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Rényi entropies for one-dimensional quantum systems with mixed boundary conditions

Benoit Estienne, Yacine Ikhlef, Andrei Rotaru

TL;DR

This work develops a general method to compute Rényi entropies $S_N^{(\alpha\beta)}$ for a one-dimensional critical system with mixed open boundaries, focusing on an interval starting at a boundary. By recasting the problem in a cyclic orbifold BCFT and leveraging null-vector constraints, the authors derive ordinary differential equations for the relevant orbifold conformal blocks, enabling exact scaling functions $\mathcal{F}_N^{(\alpha\beta)}$ and leading finite-size corrections. They provide explicit results for the second Rényi entropy in generic BCFTs, and specialize to the Ising and three-state Potts models, obtaining detailed leading and subleading scaling functions that show excellent agreement with lattice simulations for modest system sizes. The approach combines orbifold Ward identities, Frobenius-series solutions, and boundary OPE data to deliver concrete, testable predictions for boundary-affected entanglement in 1D critical systems, with clear limitations for larger $N$ due to increasing differential equation order. These results illuminate the role of boundary conditions in quantum entanglement and offer a framework for precise comparison with numerics and potential extensions to bulk intervals and larger $N$.

Abstract

We present a general method for calculating Rényi entropies in the ground state of a one-dimensional critical system with mixed open boundaries, for an interval starting at one of its ends. In the conformal field theory framework, this computation boils down to the evaluation of the correlation function of one twist field and two boundary condition changing operators in the cyclic orbifold. Exploiting null-vectors of the cyclic orbifold, we derive ordinary differential equations satisfied by these correlation functions. In particular, we obtain an explicit expression for the second Rényi entropy valid for any diagonal minimal model, but with a particular set of mixed boundary conditions. In order to compare our results with numerical data for the Ising and three-state Potts critical chains, we also identify and compute the leading finite size corrections.

Rényi entropies for one-dimensional quantum systems with mixed boundary conditions

TL;DR

This work develops a general method to compute Rényi entropies for a one-dimensional critical system with mixed open boundaries, focusing on an interval starting at a boundary. By recasting the problem in a cyclic orbifold BCFT and leveraging null-vector constraints, the authors derive ordinary differential equations for the relevant orbifold conformal blocks, enabling exact scaling functions and leading finite-size corrections. They provide explicit results for the second Rényi entropy in generic BCFTs, and specialize to the Ising and three-state Potts models, obtaining detailed leading and subleading scaling functions that show excellent agreement with lattice simulations for modest system sizes. The approach combines orbifold Ward identities, Frobenius-series solutions, and boundary OPE data to deliver concrete, testable predictions for boundary-affected entanglement in 1D critical systems, with clear limitations for larger due to increasing differential equation order. These results illuminate the role of boundary conditions in quantum entanglement and offer a framework for precise comparison with numerics and potential extensions to bulk intervals and larger .

Abstract

We present a general method for calculating Rényi entropies in the ground state of a one-dimensional critical system with mixed open boundaries, for an interval starting at one of its ends. In the conformal field theory framework, this computation boils down to the evaluation of the correlation function of one twist field and two boundary condition changing operators in the cyclic orbifold. Exploiting null-vectors of the cyclic orbifold, we derive ordinary differential equations satisfied by these correlation functions. In particular, we obtain an explicit expression for the second Rényi entropy valid for any diagonal minimal model, but with a particular set of mixed boundary conditions. In order to compare our results with numerical data for the Ising and three-state Potts critical chains, we also identify and compute the leading finite size corrections.
Paper Structure (34 sections, 219 equations, 7 figures, 3 tables)

This paper contains 34 sections, 219 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Scaling functions for Rényi entropies
  4. Generic BCFT
  5. The $N=2$ Rényi entropy.
  6. The $N=3$ Rényi entropy.
  7. The critical Ising model
  8. The critical three-state Potts model
  9. The cyclic orbifold in the presence of a boundary
  10. The cyclic orbifold construction
  11. The cyclic orbifold on the upper half-plane
  12. Differential equations for the scaling functions
  13. The function $\langle \Psi_{12}\cdot \sigma \cdot \Psi_{12} \rangle$ in a generic $\mathbb{Z}_2$ orbifold
  14. The function $\langle \Psi_{12} \cdot \sigma_{\phi} \cdot \Psi_{12} \rangle$ in a generic $\mathbb{Z}_2$ orbifold
  15. The function $\langle \Psi_{12} \cdot \sigma \cdot \Psi_{12} \rangle$ in a generic $\mathbb{Z}_3$ orbifold
  16. ...and 19 more sections

Figures (7)

  • Figure 1: An interval of length $\ell$ in a 1d critical chain with mixed BC $(\alpha\beta)$ and length $L$.
  • Figure 2: Operator insertions for the $\mathbb{Z}_N$ orbifold BCFT correlators of interest on the upper-half plane with mixed BC ($\alpha$ for $x>0$ and $\beta$ for $x<0$ ). There is another BCCO at $x=\infty$, which is not depicted here.
  • Figure 3: The second Rényi entropy of the interval $[0,m]$ in the critical Ising chain with two types of mixed BC, for a chain of size $M=26$. In both cases, the interval is adjacent to the $\alpha=+$ boundary.
  • Figure 4: The second Rényi entropy of the interval $[0,m]$ in the critical Ising chain of size $M=26$ with mixed free fixed BC. Inclusion of the subleading term in the expansion \ref{['eq_generic_subleading_corrections']} is crucial for obtaining a satisfying agreement between CFT predictions and lattice results.
  • Figure 5: The third Rényi entropy of the interval $[0,m]$ in the critical Ising chain with two types of mixed BC, for a chain of size $M=26$. In both cases, the interval is adjacent to the $\alpha=+$ boundary.
  • ...and 2 more figures