Gaussian quantum fluctuations in the superfluid–Mott-insulator phase transition

Recent advances in cooling techniques make possible the experimental study of quantum phase transitions, which are transitions near absolute zero temperature accessed by varying a control parameter. A paradigmatic example is the superfluid-Mott transition of interacting bosons on a periodic lattice. From the relativistic Ginzburg-Landau action of this superfluid-Mott transition we derive the elementary excitations of the bosonic system, which contain in the superfluid phase a gapped Higgs mode and a gapless Goldstone mode. We show that this energy spectrum is in good agreement with the available experimental data and we use it to extract, with the help of dimensional regularization, meaningful analytical formulas for the beyond-mean-field equation of state in two and three spatial dimensions. We find that, while the mean-field equation of state always gives a second-order quantum phase transition, the inclusion of Gaussian quantum fluctuations can induce a first-order quantum phase transition. This prediction is a strong benchmark for future experiments on quantum phase transitions.

Figure 1

Mean-field phase diagram of the Bose-Hubbard model at zero temperature in $D$ spatial dimensions. Here $J$ is the hopping energy, $\epsilon$ is the on-site energy, $U$ is the on-site interaction energy, and $\mu$ is the chemical potential. The tips of the Mott lobes are critical points.

Figure 2

Comparison of the predictions of our theory (solid line) for the sum $\mathrm{\Delta }$ of the two gaps with experimental data (squares with error bars) of Ref. [15]. On the horizontal axis $J_c$ is the critical value of $J$ at which the critical transition of the $n=1$ Mott lobe occurs. Within our theoretical scheme $J_c$ is given by $J_c=0.043U$.

Figure 3

Pressure $P=-\mathrm{\Omega }/V$ and its derivatives with respect to the control parameter $U$, across the critical point (tip) of the $n=1$ Mott lobe, as a function of $U$. The red dashed line represents mean-field theory and the blue solid line beyond-mean-field theory. Results were obtained for the two-dimensional bosonic system, with $J=1$ and ${[(\mu -\epsilon )/U]}_c=0.414$.

Figure 4

Pressure $P=-\mathrm{\Omega }/V$ and its derivatives with respect to the control parameter $\mu -\epsilon$, across a noncritical transition point of the $n=1$ Mott lobe, as a function of $\mu -\epsilon$. The red dashed line represents mean-field theory and the blue solid line beyond-mean-field theory. Results were obtained for the two-dimensional bosonic system, with $J=1$ and $U=25$.