SciPost Phys. 12, 107 (2022) ·
published 25 March 2022
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The detection of phase transitions in quantum many-body systems with lowest
possible prior knowledge of their details is among the most rousing goals of
the flourishing application of machine-learning techniques to physical
questions. Here, we train a Generative Adversarial Network (GAN) with the
Entanglement Spectrum of a system bipartition, as extracted by means of Matrix
Product States ans\"atze. We are able to identify gapless-to-gapped phase
transitions in different one-dimensional models by looking at the machine
inability to reconstruct outsider data with respect to the training set. We
foresee that GAN-based methods will become instrumental in anomaly detection
schemes applied to the determination of phase-diagrams.
Philipp Schmoll, Augustine Kshetrimayum, Jens Eisert, Román Orús, Matteo Rizzi
SciPost Phys. 11, 098 (2021) ·
published 29 November 2021
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The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the $O(3)$ non-linear sigma model in $1+1$ dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in $3+1$ dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an $SU(2)$ symmetry in our
two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to $\chi_E^\text{eff} \sim 1500$, a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the
literature, namely a finite-$T$ transition and asymptotic freedom, though with a slight preference for the second.
Pietro Silvi, Ferdinand Tschirsich, Matthias Gerster, Johannes Jünemann, Daniel Jaschke, Matteo Rizzi, Simone Montangero
SciPost Phys. Lect. Notes 8 (2019) ·
published 18 March 2019
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We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimension physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer's companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.
