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Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes

Pierre-Antoine Bernard, Zachary Mann, Gilles Parez, Luc Vinet

Abstract

This study investigates the scaling behavior of the ground-state entanglement entropy in a model of free fermions on folded cubes. An analytical expression is derived in the large-diameter limit, revealing a strict adherence to the area law. The absence of the logarithmic enhancement expected for free fermions is explained using a decomposition of folded cubes in chains based on its Terwilliger algebra and $\mathfrak{so}(3)_{-1}$. The entanglement Hamiltonian and its relation to Heun operators are also investigated.

Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes

Abstract

This study investigates the scaling behavior of the ground-state entanglement entropy in a model of free fermions on folded cubes. An analytical expression is derived in the large-diameter limit, revealing a strict adherence to the area law. The absence of the logarithmic enhancement expected for free fermions is explained using a decomposition of folded cubes in chains based on its Terwilliger algebra and . The entanglement Hamiltonian and its relation to Heun operators are also investigated.
Paper Structure (16 sections, 102 equations, 7 figures)

This paper contains 16 sections, 102 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Free fermions on folded cubes
  3. $d$-cubes and folded $d$-cubes
  4. Free-fermion Hamiltonian on $\square_{2d + 1}$
  5. Ground state and correlation matrix
  6. Bipartite entanglement entropy
  7. Definitions
  8. Dimensional reduction and $\mathfrak{so}(3)_{-1}$
  9. Anti-Krawtchouk chains and $S(j)$
  10. Numerical investigation of $S(j)$
  11. Large diameter limit of the folded cube
  12. Entanglement Hamiltonian and Heun operator
  13. Commuting Heun operator and $\varrho_T(t_1,t_2)$
  14. Rényi fidelities and Réyni entropies of $\varrho_A$ and $\varrho_T(t_1,t_2)$
  15. Conclusion
  16. ...and 1 more sections

Figures (7)

  • Figure 1: A $3$-cube (left) and a $4$-cube (right).
  • Figure 2: Folded $4$-cube. It is either obtained by merging antipodal vertices in a $4$-cube or by adding the red edges between the antipodal vertices of the $3$-cube in black.
  • Figure 3: Scaling of the entanglement entropy $S(j)$ for half anti-Krawtchouk chains at half-filling. Oscillations are due to sub-leading terms and vanish with large $j$. Solid lines were obtained by fitting \ref{['sc1']} with the unknown coefficient $a_1(\kappa,\xi)$.
  • Figure 4: Entanglement entropy of anti-Krawtchouk chains in the large-diameter limit with fixed filling ratio $\kappa$, aspect ratio $\Delta = L/d$ and subsystem size $\ell$. The left figure illustrates the convergence of $S(d-L-1 + \ell)$ to a value $a_2(\kappa,\Delta,\ell)$ of magnitude near $\ln(2)$ at large diameter $d$ and $\kappa = \Delta = 1/2$. The right figure presents the value of the ratio $a_2(\kappa,\Delta,\ell)/\ell \ln(2)$ at $\kappa = 1/2$ for various $\Delta$ and $\ell$.
  • Figure 5: Ratio of the entanglement entropy $S_A$ over the boundary area $|\partial A|$ for region $A$ composed of the first $L$ neighborhoods of a vertex in a folded cube, at half filling $\kappa = 1/2$. The entanglement entropy $S_A$ is obtained by numerical diagonalization of the truncated correlation matrix $C$. In the large-diameter $d$ limit, the ratio $S_A/|\partial A|$ converges near $\ln(2)$ and shows no logarithmic enhancement.
  • ...and 2 more figures