Crosscap Quench in Dirac Fermion CFT

• The crosscap quench in free Dirac fermion CFT is a protocol that initializes a maximally entangled, antipodal state to probe unique nonequilibrium dynamics.
• The analytical framework uses bosonization and the replica trick to derive closed-form expressions for time-dependent entanglement entropy.
• The study contrasts integrable and chaotic evolutions, showing how deformed Hamiltonians impact quasiparticle trajectories and entanglement revival patterns.

A crosscap quench in a free Dirac fermion conformal field theory (CFT) probes nonequilibrium dynamics and entanglement evolution arising from a global quench where the initial state encodes long-range, antipodal entanglement: the CFT is prepared in a crosscap state, which is maximally entangled between spatially antipodal points, and evolved unitarily under the standard or deformed Hamiltonian. This setting provides access to entanglement patterns and scrambling phenomena fundamentally distinct from those obtained after quenches from low-entangled “boundary” states and is crucial for understanding non-orientable topologies in quantum field theory as well as the information structure of elliptic de Sitter spacetimes (Dulac et al., 30 Nov 2025, Bai et al., 29 Oct 2025, Chalas et al., 5 Dec 2024).

1. Construction and Properties of the Crosscap State

The crosscap state, $|C⟩$, in the free Dirac fermion CFT is defined by imposing an orientation-reversing involution which glues left- and right-movers at spatially antipodal points. In the bosonized representation, the left and right Dirac fermions are $\psi_L(z) = e^{i X_L(z)}$, $\psi_R(\bar{z}) = e^{i X_R(\bar{z})}$, with mode expansions

$$X_L(z) = x_L - i p_L \ln z + i \sum_{k\ne 0} \alpha_k \frac{z^{-k}}{k}, \quad X_R(\bar{z}) = x_R - i p_R \ln \bar{z} + i \sum_{k\ne 0} \tilde{\alpha}_k \frac{\bar{z}^{-k}}{k}$$

The crosscap condition on $\mathbb{RP}^2$ enforces

$$L_n - (-1)^n \bar{L}_{-n} = 0 \implies \alpha_k |C_\pm\rangle = \pm (-1)^k \tilde{\alpha}_{-k} |C_\pm\rangle$$

giving rise to two crosscap Ishibashi states (differing by their zero modes):

$$|C_+\rangle = \exp\left[\sum_{k>0} \frac{(-1)^k}{k} \alpha_{-k} \tilde{\alpha}_{-k}\right] \!\! \sum_{m\in 2\mathbb{Z}} |m,0⟩,\quad |C_-\rangle = \exp\left[-\sum_{k>0} \frac{(-1)^k}{k} \alpha_{-k} \tilde{\alpha}_{-k}\right] \!\! \sum_{w\in 2\mathbb{Z}} |0,w⟩$$

In the fermionic basis, this is equivalently formulated as

$$(\psi_r + i \bar{\psi}_{-r}) |C⟩ = 0\ , \quad r \in \mathbb{Z}+\tfrac12$$

where $|C⟩$ is maximally entangled over antipodal points, locally corresponding to an infinite-temperature state for subsystems smaller than half the system.

2. Quench Protocol and Dynamics

The crosscap quench protocol prepares the system in a regularized crosscap state through Euclidean smearing,

$$|\Psi(0)\rangle = \mathcal{N}\, e^{-\frac{\beta}{4} H} |C⟩, \qquad \mathcal{N} = \langle C | e^{-\frac{\beta}{2} H} | C \rangle^{-1/2}$$

with the system then evolving under the (possibly deformed) Hamiltonian $H$:

$|\Psi(t)\rangle = e^{-i H t} |\Psi(0)\rangle$

For the uniform CFT:

$$H_0 = \frac{2\pi}{L}(L_0 + \bar{L}_0 - \tfrac{c}{12})$$

In the presence of spatially inhomogeneous deformations, the Hamiltonian may take the form

$$H_{\text{def}} = \frac{2\pi}{L}\Big[\sigma^0 (L_0 + \bar{L}_0) + \sigma^+ (L_{q,+} + \tilde{L}_{q,+}) + \sigma^- (L_{q,-} + \tilde{L}_{q,-})\Big] - \frac{\pi c}{6L}$$

with $L_{q,\pm} = (L_q \pm L_{-q})/2$ and similar for the right-movers, incorporating $SL^{(q)}(2,\mathbb{R})$ symmetry (Bai et al., 29 Oct 2025).

3. Entanglement Entropy: Replica Trick and Correlation Functions

The entanglement entropy of a subsystem $A$ after a crosscap quench is computed using the replica trick. On $\mathbb{RP}^2$, the $n$-th Rényi entropy involves two-point functions of twist operators with prescribed monodromy:

$$\text{Tr}\, \rho_A^n \propto \prod_{a = -\frac{n-1}{2}}^{\frac{n-1}{2}} \langle \sigma^{(a)}(z_1) \sigma^{(-a)}(z_2) \rangle_{\mathbb{RP}^2}$$

