Fermion-Parity Resolution in Majorana CFT

• Fermion-parity resolution is a method for decomposing the entanglement spectrum in fermionic systems by partitioning sectors based on parity eigenvalues.
• It employs precise Hilbert space factorization into NS and Ramond sectors to reveal the impact of boundary conditions and Majorana zero modes on entanglement.
• The approach provides actionable insights for identifying symmetry-protected topological phases, bridging theoretical models with experimental diagnostics.

Fermion-parity resolution refers to the decomposition and analysis of quantum entanglement, reduced density matrices, and associated spectra according to sectors labeled by the eigenvalues of the fermion-parity operator $F$, the generator of the $\mathbb{Z}_2^F$ symmetry in fermionic theories. In the context of the Majorana fermion conformal field theory (CFT), this perspective goes beyond standard entanglement entropy measures and allows the detailed paper of how different boundary conditions, spin structures, and the presence or absence of Majorana zero modes impact the entanglement characteristics of a given state. The concept is central both to the mathematical structure of fermionic quantum information and to physical realizations of symmetry-protected topological (SPT) phases.

1. Entanglement in Majorana BCFT and Fermion-Parity Resolution

In the Majorana fermion CFT, entanglement properties are systematically studied for distinct global states: the vacuum, the primary fermion excitation, and states built from conformal interfaces. The boundary state methodology enables explicit Hilbert space factorizations, parametrized by conformal boundary conditions, leading to two distinct factorization maps $$\iota_{(\alpha\beta)}: \mathcal{H} \to \mathcal{H}^{A}_{(\alpha\beta)} \otimes \mathcal{H}^{B}_{(\alpha\beta)}$$, with $(\alpha, \beta)$ denoting boundary conditions at the entanglement cut.

The reduced density matrix (RDM) for a subsystem $A$ is constructed via these factorizations. The standard measures, such as Rényi entropies $S_n = (1/(1-n))\log\mathrm{tr}(\rho_A^n)$ and relative entropies, are universal for the vacuum and insensitive to spin structure or parity sector. However, this universality masks substantial differences when the entanglement spectrum is further resolved by fermion parity.

In the presence of a conformal interface, realized by the operator $I(O)$ parameterized by $O(2)$ matrices, factorization of the Hilbert space leads to nontrivial entanglement spectra differing by spin structure, with results obtained via explicit uniformization of the replica geometry onto an annulus. Primary states such as the fermion excitation display measurable differences when their symmetry-resolved entanglement is computed.

2. Hilbert Space Factorization and Spin Structures

There exist two fundamentally different ways of factorizing the Hilbert space corresponding to distinct choices of boundary conditions at the entangling edges. These factorization schemes correspond, respectively, to selecting Neveu–Schwarz (NS) or Ramond (R) sectors on the subregion. Each choice produces a different partition function for the subsystem,

$$Z_{(\alpha\alpha)}(q) = Z_3(q) = \sqrt{\vartheta_3(q)/\eta(q)}, \qquad Z_{(-\alpha, \alpha)}(q) = Z_2(q) = \sqrt{\vartheta_2(q)/\eta(q)}$$

with the theta functions parametrizing the different spin structures.

A critical consequence of this distinction is the appearance of a $\sqrt{2}$ factor in the Virasoro character expansion for the $Z_2(q)$ partition function, which signals the presence of an unpaired Majorana zero mode in the associated factorization. Consequently, the entanglement spectrum for a single interval depends crucially on the chosen factorization, with the Majorana zero mode's presence or absence yielding qualitatively distinct features apparent only after fermion-parity resolution.

3. Symmetry-Resolved Entanglement and Parity Decomposition

Fermion-parity resolution is implemented by using the natural $\mathbb{Z}_2^F$ symmetry of the Majorana theory. The subsystem Hilbert space is decomposed according to the eigenvalues of the fermion-parity operator $F$,

$$\mathcal{H}^{A}_{(\alpha\beta)} = \bigoplus_{a=\pm} V_a \otimes \mathcal{H}^{A}_{(\alpha\beta)}(a)$$

with projectors $\Pi_\pm = \frac{1}{2}(1 \pm (-1)^F)$. This decomposition enables the definition of symmetry-resolved (fermion-parity-resolved) Rényi entropies,

$$S_n\big(\rho^{\phi}_{(\alpha\beta)}(a)\big) = \frac{1}{1-n}\log \frac{Z^{\phi(a|n)}_{(\alpha\beta)}}{\left[Z^{\phi(a|1)}_{(\alpha\beta)}\right]^n}$$

where $$Z^{\phi(a|n)}_{(\alpha\beta)} = \mathrm{tr}[\Pi_a\, (\rho^{\phi}_{(\alpha\beta)})^n]$$ tracks the partition function restricted to the $a$th parity sector. Importantly, these quantities are sensitive to the structure of the underlying spin sector, revealing information not captured by the aggregate (unresolved) Rényi or relative entropies.

