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The monomer-dimer models in two and three dimensions: Tensor renormalization group study

大塚, 高弘 大阪大学 DOI:10.18910/87816

2022.03.24

概要

二原子分子(dimer)を重なりを禁止した上で空間上に配置する場合の数を数え上げる問題をdimer問題と呼ぶ。dimer問題は、格子上でpure dimer model(PDM)として定義され、2次元モデルについては解析解が知られている。PDMは最も単純なモデルの一つであり、その単純さゆえに、様々な物理と関連し広く研究されている。例えば、PDMとIsingモデルの間には一対一の対応関係があることが知られている。また、PDM自身も格子の形状に応じて臨界現象を示すなど、普遍性クラスの分類という観点からも興味深い対象となっている。PDMはdimerだけでなく単原子分子(monomer)を含むmonomer-dimer model(MDM)にも拡張することができる。MDMではPDMとは異なり、2次元であっても解析解は存在せず、MDMの性質の理解は不十分な段階にある。

本論文では、DMRGと特異値分解に基づくテンソルネットワーク計算法である高次テンソル繰り込み群(HOTRG)を用いて、2次元および3次元格子上のMDMの臨界現象を研究する。臨界現象はエントロピーやモノマー密度などの物理量にべき乗からのずれとして反映される。この振る舞いを解析することで、MDMの臨界現象を明らかにし、臨界指数を決定することで普遍性クラスからMDMを分類することができる。同時に、計算手法であるHOTRGに対しても、高次元系に対するアルゴリズムの最適化を行う。また、HOTRGの精度を示す指標となる特異値分布の漸近的振る舞いについても言及する。

最初に、DMRGとHOTRGを用いて、2次元MDMの臨界的振る舞いを解析する。MDMと磁場入りイジングモデルの間には対応関係があると考えられているが、普遍性クラスの観点からは議論されていない。そこで、モノマー密度に現れる臨界的振る舞いかた臨界指数を決定し、MDMとIsingモデルの普遍性クラスが異なることを明らかにする。

次に、3次元MDMの臨界振る舞いを調べる。3次元系の例である、有限の層数の層状モデルと無限系の大きく分けて2つのモデルを解析する。層数が少ない有限層の二部格子モデルに対しては、2層系はnon-critical、3層系はcriticalであることをDMRGによって示す。3次元無限系に対しては莫大な消費メモリが必要となるため、最初に3次元HOTRGのアルゴリズムの最適化を行う。具体的には、テンソルの縮約の順序の変更とテンソル成分を異なるコアに割り当てるアルゴリズムを導入することでメモリコストの低減を行う。これらのアルゴリズムを用いることで、PDMの残留エントロピーを計算し、MDMの臨界的振る舞いを解析する。残留エントロピーの値は先行研究のTN計算よりも高い精度で計算できたが、臨界指数を決定するには精度が不十分であった。

最後に、HOTRGのスペクトル(特異値)の漸近的振る舞いをBaxterによって導入されたCTM像に基づき解析する。HOTRGのスペクトルの漸近形とDMRGのスペクトルの漸近形を比較し、非臨界領域における可積分モデルに成立する関係を導く。

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