Figure 1

Identifying the quantum critical region using deviations from the fluctuation-dissipation theorem. Panels (a) and (b) show the characteristic behavior of $R\left(T\right)$ obtained using QMC at a MF QCP and a 3D $\mathit{XY}$ QCP, respectively, and clearly distinguishes between the two universality classes. At a MF QCP, the critical behavior of $R$ is determined by the slow convergence of $\delta n_i^2$ and approaches a constant at low temperatures, while at a 3D $\mathit{XY}$ QCP, $R$'s critical behavior is dominated by the compressibility's power-law critical behavior, indicated by the dashed red line. The dashed black line is the high-temperature behavior. Panel (c) shows the $T=0$ phase diagram of the BHM with the parameters of each figure indicated.Reuse & Permissions

Figure 2

Temperature scales encoded in $R$. Panels (a) and (b): high-temperature behavior of compressibility ${\kappa }_i$ (circles) and number fluctuations $\delta n_i^2$ (squares) demonstrates that the LDF $R={\kappa }_i/\delta n_i^2$ is determined by $\delta n_i^2$. The inset in (b) replots the data as $R\left(\beta \right)$ to exhibit the low-temperature behavior. Panels (c) and (d): in the MI, ${\kappa }_i$ is exponentially suppressed as $T\to 0$, while local quantum fluctuations maintain $\delta n_i^2$ at a finite value. The corresponding LDF $R$ in the MI shows a characteristic maximum near $T_{\Delta }$ and decays exponentially at low $T$. Panels (e) and (f): in the SF, ${\kappa }_i$, $\delta n_i^2$, and $R$ all approach a finite constant as $T\to 0$. The peak in $R$ occurs at ${\beta }_{\text{max}}$ and agrees well with the inverse superfluid $T_c$ (indicated by the arrow) obtained from the superfluid density ${\rho }_s$ [squares in (f)]. In all panels, the inverse temperature ${\beta }^{*}$ indicates when $R$ deviates from the classical limit and is estimated from $R\left({\beta }^{*}\right)/{\beta }^{*}=0.95$. The dashed black line indicates the high-temperature $R\approx \beta$ limit in (b), (d), and (f) and horizontal bars indicate uncertainty in ${\beta }^{*}$ and ${\beta }_{\text{max}}$.Reuse & Permissions

Figure 3

Boson bunching and antibunching. Panel (a) shows bunching ($R>\beta$) at low temperatures for $t/U=1.0$ and $\mu /U=-0.95$. This system is not superfluid at the temperatures probed. Panel (b) shows antibunching ($R<\beta$) at low temperatures for $t/U=0.15$, $\mu /U=0.05$ and the error in ${\beta }^{*}$ is indicated by the horizontal bar. The large error in ${\beta }_{\text{max}}$ (horizontal bar) is characteristic of the superfluid in the presence of strong interactions, where the peak in $R$ is difficult to resolve. The inset in each panel shows the density for the same parameters.Reuse & Permissions

Figure 4

Extracting the particle-hole gap from $R$. The right-hand column shows $R\left(T\right)$ (circles) at $t/U=0.15$ and for several $\mu /U$ values as well as the exponential fit to extract ${\Delta }_{\text{ph}}$ (solid line). The left-hand plot shows the extracted ${\Delta }_{\text{ph}}$ (circles) and the results from $T=0$ QMC calculations by Capogrosso-Sansone et al. [34] (solid line).Reuse & Permissions

Figure 5

Determination of the phase diagram of a quantum Hamiltonian using local observables: this figure shows the density plot of LDF ratio $R$ in the inverse temperature-chemical potential ($\beta t-\mu /U$) plane at (a) $t/U=0.15$ and (b) $t/U=0.4$. We determine the phase diagram from $R\left(\beta \right)$ by marking two inverse temperature scales, ${\beta }^{*}=1/T^{*}$ (solid black squares) that indicates the onset of quantum effects and ${\beta }_{\text{max}}=1/T_{\text{max}}$ (solid gray circles) that delineate phase transitions or crossovers. ${\beta }_{\text{max}}$ agrees with the characteristic temperature scales in the superfluid $1/T_c$ (open red circles) and the MI phases $1/T_{\Delta }$ (open orange squares). These are calculated independently of $R$ by examining ${\rho }_s$ [${\rho }_s\left(T_c\right)=0.1$] and ${\kappa }_i$ [${\kappa }_i\left(T_{\Delta }\right)t=0.05$]. To highlight the peak at $R\left({\beta }_{\text{max}}\right)$, the density color scheme is normalized so that $R\left({\beta }_{\text{max}}\right)=1$ for each value of $\mu$.Reuse & Permissions

Figure 6

Finite-size effects at the superfluid-normal transition at $t/U=0.08$ and $\mu /U=0.08$. We observe significant finite-size effects in $R\left(T\right)$ (a) and superfluid density ${\rho }_s$ (b). As the system size increases, the peak in $R\left(T\right)$ becomes more pronounced and the system becomes less superfluid at a given temperature. The finite-size effects are smaller for a system further from the QCP at $\mu /U\simeq 0.084$.Reuse & Permissions

Figure 7

Finite-size effects at the mean-field quantum critical point at $t/U=0.15$ and $\mu /U=0.172$. The local number fluctuations $\delta n_i^2$ (a) do not depend on the system size. The long-range observables like the compressibility ${\kappa }_i$ (b), superfluid density ${\rho }_s$ (c), and $R$ (d) do vary with the system size before converging for ${48}^2$ and ${64}^2$ site systems. Due to simulation time limitations, the error for the ${64}^2$ site system is larger than for the other system sizes.Reuse & Permissions