Supersolid and pair correlations of the extended Jaynes-Cummings-Hubbard model on triangular lattices

We study the extended Jaynes-Cummings-Hubbard model on triangular cavity lattices and zigzag ladders. By using density-matrix renormalization-group methods, we observe various types of solids with different density patterns and find evidence for light supersolids, which exist in extended regions of the phase diagram of the zigzag ladder. Furthermore, we observe strong pair correlations in the supersolid phase due to the interplay between the atoms in the cavities and atom-photon interaction. By means of cluster mean-field simulations and a scaling of the cluster size extending our analysis to two-dimensional triangular lattices, we present evidence for the emergence of a light supersolid in this case also.

Figure 1

Photon denoted by a red symbol is tunneling between two different cavities which are labeled by $i$ and $i+1$, and $t$ is the hopping strength. In each cavity, the atom has two energy levels which are labeled by two separated horizontal lines.

Figure 2

Sketch of the extended JCH model on a triangular cavity ladder geometry.

Figure 3

Sketch of the typical density configurations (blue bullets) for the ladder for the (a) ${\mathrm{DW}}_{1/3}$ phase ($t/U=0.01,\mu /U=-0.88$), (b) ${\mathrm{DW}}_{1/2}$ phase ($t/U=0.01,\mu /U=-0.82$), and (c) ${\mathrm{DW}}_{2/3}$ phase ($t/U=0.01,\mu /U=-0.74$). The black lines indicate the strength of the nearest-neighbor density correlations $\langle n_i^b\rangle \langle n_j^b\rangle$ (one-site MF simulation; see below).

Figure 4

Phase diagram of the extended JCH model on a quasi-1D zigzag ladder (DMRG simulations) in the limit of vanishing cavity coupling $t=0$, as discussed in the text, as a function of the chemical potential $\mu$ and the nearest-neighbor interaction $V$.

Figure 5

Phase diagram of the extended JCH model on a quasi-1D zigzag ladder (DMRG simulations, $V/U=0.4$). We observe several gapped DW phases at fillings $1/3,1/2$, and $2/3$, as well as a MI region at unit filling (not shown). The green shaded regions mark the pair-supersolid (P-SS) surrounding the ${\mathrm{DW}}_{1/2}$ phase (see text).

Figure 6

$\mu -\rho$ curve for the extended JCH model on a quasi-1D zigzag ladder (DMRG simulations, $V/U=0.4,L=96$ sites) for different values of $t/U=0.002,t/U=0.008$, and $t/U=0.015$ (from left to right). The curves have been shifted by 0.1 among each other for clarity. The paired phase can be seen in this diagram by the presence of steps of $\mathrm{\Delta }N=2$. The arrow marks the transition to the SF phase with $\mathrm{\Delta }N=1$ for the $t/U=0.015$ curve.

Figure 7

$\mu -\rho$ curve for the extended JCH model on a quasi-1D zigzag ladder (DMRG simulations, $V/U=0.4,L=96$ sites, $t/U=0.008$) for the hard-core and the soft-core JCH model (allowing for the occupation of two photons per cavity). Within the given parameters both curves coincide almost perfectly.

Figure 8

Cuts through the phase diagram of Fig. 5 for a fixed density $\rho =5/12$. (a) Single- and two-particle excitation gap $\mathrm{\Delta }{E}_1$ and $\mathrm{\Delta }{E}_2$ for (top to bottom and light to dark colors) $L=48,96$, and 144 sites as well as the extrapolation to the thermodynamic limit. (b) (Single-particle) momentum distribution for different system sizes $L=48,96,144$, and 192. The lower black line is the extrapolation to the thermodynamic limit using a higher-order polynomial.

Figure 9

DMRG method detailed description of the extended JCH model on the triangular zigzag ladder with $V/U=0.4,\rho =0.416$ ($L=192$ sites) for (a) $t/U=0.004$ (P-SS phase), (b) $t/U=0.012$ (P-SS), and (c) $t/U=0.026$ (SF). While $C_a^{\sigma }\left(r\right),C_n^{\sigma }\left(r\right)$ show always a clear algebraic decay, the single-particle correlation $C_a\left(r\right)$ is only algebraic in the SF phase, while exponentially in the P-SS phases.

Figure 10

Various correlation functions in the P-SS phase of the extended JCH model on the triangular zigzag ladder ($V/U=0.4,\rho =0.416,L=192$ sites, $t/U=0.012$). DMRG data.

Figure 11

Structure factor $S\left(k\right)$ and momentum distribution $n\left(k\right)$ for the zigzag ladder for $\rho =5/12,V/U=0.4$ and (a) $t/U=0.008$, P-SS phase, and (b) $t/U=0.026$, SF phase ($L=96$ sites, DMRG simulation). For (a), in the P-SS the third local maximum of the structure factor is also observed.

