Topological phase transition induced by modulating unit cells in photonic Lieb lattice

TL;DR

This work demonstrates a hierarchy of topological phases in gyromagnetic photonic Lieb lattices, induced by broken time-reversal symmetry and deliberate unit-cell deformations. By combining uniform and deformed sublattice radii with shifted positions, the authors realize first-order Chern phases and second-order quadrupole phases, and in some cases dipole phases, with phase boundaries read from bandgap closures. Topological invariants are computed via Wilson-loop and nested Wilson-loop methods, yielding $P_x$, $P_y$, and $q_{xy}$ that confirm edge and corner state realizations under open boundaries. The findings highlight sublattice engineering as a versatile route to multifunctional, disorder-resistant photonic devices and provide design principles extendable to other lattice platforms and wave systems.

Abstract

Topological photonics was embarked from realizing the first-order chiral state in gyromagnetic media, but its higher-order states were mostly studied in dielectric lattice instead. In this paper we theoretically unveil a hierarchy of topological phases under broken time-reversal symmetry, which include the first-order Chern, and the second-order dipole, quadrupole phases. Concretely, by relaxing a certain spatial symmetry of unit cell, versatile topological phases including both edge and corner states can be established to transit around, with bandgap closures marking the phase boundaries. Our results on gyromagnetic photonic crystals may broaden the scope of sublattice engineering design for topological phase manipulation, potentially enabling multifunctional disorder-resistant waveguides and integrated photonic circuits for information communication.

Figures (12)

• Figure 1: (a) Schematic for the primitive Lieb lattice consisting of YIG rods in air. The labels A, B and C represent three disks as sublattices. (b) Unit cell for the primitive lattice, with uniform radius $r$ of the gyromagnetic sublattices. (c) Deformed unit cell with non-uniform radii $r_A, r_B, r_C$ of three sublattices. (d) Deformed unit cell with shifted sublattices A and C, for which the shifted distances are represented by $\delta d_A$ and $\delta d_C$ respectively.
• Figure 2: (a) The band gap diagram of the primitive lattice with various topological phases as the radius $r$ of three sublattices increases. The coloration for each topological phase is indicated in its legend. (b) Band structure of the gyromagnetic Lieb PhC with radius $r=0.07a$, with the Chern number of each band marked. The inset represents the unit cell structure of the primitive lattice, and the coloration for phase follows the legend of panel (a). (c) The Wannier center distributions of each band in four panels as follows. Top panel: the Wannier center $v_x$ of the first band gap with trivial dipole polarization $P_x=0$, upper panel: the polarization of the Wannier center $p_y^{v_x}$ of the first band gap with non-trivial quadrupole moment; lower panel: the Wannier center $v_x$ of the fourth band with trivial polarization, bottom panel: $v_x$ of the fifth and sixth bands with non-trivial Chern phase with $C=1$. (d) Projected band diagram of a super cell consisting of $1\times 20$ unit cells, with $r=g=0.07a$. The red curves represent the edge states in gaps. The inset shows the electric field profile $E_z$ for the point marked by the pentagram. (e) Eigenstate diagram of a super cell consisting of $10 \times10$ unit cells with $r=0.07a, g=0.12a$, bounded by four OBCs along with an air gap beneath. The inset shows the electric field $E_z$ for the corner state marked by the black hexagram.
• Figure 3: (a) The band gap diagram with different topological phases as the radius $r_A$ increases. We fix $r_C=0.12a$ and request $r_A=r_B$ to introduce non-uniform radii. The coloration for each topological phase is indicated in its legend. (b) Band structure of such deformed Lieb PhC (type I lattice) with $r_A=r_B=0.09a$ and $r_C=0.12a$, with the Chern number of each band marked. The inset represents the unit cell structure, with each sublattice colored following panel (a). (c) The Wannier center $v_x$ and $v_y$ of the first three bands and their polarization $p_y^{v_x}$ and $p_x^{v_y}$. These Wannier centers indicate trivial dipole phases, and the polarization of the Wannier centers shows a non-trivial quadrupole phase. (d) Projected band diagram of the super cell consisting of $1 \times 20$ type I lattice. There are two air gaps with $g=0.12a$ at the top and bottom boundaries. The inset shows the electric field profile corresponding to the edge state marked by the pentagram. (e) Eigenstates of the super cell consisting of $10\times10$ type I lattice with the air gaps with $g=0.12a$ at the four boundaries. And the inset shows the electric field profile corresponding to the corner state marked by the hexagram.
• Figure 4: (a) The band gap diagram with different topological phases as the radius $r_A$ increases. Here we fix $r_B=0.12a$ and define $r_A=r_C$ to introduce non-uniform radii. The coloration for each topological phase is indicated in its legend. (b) Band structure of type II lattice with radius $r_A=r_C=0.02a$ and $r_C=0.12a$, with the Chern number of each band marked. The inset represents the unit cell structure, and the coloured gaps indicate corresponding phases following panel (a). (c) The Wannier center $v_x$ distributions of each band. Top panel: the Wannier band of the first band, which shows non-trivial dipole phase; middle two panels: the Wannier band of the second and the third band, which show non-trivial Chern phases with $C=1$ and $C=-2$ respectively; Bottom panel: the Wannier band of the first three bands, which shows non-trivial Chern phase $C=-1$. (d) Projected band diagram of the super cell consisting of $1\times20$ type II lattices. There are two air gaps with $g=0.5a$ at the top and bottom boundaries respectively. The inset shows the electric field profile corresponding to the edge state marked by the pentagram. (e) Eigenstates of the super cell consisting of $10\times10$ type II lattices. The air gaps are $g=0.6a$ at the four boundaries and the inset shows the electric field profile corresponding to the corner state marked by the hexagram.
• Figure 5: (a) The band gap diagram with different topological phases as the shifted distance $\delta d_A$ increases. Here we fix the position of sublattice C, i.e. $\delta d_C=0$ and define $r_A=r_B=r_C=0.12a$. The coloration for each topological phase is indicated in its legend. (b) Band structure of the type III lattice with $\delta d_A=0.2a$, with the Chern number of each band marked. The inset represents the unit cell and the coloured gaps indicate corresponding phases following panel (a). (c) The Wannier center $v_x$ and $v_y$ distributions of the first three bands and their polarization $p_y^{v_x}$ and $p_x^{v_y}$. (d) Projected band diagram of the super cell consisting of $1\times20$ type III lattices, with $g=0.12a$ at the top and bottom boundaries. The inset shows the electric field profile corresponding to the edge state marked by the pentagram. (e) Eigenstates of the super cell consisting of $10\times10$ type III lattices, with air gaps $g=0.12a$ at the four boundaries and the inset for the electric field profile of the corner state marked by the hexagram.