4. Majorana Zero Modes and Equipartition of Parity-Resolved Entropies

A salient feature is that, when the Hilbert space factorization corresponds to a boundary condition yielding a Majorana zero mode (as signaled by the $\sqrt{2}$ factor in $Z_2(q)$), the parity-resolved Rényi entropies are exactly equipartitioned; i.e., they yield

$$S_n\big(\rho^{\psi}_{(-\alpha,\alpha)}(a)\big) = S_n\big(\rho^{\psi}_{(-\alpha,\alpha)}\big) - \log 2$$

so that the entanglement content in each sector is precisely the same at all orders in the UV cutoff $\epsilon$. The $\log 2$ factor represents the entropy of mixing between two sectors of equal weight.

By contrast, in the absence of such a zero mode (i.e., for $Z_3(q)$), this equipartition is broken, with the parity-resolved entropies differing by quantities governed by Ramond-sector data (notably a conformal weight $1/16$ and the associated degeneracy). The magnitude of the equipartition breaking encodes the properties of the Ramond sector, providing a signature for the absence of an unpaired Majorana mode.

5. Fermion Parity Stabilization and Conformal Interfaces

For conformal interface states, the interplay between the choice of factorization and the stabilization of fermion parity is essential. The interface operator $I(O)$, acting as a nonlocal transformation, may not itself preserve the $\mathbb{Z}_2^F$ symmetry. Nevertheless, the explicit construction of the reduced density matrix via boundary state methods shows that the factorization—through the appropriate placement of conformal boundary states—stabilizes the fermion parity symmetry. This stabilization enables consistent definition and computation of symmetry-resolved entropies in all states considered, including those built from nontrivial conformal interfaces.

The full entanglement spectrum and its symmetry-resolved decomposition depend on parameters of the interface (the ``interface modulus'') and on the details of the factorization, leading to modified spectra that can be captured only by this fully resolved analysis.

6. Connection to Symmetry-Protected Topological Phases

The presence of a Majorana zero mode and the resulting exact equipartition in the symmetry-resolved entanglement spectrum constitute a robust signature of a nontrivial SPT phase, such as realized in the Kitaev chain. The ``symmetry-enforced vanishing'' of partition functions or entropies, familiar from Jackiw–Rebbi index theorems and SPT literature, appears here as the equipartition of fermion-parity-resolved entanglement. The methodology therefore has direct applications as a diagnostic tool for distinguishing trivial and nontrivial SPT phases in fermionic systems.

7. Comparison with Twist Field Approaches

The twist field formalism is the standard method for computing Rényi and entanglement entropies in many CFTs, using correlation functions of branch-point twist fields. While this approach captures the universal scaling (such as logarithmic dependence on subsystem size), it fails to resolve the entanglement spectrum into fermion parity sectors, as it does not encode the specific spin structure or parity decomposition inherent to the system's Hilbert space factorization.

By contrast, the boundary state approach, as employed in the discussed work, allows for the explicit separation of bosonic (even-parity) and fermionic (odd-parity) contributions, and for identifying the correspondence with topological features such as unpaired Majorana modes. This provides insights inaccessible to the twist field method, especially in contexts sensitive to SPT structure, interface moduli, and the specifics of physical factorization.

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In summary, fermion-parity resolution of entanglement in the Majorana CFT framework uncovers structure invisible to conventional entropy measures: it reveals the impact of spin structure, boundary condition, and Majorana zero mode presence on the entanglement spectrum, precisely quantifies equipartition or its breaking, and connects these microscopic details to global topological characteristics such as SPT phase identification. This framework is broadly applicable to both theoretical analyses and numerical diagnostics in fermionic many-body systems (Northe, 3 Sep 2025).