Excitation spectra of strongly correlated lattice bosons and polaritons

Spectral properties of the Bose-Hubbard model and a recently proposed coupled-cavity model are studied by means of quantum Monte Carlo simulations in one dimension. Both models exhibit a quantum phase transition from a Mott insulator to a superfluid phase. The dynamic structure factor $S\left(k,\omega \right)$ and the single-particle spectrum $A\left(k,\omega \right)$ are calculated, focusing on the parameter region around the phase transition from the Mott insulator with density one to the superfluid phase, where correlations are important. The strongly interacting nature of the superfluid phase manifests itself in terms of additional gapped modes in the spectra. Comparison is made to recent analytical work on the Bose-Hubbard model. Despite some subtle differences due to the polaritonic particles in the cavity model, the gross features are found to be very similar to the Bose-Hubbard case. For the polariton model, emergent particle-hole symmetry near the Mott lobe tip is demonstrated and temperature and detuning effects are analyzed. A scaling analysis for the generic transition suggests mean-field exponents, in accordance with field-theory results.

Figure 1

Zero-temperature phase diagram for (a) the Bose-Hubbard model and (b) the polariton model in 1D. We only show the Mott lobes with density one. These DMRG results were obtained by (a) Kühner et al. [14] and (b) Rossini and Fazio [39].

Figure 2

(Color online) Single-boson spectral function $A\left(k,\omega \right)$ of the 1D Bose-Hubbard model, for different hoppings $t$ (and total density $n$), corresponding to (a) the MI phase, (b) just below the MI-SF transition, (c) just above the transition, and (d) the SF phase. Here, $\mu /U=0.5$, $L=64$, and $\beta U=3L$. Here and in subsequent spectra, the symbols and error bars indicate the maxima of the peaks and the associated errors obtained by the maximum entropy method. As discussed in Sec. 4A2, features with very small spectral weight are difficult to determine accurately. Solid red lines in (a) are mean-field results [63]. Solid lines in (c) and (d) are the Bogoliubov results, while dashed lines are a fourth-order approximation (see text) [62].

Figure 3

(Color online) Quasiparticle weights ${\mathsf{u}}_k$ and ${\mathsf{v}}_k$ of the gapless modes at $t/U=0.2$. The symbols are integrated intensities from QMC and maximum entropy; the lines are the predictions from Bogoliubov theory. Inset shows data at $t/U=0.14$. Again, $\mu =0.5$, $L=64$, and $\beta =3L$.

Figure 4

(Color online) Dynamic structure factor $S\left(k,\omega \right)$ of the Bose-Hubbard model for the same parameters as in Fig. 2. Panel (d) includes the same analytical approximations as Fig. 2d.

Figure 5

(Color online) Single-photon spectral function $A^{\text{ph}}\left(k,\omega \right)$ of the 1D polariton model at $\mu /g=0.4$ for different hoppings $t$, corresponding to (a) deep in the MI, (b) just below the MI-SF transition, (c) just above the transition, and (d) in the SF phase. Here, $\beta g=3L$ and $L=64$. With increasing $t$, the density plots are more and more “overexposed” to see less dominant features.

Figure 6

(Color online) Single-photon spectral function in the Mott phase for the same parameters as in Fig. 5 showing additional excitations at higher energies.

Figure 7

(Color online) Quasiparticle weights ${\mathsf{u}}_k$ and ${\mathsf{v}}_k$ of the gapless modes of the polariton model, similar to Fig. 3.

Figure 8

(Color online) Single-photon spectrum $A^{\text{ph}}\left(k,\omega \right)$ of the polariton model along the line $n_p=1$ crossing the Kosterlitz-Thouless transition. Here, $L=64$ and $\beta g=3L$.

Figure 9

(Color online) Effective particle $\left(m^{+}\right)$ and hole masses $\left(m^{-}\right)$ along the line $n_p=1$, as obtained from fits to the bands in $A^{\text{ph}}\left(k,\omega \right)$ near $k=0$.

Figure 10

(Color online) Polariton dynamic structure factor $S\left(k,\omega \right)$ for the same parameters as in Fig. 5. Insets show an extrapolation of the Mott gap at small $k$ to $L\to \infty$.

Figure 11

(Color online) Dynamic structure factor for excitons $\left(S^{\text{at}}\right)$ and photons $\left(S^{\text{ph}}\right)$ for the same parameters as Fig. 10a.

Figure 12

(Color online) Temperature dependence of $A^{\text{ph}}\left(k,\omega \right)$ in the polariton model at $\mu /g=0.4$ in the MI phase (left) and in the SF phase (right). Here again $L=64$. Corresponding results at $\beta g=192$ can be found in Figs. 5a, 5d.

Figure 13

(Color online) Single-photon spectra with detuning $\Delta =ϵ-{\omega }_0$ for the [(a) and (b)] excitonic case $\Delta /g=-2$, $\mu /g=-0.5$ and [(c) and (d)] photonic case $\Delta /g=2$, $\mu /g=0.64$ in [(a) and (b)] the MI and [(c) and (d)] in the SF. Here, $L=64$ and $\beta g=3L$.

Figure 14

(Color online) Finite-size scaling for the generic transition in the polariton model at $\mu /g=0.4$, testing the scaling hypothesis Eq. (12).