一維非厄密特量子系統之不可約化參數

Abstract

共形不變性通常出現在臨界系統中，此時關聯長度發散，這種情況在無能隙的厄米量子系統（熱力學極限下）中尤為常見。然而，當厄米性被破壞時，情況會變得更加微妙。在本工作中，我們提出了一個僅包含質量項與二階導數項、而不含一階導數（動能）項的 (1 + 1) 維非厄米自由費米子量子場論。有趣的是，該系統具有線性色散的無能隙譜。我們通過構造一組滿足 Virasoro 代數且中心荷c = −2 的無限多算符，證明該模型在低能下具有共形不變性，這反映了系統的非么正性。此外，該模型的希爾伯特空間呈現不可分解的 Jordan-cell 結構，表明它是一個對數共形場論。我們同時給出此非厄米場論的晶格實現，並從中識別出一個表徵系統不可分解性的普適量。結果顯示，我們模型的中心荷與不可分解參數與辛費米子理論完全一致，暗示兩者之間存在密切聯繫。

Conformal invariance typically emerges in critical systems where the correlation length diverges, as is generally the case for Hermitian quantum systems without a spectral gap in the thermodynamic limit. However, the situation becomes more subtle when Hermiticity is broken. In this work, we propose a (1+1)-dimensional non-Hermitian freefermion quantum field theory containing only mass and second-derivative terms, without a first-derivative (kinetic) term. Interestingly, this system possesses a gapless spectrum with linear dispersion. By constructing an infinite set of operators that satisfy the Virasoro algebra, we demonstrate that this model is conformally invariant at low energy, with a resulting central charge of −2, reflecting the system’s non-unitarity. Furthermore, the Hilbert space of this model exhibits indecomposable Jordan-cell structures, indicating that the theory is a logarithmic conformal field theory. We also give a lattice realization of our proposed non-Hermitian field theory, from which we identify a universal quantity characterizing the system’s indecomposability. It turns out that the central charge and indecomposability parameter of our model are identical to those of the symplectic fermion theory, suggesting a close connection between the two.