Rényi Correction in Quantum Systems

• Rényi Correction is defined as the subleading term in Rényi entropies, capturing universal features through logarithmic, power-law, or double-logarithmic contributions.
• It is used to probe operator content, symmetry, and scaling behavior in diverse settings including critical, massive, nonequilibrium, and holographic systems.
• Studies employ finite-size scaling, modular Hamiltonians, and twist field techniques to extract Rényi corrections, offering insights into phase transitions and quantum correlations.

A Rényi correction encompasses the subleading and non-universal contributions to Rényi entropies in quantum many-body and field theoretical systems. These corrections arise in a variety of contexts—critical systems, integrable and non-integrable models, holography, statistical mechanics, and information theory—manifesting as power-law, logarithmic, or double-logarithmic terms that encode universal information about operator spectra, symmetry, and underlying conformal or topological structure.

1. Rényi Correction in Critical Quadratic Fermion Chains

Consider the ground state of a one-dimensional translationally invariant critical quadratic free-fermion chain. The Rényi–Shannon entropy in a local occupation-number basis for a block of length $L$ exhibits an expansion

$$R_\alpha(L)=a_\alpha L+\gamma_\alpha \ln L + c_\alpha + o(1)$$

with $a_\alpha$ the leading volume-law term, $\gamma_\alpha$ the subleading logarithmic correction (Rényi correction), and $c_\alpha$ a non-universal constant (Tarighi et al., 2022). The universality and quantitatively distinct forms of $\gamma_\alpha$ depend critically on the symmetry of the chain and the structure of the dispersion polynomial zeros on the unit circle. The generic classification is as follows:

• U(1)-symmetric chains (fermion-number conserving; hopping models):

$\gamma_{\alpha} = \begin{cases} \frac{c}{8} \,, & \alpha \leq 4 \[1.5ex] \frac{\alpha}{\alpha - 1}\frac{c}{8} \,, & \alpha > 4 \end{cases}$

where $c$ is the total central charge, half the number of Fermi points.

• Non-U(1)-symmetric chains (pairing present):

$\gamma_{\alpha} = \begin{cases} \frac{b(\alpha)}{8} = \frac{\mathfrak b(\alpha) H}{8} \,, & 0 < \alpha \leq 1 \[1.5ex] \frac{\alpha}{\alpha - 1}\frac{c}{8} \,, & \alpha > 1 \end{cases}$

with $H$ the number of simple zeros on $|z|=1$, $c = H/2$, and $\mathfrak b(1) \approx 0.480016$ a numerically universal constant.

For the Ising case at $\alpha=1$,

$\gamma_1\approx 0.0600 H.$

The transitions in slope—at $\alpha=4$ (U(1)) or $\alpha=1$ (non-U(1))—mark the crossover from a Luttinger-like regime to single-configuration dominance. These coefficients have been validated using the discrete version of the Bisognano–Wichmann modular Hamiltonian, establishing their origin in operator content and central charge of the underlying CFT (Tarighi et al., 2022).

2. Corrections to Rényi Scaling in 1D and 2D Critical Models

2.1. CFT and Spin Chains

The universal scaling of $\alpha$-Rényi entropy of a block of size $\ell$ in a critical 1D chain is given by the Cardy–Calabrese formula,

$$S_\alpha(\ell) = \frac{c}{6}\Big(1 + \frac1\alpha\Big)\ln \ell + \text{const}.$$

Corrections to this scaling are classified as (Cardy et al., 2010, Xavier et al., 2011):

• Unusual n-dependent corrections due to conical singularities in the replica construction:

$\Delta S_\alpha(\ell) \sim \ell^{-2x/\alpha}$

for $x$ the scaling dimension of the operator localized at branch points (commonly the energy operator $X_\epsilon$). The exponent for the leading correction is

$$p_\alpha = \frac{2 X_\epsilon}{\alpha}, \quad (\alpha > 1)$$

with $X_\epsilon$ model-dependent. Von Neumann entropy ($\alpha=1$) has a different, typically $\ell^{-2}$, correction.

• Oscillatory corrections ($\cos(\kappa\ell)$) occur for chains with U(1) symmetry (e.g., XXZ chain) and $\alpha>1$; they are absent in chains with only discrete symmetry (Ising, Blume–Capel, three-state Potts).
• Marginally irrelevant perturbations produce corrections of the form $(\ln \ell)^{-2}$ (Cardy et al., 2010).

Finite-size studies confirm that the amplitude and scaling of these corrections allow precise extraction of operator dimensions governing low-energy physics (Xavier et al., 2011).

2.2. Mixed Boundary Conditions

For 1D critical systems with mixed open boundaries, finite-size corrections are encoded through composite twist fields $\sigma_\phi$ in the orbifold CFT, generating leading corrections of the type $L^{-2 h_{\sigma_\phi}}$ where $h_{\sigma_\phi} = h_\sigma + h_\phi/n$ (Estienne et al., 2023). This structure is universal across minimal models, including Ising and Potts chains.

