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

Iao-Fai Io, Fu-Hsiang Huang, Chang-Tse Hsieh

TL;DR

This work demonstrates that conformal invariance can emerge in a genuinely non-Hermitian 1+1D system at an exceptional point. Using a biorthogonal framework, the PT-symmetric free-fermion field theory admits a traceless energy-momentum tensor and two commuting Virasoro algebras with central charge $c=-2$, signaling a logarithmic CFT structure with indecomposable Virasoro modules. The LCFT data are shown to be universal by a lattice realization: at the EP, the lattice spectrum and correlation functions exhibit logarithmic scaling, and a refined Koo–Saleur construction reveals a lattice $c=-2$ Virasoro algebra with level-$M$ indecomposability parameters matching the continuum predictions. Finite-size spectra on the lattice reproduce the same indecomposability parameters $\beta_M$, reinforcing the field-theoretic LCFT description and illustrating a robust non-Hermitian universality class for critical points.

Abstract

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.

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

TL;DR

This work demonstrates that conformal invariance can emerge in a genuinely non-Hermitian 1+1D system at an exceptional point. Using a biorthogonal framework, the PT-symmetric free-fermion field theory admits a traceless energy-momentum tensor and two commuting Virasoro algebras with central charge , signaling a logarithmic CFT structure with indecomposable Virasoro modules. The LCFT data are shown to be universal by a lattice realization: at the EP, the lattice spectrum and correlation functions exhibit logarithmic scaling, and a refined Koo–Saleur construction reveals a lattice Virasoro algebra with level- indecomposability parameters matching the continuum predictions. Finite-size spectra on the lattice reproduce the same indecomposability parameters , reinforcing the field-theoretic LCFT description and illustrating a robust non-Hermitian universality class for critical points.

Abstract

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 . 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.
Paper Structure (15 sections, 120 equations, 1 figure, 1 table)

This paper contains 15 sections, 120 equations, 1 figure, 1 table.

Table of Contents

  1. Canonical quantization of 1+1d PT-symmetric free-fermion field theory
  2. Conformal invariance of the PT-symmetric free-fermion field theory
  3. Improved energy-momentum tensor
  4. Virasoro algebra
  5. General form of the indecomposability parameters
  6. LCFT data of the PT-symmetric free-fermion lattice model
  7. Spectrum and two-point correlation functions at EP
  8. Single-particle spectrum.
  9. Two-point correlation functions.
  10. Lattice Virasoro operators and indecomposability parameters
  11. Koo-Saleur construction.
  12. Improved lattice Virasoro operators.
  13. Indecomposability parameters on the lattice.
  14. Level 1.
  15. Level 2.

Figures (1)

  • Figure 1: Level-$M$ holomorphic staggered module.