Chiral topological phases from artificial neural networks

Motivated by recent progress in applying techniques from the field of artificial neural networks (ANNs) to quantum many-body physics, we investigate to what extent the flexibility of ANNs can be used to efficiently study systems that host chiral topological phases such as fractional quantum Hall (FQH) phases. With benchmark examples, we demonstrate that training ANNs of restricted Boltzmann machine type in the framework of variational Monte Carlo can numerically solve FQH problems to good approximation. Furthermore, we show by explicit construction how $n$-body correlations can be kept at an exact level with ANN wave functions exhibiting polynomial scaling with power $n$ in system size. Using this construction, we analytically represent the paradigmatic Laughlin wave function as an ANN state.

Figure 1

Graphical representation of the restricted Boltzmann machine (RBM) network for the variational wave function [see Eq. (1)]. The physical spins (in the blue shaded area $\mathcal{S})$ are denoted by ${\sigma }_j,j=1,...,N$, and the auxiliary variables (in the green shaded area $\mathcal{A})$ are denoted by green dots $a_j,j=1,...,M$, with $M=\alpha N$. The coupling strengths on the links between $i$ and $j$ (solid lines) are labeled $w_{ij}$, while the local fields are denoted by $b_i$ (dashed lines) for the auxiliary variables and by $c_j$ (dash-dotted lines) for the physical spins.

Figure 2

Sketch of the cluster neural network quantum state (CNQS) architecture with cluster size $n=2$ and $m=1$ [see Eq. (3)]. The physical spins (in the blue shaded area $\mathcal{S})$ are denoted by ${\sigma }_i$. The auxiliary variables (in the green shaded area $\mathcal{A})$ are denoted by $a_{ij}$. The coupling strengths on the links between $i\ne j$ (dash-dotted lines) are labeled $w_{ij}$ and ${\tilde{w}}_{ij}$, while the local fields (solid lines) at $a_{ij}$ are labeled $b_{ij}$. For $n>2$ (not shown), $m$ auxiliary variables $a_{i_1,...i_n}^{\nu },\nu =1,...m$ are associated with every cluster of $n$ distinct sites labeled $i_1,...,i_n$.

Figure 3

Energy expectation value $E=〈{\psi }_{\mathcal{W}}\left|H\right|{\psi }_{\mathcal{W}}〉/〈{\psi }_{\mathcal{W}}|{\psi }_{\mathcal{W}}〉$ as a function of the stochastic reconfiguration (SR) iterations for system size $6\times 4$. For this system size, the expectation values are calculated using a sample of 10 000 configurations drawn with a standard Metropolis algorithm. The inset shows the final SR iterations, and the horizontal red line marks the exact ground-state energy from ED.