Entanglement Spectra and Modular Spectral Data

An entanglement spectrum is the spectrum of a regulated reduced density matrix, usually displayed as modular energies. It can approximate continuum modular spectral information for controlled observables, but a finite cutoff spectrum is not the spectrum of a sharp type-III local density matrix, because no such intrinsic density matrix exists.

Required background. Use Relative Entropy for QFT States. Helpful background. Gaussian correlation-matrix entropy supplies a computable regulated example.

Eigenvalues and modular energies

For a type-I subsystem with

$$\rho_A=\sum_i\lambda_i\lvert i\rangle\langle i\rvert,$$

define the entanglement energies

$$\xi_i=-\log\lambda_i.$$

Equivalently, $K_A=-\log\rho_A$ is the modular Hamiltonian of the regulated state. The full spectrum contains more information than any one Rényi entropy because

$$\operatorname{Tr}\rho_A^n=\sum_i e^{-n\xi_i}.$$

Normalization, degeneracy counting, symmetry sectors, and whether a constant shift has been removed must be stated in a spectral plot.

The structure figure places spectra with geometric data only after the algebra and regulator have been fixed.

Entanglement energies are spectral data of a chosen type-I realization. Their low-lying scaling and symmetry organization may approach modular information, while ultraviolet levels retain cutoff and boundary sensitivity. Schematic.

Free-fermion interval

For a Gaussian fermionic state, restrict the two-point function to $A$ and diagonalize its eigenvalues $\nu_j$. The many-body density matrix factorizes into entanglement modes with single-particle modular energies

$$\varepsilon_j=\log\frac{1-\nu_j}{\nu_j}.$$

At several lattice spacings, compare low-lying gaps after fixing the physical interval and boundary prescription. Scaling stability of selected low modes can support a continuum modular interpretation. High modes near $\nu=0$ or $1$ are extremely sensitive to truncation and should not be overinterpreted.

Degeneracy patterns may reflect a symmetry sector, edge theory, or topology, but they can also arise from the cut, accidental near-degeneracy, or finite size. A phase diagnosis belongs with Hamiltonian, response, and finite-size evidence, not with this spectrum alone. Li and Haldane 2008, pp. 010504-1–010504-4 introduced the influential spectrum diagnostic in fractional Hall states.

Continuum boundary

The validity map makes the type-I/type-III distinction explicit.

Low-lying spectral scaling can be meaningful under controlled cutoff removal. Cut geometry, basis truncation, symmetry resolution, and ultraviolet tails can alter degeneracies, so a finite regulated spectrum is not an intrinsic sharp-region density-matrix spectrum. Schematic.

Report the modular-energy convention, cut, regulator, physical size, symmetry resolution, truncation, and scaling window. Compare spectral claims with entropy moments and local modular observables when available.

References

• Li, Hui, and F. Duncan M. Haldane. “Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States.” Physical Review Letters 101 (2008): 010504. DOI. Open preprint.

Further reading

• Bisognano, Joseph J., and Eyvind H. Wichmann. “On the Duality Condition for a Hermitian Scalar Field.” Journal of Mathematical Physics 16 (1975): 985–1007. DOI.