3. Rényi Corrections in Massive Integrable and Quasiparticle-Excited Systems

For a 1D integrable model near criticality, the Rényi entropy for a block in the massive phase with large correlation length $\xi$ is (Calabrese et al., 2010)

$$S_n = \frac{c}{12}\left(1+\frac{1}{n}\right)\ln \xi + A_n + B_n \xi^{-x/n} + O(\xi^{-2x/n}),$$

with $x$ the dimension of the leading relevant CFT operator. This $\xi^{-x/n}$ “unusual correction” comes from the fusion of branch-point twist fields and was confirmed in XXZ and Ising chains.

For excited quasiparticle states in quantum chains, corrections to the universal part of Rényi entropy depend crucially on subsystem geometry and momentum differences between excitations. For large momenta and well-separated excitations, a universal binomial form dominates, while finite momentum difference yields analytic corrections, which can be compactly represented through determinant/permanent formulas depending on statistics (Zhang et al., 2020).

4. Logarithmic Corrections in Holography and Black Hole Entropy

In holographic systems (CFTs with AdS duals), Rényi entropies of spherical regions are mapped to horizon entropies of topological black holes in AdS. The Wald entropy receives quantum corrections of the form $C\ln S_{\text{Wald}}$ leading to universal logarithmic terms in the Rényi entropy (Mahapatra, 2016):

$S_q = S_q^{(0)} + C\,\ln(\text{area}) + \dots$

where $C$ is determined via horizon symmetries and the Cardy formula, e.g. $C = -\frac{3}{2}$ for Einstein gravity. These corrections are present in all dimensions, including odd $d$, and appear in both entanglement and Rényi entropies. At strong coupling ($1/\lambda^{3/2}$ corrections in $\mathcal{N}=4$ SYM), similar logarithmic terms are systematically computed via stringy corrections to the AdS gravitational action (Galante et al., 2013).

5. Thermal and Finite-Temperature Rényi Corrections

Thermal corrections to Rényi entropies are controlled by the excited-state spectrum and their two-point functions on $n$-sheeted geometries. For a subsystem $A$,

$$\delta S_n = \frac{n}{1-n} \sum_i \left[\frac{\langle \psi_i(z)\psi_i(z')\rangle_n}{\langle \psi_i(z)\psi_i(z')\rangle_1}-1\right]e^{-\beta E_\psi}$$

with $E_\psi$ the energy gap to the first excited state (Herzog et al., 2014, Herzog et al., 2015). For free fields and conformal theories, these corrections can be analytically computed using image methods on conical spaces (Herzog et al., 2014) and provide explicit dependence on subsystem geometry, temperature, and Rényi index.

6. Rényi Corrections in Nonequilibrium and Statistical Systems

In non-equilibrium systems such as the totally asymmetric exclusion process (TASEP), the leading behavior of Rényi entropy is the Bernoulli measure, with the correction in the maximal-current phase exhibiting a logarithmic dependence:

$$\Delta S_2(L) = -\frac{1}{2}\ln L - \ln F(\alpha,\beta) + \dots$$

Logarithmic corrections are absent in low- and high-density phases and are tightly linked to the underlying algebraic decay of correlations (Wood et al., 2017).

7. Information-Theoretic and Operational Rényi Corrections

When superselection rules (such as particle number conservation) are imposed, the accessible (operational) Rényi entanglement entropy $S_\alpha^{\text{op}}$ exhibits a double-logarithmic correction compared to the unconstrained result:

$$S_\alpha^{\text{op}}(\rho_A) \sim c_\alpha L^{d-1} \ln L - \frac{1}{2} \ln(L^{d-1}\ln L) + \dots$$

This subleading term arises from number-sector fluctuations and has been numerically and analytically confirmed for free fermions (Barghathi et al., 2018).

References (arXiv IDs)

• Universal logarithmic correction to Rényi (Shannon) entropy: (Tarighi et al., 2022)
• Corrections to scaling in massive 1D models: (Calabrese et al., 2010)
• Unusual corrections to scaling and extrapolation: (Sahoo et al., 2015)
• Unusual corrections and operator content: (Cardy et al., 2010, Xavier et al., 2011, Estienne et al., 2023)
• Holographic and black hole corrections: (Mahapatra, 2016, Galante et al., 2013, Chen et al., 2014, Chen et al., 2015, Akers et al., 2018)
• Thermal corrections (CFT/free models): (Herzog et al., 2014, Herzog et al., 2015, Zhong, 2023)
• Corrections in excited states: (Zhang et al., 2020)
• Statistical models (TASEP): (Wood et al., 2017)
• Operational entanglement: (Barghathi et al., 2018)
• Cosmological Rényi corrections: (Fazlollahi, 2022)
• Rényi common information: (Yu et al., 2018)

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The study of Rényi corrections provides a powerful route to probe operator content, universality classes, quantum criticality, holography, and even gravitational corrections, serving as an indispensable tool in modern theoretical physics and quantum information theory.