Topological bands in two-dimensional networks of metamaterial elements

We show that topological frequency band structures emerge in two-dimensional electromagnetic lattices of metamaterial components without the application of an external magnetic field. The topological nature of the band structure manifests itself by the occurrence of exceptional points in the band structure or by the emergence of one-way guided modes. Based on an EM network with nearly flat frequency bands of nontrivial topology, we propose a coupled-cavity lattice made of superconducting transmission lines and cavity QED components which is described by the Janes-Cummings-Hubbard model and can serve as simulator of the fractional quantum Hall effect.


I. INTRODUCTION
The topological description of the quantum states of matter sets in a new paradigm in the description and classification of atomic solids. Namely, atomic solids whose energy band structure possess nontrivial topological properties constitute a new class of materials whose salient properties are robust to phase transitions which modify the symmetry order The advent of artificial electromagnetic (EM) structures such as photonic crystals and metamaterials has established over the years a continuous conveyance of ideas and methods from atomic solids to their EM counterparts. Quite naturally, the concept of topological order has been adapted to photonic crystals starting with the QHE: a two-dimensional (2D) lattice of gyromagnetic/ gyroelectric cylinders is a system with broken time-reversal symmetry 1 with frequency bands characterized by a nonzero Chern number, allowing for the emergence of unidirectional (one-way) edge states 2 in analogy with the chiral edges states in QHE systems such as a 2D electron gas or graphene nanoribbons under magnetic field.
Anomalous QHE can also be simulated with artificial chiral metamaterials of gyromagnetic components. 3 In TIs 4 and quantum spin Hall systems 5 in 2D, the presence of magnetic field is not prerequisite for the appearance of topological electron states. In analogy with atomic TIs, in certain 3D photonic crystals and metamaterials with proper design, topological frequency bands appear without comprising gyromagnetic/ gyroelectric materials which require the application of external magnetic field in order to break time-reversal symmetry. [6][7][8] In this Letter, we propose a class of 2D EM networks possessing topological frequency bands without the application of an external magnetic field. Namely, we show that topological bands emerge in 2D lattices of EM resonators connected with left-and right-handed metamaterial elements such as transmission lines or waveguides loaded with a negative refractive-index medium. The topological nature of the corresponding frequency bands is manifested by the emergence of an exceptional point for transverse electric (TE) and by the generation of one-way modes for transverse magnetic (TM) waves. In the latter case, the system can be viewed as a simulator of the FQHE for polaritons.

II. LATTICE OF COUPLED DIPOLES.
The EM crystals under study here are amenable to a photonic tight-binding description within the framework of the coupled-dipole method. 9 The latter is an exact means of solving Maxwell's equations in the presence of nonmagnetic scatterers. We consider a lattice of cavities within a lossless metallic host. The i-th cavity is represented by a dipole of moment P i = (P i;x , P i;y , P i;z ) which stems from an incident electric field E inc and the field which is scattered by all the other cavities of the lattice. This way the dipole moments of all the cavities are coupled to each other and to the external field leading to the coupled-dipole equation (1) is the electric part of the free-space Green's tensor and α i (ω) is the polarizability the i-th cavity. Eq. (1) is a 3N × 3N linear system of equations where N is the number of cavities of the system.
For a particle/cavity of electric permittivity ǫ embedded within a material host of permittivity ǫ h , the polarizability α is provided by the Clausius-Mossotti formula α = (3V /4π)(ǫ − ǫ h )/(ǫ + 2ǫ h ), where V is the volume of the particle/ cavity. For a lossless Drude-type (metallic) host i.e., ǫ h (ω) = 1 − ω 2 p /ω 2 (where ω p is the bulk plasma frequency), the polarizability α exhibits a pole at ω 0 = ω p 2/(ǫ + 2) (surface plasmon resonance). By making a Laurent expansion of α around ω 0 and keeping the leading term, 8 we may write For sufficiently high value of the permittivity of the dielectric cavity the electric field of the surface plasmon is much localized at the surface of the cavity. As a result, in a periodic lattice of cavities, the interaction of neighboring surface plasmons is very weak leading to much narrow frequency bands. By treating such a lattice in a tight binding (TB) manner, 8 we may assume that the Green's tensor G ii ′ (ω) does not vary much with frequency and therefore, G ii ′ (ω) ≃ G ii ′ (ω 0 ). In this case, Eq. (1) becomes an eigenvalue problem where F has been absorbed within the definition of G ii ′ (ω 0 ) and we have set E inc = 0 in Eq.
(1) as we are seeking the eigenmodes of the system of cavities. In the following, we will be dealing with 2D lattices of cavities. We can, therefore, treat separately the case where the electric field lies within the plane of cavities (TE modes) from the case where the electric field is perpendicular to the plane (TM modes).

