機器學習識別古典及量子自旋模型相態

Abstract

三角易辛（Ising）反鐵磁在絕對零度因幾何挫折性而不具磁性。有趣的是，具量子效應的橫向磁場可誘發易辛反鐵磁零溫基態之有序性，產生具Z6 對稱破缺的時鐘態；這個零溫有序態可由更強的橫向磁場或有限溫度破壞。在絕對零度，一量子臨界點區分弱場下的有序時鐘態與強場下的無序順磁態。而在有限溫度，一Kosterlitz-Thouless相態區隔了低溫的時鐘態及高溫的順磁態。我們以量子蒙地卡羅方法針對許多不同溫度值及橫場值產生自旋組態，接著藉機器學習技術的多層感知器和捲積神經網路訓練機器辨識自旋組態與相態的關係，再以更多的自旋組態使神經網路識別其對應的相態。上述機器學習方法可頗精確辨識古典易辛模型的簡單相態，但對我們主要考慮的三角反鐵磁相態卻無法呈現良好的辨識力。

The triangular Ising antiferromagnet has no magnetic order down to zero temperature due to geometrical frustration. Interestingly, a weak transverse field, introducing quantum fluctuations, can induce magnetic order in the triangular antiferromagnet at zero temperature, resulting in the clock phase with a broken Z6 symmetry; this ordered clock phase can be destroyed by a strong transverse field or at finite temperature. At T=0, there is a quantum critical point separating the clock phase in weak fields and a paramagnetic phase in strong fields; at finite temperature, the antiferromagnet exhibits an extended Kosterlitz-Thouless (KT) phase intervening between the clock and paramagnetic phases. We generate spin configurations of the triangular antiferromagnet at different temperatures and transverse fields by quantum Monte Carlo (QMC) simulations. We attempt to use supervised machine learning techniques via multilayer perceptrons and convolutional neural networks to classify the phases of the antiferromagnetic system, solely based on spin configurations sampled with QMC. We find that the neural network models perform the classification task with a 70% accuracy for the triangular quantum antiferromagnet, while successfully distinguishing the classical Ising states with more than 90% accuracy.