Two-dimensional topological photonic systems

Accepted Manuscript Two-dimensional topological photonic systems Xiao-Chen Sun, Cheng He, Xiao-Ping Liu, Ming-Hui Lu, Shi-Ning Zhu, Yan-Feng Chen PII:

S0079-6727(17)30029-0

DOI:

10.1016/j.pquantelec.2017.07.004

Reference:

JPQE 209

To appear in:

Progress in Quantum Electronics

Received Date: 0079-6727 0079-6727 Revised Date:

0079-6727 0079-6727

Accepted Date: 0079-6727 0079-6727

Please cite this article as: X.-C. Sun, C. He, X.-P. Liu, M.-H. Lu, S.-N. Zhu, Y.-F. Chen, Twodimensional topological photonic systems, Progress in Quantum Electronics (2017), doi: 10.1016/ j.pquantelec.2017.07.004. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Two-dimensional topological photonic systems Xiao-Chen Sun1, Cheng He1, 2, Xiao-Ping Liu, Ming-Hui Lu1, 2*, Shi-Ning Zhu1, 2, Yan-Feng Chen1, 2* 1

National Laboratory of Solid State Microstructures & Department of Materials Science and

2

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Engineering, Nanjing University, Nanjing, Jiangsu 210093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, Jiangsu

210093, China

Email: M.-H. Lu ([email protected]), and Y.-F. Chen ([email protected]).

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Abstract

The topological phase of matter, originally proposed and first demonstrated in fermionic electronic systems, has drawn considerable research attention in the past decades due to its robust transport of edge states and its potential with respect to future quantum information, communication, and

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computation. Recently, searching for such a unique material phase in bosonic systems has become a hot research topic worldwide. So far, many bosonic topological models and methods for realizing them have been discovered in photonic systems, acoustic systems, mechanical systems, etc. These

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discoveries have certainly yielded vast opportunities in designing material phases and related properties

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in the topological domain. In this review, we first focus on some of the representative photonic topological models and employ the underlying Dirac model to analyze the edge states and geometric phase. On the basis of these models, three common types of two-dimensional topological photonic systems are discussed: 1) photonic quantum Hall effect with broken time-reversal symmetry; 2) photonic topological insulator and the associated pseudo-time-reversal symmetry-protected mechanism; 3) time/space periodically modulated photonic Floquet topological insulator. Finally, we provide a summary and extension of this emerging field, including a brief introduction to the Weyl point in

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Keywords:

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Dirac equation; Photonic topological insulator; Photonic quantum Hall effect; Gapless edge mode

ACCEPTED MANUSCRIPT 1. Introduction Topology is an important branch of modern mathematics and is related to the invariant properties of space under continuous deformations1. It only concerns the relationship between neighboring elements

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and certain global properties, but not the exact shapes and sizes. Several famous topological cases in mathematics include the seven bridges of the Königsberg problem, Euler's polyhedral formula, and the four-color map problem. When objects are continuously deformed, such as by stretching and bending

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but not by tearing or gluing, certain global indices will remain constant. For example, a ball can be

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continuously changed into a bowl; a donut can be changed into a cup. However, in the absence of tearing or gluing, the ball and the donut cannot be converted into each other. In other words, the sphere and the ring have different topologies, i.e., genera. A genus can be described by the Gauss-Bonnet theorem 2 (1 − g ) = ∫

surface

Kda ( 2π ) , i.e., the surface integrals of Gaussian curvature ( K )[1]. Here, g

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is an integer that is equal to the number of holes in the object, e.g., in general cases, for a ball, g = 0 , and for a donut, g = 1 , as shown in Fig. 1.1. In condensed matter physics, the reciprocal spaces of a crystal (or momentum-energy space, energy band) are found to have different topologies. In 1984,

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Berry generalized the geometric phase in physics into the “Berry phase”[2, 3] on the basis of previous

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works[4, 5]. This discovery soon led to the further discovery of a new kind of topological phase of matter and the corresponding topological phase transition[6]. One of the most representative examples is the family of quantum Hall effects.

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Fig. 1.1 Items with different topologies, i.e., genera. The ones with the same (different) genus are

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topologically equivalent (inequivalent) and can (cannot) be continuously transformed into each other

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without tearing or gluing.[7]

In 1879, American physicist Edwin H. Hall discovered the Hall effect[8]: when a current flows through a conductor under a vertically applied magnetic field (B), there is a potential difference in the

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direction perpendicular to the current and the magnetic field (Fig. 1.2a). The Hall resistance ( RH ) increases linearly with B (dashed line in Fig. 1.2b). Approximately one hundred years later, in 1980, Klaus von Klitzing discovered the integer quantum Hall effect (QHE) in a two-dimensional (2D)

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electron gas[9]. In this case, despite the usual overall upward trend, there exists a quantized Hall plateau in the RH

and B

relation, as shown in Fig. 1.2b. Each plateau corresponds to

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RH = h ( ne 2 ) , where the prefactor n is known as the filling factor and usually can have an integer

value without electron-electron interaction. Then, in 1982, Thouless et al. noted that the quantization originates from the non-trivial topological properties of energy bands. According to the famous TKNN theory[10], the integer n that appears in the Hall effect corresponds to a type of topological quantum numbers called the first Chern numbers, which are closely related to the Berry phase. For an ordinary insulator or vacuum, the Chern number is zero. There, however, exists a gapless edge state in a

ACCEPTED MANUSCRIPT non-trivial topological system (with a non-zero Chern number) near the Fermi surface in the bulk energy gap, which connects the valence band and conduction band. Such an edge state is topologically protected and robust, characterized by a completely backscattering suppression and unidirectional

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transport against non-magnetic defects. Because the Landau continuous phase transition theory, based on the localized order parameters and simultaneously accompanied by symmetry breaking, cannot demonstrate this new material phase, it has to be described by using a global topological invariant, the

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so-called Chern number. The topological description of matter has since become a new-emerging

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research branch in condensed matter physics. In 1988, Haldane proposed a method to achieve the quantum anomalous Hall effect (QAHE) with periodic magnetic flux[11]. Although the total magnetic flux is zero, the electrons are driven by the periodic magnetic flux to form a conducting edge channel[12]. In 2005 and 2006, Kane and Zhang et al. proposed that topological states can also be

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obtained without using any external magnetic field to break the time-reversal symmetry. Instead, by introducing a material with inherent strong spin-orbit coupling, a pair of gapless edge (surface) states with conjugate electronic spins, whose robustness is protected by the time-reversal symmetry, will

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propagate in opposite directions with immunity to scattering, exhibiting the so-called quantum spin

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Hall effect (QSHE) in two dimensions or the topological insulator (TI) in three dimensions [13, 14] (Fig. 1.2c). In this case, the total Hall conductance is zero, while the spin Hall conductance is non-zero. This non-trivial topological property can be described by a Z 2 topological invariant[15] or a spin Chern number[16]. Later, in 2011, Fu proposed a new kind of topological insulator called the topological crystalline insulator (TCI), in which the topological property is constructed and protected by lattice symmetry (e.g., mirror symmetry) rather than by time-reversal symmetry[17] (Fig. 1.2d). In addition, a class of Floquet topological insulators (FTIs) can also be realized in temporally periodical

ACCEPTED MANUSCRIPT potential energy induced by electromagnetic waves[18] (Fig. 1.2e). Realizing these phenomena, however, requires overcoming many difficulties associated with the complexity of the electronic system itself, e.g., material defects and even the validity of the single electron approximation, upon

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which most of the topological description is built. For instance, the QAHE was not realized until 2013, 25 years after Haldane’s proposal[19] (Fig. 1.2f). Hence, a number of research groups have begun to

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seek other “clean” systems to study how to achieve the topological phenomena.

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Fig. 1.2 The family of quantum Hall effects. (a) Hall effect; (b) Integer quantum Hall effect[9]; (c) Quantum spin Hall effect[20]; (d) Topological crystalline insulator[17]; (e) Floquet topological insulator[21]; (f) Experiment of quantum anomalous Hall effect[19].