A. TE modes
In this case, P i = (P i;x , P i;y ) and the Green's tensor G ii ′ (ω 0 ) is given by with Since we focus our attention around the surface plasmon frequency ω 0 , we operate in the subwavelength regime where q 0 |r ii ′ | ≪ 1. In this regime, the functions C(q 0 |r ii ′ |), J(q 0 |r ii ′ |) are written as where t ii ′ and φ ii ′ are real numbers. In what follows, the cavities are connected via coupling elements, i.e., waveguides or transmission lines, in which case the phase factors φ ij are not necessarily related with the wavevector of the host medium ǫ h and can therefore be considered as independent parameters.
For a 2D lattice of cavities, we assume the Bloch ansatz for the polarization field, i.e., The cavity index i becomes composite, i ≡ nβ, where n enumerates the unit cell and β the positions of inequivalent cavities in the unit cell. Also, R n denotes the lattice vectors and k = (k x , k y ) is the Bloch wavevector. By substituting Eq. (5) into Eq.
(2) we finally obtain Solution of Eq. (6) provides the TE frequency band structure of a periodic system of cavities.
In order to seek for topological Bloch modes in a 2D lattice, we need at least two distinct frequency bands. Since, TE modes correspond to two degrees of freedom for the polarization field, i.e., (P x , P y ) we may consider a 2D lattice with one cavity per unit cell. Namely, we consider the square lattice of Fig. 1a where we consider nearest-neighbor (NN) and nextnearest-neighbor (NNN) hoppings of the EM field among the cavities. The NN hopping carries a nonzero phase t exp(±φ) whose signs are denoted by the arrows in Fig. 1a. The NNN hopping is denoted by t ′ . For the lattice of Fig. 1a, the Green's tensorG of Eq. (7) becomes

B. TM modes
Next, we assume that the polarization at each dipole is oriented in the z-axis. In this case, the eigenvalue problem of Eq.
The occurrence of topological properties such as the exceptional point in Fig. 1b and the one-way modes in Fig. 3 are a result the synthetic gauge field which is generated by the geometry of the metamaterial-based coupling elements (formation of closed flux loops of the phase of the EM field).

III. SIMULATION OF THE FQHE
Having established a nearly flat topological frequency band of the EM field for the lattice of Fig. 2a, we are able to design a system for creating an EM analog of the FQHE. The most natural choice would be to consider a coupled cavity array (CCW) wherein polaritons propagate through a hopping mechanism (as in our case) and interact strongly with the reservoir of modes when they reside within the cavity. 20,21 FQHE with magnetic field can also be simulated by atoms confined in a 2D CCW. 22 As stated above, the EM lattice of  Fig. 2c (Fig. 2d). This means that the topological bands lie in the GHz regime. Therefore, in order to simulate the FQHE for microwave photons we need to implement a cavity QED scheme in this regime. This can be achieved by considering a superconducting-circuit cavity QED system 23,24 consisting of a Cooper-pair box (CPB) coupled to a TL resonator (see the equivalent circuit of Fig. 2b). The CPB operates as an artificial atom (two-level system) 25,26 and couples to the microwave photons of a superconducting TL resonator which plays the role of an on-chip cavity reservoir. The microwave response of superconducting circuit cavity QED system is described by the Jaynes-Cummings Hamiltonian 23,24 where ω r = 1/ √ LC is the frequency of the superconducting resonator, a + i (a i ) creates (annihilates) a microwave photon in the TL resonator (cavity), σ + i (σ i ) creates (annihilates) an excitation in the CPB, g is the coupling parameter between the CPB and the TL resonator, where H T B is the tight-binding form of the Hamiltonian of the microwave photons propa-gating within the checkerboard lattice of Fig. 2a, i.e., which is the direct-space representation of the Green's tensor of Eq. (10). An important ingredient which gives rise to the FQHE is the presence of repulsive interactions among the microwave photons and is inherently present in Eq. (13) as photon blockade. 27 The latter phenomenon has been recently observed experimentally in the GHz regime for superconducting circuit cavity QED systems such as the one considered here (CPB + TL resonator). 28,29 The different FQHE phases can be calculated by direct-diagonalization of the Hamiltonian of Eq. (13). We note that the proposed quantum simulator for the FQHE differs fundamentally with previous proposals 30, 31 since it essentially constitutes a passive design requiring no externally applied electric or magnetic fields.
Some typical values of the Hamiltonian of Eq. (13) are: 23 ω r = 38 GHz, E J,max = 8GHz, 314 GHz. The latter parameter, g, is much larger than the loss rate of the TL resonator (∼ 0.005 GHz) and the decoherence rate of the CPB (∼ 0.004 GHz). The frequency ω r of the TL resonator should fall within the operating bandwidth of the LH-and RHTLs. In Fig. 2c and Fig. 2d we have considered ideal TL which have infinite bandwidth.
However, actual LH-and RHTLs have very large bandwidth which is a distinctive feature of nonresonant metamaterials compared to the resonant ones. 16 Lastly, if the superconducting QED chip has a thickness of 1mm, the coupling TLs (RH or LH) have 10mm length and the TL resonator covers an area of 30mm 2 , for the given resonator frequency (38GHz), the hopping strength t is about 0.5GHz which is also significantly larger than both the TL loss and CPB decoherence rates.
In order to probe experimentally the FQHE with the proposed structure one needs to create the phase diagram of the spectrum gap between the FQHE ground-state manifold and the lowest excited states, as a function of the coupling parameters g for NN and NNN hopping when the latter lie in the photon blockade regime. Generally speaking, in the FQHE state the spectral gap assumes much larger values than in superfluid and solid phases. 18 The frequencies of the ground and excited states (and thus their corresponding gaps) can be measured by microwave transmission experiments.

IV. CONCLUSION
In conclusion, we have shown that topological frequency bands emerge in 2D electromagnetic lattices of metamaterial components in the absence of an applied magnetic field. The topological nature of the corresponding band structures gives rise to significant phenomena such as one-way waveguiding and coalescence of EM modes. The above lattices can be the basis for realizing a simulator for the FQHE based on superconducting transmission lines and circuit cavity QED systems. * Electronic address: vyannop@upatras.gr