強關聯電子系統中傳輸與量子臨界現象之理論研究

這篇論文主要包含兩部分，分別是對於強關聯電子系統中傳輸與量子臨界現象的理論研究。在第一部分(Part I)，我們理論上研究了具有可調變透明度(tunneling trans- parency)平面金屬/超導體(N/SC)穿隧結上的安德烈夫反射(Andreev reflection)，這些穿隧結上的一般金屬具有相對論性Dirac能帶結構。藉由 Blonder-Tinkham-Klapwijk 理論，我們計算了穿隧結上的安德烈夫電導 (Andreev conductance)。一般認知上，由於克萊因隧道效應(Klein tunneling)的影響，二維相對論性材料接面處的雜質傾向於增強(而不是抑制)電子的穿隧效應。然而直觀上，若接面處並非均質的，電子的傳輸會被此非均質接面所抑制，因此直接將接面處的效應等效為雜質並不適合用來模擬真實系統中的穿隧結。在這個研究裡，我們藉由在接面處施加狹窄且均勻的局部應力來模擬真實系統中的穿隧結，此局部應力在接面處可視為一個等效的局部位能，使得電子在穿過穿隧結接面時產生散射。此外，穿隧結上的位能強度可通過改變外加應力的大小來調整，間接調整穿隧結的反射程度。隨著局部應力強度增加時，我們確實在以石墨烯為基底的金屬/超導體穿隧結內觀察到安德烈夫電導逐漸被抑制。接下來，我們還在拓撲穿隧結中探索兩個不等價狄拉克點的安德烈夫電導，並預測了手性 (chiral)至螺旋性(helical)拓撲相變過程中電導的獨特行為。在此論文的第二部分(Part II)，我們提供了重費米子材料中與量子臨界現象、奇異金屬態(strange metal state)和奇異超導態(strange superconductivity)相關的理論研究。在具有非費米液體特性(non-Fermi liquid)的非常規金屬態(又稱為奇異金屬態) 被認為是重費米子 (heavy fermion) 系統中量子相變點附近的一般特徵。目前在微觀上，此特徵仍缺乏完整的理論解釋。重費米子材料中的一個典型例子是重費米子金屬 (heavy fermion metal) YbRh2Si2，此種材料可經由改變磁場來誘發量子相變。當利用少部分的鍺元素(Ge)取代矽(Si)時，實驗顯示有限範圍的磁場下，奇異金屬態在極低溫下(幾乎接近絕對零度時)仍然存在，而非一個點 ; 而在大於臨界磁場時，奇異金屬態被重費米液體態(heavy Fermi liquid)所取代。非費米液體現像，其中最值得注意的包括電阻率與溫度成線性關係，比熱係數(specific heat coefficient)在高溫時與溫度成對數關係而在低溫時與溫度成冪次律(power-law)的關係，目前仍然缺乏明確的理解。在這研究中，我們提出一個機制來解釋實驗觀察到的結果:在接近量子臨界點附近時，此種非費米液體行為是由於準二維(quasi-two dimensional)的 short-range resonating- valence-bond spin-liquid 與準二維的 Kondo correlation 這兩種物理量的擾動彼此間的競爭所造成，根據此想法，我們從large-N [Sp(N)] 反鐵磁 Kondo-Heisenberg 模型中建構了一個在相變點附近的有效場論，並運用微擾重整化群(perturbative renormalization group)分析此有效場論的臨界現象。由重整化群的分析中，我們找到了一個 critical fixed point，此 fixed point 剛好對應到實驗中發現的量子臨界點。此外，我們也計算了在此 critical fixed point 附近一些重要的物理量，例如:crossover scales、電阻率與比熱係數等。理論結果與實驗所獲得奇異金屬態的行為在定性上吻合。此外，傳統上一般都認為標度理論(scaling theory)只存在於臨界維度(upper critical dimension)小於四 (d + z > 4) 的有效場論當中，然而研究結果顯示標度理論有機會適用於我們這個臨界維度(upper critical dimension)大於四 (d + z > 4) 的有效場論中。我們將標度理論的適用性歸因於此有效場論中存在著一個 interacting Gaussian fixed point，而此 fixed point 的存在是因為有效場論中包含了玻色子-費米子(Yukawa)交互作用項所致。在這個研究計畫中，我們的理論對於解決重費米子系統中量子臨界現象與奇異金屬態這個議題提供了新的基礎。接下來，我們將目光轉向至奇異超導態的主題上，這是在許多重費米子超導體材料中觀察到最為奇特的現象之一。奇異超導態中的一個典型例子是“115族”重費米子材料 CeMIn5 (M = Co，Rh，Ir)。奇異超導態通常出現在 CeMIn5 反鐵磁量子臨界點附近，並且具有非常規d波庫柏電子對(Cooper pair)，在轉變成奇異超導態前為奇異金屬態。奇異超導態的微觀原理及其與反鐵磁量子臨界性間的關聯仍然是長期存在的謎團之一。最近的實驗表明，Kondo hybridization 與 f 軌域電子中反鐵磁擾動對於 CeCoIn5 中超導電態的形成扮演了重要角色。根據實驗結果，我們提出奇異超導體的微觀形成機制，並藉由重整化群分析二維 large-N Kondo-Heisenberg 模型的有效場論。我們得出此機制與上述的奇異金屬態形成機制有多處雷同，均與準二維的 Kondo correlation 與準二維的 short-range resonating-valence-bond spin-liquid 的擾動有關，然而我們發現奇異超導態在量子相變點附近的形成原因不僅僅是由於上述兩種物理量擾動之間的競爭，這兩種擾動的共存也扮演了重要的作用。最後，我們總結出這兩種相互作用間彼此的競爭與共存提供了對奇異超導性如何從奇異金屬態中誕生的理解，並且解釋了部分實驗上觀察到的結果。