Optical phenomena in periodic structures have been studied since 1887[22]. However, taking such studies a step forward to develop the seminal concept of “photonic crystals (PC)” occurred a hundred years later, in 1987, after Eli Yablonovitch and Sajeev John published two milestone papers[23, 24]. Photonic crystals have since become a key platform upon which to study a broad spectrum of energy

ACCEPTED MANUSCRIPT band-related physics. However, whether the very successful topological description in electronic systems can actually be applied to photonic systems has remained an open question until recently due to their differences in statistics. Researchers now understand that the appearance of quantized

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phenomena in the Quantum Hall effect occurs solely due to the topological property of the energy band and is unrelated to the statistics of the underlying particles. More importantly, there are many advantages that result from studying the topological properties of photons instead of studying those of

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electrons: 1) Without the limitation of Fermi levels, we can choose the most appropriate parts of the

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spectrum to analyze. 2) It is more convenient and accurate to introduce a defect or disorder into a photonic system, while this is difficult to do in an electronic system. 3) From an experimental point of view, a photonic system has a high-resolution time and space scale compared to that of an electronic system, and it is much easier to conduct exact experimental measurement and phase control[25]. In this

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direction, in 2005, D. Haldane and S. Raghu were the first to show a possible way to realize the photonic analogue of QHE by using magneto-optic photonic crystals[26]. In 2008, Wang et al. proposed a practical model to realize the integer photonic QHE and then observed the topological

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one-way edged mode for the first time in a later experiment[27]. After that, this emerging field of

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optics entered into a period of rapid development, which also had a considerable impact on other fields, e.g., mechanics and acoustics. The design of all these bosonic systems in topological space not only is of great interest for fundamental research but also offers great potential for future applications, where the unique topologically protected one-way transport of edge states might be leveraged to provide unrivalled functionalities. In reviewing the topological phenomena and properties of the energy band, we here use the Dirac model as a guide to correlate them[12]. The Dirac model was originally developed from relativistic

ACCEPTED MANUSCRIPT quantum mechanics and was used to describe the equation of motion for electrons. It also has a very important role in the field of topological materials for the following reasons: 1) Most TIs need strong spin-orbit coupling, which corresponds to a Dirac model at the non-relativistic limit. 2) The (effective)

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Hamiltonian of the QSHE and TIs is a Dirac-like equation. A description of the electron spins and photonic pseudospins requires an even number of degrees of freedom (i.e., 2 by 2 matrix for 2D cases), which is also governed by a Dirac-like equation.

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In this paper, we focus on the study of 2D photonic topological states. In Chapter 2, we will begin

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with the Dirac equation and give a physical explanation of the robust edge state and the relationship between topological invariants and symmetry. In Chapter 3, we will introduce the time-reversal symmetry-breaking photonic integer QHE. In Chapter 4, we will focus on time-reversal symmetry and the QSHE. In Chapter 5, we will introduce some photonic models of the FTI. The final part presents

2. Dirac equation

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our discussion and outlook.

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In this chapter, we will transform the Maxwell equations into the Schrödinger equation and then use the

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second quantization method to get the Dirac equation. We will briefly analyze the basic form of the Dirac equation and deduce the edge state under the non-trivial situation. Finally, we will analyze the non-trivial topological invariant obtained by using the Dirac equation and will compare it with that obtained using the TKNN method. 2.1 From Maxwell equations to Dirac equation via tight binding approximation and the second quantization methods Taking the electric field as an example, the Maxwell equations can be written as

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{

t −1

( )} = ωc E ( r , t )

sr −1

ur r

2

ε ∇ × µ ∇ × E r, t

ur r

(1)

2

sr t where ε and µ are the relative permittivity and permeability, respectively. For uniaxial media, t

sr

ε = diag {ε  , ε  , ε ⊥ } , µ = diag {µ , µ , µ ⊥ } . Let us only consider the TE mode (electric field out of the

  1   1   ω2 ∂ x   ∂ x + ∂ y   ∂ y  Ez = ε ⊥ 2 Ez c  µ     µ 

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plane) in a 2D case; Eq. (1) can be written as

(2)

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It should be emphasized that we choose the mode whose electric field only has the Ez component as the TE mode throughout this paper. We regard E z as a wave function φn ; the operator before Ez as

ω2

as an eigen energy En . We can get an equivalent

c2

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a Hamiltonian H ( x ) ; and the eigenvalue

single-electron-like Hamiltonian (stationary Schrödinger equation)

H ( x ) φn ( x ) = Enφn ( x )

(3)

The wave function of the system can be expressed as

ψ ( x ) = ∑ cnφn ( x )

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(4)

n

After the second quantization, the total Hamiltonian reads

 = ψ † ( x ) H ( x )ψ ( x ) dx = c$ †n c$ m φ ( x ) H ( x ) φ ( x ) dx = c$ †n c$ m t H ∑ ∫ n ∑ m nm ∫

(5)

n,m

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n,m

We take the eigenfunction as the intrinsic wave function within each cell and only keep the nearest

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neighboring term to obtain the effective Hamiltonian  = H

† i

ε c$ α c$ β + ∑ t α ∑ α β α β 0

i, ,

i , jβ

i

† c$ iα c$ j β

(6)

i, j , ,

The first term indicates the on-site term tn = m → ε0 , and the second corresponds to the nearest neighboring coupling t n , m → tiα , jβ .

i, j

denotes the nearest lattice site, and α , β denotes the

different orbitals. For example, in a graphene structure, α , β represents the wave functions of two different atoms in the same lattice (in the quadratic case, they can be used to represent the orbital

ACCEPTED MANUSCRIPT angular momentum s wave and p wave). After Fourier transform c$ iα = ∑ c$ kα e − ik ⋅ Ri , the Hamiltonian is k

 = H

† k

αβ

c$ α H αβ c$ β , H αβ = ∑ H δ ∑ α β δ k

k

k

αβ

e − ikδ , H δ = ε0 + ∑ tδαβ eik ⋅δ

where δ represents the coupling of the nearest neighbors

(7)

δ

k, ,

i, j . We take the one-dimensional (1D)

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system as an example to show how it can be transformed into the Dirac equation. The schematic of 1D atomic chain coupling is shown in Fig. 2.1. We can get

H kss = εs + 2tss cos k H kpp = ε p + 2t pp cos k = −2itsp sin k

where

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εs = s, m H s, m

(8)

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H

sp k

ε p = p, m H p, m

tss = s, m H s, m − 1 = s, m H s, m + 1

t pp = p, m H p, m − 1 = p, m H p, m + 1

tsp = s, m + 1 H p, m = − s, m H p, m − 1

linear term, which leads to

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If the band gap is around the k = 0 point, we can expand the Hamiltonian around k = 0 up to a

Where ε0 =

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 εs + 2tss H =  2itsp k

−2itsp k  2  = ε + m0 v f τ z + hv f kτ y εp + 2t pp  0

(9)

1 (εs + 2tss +εp + 2t pp ) , m0 v2f = 12 (εs + 2tss − εp − 2t pp ) and hv f = 2tsp This is just the 2

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basic form of the Dirac equation.

ACCEPTED MANUSCRIPT Fig. 2.1 The schematic of 1D atomic chain coupling. [28] (a) Two kinds of states in a single atom: s wave and p wave; (b) Four kinds of nearest coupling: s vs. s, p vs. p, s vs. p and p vs. s.

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2.2 Modified Dirac equation and edge states In 1928, Dirac wrote a relativistic quantum mechanics wave equation that describes the motion of electrons[29].

ur ur H = c p ⋅ α + mc 2 β

(10)

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ur ur where c represents light speed, p is momentum, and m is the mass term. α and β are Dirac

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matrices that must obey the following anti-commutation relations

α i2 = β 2 = 1 α iα j = −α jα i

(11)

α i β = − βα i

Obviously, the Pauli matrix satisfies the above relation

0 1  0 −i  1 0   ,σ y =   ,σ z =   1 0 i 0   0 −1 

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σx = 

{σ ,σ } = 2δ i

j

(12)

ij

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we choose

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In 1D and 2D cases, we can choose Pauli matrices as Dirac matrices. For convenience, in the 1D case,

αx = σ x , β = σ z

(13)

α x = σ x ,α y = σ y , β = σ z

(14)

and in the 2D case, we choose

However, the basic Dirac equation is not sufficient to solve our problem because of the following reasons. 1) Taking the vacuum boundary condition as an example (Fig. 2.2a), for a real material system, we need open boundary conditions ψ ( x = 0 ) = 0

and infinite distance boundary conditions

ψ ( x → ∞ ) = 0 . However, the solution of the Dirac equation cannot satisfy the above requirements

ACCEPTED MANUSCRIPT simultaneously. 2) After unitary transformation, the sign of the mass item m in the basic Dirac equation is changed. This means that trivial or non-trivial topology is not only dependent on m but also on another parameter, which does not satisfy the following expectation: "topology is part of the

equation[12], i.e.,

(

)

ur ur ur 2 H = v f p ⋅ α + mv 2f − B p β

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intrinsic nature of the material". Thus, we need to add a quadratic momentum term to the basic Dirac

(15)

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Here, we change the speed of light c to the particle velocity v f . This term obviously breaks the m

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symmetry. We will see later that the new equation will fully satisfy the boundary conditions mentioned

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above.

