單缺陷非厄米系統與邊界共形場論

Abstract

共形場論對描述臨界現象中的普適性行為提供了一套精確的分析框架，而邊界共形場論則在理解臨界系統中的量子雜質問題方面扮演重要的角色。另一方面，非厄米物理源於放寬厄米性條件，能有效刻畫具有增益、耗散與衰減等現象的開放量子系統的動力學。受這些發展的啟發，我們探討非么正邊界共形場論與非厄米量子雜質臨界系統之間的關聯。我們將邊界共形場論的框架擴展至包含非厄米雜質的情況，並利用折疊技巧將晶格模型中的雜質強度對應到一個有效的位勢參數，從而建立晶格散射矩陣與邊界共形場論黏合矩陣之間的對應關係。在此對應中，非厄米模型中機率守恆的違反體現在邊界共形場論中非么正的邊界條件。我們進一步顯示，晶格模型在費米點所產生的雜質能量位移，與邊界共形場論中基態能量的有限尺寸修正相符；該修正隨系統尺寸呈反比縮放，並由費米動量下的傳輸振幅所決定。這項結果推廣了對厄米雜質系統的既有研究成果，例如 [1,2] 中所提出的。總結而言，我們的研究表明，邊界共形場論的技術可以有效應用於非厄米量子雜質問題的分析，為非厄米性與邊界或缺陷臨界性之間的關係提供了新的見解。

Conformal field theory (CFT) offers an exact analytical framework for describing the universal features of critical phenomena, while boundary conformal field theory (BCFT) plays a central role in understanding quantum impurity problems at criticality. Meanwhile, non-Hermitian physics, which arises from relaxing the Hermiticity condition, effectively captures the dynamics of open quantum systems with gain, loss, and decay. Motivated by these developments, we explore the connection between non-unitary BCFT and non-Hermitian quantum impurity critical systems. Extending the BCFT framework to include non-Hermitian impurities, we map the lattice impurity strength to an effective barrier parameter using the folding trick, establishing a direct correspondence between the lattice scattering matrix and the BCFT gluing matrix. In this correspondence, the violation of probability conservation in the non-Hermitian model manifests as non-unitary boundary conditions in BCFT. We further show that the impurity-induced energy shift at the Fermi point(s) in the lattice model matches the finite-size correction to the ground-state energy in BCFT, which scales inversely with the system size and is determined by the transmission amplitude at the Fermi momentum. This extends the known results for Hermitian impurity systems, e.g., presented in [1,2]. Our results demonstrate that BCFT techniques can be fruitfully applied to the study of non-Hermitian quantum impurity problems, providing new insights into the interplay between non-Hermiticity and boundary or defect criticality.