Machine learning and optimization-based approaches to duality in statistical physics

TL;DR

It is shown that this framework can rediscover the celebrated Kramers-Wannier duality for the 2d Ising model, reconstructing the known mapping of temperatures and an alternative approach which uses known features of the mapping of topological lines to reduce the problem to optimizing the couplings in a dual Hamiltonian.

Abstract

The notion of duality -- that a given physical system can have two different mathematical descriptions -- is a key idea in modern theoretical physics. Establishing a duality in lattice statistical mechanics models requires the construction of a dual Hamiltonian and a map from the original to the dual observables. By using simple neural networks to parameterize these maps and introducing a loss function that penalises the difference between correlation functions in original and dual models, we formulate the process of duality discovery as an optimization problem. We numerically solve this problem and show that our framework can rediscover the celebrated Kramers-Wannier duality for the 2d Ising model, reconstructing the known mapping of temperatures. We also discuss an alternative approach which uses known features of the mapping of topological lines to reduce the problem to optimizing the couplings in a dual Hamiltonian, and explore next-to-nearest neighbour deformations of the 2d Ising duality. We discuss future directions and prospects for discovering new dualities within this framework.

Figures (12)

• Figure 1: The two-point product of spins in the original frame $\sigma_i \sigma_j$ is related to the product of spins $\tilde{\sigma}_{i^*} \tilde{\sigma}_{j^*}$ in the dual frame, where $i^*,j^*$ are related to $ij$ as shown.
• Figure 2: We parametrize $G$ as a neural network that takes neighbouring links of a given link (in this case # 6) as its input. The assignment on horizontal links is related to that on vertical ones by a rotation and reflection.
• Figure 3: Examples of three features showing link products considered.
• Figure 4: Final $\tilde{\beta}$ as found by the deep learning framework closely matches that of the theoretical results. Points are scaled by the negative logarithm of the best loss such that the size of the points is inversely proportional to the loss. We cap the minimum size so that smaller points are visible. The loss is a minimum along two fronts, i.e, original frame $\beta = \pm \tilde{\beta}$ and the dual frame along the lines $\sinh(2\beta)\sinh(2\tilde{\beta}) = 1$.
• Figure 5: Emergence of dual lattice: e.g. if four original links (marked by $6$) form a square, the corresponding four links that are referenced by the neighbour mapping (marked by $2$) in Figure \ref{['fig:linkMap']} form a cross, as expected for the dual lattice.