Fig. 2.2 (a) The wave function on the boundary of materials in a vacuum. Two helical edge states

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propagate along the boundary, and their dispersion relations are shown in (b).

Using the 1D case as an example, H1MD = ( mv 2f − Bpx2 )τ z + v f pxτ x

Consider the situation in which the eigen solution ψ

(16)

corresponds to the energy E , i.e.,

H1MDψ = Eψ . Obviously, H1MDτ yψ = −τ y H1MDψ = −Eτ yψ , i.e., ψ ' = τ yψ will be the eigen solution of − E . Therefore, there will be a single zero-energy state ( E = 0 ), the so-called zero mode. It is quite

ACCEPTED MANUSCRIPT special since it cannot be removed without breaking the chiral symmetry. Now let us focus on this particular zero-energy solution. (In fact, this zero mode is also the boundary state continuity requirement for the basic Dirac equation situation.) By substituting ψ ( x ) ~ eλ xξ , where ξ is a

( mv

+ Bh2 λ 2 ) ξ = λ hv f τ yξ

2 f

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two-component wave function independent of x , we can get (17)

solved as

λs , ± =

vf 2 Bh

(s ±

1 − 4 Bm

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Thus, ξ must be the eigen wave vector of τ y : τ y ξ s = sξ ( s = ±1) . The corresponding λ can be

)

(18)

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Using the boundary conditions ψ ( x = 0 ) = 0 , the general form of the solution can be written as

ψ ( x ) = ∑ cs ( eλ x -eλ s+

s

s− x

)ξ

s

(19)

With boundary conditions ψ ( x → ∞ ) → 0 , we require that λs,± < 0 or λs,± > 0 ( λ−s ,± < 0 ). This is

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equivalent to λs ,+ λs ,− ~ m B ~ mB > 0 or to the concept of “band inversion”. With this existence condition of the solution, we can get

vf

2Bh

( -1 ±

λ+ x

 sgn ( B )  -eλ− x )    i 

(20)

)

1 − 4 Bm , η = sgn ( B ) . Notice that we have assumed that hv f > 0 .

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where λ± =

(e N

1

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ψη ( x ) =

In a 2D case, we will first discuss the QAHE model; the QSHE model can be considered as simply

two copies of the QAHE model. H QAH = ( mv 2f − Bp 2 )τ z + v f ( pxτ x + p yτ y )

(21)

We separate the eigen equation H QAHψ ( x ) = Eψ ( x ) into two equations

( mv

2 f

− Bh 2 k y2 + Bh 2 ∂ 2x )τ zψ ( x ) + ( −ihv f ∂ x )τ xψ ( x ) = 0

hv f k yτ yψ ( x ) = Eψ ( x )

(22)

The first equation above is exactly the same as that of the 1D case, and the existence condition is as

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B ( mv 2f − Bk y2 ) > 0

(23)

However, at this time, we must ensure that the first and the second equations can be satisfied at the

the solution in the 1D case can be used in the 2D case.

( −1 ± 2Bh vf

λ+ x

 sgn ( B )  ip y y -e λ− x  e i  

)

h

(24)

)

1 − 4 Bm+ 4 B 2 p y2 v 2f . The edge state of the quantum anomalous Hall effect

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where λ± =

(e N

1

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ψη ( x) =

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same time. Fortunately, both equations require ψ to be the intrinsic wave function of σ y . Therefore,

is the so-called chiral edge state since it only propagates along one direction. Next, we can generalize our zero-mode calculation to the QSHE case. The QSHE model is exactly two copies of the QAHE model with time-reversal symmetry.

σ0

(25)

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 H QAH ( k ) 0  2 2 H QSH =   = ( mv f − Bp )τ z ⊗ σ 0 + v f ( k yτ y ⊗ σ 0 + k xτ x ⊗ σ z ) * 0 H QAH ( − k )  

and τ 0 represent the identity matrix. It should be noted that the time-reversal symmetry can be

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written as T = iτ 0 ⊗ iσ y K . Then, the corresponding solutions are

ψ ↑ ( x) =

(e N

1

λ+ x

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ψ ↓ = Tψ ↑ ( x ) = −

 sgn ( B )  ip y y -eλ− x  e  i 

)

1 N

(e

λ+ x

-e

λ− x

)

h

↑

 sgn ( B )  −ip y y  e  −i 

(26) h

↓

The dispersion relations are

E↑ = − h v f k y E↓ = h v f k y

(27)

As shown in Fig. 2.2b. This kind of edge state is the so-called “helical edge state”, as its propagation direction is locked to the spin.

ACCEPTED MANUSCRIPT 2.3 Berry phase from Dirac equation We have dealt with the Dirac equation to obtain chiral and helical edge states. In this part, we will discuss the topological properties of chiral and helical edge states using the TKNN method and Dirac

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equation, respectively. For the periodic lattice structure, we have

() ( ()

)

r r ur H r =H r+R rr r r ψ n, kr r = ei k ⋅r un, kr r

(

)

()

()

r r ur r where un , kr r = un , kr r + R . Expanding H r

()

()

rr r r rr r to k space such that H kr r = e −i k ⋅r H r ei k ⋅r

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()

(28)

()

()

()

r = En, kr un , kr r

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r r H kr r un , kr r

(29)

Using the TKNN method, the Hall conductivity can be expressed as[10, 30]

σ αβ = i

e2 h

d 2k

∫ ( 2π ) ∑  ∂ 2

BZ

εn < εF

kα

un ∂ kβ un − ∂ kβ un ∂ kα un  

(30)

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ur where α , β represent directions. We introduce a vector potential A

()

ur r A k = u n ∇ k un

(31)

The corresponding field strength (Berry curvature) is

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ur ur F = ∇k × A

(32)

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The Hall conductivity can then be expressed as

σ xy = i

e2 h

d 2k

∫ ( 2π ) BZ

2

Fxy = i

e2 1 h 2π

(

)

ur e2 1 d 2 k ∇ k × A =i Ω z h 2π

∫

∫

∂Ω

r ur dk ⋅ A

(33)

In mathematics, the integral term happens to have the form of the first Chern number C≡

i 2π

∫

Ω

∂Aµ ∂kν

dkν ∧ dk µ

(34)

The Hall conductivity is

σ xy =

e2 C h

However, if the Hamiltonian is the Dirac equation, the Hall conductivity can be expressed as

(35)

ACCEPTED MANUSCRIPT σ xy

ur ur e2 d 2 k ur  ∂ d ∂ d d ⋅ = × 2h ∫Ω ( 2π )2  ∂k x ∂k y

  

(36)

ur where d denotes the components of the Pauli matrix in the Dirac equation ur r H = d ⋅τ

(37)

Further derivation gives[31, 32]

σ xy =

e2 C , C = ± ( sgn ( m ) + sgn ( B ) ) 2 h

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d x = hv f k x , d y = hv f k y , d z = mv 2f − Bh 2 k 2

(38)

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When mB > 0 , C ≠ 0 , there will be a boundary state with a non-trivial topology; when mB < 0 ,

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C = 0 , the system will be as in the topological trivial case. The above derivation is in good agreement with the existing boundary conditions. Thus, from the Dirac equation, we can also get the same non-zero Hall conductivity as that obtained using the TKNN method. Recently, in addition to the theoretical approach to determining the topological invariant, some ideas about measuring topological

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invariants directly have also been proposed[33, 34].

