Random singlets in an amorphous valence-bond solid: a strong-disorder renormalization group study

處於絕對零溫時，自旋 1/2 海森堡反鐵磁鍊加上額外的多自旋耦合（稱為 J-Q 鍊）可自發性地失去晶格對稱性而形成價鍵固態，此形成條件為強且均質的多自旋Q 耦合。價鍵固態性質與標準海森堡反鐵磁鍊（又稱 J 鏈）之基態性質截然不同，後者屬臨界態且具晶格對稱性。根據強無序重整化群法分析結果，具非均質無序耦合的海森堡反鐵磁鏈基態為一所謂的「隨機單態」，可視為一組具任意長度的價鍵（雙自旋單態）之組合，此價鍵結構造成無序自旋鏈獨特的低溫性質，包含非尋常的能量－長度關係，以及平均自旋關聯函數之冪次下降行為。藉價鍵態的概念，我們於本論文推導適用於無序 J-Q 鍊的強無序重整化群法則。針對零 J 極限的計算結果顯示：完美價鍵固態在非均質耦合環境下將破裂成無序交錯的二聚化區域,兩相鄰區域間的域壁為一帶自旋 1/2 的旋子，且跨越二 聚化區域的旋子兩兩以微弱價鍵連結。此種「非晶形價鍵固態」於長距離範疇亦雷同無序海森堡反鐵磁鏈之基態屬隨機單態，也就是說，原呈現於均質系統的價 鍵固態相變不存在於無序 J-Q 鍊。此外，我們發現平均四自旋關聯函數在隨機單態中亦呈現冪次下降的形式。The ground state of the antiferromagnetic spin-1/2 Heisenberg chain with additional multi-spin couplings (the so-called J-Q chain) in the absence of disorder and in the limit of strong multi-spin (Q) couplings is a valence-bond solid (VBS) with spontaneous dimerization; this VBS ground state is different from the critical ground state of the standard Heisenberg chain with nearest-neighbor antiferromagnetic (J) couplings. In the presence of bond randomness, the ground state of the standard Heisenberg chain solved by a strong-disorder renormalization group (SDRG) method was suggested to be in the random-singlet phase consisting of a set of singlets (valence bonds) in a random fashion. This valence-bond (VB) structure leads to unique low-energy properties of the disordered chain, including unconventional energy-length scaling and a power-law decay of the mean spin correlation function. In this thesis we introduce an SDRG scheme using the concepts of the valence-bond basis to study the J-Q chain with random couplings. Our results show that the VBS state breaks into alternating dimerized domains with random singlets formed between spinons localized at domain walls. This amorphous valence-bond solid at long distances is also asymptotically a random-singlet state. Thus, in the random J-Q chain, we do not expect any dimerization phase transition as in the clean system. In addition, we find that the mean dimer correlation function decays algebraically in the random-singlet phase