使用卷積神經網路解決量子多體物理之新方法

量子多體系統的特徵值問題一直是一個富有挑戰性的課題。原因是，當我們提升所要計算的系統大小，希爾伯特空間的維度與所需要的計算記憶體資源將會指數型成長。在這篇碩士論文中，我們著重於機器學習的方法對於此問題的應用，特別是避免指數型成長的希爾伯特空間問題的策略。我們提出了一個基於卷積神經網路模式辨別的新方法，多體系統的基態與低階激發態能量將藉由隨機取樣從一個多體哈密頓量取出。我們用一維在橫向場中的易辛模型、一維Su-Schrieffer-Heeger(SSH)模型、一維費米-哈伯模型與一維XXZ模型作為例子，並展示在一維易辛模型中，基態簡併性的預測以及其在經過量子相變點後是如何消失。SSH模型中，象徵拓樸相變的能量零模的預測，以及費米-哈伯模型和XXZ模型的洞子和旋子激發光譜的預測，即使這些卷積網路是由可用微擾理論的參數範圍內的數據訓練而成。這些結果表明了這種新策略的潛力和靈活性，並且或許可以用在檢查拓撲相變和二階相變的臨界行為以及普適類的分類。

關鍵字

機器學習；多體物理；相變； Ising 模型； SSH 模型；費米-哈伯模型； XXZ 模型

並列摘要

The eigenvalue problem of a quantum many-body system is a challenging topic since the dimension of Hilbert space (and hence computational memory) grows exponentially as the system size increases. In this thesis, we focus on the application of machine learning method in this problem, in particular the strategy to avoid exponentially-growth Hilbert-space problem, where we propose a completely new method based on the pattern recognition of Convolutional Neural Networks, where the ground state and lowly excited state energies are extracted from a large random sampling of the many-body Hamiltonian. We use 1D Ising Model with a transverse field (TFIM), 1D SSH model, 1D Fermi-Hubbard model and 1D XXZ model as examples, and show how the ground state degeneracy could be predicted and lifted near the quantum phase transition point for 1D TFIM, the emergence of energy zero modes which indicate topological phase transition for 1D SSH model, the holon and spinon excitation spectrum for Fermi-Hubbard model and XXZ model, even though the model is trained by data in the perturbative regime. These results show the potential and flexibility of this new strategy, which should be useful for examining the critical behavior of both topological phase transition and second-order phase transition and also the classification of universality classes.

並列關鍵字

Machine Learning ； Many-body physics ； phase transition ； Ising model ； SSH model ； Fermi-Hubbard model ； XXZ model

參考文獻

[1] McCulloch, Warren S.; Pitts, Walter. “A logical calculus of the ideas immanent in nervous activity. The bulletin of mathematical biophysics.” 5 (4): 115 133 (1943)