3. Photonic quantum Hall effect

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In 2005, Haldane and Raghu proposed the photonic QHE for the first time by using a hexagonal-lattice

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gyromagnetic PC[26]. In 2008, Wang et al. proposed another specific scheme for realizing photonic QHE by using a square-lattice gyroelectric PC[27]. It is worth noting that both proposals used the Dirac equation with the mass term to achieve a non-zero Chern number. In the following, we will discuss how to construct the QHE in photonic systems with a non-zero Chern number.

3.1 Dirac equation in a hexagonal lattice and the broken time-reversal symmetry A hexagonal lattice of a photonic crystal is shown in Fig. 3.1a. For the TM mode (magnetic field out of the plane), we still use the formula given in Eq. (7),

ACCEPTED MANUSCRIPT  = H

† k

c$ α H αβ c$ β , H αβ = ∑ H δ ∑ α β δ

αβ

k

k

k

e − ikδ

(39)

k, ,

where

ur

3r 1r ex + e y 2 2 (40) ur r 3 1r ex − e y δ2= 2 2 uur r r r uur After we expand the equation by δ k = k − K near the point K = − 4π 3 ⋅ e y (Dirac point), the

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δ 1=

massless Dirac equation is obtained

  3 0 − γ 0 ( iδ k x − δ k y )  r  2  = hv f ( δ k x σ y + δ k y σ x ) H0 k =    3 γ 0 ( −iδ k x − δ k y ) 0 −   2 

SC

()

(41)

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where hv f = 3 2 γ 0 , γ 0 = ∫ drφ A* ( r ) H φB ( r ) presents the nearest coupling between two atoms in one unit cell. The dispersion relation is E = hv f δ k , as shown in Fig. 3.1b. To break the degeneracy, the Faraday term is introduced

ε xy ε

TE D

 ε ε = diag {ε  , ε  , ε ⊥ } →  ε xy  0 

0

0  0 ε ⊥ 

(42)

AC C

becomes[26]

EP

where ε xy = −ε yx = iε 0ε . The system becomes anisotropic, and the corresponding Hamiltonian

H = H 0 + mv 2f σ z

(43)

where mv 2f ∝ ε . The dispersion relation is 2

E = hv f

(

 mv  2 δ k +  f  =hv f δ k + κ 2  h  2

)

2

(44)

where κ = mv f h . For small κ , the Berry curvature near the Dirac point can be written as

(

1 3 F±xy (δ k ) = ± κ δ k + κ 2 2

)

−3 2

(45)

The corresponding geometry phase is ±π . In addition, the Chern number is ±1 for split bands. For a system that maintains spatial inversion symmetry, Fnab ( −k ) = Fnab ( k ) , whereas for a system that

ACCEPTED MANUSCRIPT maintains time symmetry, Fnab ( −k ) = − Fnab ( k ) , as shown in Fig. 3.1b. Thus, when and only when the time-reversal symmetry is broken, the Berry curvatures at two different Dirac points will have the same sign and yield a non-zero Chern number. In a case with broken spatial inversion symmetry but

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unbroken time-reversal symmetry, the curvatures will have opposite signs, and the Chern number will

TE D

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SC

vanish.

Fig. 3.1 (a) A hexagonal lattice and its corresponding reciprocal space. δ represents the coupling of

EP

the nearest neighbors. (b) Both T and P symmetry breaking can split the Dirac point. Broken P leads to trivial topology, i.e., Fnab ( −k ) = − Fnab ( k ) with C = 0 , while broken T leads to non-trivial

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topology, i.e., Fnab ( −k ) = Fnab ( k ) with C ≠ 0 .

3.2 Dirac equation in a square lattice According to the above hexagonal-lattice case, the usual procedure for obtaining a non-trivial photonic crystal is first to construct Dirac degeneracy and next to lift the degeneracy by breaking the time-reversal symmetry. The question is whether it is necessary to form Dirac degeneracy. In fact, in 2008, Wang et al. used quadratic degeneracy instead of Dirac degeneracy (linear) in a square lattice to

ACCEPTED MANUSCRIPT realize photonic QHE[27] (Fig. 3.2a). A year later, they experimentally realized the photonic QHE for the first time[35]. Similar to Haldane's model, Wang et al. introduced virtual diagonal terms in permeability using

 µ µ =  −iκ  0

iκ

sr

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gyromagnetic materials

0 0  µ0 

µ 0

(46)

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There are three different Bloch modes of TE states ( E z component) in every unit cell. As a result, the

px

and

p y . The interaction between

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basis for the tight-binding approximation (TBA) model contains three modes corresponding to s ,

px

and

only contains on-site coupling, while the

py

other interactions contain two parts: on-site coupling and nearest neighboring coupling. The on-site interaction between

px

and

px

can be calculated as[36] r

(

uur *

uur

ω0 ∫ dsκ e z ⋅ H p × H p

TE D

i iV p = 2

uur ∫ ds µ0 H px

2

x

y

)

uur ∫ ds µ0 H py

(47)

2

uur uur where ω0 is the frequency of the resonance in the absence of κ . H px and H p y represent the px

and

p y , respectively. For the other interactions, the on-site coupling and

EP

magnetic fields of

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the nearest neighboring coupling can be defined as follows, according to the analysis of symmetry. s , m H s, m = ε s px , m H px , m = p y , m H p y , m = ε p s , m H s , m ± x = s, m H s , m ± y = t s px , m H p x , m ± x = p y , m H p y , m ± y = t pσ

(48)

px , m H p x , m ± y = p y , m H p y , m ± x = t pπ s, m H p x , m + x = − s, m H px , m − x = tsp s , m H p x , m ± y = s, m H p y , m ± x = 0 m , m ± x and m ± y represent the neighboring sites. The Hamiltonian based on s ,

py

is given by

px

and

ACCEPTED MANUSCRIPT

()

r  Es k  H =  2itsp sin k x    −2itsp sin k y

−2itsp sin k x r E px k

()

−iV p

2itsp sin k y    iV p  r  E py k  

(49)

()

where

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Es = ε s + 2t s ( cos k x + cos k y ) E px = ε p + 2t pσ cos k x + 2t pπ cos k y E py = ε p + 2t pπ cos k x + 2t pσ cos k y

± i py

) . On the

 2itsp ( sin k x + i sin k y ) − 2itsp ( sin k x − i sin k y )   1 1 E + E − V − E − E ( px py ) p ( px py )  2 2  1 1 − ( E px − E py ) E px + E py ) + V p  (  2 2

(50)

p+

1

2

(p

x

p− , the Hamiltonian takes the form of

and

 Es   H =  − 2itsp ( sin k x − i sin k y )    2itsp ( sin k x + i sin k y ) 

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basis of s ,

p± = m

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Then, the linear superposition of the two p orbits is described as

i.e.,

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We can divide the Hamiltonian H into two diagonal parts H A and H B and a coupling part H AB ,

H H =  †A  H AB

(φ A , φ B )

(51)

denotes the eigenfunction with the eigenvalue E , and the equation can be divided into two

EP

T

H AB   HB 

† parts: H AφA + H ABφB = EφA and H ABφA + H BφB = EφB . Under ideal conditions, we can remove φB ,

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and the effective Hamiltonian that only has the component φ A can be written as † H A,eff φ A = Eφ A , H A, eff = H A + H AB ( E − H B ) H AB −1

(52)

This effective Hamiltonian H A,eff depends on the eigen energy E , which means that it is a self-consistent equation. The eigen energies of H A and H B should have considerable difference such that we can use the zero-order eigen energy of H A , denoted as EA,0 , instead of E . This is equivalent to the second-order perturbation theory. In our case, H A , H B and H AB are given by

ACCEPTED MANUSCRIPT  Es 2it sp ( sin k x + i sin k y )    HA =  1  E px + E py ) − V p  (  − 2itsp ( sin k x − i sin k y ) 2  1 H B = ( E px + E py ) + V p 2  − 2itsp ( sin k x − i sin k y )    H AB =  1  − ( E px − E py )   2

near the origin point kx = ky = 0 . In addition,

(E − HB )

