Rényi-like entanglement probe of the chiral central charge

1. Problem Statement

The chiral central charge ($c_-$) is a fundamental topological invariant characterizing two-dimensional (2D) gapped quantum many-body systems. It determines the thermal Hall conductance at low temperatures. While $c_-$ is traditionally defined via transport properties, there is a strong theoretical interest in computing it directly from a ground state wavefunction on an infinite 2D plane without relying on thermal properties.

Previous work proposed the modular commutator ($J$) as a bulk formula to extract $c_-$:
$$J = i\langle [\ln \rho_{AB}, \ln \rho_{BC}] \rangle$$
where $\rho_{AB}$ and $\rho_{BC}$ are reduced density matrices of adjacent regions. While $J$ has been proven for specific cases (non-interacting fermions, entanglement bootstrap axioms), it involves logarithms of density matrices, which are difficult to compute analytically for interacting systems and challenging to measure in numerical simulations or experiments.

The Goal: The authors propose a generalization of the modular commutator, termed the Rényi modular commutator ($\omega_{\alpha,\beta}$), which avoids logarithms, allows for integer parameters suitable for replica methods, and serves as a probe for $c_-$.

2. Methodology

The authors define the Rényi modular commutator $\omega_{\alpha,\beta}$ for two positive real parameters $\alpha, \beta$ and three non-overlapping regions $A, B, C$ arranged in a specific geometry (Fig. 1):
$$\omega_{\alpha,\beta} = \frac{\langle \rho_{AB}^\alpha \rho_{BC}^\beta \rangle}{|\langle \rho_{AB}^\alpha \rho_{BC}^\beta \rangle|}$$
This quantity represents the phase of the expectation value of the product of powers of reduced density matrices.

Key Methodological Steps:

1. Replica Representation: For integer $\alpha=m, \beta=n$, the authors show that $\omega_{m,n}$ can be expressed as the expectation value of a permutation operator acting on a system of $m+n+1$ replicas. This connects the quantity to standard numerical techniques (like Quantum Monte Carlo) and experimental protocols.
2. Analytic Computation for Non-Interacting Fermions:
   • They utilize the quasidiagonality property of the spectral projector $P$ for gapped systems (matrix elements decay exponentially with distance).
   • They derive a single-particle determinant formula for $\langle \rho_{AB}^\alpha \rho_{BC}^\beta \rangle$.
   • Using the polar decomposition $M = U|M|$, they isolate the unitary phase $U$.
   • They employ a local truncation argument, showing that the determinant of $U$ receives contributions only from the "triple points" where regions $A, B, C$ meet.
   • They reduce the problem to computing the trace of commutators of operators supported on these triple points, relating the result to the real-space Chern number $\nu(P)$.
3. Analytic Computation for String-Net Models:
   • They analyze exactly solvable string-net models (which realize topological phases with $c_-=0$).
   • They utilize the specific structure of the reduced density matrix for string-net ground states, which factorizes into a boundary operator $X_R$ and a bulk projector $P_R$.
   • They demonstrate that the expectation value $\langle \rho_{AB}^\alpha \rho_{BC}^\beta \rangle$ is strictly real and positive, implying the phase is trivial.

3. Key Contributions and Results

A. Universal Formula

The primary result is the derivation of a universal formula for $\omega_{\alpha,\beta}$ in the limit of large regions $A, B, C$:
$$\omega_{\alpha,\beta} = \exp\left[ -\frac{\pi i}{12} q(\alpha, \beta) c_- \right]$$
where the function $q(\alpha, \beta)$ is given by:
$$q(\alpha, \beta) = \frac{\alpha}{\alpha + 1} + \frac{\beta}{\beta + 1} - \frac{\alpha + \beta}{\alpha + \beta + 1}$$

B. Verification in Specific Models

1. Non-Interacting Fermions:
   • For complex fermions, the authors prove $\omega_{\alpha,\beta} = \exp[-\frac{\pi i}{24} q(\alpha, \beta) \nu(P)]$, where $\nu(P)$ is the Chern number. Since $c_- = \nu(P)/2$ (or $\nu(P)$ depending on convention, the paper clarifies $c_- = \nu(P)$ for complex fermions in their specific normalization context, though standardly $c_- = \nu/2$ for Dirac fermions; the paper states $c_- = \nu(P)/2$ for Majorana and $c_- = \nu(P)$ for complex, leading to the unified formula).
   • The result holds for both charge-conserving (complex) and Majorana fermion models.
2. String-Net Models:
   • For string-net ground states, the authors prove $\omega_{\alpha,\beta} = 1$.
   • This is consistent with the formula because string-net models have a vanishing chiral central charge ($c_- = 0$).

C. Consistency Checks

The proposed quantity satisfies necessary physical properties for a topological invariant:

• Multiplicativity: $\omega_{\alpha,\beta}(\rho \otimes \rho') = \omega_{\alpha,\beta}(\rho) \cdot \omega_{\alpha,\beta}(\rho')$.
• Time-Reversal Symmetry: It is odd under time reversal ($\omega_{\alpha,\beta}(T\rho T^{-1}) = \omega_{\alpha,\beta}(\rho)^*$).
• Triviality for Product States: It equals 1 for any tripartite pure state supported on only three of the four regions ($A, B, C, D$).
• Limit Recovery: As $\alpha, \beta \to 0$, $\omega_{\alpha,\beta}$ recovers the original modular commutator $J = \frac{\pi}{3} c_-$.

D. Limitations

The authors acknowledge that the formula is not universally true for all gapped Hamiltonians. There exist fine-tuned counterexamples (similar to those for the original modular commutator) where $\omega_{\alpha,\beta}$ yields "spurious" values unrelated to $c_-$. However, they argue it holds for generic (non-fine-tuned) gapped ground states.

4. Significance

• Experimental and Numerical Accessibility: The most significant advantage of $\omega_{\alpha,\beta}$ over the original modular commutator is that for integer $\alpha, \beta$, it can be measured as an expectation value of a permutation operator in a replica system. This makes it directly accessible to Quantum Monte Carlo simulations and potential experimental protocols (e.g., using interferometry or swap tests), which struggle with the logarithm of density matrices required for $J$.
• Analytic Tractability: The formulation allows for analytic derivation in interacting systems (like string-nets) where the logarithm of the density matrix is intractable.
• Generalization of Topological Probes: It establishes a family of entanglement probes parameterized by $\alpha, \beta$, suggesting that replica permutation operators can extract a broader class of topological data beyond just the central charge (potentially including topological spins and quantum dimensions).

Conclusion

Gass and Levin successfully generalize the modular commutator into a Rényi-like quantity $\omega_{\alpha,\beta}$. They prove that this quantity yields a universal phase directly proportional to the chiral central charge $c_-$ for generic gapped systems, including non-interacting fermions and string-net models. By avoiding logarithms and utilizing replica permutation operators, this probe offers a more practical route for extracting topological invariants in both theoretical calculations and experimental/numerical settings.