關鍵字

安德烈夫反射；量子相變；重費米子；奇異金屬態；奇異超導態

並列摘要

This thesis consists of two parts relevant to theoretical studies on strongly correlated electron systems. In part I, we theoretically study the Andreev reflection (AR) across several realistic two- dimensional (2D) normal-metal (N)/superconductor (SC) planer junctions with tunable tunnelling transparency within the Blonder-Tinkham-Klapwijk formalism. It is worthwhile mentioning that the normal-metal regions of the planar junctions of our interest feature relativistic linear- in-momentum Dirac spectrum. It is known that due to the effect of Klein tunneling, impurity potentials at the interface of 2D relativistic junctions will enhance (not suppress) the tunneling and therefore are not suitable to model a realistic tunnel junction of these materials. Here, we propose a way to construct a more realistic tunnel junction by adding a narrow, homogeneous local strain, which effectively generates a delta-gauge potential and variations of electron hopping at the interface, to adjust the transparency of the N/SC junction. Remarkable suppression of the Andreev conductance is indeed observed in the graphene N/SC junction as the strength of the local strain increases. We also explore the Andreev conductance in a topological N/SC junction at the two inequivalent Dirac points and predict the distinctive behaviors for the conductance across the chiral-to-helical topological phase transition. The relevance of our results for the adatom-doped graphene is discussed. In Part II, we focus on the theoretical investigations on the quantum criticality, the strange metal state, and the strange superconducting phase in heavy fermion compounds. Unconven- tional metallic or strange metal (SM) behavior with non-Fermi liquid (NFL) properties, generic features of heavy fermion systems near quantum phase transitions, are yet to be understood microscopically. A paradigmatic example is the magnetic field-tuned quantum critical heavy fermion metal YbRh2Si2, revealing a possible SM state over a finite range of fields at low temperatures when substituted with Ge. Above a critical field, the SM state gives way to a heavy Fermi liquid with Kondo correlation. The NFL behavior, most notably a linear-in-temperature electrical resistivity and a logarithmic-in-temperature followed by a power-law singularity in the specific heat coefficient at low temperatures, still lacks a definite understanding. We propose a promising mechanism as origin to explain the experimentally observed behavior: a quasi-2d fluctuating short-ranged resonating-valence-bond spin liquid competing with the Kondo corre- lation near criticality. Applying a field-theoretical renormalization group analysis on an effec- tive field theory beyond a large-N [Sp(N)] approach to an antiferromagnetic Kondo-Heisenberg model, we identify the critical point, and explain well the SM behavior. In addition, we found that the scaling ansatz, which originally exists below the upper critical dimension d + z < 4, will have a chance to survive in our effective field theory with d + z > 4. We attribute the applicability of scaling ansatz to the existence of a nontrivial interacting Gaussian fixed point due to the presence of a boson-fermion (Yukawa) interaction in the field theory. Our theory goes beyond the well-established framework of quantum phase transitions and serves as a new basis to address open issues in quantum critical heavy fermion systems. Next, we turn our attention to the subject of strange superconductivity, one of the most exotic phenomena being observed in numerous heavy fermion superconductors. The heavy fermion systems CeMIn5 with M = Co, Rh, Ir, the “115” family, provide a prototypical example of strange superconductivity with unconventional d-wave pairing and strange metal normal state, emerged near an antiferromagnetic quantum critical point. The microscopic origin of strange superconductivity and its link to antiferromagnetic quantum criticality of the strange metal state are still long-standing open issues. Recent experiments show evidence that both Kondo hybridization and antiferromagnetic fluctuations among f -electrons play important roles in the formation of superconductivity in heavy fermion materials, especially for CeCoIn5 near its antiferromagnetic QCP. Based on the experimental findings, we propose a microscopic mechanism for strange superconductor, based on the coexistence and competition between the Kondo correlation and the quasi-2d short-ranged antiferromagnetic resonating-valence-bond spin-liquid near the antiferromagnetic quantum critical point via a large-N Kondo-Heisenberg model and renormalization group analysis beyond the mean-field level. The interplay of these two effects provides a qualitative understanding on how superconductivity emerges from the strange metal state and the observed superconducting phase diagrams for CeMIn5.