† H AB ( E − H B ) H AB −1

1 . The second term in Eq. (52) is 2V p

 2 itsp ( E px − E py )( sin k x − i sin k y )  2   2 1 E px − E py )  ( 4 

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 2tsp2 ( sin 2 k x + sin 2 k y )  =  2 itsp ( E px − E py )( sin k x + i sin k y ) −  2

≈

1 ( E px + E py ) − Vp  12 ( E px + E py ) + Vp 2

SC

given by

−1

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As an approximation, we can address this case using Es ~

(53)

(54)

All the terms in the second-order perturbation correspond to the long-range hopping, which can be

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neglected in our case. Finally, only the lowest-order term H A remains. After expanding the Hamiltonian H A to the linear k term, we can obtain the final result. For the sake of simplicity, we can rewrite the Hamiltonian by renaming the mixed basis

p

instead of

p+

as

EP

  Es ( 0 ) 2itsp ( k x + ik y )   = 1   − 2itsp ( k x − ik y ) 2 ( E px ( 0 ) + E py ( 0 ) ) − V p 

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H eff

where ε 0 =

(55)

= ε 0 + m0τ z + hv f ( k yτ x + k xτ y ) → H 0

1 1  1 1   Es ( 0 ) +  ( E px ( 0 ) + E px ( 0 ) ) − V p   , m0 =  Es ( 0 ) −  E px ( 0 ) + E px ( 0 ) − Vp   , 2 2 2  2 

(

)

hv f = − 2t sp .

Thus, we have obtained the Dirac-like equation in a square lattice. As a result, there exist robust one-way edge states, as shown in Fig. 3.2c.

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ACCEPTED MANUSCRIPT

Fig. 3.2 (a) Quadratic degeneracy in the square lattice is broken to realize a non-trivial photonic energy

TE D

band[27]. (b) TE states in a single unit cell. (c) Experiment realization of photonic QHE[35].

3.3 Realization of photonic quantum Hall effect

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EP

The first step to realizing photonic QHE is to construct the permeability tensors as shown in Eq. (46)

 µ sr  µ =  −iκ  0

iκ

µ 0

0 0  µ0 

These parameters can be deduced from a microscopic view of the atom[37] and expressed as 

µ = µ0 1 + 

ω 0ω m   ω 02 − ω 2 

gyromagnetic

ratio,

and

κ = µ0

which

ωωm , where ω02 − ω 2

depends

on

the

ω0 = µ0γ H 0 , ωm = µ0γ M s ,

charge

and

mass

of

γ

the

is

the

electron

γ = q me = 1.759 ×1011 C kg . H 0 is the bias field, and M s is the saturation magnetization. ω

is the frequency of the

ACCEPTED MANUSCRIPT time-harmonic AC magnetic field. It should be mentioned that due to the restriction on the gyromagnetic resonance of ferromagnetic materials, it is difficult to extend the effective frequency to optical frequency[38]. It should be mentioned that, we can use similar relative permittivity with

band( 10µ m ). However, this design is still in theory research stage.

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gyroelectric medium to realize one-way edge modes[39] , and the effective frequency can be infrared

Following the gyromagnetic principle above, Wang et al. carried out the photonic QHE experiment in

SC

2009, observing the one-way propagating topological photonic edge state with backscattering

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suppression for the first time[35]. The experiment involved using ferrite cylinders made with a vanadium-doped Ca-Fe-based garnet. This material has a high saturation magnetization 5 −1 ( M s = 1.52 ×10 Am ) and a low loss (a loss tangent of tan δ = 0.00010 ) under a DC external

magnetic field of H 0 = 0.20T . When the frequency is approximately 4.5 GHz (in the second band

TE D

gap), the contrast ratio between the forward and backward propagation of the chiral boundary states is greater than 50 dB, indicating that the backward propagation is suppressed. Additionally, the robustness of such unidirectional propagation against obstacles was investigated.

EP

On the other hand, it was shown that such edge states may appear outside the air line and become

AC C

surface modes[40]. They used the yttrium iron garnet (YIG) ferrite as the photonic atom. The saturation magnetization was 4π M s = 1884G , and the bias field was H 0 = 800Oe . In addition, in their work, a gap between 7.75 and 8.42 GHz was identified. Moreover, if the number of degeneracy points exceeds one, multiple gapless topological edge patterns will appear, and the number of gapless edge states can be determined based on the value of the Chern number (|C|)[41]. This phenomenon also depends on the gyromagnetic resonance effect, as the effective frequency is still confined to the GHz scale.

ACCEPTED MANUSCRIPT Some other models for realizing non-trivial photonic crystals have been proposed[42]. The findings related to the photonic QHE have inspired the design of many interesting devices including reflectionless optical waveguide bends and splitters (optical diode)[43, 44], unidirectional delay lines,

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directional filters[45, 46], magnetically controllable devices[47], binary digital generators, slow-light

4. Photonic quantum spin Hall effect

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devices[48], broadband circulators[49], signal switches[50], topological circuits[51], etc.

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As discussed above, the QSHE can be regarded as a combination of QAHEs. For electronic systems, an electron’s spin-up and spin-down states form a pair of conjugate states, which are degenerate at the origin point under time-reversal symmetry, resulting in Kramers degeneracy[52]. For photonic systems, under time-reversal symmetry, Kramers degeneracy cannot be established. However, the time-reversal

TE D

symmetry is not so necessary, but constructing a pair of conjugate states as a pseudospin to satisfy Kramers degeneracy under certain artificial gauge symmetries is a vital step towards realizing and finally understanding the photonic QSHE and photonic TI.

EP

4.1 From quantum anomalous Hall effect to quantum spin Hall effect

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In electronic systems, the QSH Hamiltonian can be expressed as

0   H (k ) 0   H QAH ( k ) H QSH =  ↑   =  * 0 H QAH ( −k )  H↓ ( k )    0

= ( mv − Bp )τ z ⊗ σ 0 + v f ( k yτ y ⊗ σ 0 + k xτ x ⊗ σ z ) 2 f

(56)

2

For the spin-up and spin-down electrons, the dispersion relation is

E↑ = −hv f k y E↓ = hv f k y In this case, the time-reversal operator can be expressed as T = iτ 0 ⊗ iσ y K . Obviously, effect on spin is such that

(57)

[ H , T ] =0 . Its

ACCEPTED MANUSCRIPT T ↑, k = − ↓, − k

(58)

T ↓, k = ↑, − k

which means that T 2 = −1 . Since the system preserves time-reversal symmetry, if there is an eigenstate ↑, k , it must be accompanied by the existence of an eigenstate T ↑, k = − ↓, −k . ↑, k

and T ↑, k

will degenerate at k = 0 when

[ H ,T ] = 0 .

The only problem is

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Clearly,

whether “they” are a single state. Supposing that “they” are indeed a single state and the relation is

(

)

SC

T ↑, 0 = eiα ↑, 0 , we will get T 2 ↑, 0 = T eiα ↑, 0 = e −iα T ↑, 0 = ↑, 0 , which contradicts

T 2 = −1 , suggesting that “they” must be two states and there must be a degeneracy, i.e., Kramers

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degeneracy, which is protected by the time-reversal symmetry.

For the photonic counterpart, pseudo-spin in its simplest form can be chosen as either TE/TM or LCP/RCP. However, photons are significantly different from electrons. Their time-reversal symmetries satisfy T 2 = 1 , which implies that Kramers degeneracy under this time-reversal symmetry cannot be

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attained. This can be illustrated by using the following example, where T 2 = 1 .

T TE = TE T TM = − TM

(59)

EP

Clearly, this kind of time-reversal symmetry is not appropriate for constructing the photonic QSHE or

AC C

the photonic TI. To find the correct symmetry protection mechanism, let us go back to the Dirac-QSH model. We notice that in this model, the time-reversal operator has the mathematic form T = iσ y K , which can also be constructed in photonic systems. The effect of this newly constructed pseudo-time-reversal operator Tp = iσ y K on the pseudospin is to transform, for example, TE

into

TM , with the π phase, as shown in Fig. 4.1. This pseudo-time-reversal symmetry Tp , as we will show next, plays a key role in realizing the photonic QSHE or the photonic TI.

