Machine learning holographic mapping by neural network renormalization group

Exact holographic mapping (EHM) provides an explicit duality map between a conformal field theory (CFT) configuration and a massive field propagating on an emergent classical geometry. However, designing the optimal holographic mapping is challenging. Here we introduce the neural network renormalization group as a universal approach to design generic EHM for interacting field theories. Given a field theory action, we train a flow-based hierarchical deep generative neural network to reproduce the boundary field ensemble from uncorrelated bulk field fluctuations. In this way, the neural network develops the optimal renormalization-group transformations. Using the machine-designed EHM to map the CFT back to a bulk effective action, we determine the bulk geodesic distance from the residual mutual information. We have shown that the geometry measured in this way is the classical saddle-point geometry. We apply this approach to the complex ${\varphi }^4$ theory in two-dimensional Euclidian space-time in its critical phase, and show that the emergent bulk geometry matches the three-dimensional hyperbolic geometry when geometric fluctuation is neglected.

Figure 1

Relation between (a) RG and (b) generative model. The inverse RG can be viewed as a generative model that generates the ensemble of field configurations from random sources. The random sources are supplied at different RG scales (coordinated by $z$), which can be viewed as a field $\zeta (x,z)$ living in the holographic bulk with one more dimension. The original field $\varphi \left(x\right)$ is generated on the holographic boundary.

Figure 2

(a) Side view of the neural-RG network. $x$ is the spatial dimension(s) and $z$ corresponds to the RG scale. There are two types of blocks: disentanglers (dark green) and decimators (light yellow). The network forms an EHM between the boundary variables (blue dots) and the bulk variables (red crosses). (b) Top view of one RG layer in the network. Disentanglers and decimators interweave in the space-time (taking two-dimensional space-time, for example). Each decimator pushes the coarse-grained variable (black dot) to the higher layer and leaves the decimated variables (red crosses) in the holographic bulk. (c) The training contains two stages. In the first stage, we fix the prior distribution $P\left[\zeta \right]$ to be uncorrelated Gaussian and train the EHM $G$ to bring it to the Boltzmann distribution of the CFT. In the second stage, we learn the prior distribution with the trained EHM held fixed. (d) The behavior of the loss function $\mathcal{L}$ in the two training stages.

Figure 3

Neural network architecture within a decimator block (the disentangler block shares the same architecture). Starting from the renormalized variable ${\varphi }^{\prime }$ and the bulk noise ${\zeta }_1, {\zeta }_2, {\zeta }_3$ as complex variables, the $\mathrm{Re}$ and $Im$ channels are first separated, then $\mathcal{S}$ applies the scaling separately to the four variables within each channel and $\mathcal{O}$ implements the $\mathrm{O}\left(4\right)$ transformation that mixes the four variables together. $\mathcal{S}$ and $\mathcal{O}$ are identical for $\mathrm{Re}$ and $Im$ channels to preserve the $\mathrm{U}\left(1\right)$ symmetry. Then the channels merge into complex variables followed by element-wise nonlinear activation describe by an invertible $\mathrm{U}\left(1\right)$-symmetric map ${\varphi }_i\mapsto ({\varphi }_i/|{\varphi }_i\left|\right)sinh|{\varphi }_i|$.

Figure 4

Performance of the trained EHM for the complex ${\varphi }^4$ theory. (a) Order parameter $\langle \varphi \rangle$ vs temperature $T$. Different models are trained separately at different temperature. For finite-sized system, $\langle \varphi \rangle$ crosses over to zero around the KT transition. Correlation function $\langle {\varphi }_i^{*}{\varphi }_j\rangle$ scaling in log-log plot (b) and log-linear plot (c). Distribution of ${\varphi }_i$ in a single sample generated by the neural network trained in (d) the algebraic liquid phase and (e) the disordered phase.

Figure 5

The boundary field configuration $\varphi$ before (left) and after (right) a local update in the most IR layer of the bulk field $\zeta$. The complex field ${\varphi }_i$ is represented by the small arrow on each site. The background color represents the vorticity. The inset shows the distribution of ${\varphi }_i$ in the complex plane.

Figure 6

(a) Distance matrix $D(A:B)$, indexed by the decimator indices $A,B$, obtained based on Eq. (17). (b) Visualization of the bulk geometry by multidimensional scaling projected to the leading three principle dimensions. Each point represents a decimator in the neural network, colored according to layers from UV to IR. The neighboring UV-IR links are add to guide the eye. Panels (c) and (d) show the distance scaling along (c) the angular and (d) the radius direction.

Figure 7

Functional dependence of variables in the neural-RG network. Each block represents a bijective map.

Figure 8

Orthogonal transformation.

Figure 9

Hamiltonian Monte Carlo (HMC) simulations in the latent space and physical space.