Effective action approach for quantum phase transitions in bosonic lattices

Based on standard field-theoretic considerations, we develop an effective action approach for investigating quantum phase transitions in lattice Bose systems at arbitrary temperature. We begin by adding to the Hamiltonian of interest a symmetry breaking source term. Using time-dependent perturbation theory, we then expand the grand-canonical free energy as a double power series in both the tunneling and the source term. From here, an order parameter field is introduced in the standard way and the underlying effective action is derived via a Legendre transformation. Determining the Ginzburg-Landau expansion to first order in the tunneling term, expressions for the Mott insulator-superfluid phase boundary, condensate density, average particle number, and compressibility are derived and analyzed in detail. Additionally, excitation spectra in the ordered phase are found by considering both longitudinal and transverse variations of the order parameter. Finally, these results are applied to the concrete case of the Bose-Hubbard Hamiltonian on a three-dimensional cubic lattice, and compared with the corresponding results from mean-field theory. Although both approaches yield the same Mott insulator–superfluid phase boundary to first order in the tunneling, the predictions of our effective action theory turn out to be superior to the mean-field results deeper into the superfluid phase.

Figure 1

(Color online) Plot of the critical value of the hopping parameter $t$ versus the chemical potential $\mu$, both scaled by the interaction energy $U$. The phase boundaries for two different temperatures are shown. The solid blue curve is $T∕U=0$ and the dashed red curve is $T∕U=0.1∕k_B$.

Figure 2

(Color online) Plot of the condensate density as a function of the hopping parameter $t∕U$ for both the mean-field theory (left) and the effective action theory (right), with fixed $\mu ∕U=0.9$. The solid blue curves are $T∕U=0$ and the dashed red curves are $T∕U=0.1∕k_B$.

Figure 3

(Color online) Plot of the average number of particles per site as a function of the chemical potential $\mu ∕U$ with fixed $t∕U=0.025$. The left figure shows the mean-field prediction, while the right figure shows the field-theoretic prediction. The solid blue curves are $T∕U=0$ and the dashed red curves are $T∕U=0.1∕k_B$.

Figure 4

(Color online) Plot of the compressibility $\kappa U$ as a function of the chemical potential $\mu ∕U$ with fixed $t∕U=0.025$. The left figure shows the mean-field prediction while the right figure shows the field-theoretic prediction. The solid blue curves are $T∕U=0$ and the dashed red curves are $T∕U=0.1∕k_B$.

Figure 5

(Color online) Contours of constant $⟨n⟩$ in parameter space for both the mean-field and effective action theories at $T=0$. The dotted blue curve shows the first three lobes of the phase boundary. The solid blue curves are the predictions of the effective action theory, while the dotted red curves show the predictions of the mean-field theory.

Figure 6

(Color online) Plots of the zero-temperature dispersion relations ${\omega }_A\left(\vec{k}\right)$ (left) and ${\omega }_{\theta }\left(\vec{k}\right)$ (right) for various values of the hopping $t$ with fixed $n=1$, $\mu ∕U=\sqrt{2}-1$ and with $\vec{k}=(1,1,1)k∕\sqrt{3}$. The solid blue lines corresponds to $t=t_c\approx 0.028U$, which for these values of $\mu$ and $n$ is at the tip of the first Mott lobe. The dotted yellow lines corresponds to $t=0.03U$, and the dashed red lines corresponds to $t=0.035U$. Note that amplitude excitations exhibit a $t$-dependent energy gap, while the phase excitations are gapless in accordance with Goldstone’s theorem.

Figure 7

(Color online) Plot of the second sound velocity $c$ as a function of $t∕U$ for $t>t_c\approx 0.028U$ with $n=1$ and $\mu ∕U=\sqrt{2}-1$.