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ACCEPTED MANUSCRIPT

Fig. 4.1 The effects of time-reversal symmetry and pseudo-time-reversal symmetry on electron spins or

SC

photon pseudospins.

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4.2 Photonic topological insulator and pseudo-time-reversal symmetry The first step to realizing photonic TI is to find two photonic pseudospins, which can be conveniently chosen as photonic polarizations, as shown in Fig. 4.3a. In 2013, Khanikaev et al. proposed a theoretic

TE D

method, by using a hexagonal lattice of double bi-anisotropic materials with electromagnetic coupling, to achieve a 2D photonic TI, as shown in Fig. 4.2a[53], for TE+TE/TE-TM polarization-based

EP

pseudospins, i.e.,

ψ + = Ez + H z

ψ − = Ez − H z

(60)

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The constitutive equations in this case are uur t ur sruur D = ε E + iχ H ur srur sruur B = −i χ E + µ H

(61)

To construct degeneracy based on the electromagnetic duality, the parameters need to satisfy

t

sr

ε =diag {ε ⊥ , ε ⊥ , ε zz } = µ = diag {µ⊥ , µ⊥ , µ zz }  0 χ =  − χ xy  0 

sr

χ xy 0 0

0  0 0 

± With ψ = Ez ± H z as eigenstates, the motion equations of the system can be expressed as

(62)

ACCEPTED MANUSCRIPT   −i χ xy    2  1 ± ∇ ⊥ ψ ± = ± ∇ ⊥   × ∇ ⊥ψ   k0 ε zz + ∇ ⊥ µ⊥   ε ⊥ µ⊥   z  

(63)

After the second quantization, the form of the Dirac equation is yielded

H ± (δ k ) = vD (τ zσ xδ k x + σ yδ k y ) ± ζτ zσ z

ψ +   L − iL1 H0  −  =  0 ψ   0

0  ψ +    = 0 L0 + iL1  ψ − 

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We re-write the equation in a 2 by 2 matrix form to analyze the symmetry

(64)

(65)

SC

 − χ xy    1 where L0 =  k02ε zz + ∇ ⊥ ∇ ⊥  , L1 = ∇ ⊥   × ∇ ⊥ . This means that H 0 = L0 I + L1 ( −iσ z ) . Apart µ⊥    ε ⊥ µ⊥ 

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from the trivial part L0 I , we can see that H 0 ∝ iσ z . The effect of the time-reversal operator

T = Tx = σ x K on the spin eigenstates is

*

ψ +  ψ −  T −= + ψ  ψ 

(66)

TE D

while the effect of pseudo time reversal Tp = iσ y K is

*

ψ +   −ψ −  Tp  −  =  +  ψ   ψ 

and Tp

operators is maintained such that

[H0 ,T ] = 0

and

EP

The symmetry under the T

(67)

 H 0 , Tp  = 0 . To clarify the robust features, two kinds of defects are mentioned: 1) a cavity obstacle,

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which corresponds to a trivial case and for which the Hamiltonian can be taken as H 1 = I ; 2) a strongly lattice disordered domain in both of the adjacent crystals, which has the same constructive parameters as the designed photonic crystal and for which the Hamiltonian can be taken as H 2 = iσ z , as indicated in Fig. 4.2b. In both cases, time-reversal symmetry is preserved:  H1,2 , T  = 0 . However, as we will determine in the next section, a solid conclusion, for example, T is responsible for topological protection, cannot be drawn merely based on this finding since pseudo-time-reversal symmetry is also preserved  H1,2 , Tp  = 0 in both cases. Later, Dong’s research group proposed

ACCEPTED MANUSCRIPT constructing a photonic TI using permittivity/permeability-matched meta-materials[54], i.e., star-shaped and disc-shaped meta-crystals in the microwave region, where the electromagnetic coupling may be derived from the TEM mode in the waveguide. Similarly, bi-anisotropic

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meta-materials with hyperbolic energy bands can also be used to achieve the topological state[55].

Fig. 4.2[53] (a) Polarization TE+TM/TE-TM is used to realize the photonic TI. (b) Cavity and disorder

TE D

are chosen to demonstrate the robustness of the edge states.

In contrast to the T-invariant photonic TI model mentioned above, Lu and Chen’s research group proposed another kind of photonic TI[56] by using the piezoelectric/piezomagnetic (PE/PM)

EP

superlattice PC and taking LCP/RCP (left-circular polarization/right-circular polarization) as the

AC C

pseudospins to realize the photonic TI even without T symmetry. In a 2D case, the LCP /RCP can be expressed as

ψ LCP = Ez + iH z ψ RCP = iEz + H z

(68)

The effective constitutive equation for the proposed PE/PM superlattice is

uur tur t uur D = ε E +ξ H ur srur sruur B = ζ E + µH where

(69)

ACCEPTED MANUSCRIPT t

ε = diag {ε xx , ε xx , ε zz } sr

µ = diag {µ xx , µ xx , µ zz }

 0 T  $  and ξ = ζ =  −ξ xy  0 

ξ xy

0  0 0 

0 0

(70)

It should be noted that ξxy is a real number implying the non-reciprocity of the superlattice. Then, the

Here,

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eigen equation can be written as 0  ψ LCP  ψ   L − iL1 (71) H 0 =  LCP  =  0  =0 L0 + iL1  ψ RCP  ψ RCP   0 1 1 L0 = k02 ε zz + ∂ x ∂x + ∂y ∂ y , L1 = ∂ x κ∂ y − ∂ y κ∂ x , µ = ( µ xx ε xx − ξ xy2 ) / ε xx , and

µ

µ

SC

κ = −ξ xy / ( µ xx ε xx − ξ xy2 ) . It can be found that H0 ∝ iσ z . Although the mathematical form is similar to

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that of Khanikaev's Hamiltonian, the physical insight is completely different. Note that the eigenvector ± of the system is ψ LCP / RCP instead ψ . In this case, the time-reversal operator is T = IK , under

which the photonic pseudospins satisfy

ψ  ψ  T  LCP  =  LCP  ψ ψ  RCP   RCP 

*

(72)

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while the pseudo-time-reversal symmetry is Tp = iσ y K , giving rise to ψ   −ψ  Tp  LCP  =  RCP  ψ RCP   ψ LCP 

*

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As for the commutation relation of the Hamiltonian, we can obtain

(73)

[ H 0 , T ] ≠ 0,  H 0 , Tp  = 0 , which

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means that such a model breaks the time-reversal symmetry but keeps the pseudo-time-reversal symmetry. As shown in Figs. 4.3b-e, by checking the robustness of edge states against a comprehensive set of impurities with all possible combinations of T and Tp symmetries, i.e., (i) unbroken T and Tp (e.g., uniaxial dielectric); (ii) broken T and broken Tp (e.g., Tellegen); (iii) unbroken T but broken Tp (e.g., chiral); (iv) unbroken T and unbroken Tp (e.g., T p -invariant chiral[53]), we find that only the Tp invariant perturbation can guarantee the pseudospin-momentum locking propagation, gapless dispersion of the edge states and robust propagation, suggesting that Tp ,

ACCEPTED MANUSCRIPT as opposed to the commonly believed T, should be the ultimate symmetry protection mechanism in this kind of photonic topological insulator. The apparent reason behind this observation is that pseudospin states, LCP and RCP here, can satisfy Kramers degeneracy only under the fermionic-like

in this case commutes with Tp :  H 0 , Tp  = 0 rather than T :

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* * pseudo-time-reversal symmetry Tp , i.e., Tpψ LCP = ψ RCP and Tpψ RCP = −ψ LCP , and the Hamiltonian

[H0 ,T ] ≠ 0

. Therefore, the

Kramers-like degeneracy can be constructed using some symmetry, such as electric-magnetic duality,

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other than time-reversal symmetry. In this sense, the band topology is still irrelevant with respect to the

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statistics of the underlying particles. This striking finding has a profound meaning because the bosonic time-reversal symmetry may not be considered as a limiting factor when designing photonic TIs, and more degrees of freedom could in principle be introduced to construct pseudospins and the corresponding pseudo-time-reversal symmetry Tp to fulfill Kramers degeneracy. Therefore, this

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EP

TE D

finding will benefit the design of photonic TIs without the limitation of bosonic time reversal [57-60].