Figure 12

Cuts through the phase diagram of Fig. 5 for a fixed density $\rho =5/12$. (a) DW order in the SS phase. Scaling of the peak maximum of the structure factor in Eq. (9) $S\left(k_{max}\right)$ for different system sizes $L=48,96,144$, and 192. The lower black line is the extrapolation to the thermodynamic limit using a higher-order polynomial. (b) Scaling of the fidelity susceptibility ${\chi }_{\mathrm{FS}}/L$ according to Eq. (11) across the P-SS to SF phase transition.

Figure 13

Scaling of the peak maximum of the structure factor in Eq. (9) $S\left(k_{max}\right)$ for different system sizes $L=12\times 3,24\times 3$, and $36\times 3$ sites of the three-leg ladder JCH model for a fixed density $\rho =5/12$ ($U=1,V=0.4$). The lower black line is the extrapolation to the thermodynamic limit using a higher-order polynomial.

Figure 14

Examples of different cluster geometries employed in the simulations: single site MF cluster (a) with $nc=1$ and $N_c=36$ one-site MF sites, triangular clusters [(b) with $nc=3$ and (c) with $nc=6]$ with $N_c=2$, and square tessellations [(d) with $nc=6$ and (e) with $nc=12$ sites] with $N_c=3$ independent clusters.

Figure 15

Sketch of the typical density configurations (blue bullets) for (from one-site method) the (a) ${\mathrm{DW}}_{1/3}$ phase ($t/U=0.015,\mu /U=-0.9$) and (b) ${\mathrm{DW}}_{2/3}$ phase ($t/U=0.015,\mu /U=-0.7$). The black lines indicate the strength of the nearest-neighbor density correlations $\langle n_i^b\rangle \langle n_j^b\rangle$.

Figure 16

Schematic phase diagram of the JCH model as obtained from the one-site MF approach ($N_c=6\times 6$) in the $\mu -t$ plane for $V/U=0.4$. The colors depict the (a) superfluid order parameter $\mathrm{\Psi }$ and (b) supersolid order $\delta \mathrm{\Psi }$.

Figure 17

Sketch of the typical density configurations (blue bullets) for (from one-site MF method) the (a) ${\mathrm{SS}}_{1/3}$ phase ($t/U=0.022,\mu /U=-0.79$) and (b) ${\mathrm{SS}}_{2/3}$ phase ($t/U=0.022,\mu /U=-0.74$). The red lines indicate the strength of the bond kinetic energy $|\langle a_i^{†}\rangle \langle a_j\rangle |$.

Figure 18

CMF method detailed description of the (a) total density $\rho$, and density of photons and excitations ${\rho }^a$ and ${\rho }^{\sigma }$, (b) SF order $\mathrm{\Psi },{\mathrm{\Psi }}^a$ and ${\mathrm{\Psi }}^{\sigma }$, and (c) (super)solid order $\delta \rho$ and $\delta \mathrm{\Psi }$ vs $\mu /U$ with $V/U=0.4,t/U=0.018$ on the triangular lattices. CMF data for clusters of three-site triangular shape (B in Fig. 14).

Figure 19

Comparison and scaling of the CMF results for various cluster sizes for (a) the density $\rho$, (b) the SF order $\mathrm{\Psi }$, and (c) the supersolid order $\delta \mathrm{\Psi }$ vs $\mu /U$ with $V/U=0.4,t/U=0.015$ on the triangular lattice. The different curves show the one-site MF simulation (simulation of a $N_c=36$ site-unit cell), different clusters of triangular shape (B and C in Fig. 14 with $nc=3$ and $nc=6$ sites, $N_c=2$), and square parketting (D and E in Fig. 14 with $nc=6$ and $nc=12$ sites, $N_c=3$).

Figure 20

Sketch of the typical density configurations (blue bullets) for (from the 12-site MF method) the (a) and (d) ${\mathrm{DW}}_{2/3}$ phase ($t/U=0.015,\mu /U=-0.72$), (b) and (e) ${\mathrm{SS}}_{2/3}$ phase ($t/U=0.015,\mu /U=-0.752$), and (c) and (f) ${\mathrm{SS}}_{1/3}$ phase ($t/U=0.015,\mu /U=-0.784$). The black lines indicate the strength of the nearest-neighbor density correlations and bond kinetic energy: (a)–(c) the disconnected part $\langle n_i^b\rangle \langle n_j^b\rangle$ and $|\langle a_i^{†}\rangle \langle a_j\rangle |$ and (d)–(f) the connected two-site correlators in the clusters $\langle n_i^b{n}_j^b\rangle$ and $|\langle a_i^{†}{a}_j\rangle |$.