted Dirac fermions (Figure 2). We can note that the type-II Dirac fermions do not appear in the present deformation as long as the parameter $q$ is real.

However, there remains an important question: Are the generalized chiral-symmetric systems generated by the above deformation exhaust the possible cases that respect the generalized chiral symmetry? In other words, it is not obvious that the generalized chiral symmetric lattice model can alway be transformed back to a lattice model with the conventional chiral symmetry. In the present paper, we precisely consider this question, and we shall show for two-band lattice models that a model with the generalized chiral symmetry can indeed be always transformed back to a lattice model with the conventional chiral symmetry. We also show that the number of zero modes is an invariant of the transformation, leading to the topological protection of the doubling of tilted and massless Dirac fermions in two dimensions.

III Generalized Chiral Operator

For discussing consequences of the generalized chiral symmetry, a general form of the generalized chiral operator $\gamma$ is crucial. If the system described by the Hamiltonian $H$ is generalized chiral symmetric, there exists an operator $\gamma$, not necessarily hermitian, that satisfies

[TABLE]

with $\gamma^{2}=1$. Here let us confine ourselves to two-band models where the above equation is described by 2 by 2 matrices in the momentum space. The general complex 2 by 2 matrix $\gamma$ can be expressed as

[TABLE]

where $\{a_{ij}\}\;(i,j=1,2)$ are complex numbers, $w_{0}=(a_{11}+a_{22})/2$, $w_{1}=(a_{12}+a_{21})/2$, $w_{2}=i(a_{12}-a_{21})/2$, $w_{3}=(a_{11}-a_{22})/2$, and $I_{2}$ stands for the 2 by 2 identity matrix. Here we assume that the elements $\{a_{ij}\}$ for the generalized chiral operator $\gamma$ are (complex) constants independent of the momentum. Since the matrix $\gamma$ must satisfy $\gamma^{2}=1$, we have

[TABLE]

where we have used a relation $\sigma_{i}\sigma_{j}=-\sigma_{i}\sigma_{j}$ for $i\neq j$. If $w_{0}\neq 0$, then the only solution is trivial $\gamma=\pm I_{2}$, so that we can set $w_{0}=0$ without loss of generality. The above condition is then reduced to

[TABLE]

If we decompose the complex coefficients $w_{i}(i=x,y,z)$ into real and imaginary parts as

[TABLE]

we end up with two equations,

[TABLE]

and

[TABLE]

where $\bm{w}_{R}=(w_{R}^{x},w_{R}^{y},w_{R}^{z})$ and $\bm{w}_{I}=(w_{I}^{x},w_{I}^{y},w_{I}^{z})$ are three dimensional real vectors. Introducing a real parameter $q$ as

[TABLE]

we arrive at

[TABLE]

with $\bm{n}_{R}^{2}=\bm{n}_{I}^{2}=1$. With these parameters, the generalized chiral operator $\gamma$ can be expressed as

[TABLE]

This can be rewritten as

[TABLE]

with

[TABLE]

because

[TABLE]

where we have used the relations $\bm{n}_{R}\times\bm{\tau}=-\bm{n}_{I}$ and $(\bm{n}_{R}\cdot\bm{\sigma})(\bm{\tau}\cdot\bm{\sigma})=i(\bm{n}_{R}\times\bm{\tau})\cdot\bm{\sigma}$. Thus the generalized chiral operator can be written, without loss of generality, as

[TABLE]

with

[TABLE]

IV Transformation to Chiral Symmetric Lattice

With this representation of the generalized chiral operator $\gamma$, we can define a deformation from $H$ back to a conventionally chiral-symmetric $H_{0}$. Indeed, if we define $H_{0}$ as

[TABLE]

it is readily verified that

[TABLE]

which means that the Hamiltonian $H_{0}$ respects the conventional chiral symmetry with the chiral operator given by $\Gamma=(\bm{n}_{R}\cdot\bm{\sigma})$. For two-band models, the generalized chiral symmetric lattice models can thus always be transformed back to the chiral symmetric models by an algebraic transformation. It is to be noted that the operator $T_{\bm{\tau}}(q)$ satisfies $\det T_{\bm{\tau}}(q)=1$ and the transformation back to the conventional chiral symmetry preserves the zero-modes of the original Hamiltonian $H$. For instance, if we have an eigenstate $\psi$ with $H\psi=0$, then $\psi^{\prime}=T_{\bm{\tau}}(q)^{-1}\psi$ is another zero-eigenstate of $H_{0}$ with $H_{0}\psi^{\prime}=0$. The number of zero modes is therefore an invariant of the transformation. Since the number of zero modes is equivalent to the number of massless Dirac fermions in the present case, this means that the number of massless Dirac fermions in $H$ has to be the same as that of $H_{0}$.