Fig. 4.3[56] (a) The polarization on the Poincaré sphere and the LCP/RCP states. Robustness against four types of impurities: (b) uniaxial dielectric impurity ( ε zz = 4 ), (c) Tellegen impurity ( ξ zz = ζ zz = 1 ),

ACCEPTED MANUSCRIPT (d) chiral impurity ( ξ zz = −ζ zz = i ), and (e) chiral impurity with Tp invariance ( ξ xy = −ξ yx = ζ xy = −ζ yx = 0.3i ).

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4.3 Photonic topological crystalline insulator In addition to the electromagnetic duality-based polarization pseudospins and T p , photonic Bloch modes and spatial point group symmetry can also be leveraged to create pseudospins and Tp . A

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pioneering work was done by Hu's research group, in which a pair of photonic pseudospins was formed

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by hybridizing degenerated Bloch modes as a result of the C6 rotational symmetry of a hexagonal lattice[61]. The associated pseudo-time-reversal symmetry comes from the point group symmetry of the

crystal.

For

instance,

U =  R (π 3 ) + R ( 2π 3 ) 

the

combination

of

60°

and

120°

rotation,

3 = −iσ y , gives rise to Tp = UK . The band folding mechanism enforces

TE D

degeneracy between the K and K ' Dirac points and thus leads to a double Dirac cone (with four-fold degeneracy) at the center of the Brillouin zone after band folding, as shown in Figs. 4.4a and 4.4b. The biggest advantage of this kind of scheme is that the degeneracy lifting and energy band

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inversion between the p-band and the d-band required to induce a topological transition can be simply

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achieved by stretching or compressing the lattice, as shown in Fig. 4.4b-c. The topologically protected spin-dependent helical edge states can be formed at the boundary between the topological photonic crystal and trivial photonic crystal. Very recently, such kind of photonic topological insulators made of dielectric materials were experimentally demonstrated[62].

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ACCEPTED MANUSCRIPT

Fig. 4.4[61] (a) Triangular-latticed photonic TCI. (b) The Dirac cones at K and K ' are folded to form a double Dirac cone at the Γ point. Lifting the band degeneracy and creating energy band

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inversion between the p-band and the d-band (c) can be achieved by stretching the lattice.

This photonic topological model is due to the mirror symmetry of symmorphic group C6 in the 2D plane, which can be regarded as a kind of photonic TCI in the 2D case[17]. Note that under the 2D

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condition, the Bloch states of the left and right sides of the topological boundary waveguide are not

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exactly the same; thus, the pseudo-time-reversal symmetry is slightly broken. As a result, the edge states are not completely gapless, with a theoretical mini-gap. Thus, strictly speaking, the backscattering of edge states cannot be completely avoided but can be largely suppressed. In practice, the mini-gap can be deliberately adjusted to an infinitesimal size to minimize its influence on the real-world performance.

4.4 Realization of the photonic quantum spin Hall effect Based on the analysis above, Tp is the key parameter necessary to construct the QSHE. Considering

ACCEPTED MANUSCRIPT that an optical system invariant is always under the action of the composition of the parity, time reversal, and duality operators (P, T, D)[63], there are two ways to realize the photonic QSHE, which correspond to the two models introduced above.

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The first is to combine D with T. In 2014, Dong and Chan’s group experimentally realized a photonic topological insulator by embedding a non-bi-anisotropic and non-resonant metacrystal into a waveguide[54] The gapped edge states is around 2.5~3GHz, due to the non-resonant feature of their

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meta-atom. In 2016, Khanikaev et al. realized a controllable bi-anisotropic material[64]. The

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topological structure is formed between two parallel copper plates through which holes are drilled to support a periodic triangular array of copper rods with ring collars. By moving the rods up or down, the reconfigurable topological domain walls are realized, yielding the robust propagation of electromagnetic edge states around 20~21GHz. Taking ψ ± =Ez ± H z as the pseudospins, the two

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models are both Tp - invariant. However, their Tp - invariance is constructed based on the duality of impedance matching materials. If this condition is broken, for example, taking anisotropic material as the impurities, the robustness will be broken.

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The other way is to combine P with T. Hu et al. realized their model in 2016 by using a hexagonal

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cluster of six neighboring dielectric rods[62]. The unidirectional backscattering immune optical pathway was demonstrated through microwave experiments around 7.5GHz. As mentioned above, the

Tp of their work is based on C6 symmetry; hence, the robustness will be broken by introducing the C6 -broken disorder. In addition, there will be a mini-gap in the edge states band.

This surprisingly simple yet very effective approach also obeys the Tp principle and offers a completely new and practically viable way to construct photonic topological phases with only common

ACCEPTED MANUSCRIPT dielectric materials. This same method or a variation of this method (e.g., lifting degeneracy by breaking the z-direction (out of 2D plane) symmetry) can also be generalized and applied to other optical spectra as well as to other bosonic systems, e.g., elastic-wave TIs[65], mechanics[66, 67] and

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acoustics[68].

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5. Photonic Floquet topological insulators

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In the photonic QHE, an external magnetic field is required to break the time-reversal symmetry and introduce vector potential[69]. There are, however, various ways of introducing an artificially synthetic effective gauge potential and effective magnetic field instead of an external magnetic field. A particular example is a temporally periodic model, with which a so-called photonic FTI can be constructed.

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5.1 Effective magnetic field and Dirac equation

In 2012, Fan’s research group theoretically proposed a photonic phase modulation method, which can be used to synthesize an effective magnetic field for the purpose of breaking time-reversal symmetry

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[70], as shown in Fig. 5.1a. In their work, the coupling phase of two resonators with different resonant

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frequencies was deliberately manipulated in the temporal domain. The coupling phase in the x-direction was 0, while it increased linearly with the number of cycles in the y-direction. After going around a plaquette, light obtained an effective phase, i.e., it experienced an effective gauge potential, which led to an effective magnetic field in the system

Beff =

1 a2

∫ A

eff

dl =

φ a2

(74)

For this effective magnetic field presented in such a system, we next explore how to modify its Hamiltonian. Recall Eq. (49) from Section 3.2 (corresponding to the case without magnetoelectric

ACCEPTED MANUSCRIPT px

and

as the base vectors, the system Hamiltonian can be expressed as

py

()

r  Es k  H 0 =  2itsp sin k x   −2itsp sin k y  Here, the

p

−2itsp sin k x r E px k

()

0

base vectors are again hybridized such that

2itsp sin k y    0  r  E py k  

()

p± = m

system Hamiltonian becomes

H0 =

1

2

(p

x

± i py

) , and then the

 2it sp ( sin k x + i sin k y ) − 2itsp ( sin k x − i sin k y )   1 1  E px + E py ) − ( E px − E py ) (  2 2  1 1  − ( E px − E py ) E px + E py ) ( 2 2 

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 Es    − 2it ( sin k − i sin k ) sp x y    2itsp ( sin k x + i sin k y ) 

(75)

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coupling). With s ,

(76)

Due to the absence of the term Vp , the system cannot be transformed into a 2 by 2 matrix. However, the system is now under the influence of an effective magnetic field, which can cause an equivalent

p± = m

1 2

(p

x

± i py

)

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Zeeman Effect. The corresponding eigenvalue of the angular momentum of

is ±1 , which means that Lz p± = ± p± . In this case, Lz is the z-direction angular momentum operator (note that h is normalized to be 1 for simplicity). The effect of the magnetic field is

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ur ur µ gL ∆H = − µ ⋅ B = B z Beff , µB is the Bohr magneton, and g is the Landau g-factor. Thus, the h

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corresponding Hamiltonian with

p±

as the basis can be written as

H zee

where ∆ z = µB gBeff h .

Then, the Hamiltonian can be written as

0 0  = ∆ z Lz =  0 ∆ z 0 0 

0   0  -∆ z 

(77)

ACCEPTED MANUSCRIPT H = H 0 + H zee  2it sp ( sin k x + i sin k y ) − 2itsp ( sin k x − i sin k y )   1 1  E px + E py ) + ∆ z − ( E px − E py ) (  2 2  1 1 − ( E px − E py ) E px + E py ) − ∆ z  (  2 2

(78)

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 Es    − 2it ( sin k − i sin k ) sp x y    2itsp ( sin k x + i sin k y ) 

Eq. (78) is exactly the same as Eq. (50), with which an equivalent Dirac equation can be derived, as

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shown in Section 3.2.

