Topological Higgs amplitude modes in strongly interacting superfluids

By studying the two-dimensional Su-Schrieffer-Heeger-Bose-Hubbard model, we show the existence of topological Higgs amplitude modes in the strongly interacting superfluid phase. Using the slave boson approach, we find that, in the large filling limit, the Higgs excitations and the Goldstone excitations above the ground state are well decoupled, and both of them exhibit nontrivial topology inherited from the underlying noninteracting bands. At finite fillings, they become coupled at high energies; nevertheless, the topology of these modes remains unchanged. Based on an effective action analysis, we further provide a universal physical picture for both their topological and phase-amplitude character at both infinite and finite fillings in a unified way. The discovery of topological Higgs amplitude modes in this paper opens the path to investigations in various systems, such as superconductors and quantum magnets.

Figure 1

(a) Two-dimensional SSH-BHM on a square lattice. A unit cell is enclosed by a gray dashed square with four sublattices labeled by an index $\eta =1,\cdots ,4$. The intracell (intercell) hopping strength is $-t_1(-t_2)$, shown in the blue (red) color. Black arrows are two primitive lattice vectors ${\mathbf{a}}_{1,2}$. We set $|{\mathbf{a}}_1|=|{\mathbf{a}}_2|=1$ as the length unit. (b) The first Brillouin zone (BZ) of the model. Four inversion symmetric points are shown explicitly. The red vectors denote the high-symmetric path used in Figs. 3 and 3.

Figure 2

Excitation spectrum of the 2D SSH-BHM in a ribbon geometry in the large filling limit for $\tilde{t}=3U/16$. (a) ($t_1/t_2=1/8$) and (c) ($t_1/t_2=8$) are the Goldstone phase modes. (b) ($t_1/t_2=1/8$) and (d) ($t_1/t_2=8$) are the Higgs amplitude modes. Bulk modes are shown by solid dots, whereas topological edge modes are shown by triangles. All modes are doubly degenerate due to inversion symmetry. (e) and (f) are typical wave functions of topological edge modes for phase modes and amplitude modes, respectively. And the gray lines are their inversion symmetric partners.

Figure 3

Excitation spectrum of the 2D SSH-BHM under PBCs (a) and (d) and in a ribbon geometry (b), (c), (e), and (f) at filling $q=1$ for (a)–(c) and $q=50$ for (d)–(f) for $\tilde{t}=3U/16$ with color code indicating the flatness defined in Eq. (7). Bulk (topological edge) modes are plotted by dots (triangles) with typical wave functions shown in (i) and (j) (only one of the two degenerate modes is plotted). In (g) and (h) we zoom in on the excitation spectrums at high energies for (c) and (f), respectively.

Figure 4

Phase diagram of the $dD$ SSH-BHM with blue lines separating the MI phase and the SF phase. It is the same as the standard BH model with the hopping parameter $t=(t_1+t_2)/2$. Here the coordination number $z=2d=4$ for the 2D SSH-BHM. The black dotted horizontal lines start from each middle of the lobe. The black dashed horizontal lines start from each tip of the lobe. The black solid lines are integer filling lines in the SF phase obtained from Eq. (B7). We always set the chemical potential $\mu$ on these black solid lines where the excitations have the most visible phase-amplitude character. In particular, the red dot corresponds to the case in Figs. 3– 3 the main text.