Automatic Differentiable Numerical Renormalization Group

TL;DR

This paper introduces a differentiable programming approach to the numerical renormalization group (NRG), enabling efficient computation of derivatives of quantum many-body physics outputs with respect to parameters, facilitating new physical insights.

Contribution

It applies automatic differentiation to NRG, allowing precise and efficient calculation of derivatives of physical quantities with respect to Hamiltonian parameters.

Findings

Derivatives of NRG outputs can be computed accurately using AD.

Physical observables like thermodynamic quantities are obtainable via derivatives.

The method enables calculation of susceptibilities and potential dynamical derivatives.

Abstract

Machine learning techniques have recently gained prominence in physics, yielding a host of new results and insights. One key concept is that of backpropagation, which computes the exact gradient of any output of a program with respect to any input. This is achieved efficiently within the differentiable programming paradigm, which utilizes automatic differentiation (AD) of each step of a computer program and the chain rule. A classic application is in training neural networks. Here, we apply this methodology instead to the numerical renormalization group (NRG), a powerful technique in computational quantum many-body physics. We demonstrate how derivatives of NRG outputs with respect to Hamiltonian parameters can be accurately and efficiently obtained. Physical properties can be calculated using this differentiable NRG scheme--for example, thermodynamic observables from derivatives of the free energy. Susceptibilities can be computed by adding source terms to the Hamiltonian, but still evaluated with AD at precisely zero field. As an outlook, we briefly discuss the derivatives of dynamical quantities and a possible route to the vertex.