V Fermion Doubling

Following the argument in Ref. Hatsugai1 , we can move on to the fermion doubling for the chiral symmetric Hamiltonian $H_{0}$. When the Hamiltonian $H_{0}$ is expressed as

[TABLE]

with $\bm{k}=(k_{x},k_{y})$ being the two-dimensional wave vector, the real three-dimensional coefficient $\bm{R}(\bm{k})$ forms in general a three-dimensional surface (closed surface, since the two-dimensional Brillouin zone is a closed (torus) space) in the space of $(R_{x}(\bm{k}),R_{y}(\bm{k}),R_{z}(\bm{k}))$. If $H_{0}$ anti-commutes with a chiral operator $(\bm{n}_{R}\cdot\bm{\sigma})$, this condition reads

[TABLE]

which implies that $\bm{R}(\bm{k})$ is “flattened” onto a plane normal to $\bm{n}_{R}$ under this condition (see Fig.2). The energy dispersion is given by

[TABLE]

so that the contact points $E(\bm{k}_{0})=0$ of the massless Dirac fermions is determined by

[TABLE]

namely, the origin of the $\bm{R}$ space. Now, for the flattened $\bm{R}$ in the presence of the chiral symmetry, for each point $\bm{R}_{0}$ on the flattened object we have always an even number of wave vectors satisfying $\bm{R}(\bm{k})=\bm{R}_{0}$. In particular, if the origin is contained in the object, we have an even number of wave vectors satisfying $\bm{R}(\bm{k})=0$ (Fig.3), which is nothing but the doubling of massless Dirac fermions. These doubled Dirac fermions are topologically protected, since massless Dirac fermions cannot be annihilated as long as the origin is included in the area, while we have no massless Dirac fermions if the origin is outside the area.

Now we discuss the fermion doubling for the Hamiltonian that respects the generalized chiral symmetry. As shown in the previous section, a generalized chiral-symmetric Hamiltonian $H$ can always be transformed algebraically back to the Hamiltonian $H_{0}$ having the conventional chiral symmetry. Since the number of zero modes is invariant of the transformation between $H$ and $H_{0}=T_{\bm{\tau}}(q)HT_{\bm{\tau}}(q)$, the number of massless Dirac fermions in $H$ has to be the same as that in $H_{0}$, which is guaranteed to be an even number due to the conventional chiral symmetry. Hence we can conclude that any two-band lattice model that respects the generalized chiral symmetry always has doubled tilted Dirac fermions.

VI Summary

To understand the stability of the doubled and tilted Dirac fermions as in organic compounds, we have investigated the role of the generalized chiral symmetry on the fermion doubling in two dimensions, in particular, for the case where the conventional chiral symmetry is broken. We have shown for two-band lattice models that the generalized chiral symmetry, defined by the existence of a constant 2 by 2 matrix $\gamma$ satisfying $\gamma^{\dagger}H\gamma=-H$ with the Hamiltonian $H$ in the momentun space, protects the topological stability of the doubling of the tilted and massless Dirac fermions in two dimensions. We have therefore relaxed the condition for the topological protection of the fermion doubling from the conventional chiral symmetry to the generalized chiral symmetry, which may be thought of an extension of Nielsen-Ninomiya’s theorem to a certain type of tilted Dirac fermions in two dimensions. The present approach to the doubling of Dirac fermions is not applicable to the type-II Dirac fermions because the generalized chiral symmetry is no longer available there.

Acknowledgements.

The work was partly supported by JSPS KAKENHI grant numbers JP15K05218 (TK), JP16K13845 (YH) and JP17H06138.