Through Fourier transformation and bosonization, each replica sector decouples and the twist operators $$\sigma^{(a,\pm)}(z,\bar{z}) = \exp[i(a/n)(X_L(z) \pm X_R(\bar{z}))]$$ reduce to free boson vertex operators. The normalized two-point function on the flat $\mathbb{RP}^2$ is

$$\langle \sigma^{(a)}(z_1,\bar{z}_1)\, \sigma^{(-a)}(z_2,\bar{z}_2)\rangle_{\mathbb{RP}^2} = \left[ \frac{|z_2 \bar{z}_1 + 1|\,|z_1 \bar{z}_2 + 1|}{|z_1 - z_2|^2\,|z_1 \bar{z}_1 + 1|\,|z_2 \bar{z}_2 + 1|} \right]^{(a/n)^2}$$

After including the appropriate conformal factors, the interval entropy for an arc $A=[-l/2, l/2]$ at time $t$ is given by the closed-form theta-function expression

$$S_A(t, l) = \frac{1}{6} \log \frac{1}{\varepsilon_{\rm UV}^2} \frac{|\theta_1(l/2\pi\,|\,i\beta/2\pi)|^2\,|\theta_1(t/\pi + 1/2 - i\beta/4\pi\,|\,i\beta/2\pi)|^2} {\eta(i\beta/2\pi)^6\,|\theta_1((t+\pi/2\pm l/2)/\pi - i\beta/4\pi\,|\,i\beta/2\pi)|}$$

where $\theta_1$ is the Jacobi theta function, $\eta$ is the Dedekind eta function, and $\beta$ is the Euclidean regularization parameter (Dulac et al., 30 Nov 2025).

4. Universal Time Evolution and Quasiparticle Picture

At $t=0$, the crosscap state exhibits maximal local entanglement: $S_A(0,l) \sim (\pi l)/(3\beta)$ for $l \gg \beta$ (volume law), peaking at $l = \pi$ (half system), reminiscent of the Page curve. Under time evolution:

• For $l<\pi$, $S_A(t, l)$ is constant until $t = (\pi - l)/2$, then falls linearly to a minimum at $t = \pi/2$, and revives to its initial value at $t = (\pi + l)/2$.
• For $l=\pi$, $S_A(t)$ decreases linearly until $t = \pi/2$, then rises and oscillates with period $\pi$.

This piecewise-linear and oscillatory behavior is explained by a refined quasiparticle picture: each initial EPR pair connects antipodal points and moves at the speed of light in opposite directions. Entanglement drops when both members of a pair enter the same subsystem, with the minimum entropy at $t=\pi/2$. The slope of entanglement decay matches that of the standard boundary-state quench ($c\pi/3\beta$), yet the overall pattern is non-monotonic—there is no conventional entanglement growth, only “scrambling” and periodic revivals (Dulac et al., 30 Nov 2025, Chalas et al., 5 Dec 2024).

5. Comparison with Standard and Deformed Quenches

Unlike global quenches from low-entangled boundary states—which display linear entanglement growth until saturation—the crosscap quench’s antipodal long-range correlations imprint a maximal initial entropy, followed by periodic depletion and revival (Chalas et al., 5 Dec 2024). When generalized to spatially inhomogeneous Hamiltonians $H_{\rm def}$ (e.g., Möbius, sine-square, or displacement deformations), the entanglement evolution can interpolate between strict periodicity (integrable Möbius case) and late-time “graph-like” patterns, with the partitioning of entangled clusters dictated by the spatial profile of the deformation. In the sine-square and displacement cases, zeroes of the deformation profile act as fixed points trapping quasiparticle pairs, leading to universal, profile-dependent mutual information “graphs” (Bai et al., 29 Oct 2025).

For integrable dynamics (free Dirac CFT or dual-unitary circuits), entanglement and mutual information exhibit delayed linear decreases and exact revivals. In chaotic (non-integrable) evolutions, the mutual information decays to zero and the entropy remains constant after the initial drop, reflecting the absence of coherent quasiparticle trajectories.

6. Analytical Frameworks and Extensions

The crosscap quench admits a complete analytic description: all time-resolved Rényi and von Neumann entropies reduce to two-point twist correlators on $\mathbb{RP}^2$, computable in closed form for free fermions by bosonization. The exact formulas match quasiparticle and membrane (domain-wall) picture predictions, but with boundary-minimization rules reflecting the highly entangled crosscap state: membranes must cut horizontally (constant time), leading to absence of initial entropy growth. The only $O(1)$ or subleading disparities arise from microscopic details (e.g., spin structure), but all macroscopic and graph-structure features are fixed solely by the deformation profile of the Hamiltonian (Dulac et al., 30 Nov 2025, Bai et al., 29 Oct 2025, Chalas et al., 5 Dec 2024).

7. Physical Significance and Broader Context

The crosscap quench in free Dirac fermion CFTs elucidates entanglement dynamics in systems with non-orientable topologies and nontrivial (antipodal) quantum correlations, relevant to elliptic de Sitter spaces where global Hilbert spaces become trivial but observer Hilbert spaces remain Fock-like (Dulac et al., 30 Nov 2025). It sharpens the understanding of how integrable and chaotic evolutions respond to long-range correlated initial conditions. Furthermore, the framework establishes deep connections between topological involution, quantum circuit dynamics, and information spreading, with analytic control that offers a rigorous benchmark for more general (interacting or chaotic) systems and for quantum simulations of long-range entangled initial conditions.