5.2 Time/space periodically modulated photonic Floquet topological insulator the

system

mentioned

above,

the

coupling

coefficient

is

a

time

harmonic,

i.e.,

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In

V ( t ) = V cos ( Ωt + φij ) , where φij is the phase factor for individual resonators. The Hamiltonian satisfies H ( t + T ) = H ( t ) , T = 2π Ω . Applying Floquet's theorem, the wave function can be written as ψ ( t ) = e −iε t φ ( t ) , with φ ( t ) = φ ( t + T ) , a periodic function, which is analogous to that derived from

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Bloch's theorem in the space domain. Here, ε can be regarded as quasi-energy, which satisfies

( H ( t ) − iI ∂ ) φ (t ) = εφ ( t ) . t

The energy band of the quasi-energy-momentum space can then be

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obtained by solving this eigen equation.

Utilizing time-dependent modulation to induce an effective gauge field to break the time-reversal

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symmetry is conceptually novel. Implementing it in experiments, however, is very difficult because of many technological issues, e.g., global synchronization and dynamical noises[71]. In 2013, Rechtsman et al. proposed a new scheme that uses spatial periodic modulation (z-axis) as opposed to the temporal periodic modulation, and they successfully demonstrated the photonic FTI[72, 73] (Fig. 5.1b). In their work, the photonic lattice was formed in the xy-plane, and the wave was injected along the z-axis. In this case, the wave equation for the propagation of the paraxial ray along the z-axis can be written in the form of a time-dependent Schrödinger equation:

ACCEPTED MANUSCRIPT i∂ zψ = −

k ∆n 1 2 ∇ψ − 0 ψ 2k0 n0

(79)

where the envelope function ψ of the electric field is defined as E ( x, y, z ) = ψ ( x, y, z ) ei ( k0 z −ωt ) x$ . The z-axis spatial symmetry is broken due to the helicity of the modulation structure, as shown in Fig.

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5.1b. Using the coordinate transformation x ' = x + R cos ( 2π z Z ) ; y' = y + R sin ( 2π z Z ) ; z ' = z (in spiral coordinates, R is the radius and Z is the periodicity along the z-axis), Eq. (79) can be written as

Here, the vector potential

2 ur 2 k R 2 ( 2π Z ) ' k 0 ∆n ' 1 ∇ ' + A ( z ' )  ψ ' − 0 ψ − ψ  n0 2k0  2

(80)

ur ' A ( z ) = k0 R ( 2π Z ) sin ( 2π z ' Z ) , − cos ( 2π z ' Z ) , 0  , the effective

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i∂ zψ ' = −

( )

( )

as ψ ' z ' = e −iε z φ ' z ' '

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ur ur magnetic field B eff = ∇ × A , and ψ ' is a periodic function about the z-axis, which can be expressed

using the Floquet theorem. Similarly, the quasi-energy can be expressed as a

function of the Bloch wave vector

(k , k ) x

y

to form the Floquet band diagram. Subsequently, the

gapless topological edge state can be created. The experimental findings show that light (633 nm) can

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propagate in the z-direction around the photonic crystal’s edge without backscattering, giving rise to a

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unidirectional spiral propagation pattern.

5.3 Robust optical delay lines via topological protection

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In 2011, M. Hafezi et al. designed a robust optical delay line[74] based on the coupled resonator optical waveguides (with the inter-coupling between two site resonators). Later, they demonstrated, in a silicon photonic platform, one-way boundary propagation states locked to the clockwise and counterclockwise wave circulation in the microrings[75]. As shown in Figs. 5.1c and 5.1d, light making a clockwise or counterclockwise roundtrip in a lattice unit cell experiences an opposite effective gauge potential. These two states of light can then be used to mimic an electron’s spins and to design photonic topological structures. In their work, one-way robust and back-reflection-free light (1539 nm)

ACCEPTED MANUSCRIPT transportation against defects was experimentally demonstrated. Later studies showed that this type of lattice of resonant cavities can be treated as a network model of photonic FTIs[76], and the gapless topological boundary state is dependent on the coupling strength. Recently, a similar topological

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concept was also investigated and demonstrated in a plasmonic system[77]. In addition, some other FTIs based on the effective field model or network model have also been proposed[78-88]. In fact, we

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can even exploit the property of microcavity polaritons for a natural magnetic field[89].

Fig. 5.1 (a) Effective potential introduced by time-dependent modulation[70]. (b) Effective potential

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introduced by spatial modulation in photonic FTIs[73]. (c) Theoretical design of robust optical delay line based on the coupled resonator optical waveguides[74]. (d) Clockwise and counterclockwise robust edge propagation for the model in (c)[75].

6. Summary and outlook The main focus of this work is limited to reviewing photonic topological states in 2D cases. There are, however, many studies concerning 1D topological phases[90], where the SSH (Su-Schrieffer-Heeger) model[91] was usually adopted to describe the geometric phase[92], named the Zak phase[93, 94], and

ACCEPTED MANUSCRIPT topological properties. In addition, the study of 3D photonic topological models, such as the photonic Weyl point[95-97] and photonic TCIs[98, 99], has recently been gaining momentum. Take the photonic Weyl point as an example. It is a degenerate point formed by the touching of two energy

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bands with a linear dispersion in all three directions. This Weyl point can be regarded as the split of the Dirac point, and the corresponding equation is H ( k ) = vx k xσ x + v y k yσ y + vz k zσ z . The topological properties of the Weyl point in the 3D Brillouin zone are stable: it can be regarded as the monopole of

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the Berry flux in the momentum space, closely related to the topological invariant Chern number.

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Recently, Lu et al. proposed and demonstrated the Weyl point by introducing parity-breaking in gyroid photonic crystals. In addition to the periodical crystal system, the topological photonic quasi-crystal has also attracted particular research interests. In these cases, the quasi-period can be treated as the projection of the high dimension on an irrational surface (line), and the relevant photonic topological

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properties can then be discussed[100-102].

The field of topological photonics is currently undergoing rapid development. There are many fundamentally as well as practically significant issues waiting to be solved. Most of the current

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demonstrations are for the purpose of illustrating the physical concepts. For example, the gyromagnetic

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medium-based photonic QHE is realized only in the microwave-frequency region for just one polarization. It is still a big challenge to conduct such kind of experiment in the visible as well as the near-infrared spectrum and for all polarization states. In addition, the effect of an important mechanism, i.e., the photon-photon interaction in the nonlinear region, on topological properties has not yet been fully explored. If the knowledge related to the electronics systems applies, it is likely that this mechanism can even lead to the photonic fractional QHE[103]. Moreover, placing the photonic topological effect in other physical systems may benefit a large scope of unexplored fields, e.g., the

ACCEPTED MANUSCRIPT fractional geometric phase in the non-Hermitian parity-time symmetric system[104]. Finally, the topological concept and theoretical model developed in photonics can be well generalized to include other bosonic systems. All the knowledge and experience learned by using the photonic system could

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be leveraged to pave a way for researching topological phases in other classic wave systems, such as acoustics[66, 67], cold atoms[105], and plasmons[106, 107]. From the application point of view, the progress of topological photonics will revolutionize current photonic devices, especially with respect to

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the property of robust backscattering-immune light propagation. This progress will benefit

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long-distance communication, reducing losses[40, 82], input power[108], device size[109] and noise. It may also have important impacts on future quantum computing and quantum simulation.

Acknowledgements

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The work was jointly supported by the National Basic Research Program of China (Grant No. 2012CB921503, 2013CB632904 and 2013CB632702) and the National Nature Science Foundation of China (Grant No. 11134006, No. 11474158, No. 11404164 ， and No. 11625418). M.H.L. also

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acknowledges the support of Natural Science Foundation of Jiangsu Province (BK20140019) and the

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support from Academic Program Development of Jiangsu Higher Education (PAPD). X.C.S also acknowledges the supported of the program A for Outstanding PhD candidate of Nanjing University (201602A013).

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