Probing the Bond Order Wave Phase Transitions of the Ionic Hubbard Model by Superlattice Modulation Spectroscopy

An exotic phase, the bond order wave, characterized by the spontaneous dimerization of the hopping, has been predicted to exist sandwiched between the band and Mott insulators in systems described by the ionic Hubbard model. Despite growing theoretical evidence, this phase still evades experimental detection. Given the recent realization of the ionic Hubbard model in ultracold atomic gases, we propose here to detect the bond order wave using superlattice modulation spectroscopy. We demonstrate, with the help of time-dependent density-matrix renormalization group and bosonization, that this spectroscopic approach reveals characteristics of both the Ising and Kosterlitz-Thouless transitions signaling the presence of the bond order wave phase. This scheme also provides insights into the excitation spectra of both the band and Mott insulators.

Figure 1

(a) Phase diagram of the 1D ionic Hubbard model as a function of $U$ for fixed $\mathrm{\Delta }$ and $J$. ${\mathrm{\Delta }}_c$ and ${\mathrm{\Delta }}_s$ are the charge and spin gaps. (b) The bond order parameter $|⟨\widehat{B}⟩|$ at fixed $\mathrm{\Delta }=50J$ for different system sizes calculated from DMRG keeping up to 400 states. The dashed vertical lines mark the position of the maximum of the derivative located at $U_{c1}\sim 51.2J$, $51.26J$, $51.28J$, $51.3J$ for system sizes $L=64$, 96, 128, 192, respectively. (c)–(e) Energy absorption rate for different regions of the phase diagram at fixed $\mathrm{\Delta }=50J$: (c) Deep in the band insulator for $U=10J$, $A=0.001J$, dashed vertical lines mark the width of the band of excitations; (d) At the Ising critical point for different system sizes and amplitudes $A=0.001J$ (filled symbols) and $A=0.0005J$ (open symbols). The Gaussian fits (solid and dashed lines) are guides to the eye, where the maxima are fixed to $\lambda (\hslash \omega /J{)}^{-3/4}$ with $\lambda$ chosen to agree with the $L=128$ peaks; (e) In the bond order wave phase near the KT transition for $L=64$, $U=52J$, and $A=0.005J$. Vertical dashed lines indicate multiples of the spin gap.

Figure 2

The position of the maximum of the absorption peak $\hslash {\omega }_{\mathrm{peak}}$ corresponds approximately to the charge gap on the band-insulating side ($\mathrm{\Delta }>U$) at fixed $\mathrm{\Delta }=50J$ and $L=64$. Inset: close to the transition the deviation from the naive expectation $\mathrm{\Delta }-U$ (dashed line) increases.

Figure 3

(a) Width of the low energy band of spin excitations in the Mott insulator, extracted from the energy absorption rate, as a function of the effective spin coupling $J_{XY}$ for $\mathrm{\Delta }<U$ at fixed $\mathrm{\Delta }=50J$ and $L=64$, $A=0.005J$. The dashed line is a linear fit to the data. (b) Energy absorption rate as a function of $\hslash \omega$ ($\mathrm{\Delta }=50J$, $L=64$, $A=0.005J$).