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Rényi-like entanglement probe of the chiral central charge

Julian Gass, Michael Levin

TL;DR

This work introduces a Rényi-like entanglement probe ω_{α,β} for 2D gapped ground states, defined as the phase of ⟨ρ_{AB}^α ρ_{BC}^β⟩, to extract the chiral central charge $c_-$. It develops a general framework with replica-based intuition, proves a single-particle determinant formula for non-interacting fermions, and shows a universal relation ω_{α,β} = exp[−(π i/12) q(α,β) c_-] (with q(α,β) = α/(α+1) + β/(β+1) − (α+β)/(α+β+1)) in the large-region limit; for complex fermions, c_- = ν(P)/2, while for Majorana fermions, ω_{α,β} = exp[−(π i/24) q(α,β) ν(P)]. In string-net models with $c_-=0$, ω_{α,β} = 1, consistent with the same formula. The paper also discusses a replica representation for integer α,β and notes caveats: ω_{α,β} may yield spurious values in fine-tuned cases, and thus may best capture generic gapped states. Overall, the Renyi modular framework provides a practical route—analytically, numerically, and potentially experimentally—to access topological chiral data from ground-state wavefunctions.

Abstract

We propose a ground state entanglement probe for gapped, two-dimensional quantum many-body systems that involves taking powers of reduced density matrices in a particular geometric configuration. This quantity, which we denote by $ω_{α,β}$, is parameterized by two positive real numbers $α, β$, and can be seen as a ``Rényi-like" generalization of the modular commutator -- another entanglement probe proposed as a way to compute the chiral central charge from a bulk wave function. We obtain analytic expressions for $ω_{α,β}$ for gapped ground states of non-interacting fermion Hamiltonians as well as ground states of string-net models. In both cases, we find that $ω_{α,β}$ takes a universal value related to the chiral central charge. For integer values of $α$ and $β$, our quantity $ω_{α,β}$ can be expressed as an expectation value of permutation operators acting on an appropriate replica system, providing a natural route to measuring $ω_{α,β}$ in numerical simulations and potentially, experiments.

Rényi-like entanglement probe of the chiral central charge

TL;DR

This work introduces a Rényi-like entanglement probe ω_{α,β} for 2D gapped ground states, defined as the phase of ⟨ρ_{AB}^α ρ_{BC}^β⟩, to extract the chiral central charge . It develops a general framework with replica-based intuition, proves a single-particle determinant formula for non-interacting fermions, and shows a universal relation ω_{α,β} = exp[−(π i/12) q(α,β) c_-] (with q(α,β) = α/(α+1) + β/(β+1) − (α+β)/(α+β+1)) in the large-region limit; for complex fermions, c_- = ν(P)/2, while for Majorana fermions, ω_{α,β} = exp[−(π i/24) q(α,β) ν(P)]. In string-net models with , ω_{α,β} = 1, consistent with the same formula. The paper also discusses a replica representation for integer α,β and notes caveats: ω_{α,β} may yield spurious values in fine-tuned cases, and thus may best capture generic gapped states. Overall, the Renyi modular framework provides a practical route—analytically, numerically, and potentially experimentally—to access topological chiral data from ground-state wavefunctions.

Abstract

We propose a ground state entanglement probe for gapped, two-dimensional quantum many-body systems that involves taking powers of reduced density matrices in a particular geometric configuration. This quantity, which we denote by , is parameterized by two positive real numbers , and can be seen as a ``Rényi-like" generalization of the modular commutator -- another entanglement probe proposed as a way to compute the chiral central charge from a bulk wave function. We obtain analytic expressions for for gapped ground states of non-interacting fermion Hamiltonians as well as ground states of string-net models. In both cases, we find that takes a universal value related to the chiral central charge. For integer values of and , our quantity can be expressed as an expectation value of permutation operators acting on an appropriate replica system, providing a natural route to measuring in numerical simulations and potentially, experiments.
Paper Structure (19 sections, 1 theorem, 112 equations, 6 figures)

This paper contains 19 sections, 1 theorem, 112 equations, 6 figures.

Table of Contents

  1. Introduction
  2. General properties of $\omega_{\alpha,\beta}$
  3. Consistency checks
  4. Replica representation
  5. Computation for non-interacting fermions
  6. Quasidiagonality and the real-space Chern number
  7. Computing Lg
  8. Majorana fermions
  9. Computation for string-net models
  10. Review of string-net models
  11. Doubling the link degrees of freedom
  12. Structure of $\rho_R$ for string-net ground states
  13. Computing $\omega_{\alpha,\beta}$
  14. Discussion
  15. Properties of quasidiagonal operators
  16. ...and 4 more sections

Key Result

Lemma D.1

Let $X$ and $Y$ be bounded operators with trace class commutator. For any holomorphic functions $f$ and $g$ for which we can define $f(X)$ and $g(Y)$ via the holomorphic functional calculus, $[f(X),g(Y)]$ is also trace class.

Figures (6)

  • Figure 1: Geometry used to compute the modular commutator $J$ and its Rényi generalization $\omega_{\alpha,\beta}$.
  • Figure 2: Tripartition of the lattice used in the definition of the real-space Chern number \ref{['nu_kitaev']}.
  • Figure 3: Depiction of the spatial regions $\Pi_i$ used in Eq. \ref{['u_truncation']}, which surround each of the four triple points in the $ABCD$ partition. Outside of these regions, $U$ is approximately equal to the identity.
  • Figure 4: Tripartitions of the plane used to compute (a) $\det\nolimits_{\Pi_1}U$ and (b) $\det\nolimits_{\Pi_2}U$. The dotted lines show the boundaries of the original $ABC$ region.
  • Figure 5: The two terms in the string-net Hamiltonian (\ref{['snet-Ham']}). The regions of support of the $Q_v$ and $B_p$ operators are indicated by filled circles.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma D.1
  • proof