Entanglement Hamiltonian and orthogonal polynomials

TL;DR

The paper studies EHs in inhomogeneous free-fermion chains whose single-particle states are tied to discrete orthogonal polynomials of the Askey scheme. By exploiting bispectrality, it constructs a tridiagonal operator $T_{ ext{A}}$ that commutes with the lattice EH and deforms the Hamiltonian linearly or parabolically, with the deformation controlled by the dual polynomial spectrum. A curved-space CFT analysis identifies the corresponding space-dependent inverse temperature $ ilde{eta}(x)$, which fixes the energy scale and explains why the rescaled $T_{ ext{A}}$ spectrum closely approximates the entanglement spectrum and entropy. The method is illustrated across Krawtchouk, dual Hahn, Hahn, and Racah chains, showing excellent agreement in the low-lying EH spectrum and offering a framework for connecting lattice models to BW-type EH via conformal data.

Abstract

We study the entanglement Hamiltonian for free-fermion chains with a particular form of inhomogeneity. The hopping amplitudes and chemical potentials are chosen such that the single-particle eigenstates are related to discrete orthogonal polynomials of the Askey scheme. Due to the bispectral properties of these functions, one can construct an operator which commutes exactly with the entanglement Hamiltonian and corresponds to a linear or parabolic deformation of the physical one. We show that this deformation is interpreted as a local inverse temperature and can be obtained in the continuum limit via methods of conformal field theory. Using this prediction, the properly rescaled eigenvalues of the commuting operator are found to provide a very good approximation of the entanglement spectrum and entropy.

Figures (5)

• Figure 1: Conformal mapping to the annulus.
• Figure 2: Comparison between the entanglement spectrum $\varepsilon_k$ (full symbols) and the scaled eigenvalues $2\pi\lambda_k/(N v_F(x_0))$ of the $T_{\mathcal{A}}$ matrix (empty symbols) for the Krawtchouck chain in the symmetric $p=1/2$ case at different $\ell$ (left), and for $p=1/4,1/3$ with fixed $\ell=20$ (right). The data are obtained numerically for $N=60$ and at half filling. The insets show the corresponding difference $\delta_k=\varepsilon_k-2\pi \lambda_k/(N v_F(x_0))$ between the two spectra, as function of $\varepsilon_k$.
• Figure 3: Fermi velocity \ref{['eq:vFHahn']} of the half-filled Hahn chain for different values of $x_1$.
• Figure 4: Comparison of the entropies $S$ and $S_{CFT}$ for the Hahn chain as function of $\ell$, computed from \ref{['eq:S']} using the eigenvalues $\varepsilon_k$ and $2\pi \lambda_k/ (N^2 (2x_0+2x_1)v_F(x_0))$, respectively. The data are obtained at half filling with $N=60$ and $x_1=x_2=8/N$. The inset shows the difference $\delta S=S-S_{CFT}$.
• Figure 5: Fermi velocity of the half-filled Racah chain with $x_1 = 3$ and $x_2 = 1$, for different values of $x_3$. This chain reduces to the Hahn case with $x_1 = x_2 =1$ in the $x_3 \rightarrow \infty$ limit.