Precision lattice test of the gauge/gravity duality at large $N$

We perform a systematic, large-scale lattice simulation of D0-brane quantum mechanics. The large-$N$ and continuum limits of the gauge theory are taken for the first time at various temperatures $0.4\le T\le 1.0$. As a way to test the gauge/gravity duality conjecture we compute the internal energy of the black hole as a function of the temperature directly from the gauge theory. We obtain a leading behavior that is compatible with the supergravity result $E/N^2=7.41T^{14/5}$: the coefficient is estimated to be $7.4\pm 0.5$ when the exponent is fixed and stringy corrections are included. This is the first confirmation of the supergravity prediction for the internal energy of a black hole at finite temperature coming directly from the dual gauge theory. We also constrain stringy corrections to the internal energy.

Figure 1

An intuitive interpretation of the matrices $X_M$. The diagonal elements correspond to positions of D0-branes and the off-diagonal elements correspond to the open strings connecting them. This figure is taken from Ref. [9].

Figure 2

The correlation between $E/N^2$ and $|P|$ at $N=16$, $L=16$ and $T=0.5$ is shown as a two-dimensional histogram where a darker color corresponds to a higher count. The blue and red histograms (three per panel) represent the normalized distribution of $E/N^2$ and $|P|$, respectively, within the slices on the two-dimensional plot bounded by dashed lines. The histograms coming from different slices are almost indistinguishable.

Figure 3

The Monte Carlo history and a corresponding histogram for the energy $E/N^2$, the Polyakov loop $|P|$, $R^2$, and $F^2$ of the $T=0.5$ $N=24$ $L=32$ ensemble. For each observable, one can see fluctuations that span many Monte Carlo steps.

Figure 4

A study of the statistical stability of $E/N^2$ for the ensemble $N=16$, $L=32$, $T=0.5$. In the left panel we show different thermalization cuts, measuring $E/N^2$ on the rest of the configurations. In the right panel we show the importance of large statistical samples by measuring on consecutive disjoint sets of trajectories. As the statistical sample grows from 1000 configurations (red squares) to 6000 configurations (blue circles), the central values and uncertainties between sets of configurations become more and more stable and compatible. In both panels we perform the analysis with bins of 50 configurations. For comparison, we also show our final analysis and its uncertainty as a black star in both panels, with its error bar displayed as dashed lines on the right panel.

Figure 5

A continuum extrapolation for $T=0.7$ with fixed $N=16$ with the improved action. Black stars represent our measurements, open circles our extrapolation, and open squares the extrapolation from Ref. [22]. Error bars and the error bands on the extrapolated curves represent $1\sigma$ errors. Our linear extrapolation is only fit to the data at $L\ge 16$. The divergence of the linear and quadratic extrapolations indicate that for this ensemble, linear extrapolations of lattice data that include data taken at $L\lesssim 16$ will be systematically biased.

Figure 6

A comparison between taking the continuum limit for the unimproved and improved actions for $T=1.0$, $N=16$. Error bars and the error bands on the extrapolated curves represent $1\sigma$ errors.

Figure 7

A simultaneous continuum and large-$N$ extrapolation for $T=0.5$ with the improved action. All the data points are fit to a single two-dimensional surface given by Eq. (23). In the right panel, we show all the data points (slightly offset for visual clarity) and the (black) $N=\infty$ slice of the fitted surface. In the left panel, we show all the data points and the (black) continuum extrapolations at each $N$ together with their subsequent large-$N$ limit as the dashed line with uncertainties given by the dotted band. We also show the fixed-$L$ slices of the two-dimensional fit, as well as the (black) $L=\infty$ slice. In both panels, the black circle represents the result of taking the large-$N$ limit of the continuum extrapolations at each $N$, while the best-fit simultaneous continuum- and large-$N$ limit $e_{00}$ is shown as a black diamond. Error bars, the error band on the extrapolated surface, and the dotted error band represent $1\sigma$ uncertainties.

Figure 8

Our best fits of $E_0$ to the data points $e_{00}$ shown as black diamonds, including the first two/three/four terms as a cyan dotted line/blue solid line/purple dot-dot-dashed line with $1\sigma$ error bands. We also show the results from Ref. [13] and Ref. [12] as green dashed and red dot-dashed lines, respectively. The SUGRA result is shown in black.

Figure 9

The same as in Fig. 8, but with $a_0$ fixed to its known SUGRA value, rather than fit.

Figure 10

Two fits of our continuum large-$N$ values $e_{00}$ (black diamonds) for $E_0(T)$. The solid blue line is a fit to Eq. (26) over $0.4\le T\le 0.8$, while the dotted cyan line is a fit to Eq. (27) over $0.4\le T\le 0.9$, with their respective (small) error bands. We also show the results from Hanada et al. [19] and Kadoh and Kamata [13] as red dot-dashed and green dashed lines, respectively, with their $1\sigma$ uncertainty band as explained in the main text. The SUGRA result is shown in black.

Figure 11

Four fits of $E_1$ to our values for $e_{10}$. In each panel, we show our measurements as black diamonds with $1\sigma$ error bars, the fit as the solid blue curve with a $1\sigma$ error band, and the known low-temperature behavior $b_0=-5.77$ as the black curve. In the top left panel we fit just $b_0$, with $b_1=0$. In the top right panel we fix $b_0=-5.77$ and fit $b_1$. In the bottom left panel we fit both $b_0$ and $b_1$. In the bottom right panel we fit $b_0{T}^p$. All fits have $0.9<{\chi }^2/\mathrm{DOF}<1.3$, and cannot be meaningfully distinguished by our data due to the large uncertainties.

Figure 12

The distributions of $|P|$ at $N=16$, $L=16$ and $T=0.8$ for different energy intervals reported in the legend. These energy intervals are close to the average value of $E/N^2$ for this ensemble. The upper panel shows the normalized probability distribution, while the lower panel shows the cumulative distribution as a proxy for the empirical distribution function used in the KS test. The value of the KS statistic $D$ and its associated $p$-value are also shown, giving confidence that the two underlying distributions are the same.

Figure 13

Same as Fig. 12, but for energy intervals at the tail of the energy distribution, instead of around the average. The KS statistic is larger and the $p$-value is considerably lower.