Dynamics and conductivity near quantum criticality

Relativistic O($N$) field theories are studied near the quantum-critical point in two space dimensions. We compute dynamical correlations by large-scale Monte Carlo simulations and numerical analytic continuation. In the ordered side, the scalar spectral function exhibits a universal peak at the Higgs mass. For $N=3$ and 4, we confirm its ${\omega }^3$ rise at low frequency. On the disordered side, the spectral function exhibits a sharp gap. For $N=2$, the dynamical conductivity rises above a threshold at the Higgs mass (density gap), in the superfluid (Mott insulator) phase. For charged bosons (Josephson arrays), the power-law rise above the Higgs mass increases from two to four. Approximate charge-vortex duality is reflected in the ratio of imaginary conductivities on either side of the transition. We determine the critical conductivity to be ${\sigma }_{\mathrm{c}}^{*}=0.3(\pm 0.1)\times 4e^2/h$ and describe a generalization of the worm algorithm to $N>2$. We use a singular value decomposition error analysis for the numerical analytic continuation.

Figure 1

Curves of $L{\rm Y}$ for a sequence of increasing system size $L$ for O$(N=2,3)$ models. The curves cross at a single point, from which we determine the value of $g_{\mathrm{c}}$. Here, we take $\mu =-0.5$ and $g_{\mathrm{c}}=2.568\left(2\right)$ for the $N=2$ case and $g_{\mathrm{c}}=1.912\left(2\right)$ for $N=3$.

Figure 2

Scaling of the gap $\Delta \left(\delta g\right)$ in the disordered phase for $N=2,3$ and $\mu =-0.5$. Fitting to the scaling form $\Delta ={\Delta }_0{\left(\delta g\right)}^{\nu }$ gives ${\Delta }_0$=1.86(1) for $N=2,\mu =-0.5$, and ${\Delta }_0$ $=$ 1.96(1) for $N=3,\mu =-0.5$. Error bars are smaller than the symbols.

Figure 3

Scaling of the helicity modulus ${\rm Y}\left(\delta g\right)$ in the ordered phase for $N=2,3$ and $\mu =-0.5$. Fitting to the scaling form ${\rm Y}={{\rm Y}}_0{\left(\delta g\right)}^{\nu }$ gives ${{\rm Y}}_0$ $=$ 0.83(1) for $N=2,\mu =-0.5$ and ${{\rm Y}}_0$ $=$ 0.67(1) for $N=3,\mu =-0.5$. Error bars are smaller than the symbols.

Figure 4

(a) The scalar susceptibility ${\chi }_s\left(i{\omega }_m\right)$ for $N=2$. The curves correspond to different values of $\delta g$ below and above the phase transition. (b), (c) Universal scaling function after rescaling for $N=2,3$. In (b) we show the scaling function for two crossing points of the phase transition. The two rescaled curves agree very well, especially at low frequencies. Simulations were performed with $\mu =0.5$ and 2 for $N=2$ and $\mu =0.5$ for $N=3$.

Figure 5

${\chi }_s^{\prime \prime }\left(\omega \right)$ in the ordered phase for $N=2$ and 3. We scale the curves according to Eq. (4) for a range of tuning parameters $\delta g$ near the critical point.

Figure 6

${\chi }_s^{\prime \prime }\left(\omega \right)$ in the disordered phase for $N=2$. We scale the curves according to Eq. (4) for a range of tuning parameters $\delta g$ near the critical point.

Figure 7

Comparison of the scalar susceptibility line shape ${\chi }_s^{\prime \prime }\left(\omega \right)$ on mirror points across the phase transition for $N=2$. The blue green curve corresponds to disordered phase and the green curve to the ordered phase.

Figure 8

Log-log scale plot for ${\chi }_s\left(\tau \right)$ in the ordered phase. For $N=3,4$, we indeed find the asymptotic behavior ${\chi }_s\left(\tau \right)\sim 1/{\tau }^4$ to agree within the error bars.

Figure 9

${\chi }_s\left(\tau \right)$ in the ordered phase for $N=2$, plotted on a log-log scale in panel (a) and a semilog scale in panel (b). The curve deviates significantly from the expected $1/{\tau }^4$ power-law form. Instead, the curve fits better to an exponential decay as in the disordered phase.

Figure 10

The optical conductivity $\mathrm{Re}\sigma \left(\omega \right)$ in the ordered and disordered phases for $N=2$. Curves are scaled according to Eq. (11) for several values of the quantum tuning parameter $\delta g$ near the critical point. The solid black curves show the analytic results from Refs. 5 and 23.

Figure 11

The real and imaginary parts of the optical conductivity in the disordered phase. Results are shown from a one-loop calculation in Appendix .

Figure 12

The real and imaginary parts of the optical conductivity in the ordered phase. The green curve displays the results of a one-loop calculation carried in Appendix . The values of $m_H/\Delta$ and ${\rho }_s/\Delta$ were taken from the QMC simulation. The blue curve depicts the optical conductivity obtained from the duality relation in Eq. (26). This curve is multiplied by $0.15$ for comparison reasons.

Figure 13

The first five vectors $v_n\left(\nu \right)$ corresponding to the largest singular values in $W$.

Figure 14

SVD analysis of the numerical analytical continuation. $n$ labels the SVD eigenmodes. The filled circles are the rapidly decreasing SVD eigenvalues of $K$, denoted $w_n$. Magnitudes of projections of noisy data, for the test model [Eq. (B13)], are denoted by $|{\tilde{p}}_n|$. $\sigma$ is the variance of the artificial noise added to the Matsubara data. The breakpoints $n^{*}$ denote the mode index where noise dominates the signal, and the projections start to flatten. The values of $n^{*}$ increase when the noise level decreases.

Figure 15

Comparison between the projections $p_n$ (solid lines) and the singular values of the kernel $w_n$ (circles) for the high-quality QMC data for the O(2) model. The linear system size is $L=120$, and coupling constant is $\delta g=1.17\%$. We see that due to the effect of the noise, ${\tilde{p}}_n$ flattens at the breakpoint at $n^{*}\approx 11$.

Figure 16

Analytic continuation obtained from the first $n$ singular values, for the QMC data of Fig. 15. We see that spectral functions converge to until $n\le 11$, in agreement with assigning $n^{*}=11$ where ${\tilde{p}}_n$ starts to flatten in Fig. 15. For $n_{\mathrm{SVD}}=12$ (dashed line), the condition ${\tilde{p}}_n<w_n$ is violated, and the resulting spectral function wildly differs from the converged function since it is dominated by random amplified errors.

Figure 17

Comparison of different regularization methods. Note that the Higgs peak position varies only slightly between the different methods, described in Sec. 9c.