Abstract
State-of-the-art machine learning techniques promise to become a powerful tool in statistical mechanics via their capacity to distinguish different phases of matter in an automated way. Here we demonstrate that convolutional neural networks (CNN) can be optimized for quantum many-fermion systems such that they correctly identify and locate quantum phase transitions in such systems. Using auxiliary-field quantum Monte Carlo (QMC) simulations to sample the many-fermion system, we show that the Green’s function holds sufficient information to allow for the distinction of different fermionic phases via a CNN. We demonstrate that this QMC + machine learning approach works even for systems exhibiting a severe fermion sign problem where conventional approaches to extract information from the Green’s function, e.g. in the form of equal-time correlation functions, fail.
Highlights
In quantum statistical physics, the sign problem refers to the generic inability of quantum Monte Carlo (QMC) approaches to tackle fermionic systems with the same unparalleled efficiency it exhibits for unfrustrated bosonic systems
Such a machine learning approach to the QMC sampling of many-fermion systems allows one to determine whether crucial information about the ground state of the many-fermion system is truly lost in the sampling procedure, or whether it can be retrieved in physical entities beyond statistical estimators, enabling a supervised learning of phases despite the presence of the sign problem
For a given finite system size L, we identify the location of the phase transition with the parameter U for which the averaged state function F is 1/2, i.e. the parameter for which the convolutional neural networks (CNN) cannot make any distinction between the two phases and assigns equal probability to both phases
Summary
Artificial neural networks have for some time been identified as the key ingredient of powerful pattern recognition and machine learning algorithms[16, 17]. CNNs, in particular, are nonlinear functions which are optimized (in an initial “training” step) such that the resulting function F allows for the extraction of patterns (or “features”) present in the data. We take this approach to construct a function F, represented as a deep CNN, that allows the classification of many-fermion phases as outlined in the previous section. In the initial training step, we optimize the CNN on a set of (typically) 2 × 8192 representative configurations sampled deep in the two fermionic phases. The question of which fundamental features, contained in the Monte Carlo configurations, are used in the resulting function F to characterize the phases under consideration, is automatically discovered during the training procedure (and beyond our direct influence)
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