Non-Hermitian free-fermion critical systems and logarithmic conformal field theory

Conformal invariance often accompanies criticality in Hermitian systems. However, its fate in non-Hermitian settings is less clear, especially near exceptional points where the Hamiltonian becomes non-diagonalizable. Here we investigate whether a 1+1-dimensional gapless non-Hermitian system can admit a conformal description, focusing on a PT-symmetric free-fermion field theory. Working in the biorthogonal formalism, we identify the conformal structure of this theory by constructing a traceless energy-momentum tensor whose Fourier modes generate a Virasoro algebra with central charge $c=-2$. This yields a non-Hermitian, biorthogonal realization of a logarithmic conformal field theory, in which correlation functions exhibit logarithmic scaling and the spectrum forms Virasoro staggered modules that are characterized by universal indecomposability parameters. We further present a microscopic construction and show how the same conformal data (with finite-size corrections) can be extracted from the lattice model at exceptional-point criticality, thereby supporting the field-theory prediction.

💡 Research Summary

The paper investigates whether a one‑dimensional non‑Hermitian, PT‑symmetric free‑fermion field theory can possess a full two‑dimensional conformal symmetry, and it establishes that it does, but in the form of a logarithmic conformal field theory (LCFT) with central charge (c=-2). Starting from the action (1) containing non‑Hermitian mass ((\Delta)) and kinetic ((u)) terms, the authors employ a bi‑orthogonal quantization scheme that treats left‑ and right‑moving fields (\psi_{L}^{\pm}) and (\psi_{R}^{\pm}) as independent. Canonical anti‑commutation relations are imposed between left and right fields, leading to a Hamiltonian (4) with an effective Fermi velocity (v_{F}=\pm\sqrt{v^{2}+u\Delta}). When (\Delta\neq0) the zero‑mode sector becomes non‑diagonalizable, signalling an exceptional point (EP) and a gapless, linearly dispersing spectrum.

To reveal conformal invariance the authors construct an improved, traceless energy‑momentum tensor by adding a total‑derivative term involving a conserved current (I_{\nu}). Fourier modes of the tensor define holomorphic and antiholomorphic generators (L_{n}) and (\bar L_{n}) (7‑9). Explicit calculation shows that these obey the Virasoro algebra with central charge (c=\bar c=-2) (10). The negative central charge reflects the intrinsic non‑unitarity of the theory; in the Hermitian limit ((\Delta=u=0)) the model reduces to a Dirac CFT with (c=1).

The LCFT nature emerges from the Jordan‑block structure of the dilatation operator (L_{0}). Two ground‑state vectors (|\phi_{R0}\rangle) and (|\psi_{R0}\rangle) form a rank‑1 Jordan cell, and higher‑level staggered modules are built from fermionic mode operators (13). The indecomposability parameters (\beta_{M}=a_{M}b_{M}) are computed analytically (20) and match those of the well‑studied (c=-2) symplectic‑fermion LCFT, confirming that the representation theory of the Virasoro algebra is identical even though the underlying fields differ (spinors versus anticommuting scalars). Correlation functions exhibit logarithmic scaling, e.g. (\langle\psi_{R}^{-}(x)\psi_{L}^{+}(y)\rangle\sim\ln|x-y|) (11), a hallmark of non‑diagonalizable dilatations.

A concrete lattice realization is provided via a tight‑binding chain with alternating hoppings and a staggered on‑site potential (21). Two PT‑symmetric implementations (site‑centered and bond‑centered) correspond to the two PT actions in the continuum. At the EP ((\mu_{s}=\pm2i\delta)) the Bloch Hamiltonian becomes defective, and expanding around this point yields the continuum parameters (v=2t), (\Delta=4i\delta), (u=i\delta) (22). The lattice two‑point function displays the same logarithmic term as the field theory (23). Using the Koo‑Saleur construction, lattice analogues of the Virasoro generators are defined from the local Hamiltonian density and momentum density (24). Finite‑size spectra extracted from these generators reproduce the central charge (-2) and the scaling dimensions predicted by the LCFT, providing a quantitative bridge between the microscopic model and the continuum description.

In summary, the work demonstrates that non‑Hermitian free‑fermion systems at exceptional‑point criticality realize a bi‑orthogonal LCFT with (c=-2). It establishes the full conformal structure both analytically in the field theory and numerically in a lattice model, highlighting logarithmic scaling, Jordan‑cell representations, and universal indecomposability parameters as robust signatures of non‑Hermitian criticality. This advances our understanding of how conformal invariance can survive beyond Hermiticity and opens avenues for exploring LCFT physics in experimentally relevant non‑Hermitian platforms.