$p$-Body $\simeq$ Range $p-1$: Exact Order-Range Mapping and Dual-Unitarity

Technical Summary: p-Body $\simeq$ Range $p-1$: Exact Order–Range Mapping and Dual-Unitarity

Problem Statement
Understanding the non-equilibrium dynamics of interacting quantum many-body systems is a central challenge due to the exponential growth of the Hilbert space dimension. While analytically tractable models based on space-time symmetries, specifically dual-unitarity, have provided exact solutions for correlation functions, entanglement spreading, and spectral form factors, existing frameworks largely rely on two-body interactions. Although genuine multi-body interactions arise naturally in lattice gauge theories, effective spin Hamiltonians with ring-exchange processes, and quantum information models, there is a lack of systematic methods to construct and analyze dual-unitary Floquet models with contiguous $p$-body interactions.

Methodology
The authors introduce a periodically kicked version of a one-dimensional spin-1/2 Ising model with contiguous $p$-body interactions. The system is defined by the Hamiltonian:
$$H^{(pB)} = \sum_{j=1}^L J \sigma_j^z \sigma_{j+1}^z \cdots \sigma_{j+p-1}^z + \sum_{j=1}^L b \sigma_j^x + \sum_{j=1}^L h_j \sigma_j^z$$
subject to periodic boundary conditions. The dynamics are governed by the Floquet evolution operator $U^{(pB)}_{KFIM} = U_K U_I$, where $U_K = e^{-iH_K}$ and $U_I = e^{-iH_I}$.

The core methodology involves establishing an exact interaction order–range mapping. By tuning the interaction strength $J$ to specific values ($J = n\pi/4$ for integer $n$), the authors utilize algebraic identities involving Pauli matrices to decompose the $p$-body interaction term. Specifically, they prove that for these specific strengths, the exponential of the $p$-body interaction operator can be exactly mapped to an operator containing only 2-body interactions with a maximum range of $p-1$, up to a global phase.

Key Contributions and Results

1. Exact Order–Range Mapping Theorem:
   The primary result is Theorem 1, which states that for a $p$-body Ising model with $J = n\pi/4$, the Floquet evolution operator $U^{(pB)}_{KFIM}$ is exactly equivalent to the Floquet evolution operator of a spin-1/2 model with 2-body interactions of maximum range $p-1$ ($U^{(p-1)R}$).

   • The mapped Hamiltonian $H^{((p-1)R)}_{KFIM}$ retains the kicked structure but features a specific coupling profile: nearest-neighbor couplings are weighted by $(p-1)$, next-nearest by $(p-2)$, etc., up to range $p-1$.
   • This mapping is independent of the kicking strength $b$ and holds for generic values of the local fields $h_j$.

2. Construction of $p$-Body Dual-Unitary Models:
   By further tuning the kicking strength to $|b| = \pi/4$ and interaction strength to $|J| = \pi/4$, the mapped range-$(p-1)$ model satisfies the conditions for dual-unitarity (as established in prior literature for long-range kicked Ising chains). Consequently, the original $p$-body model is proven to be dual-unitary. This provides the first concrete realization of a contiguous $p$-body kicked-Ising Floquet model that is dual-unitary.

3. Analysis of the $p=3$ Case and Chain Parity:
   The authors analyze the minimal case of $p=3$ (3-body interactions).

   • Even System Size ($L$): The mapped model decouples into two independent chains (one on odd sites, one on even sites), each with nearest-neighbor interactions. This leads to Poissonian level spacing statistics, indicative of integrable dynamics within the decoupled sectors.
   • Odd System Size ($L$): Due to periodic boundary conditions, the two sectors couple, resulting in a single connected chain equivalent to the standard kicked Ising model (KFIM) after site relabeling. This yields Wigner-Dyson level spacing statistics (Circular Orthogonal Ensemble), indicative of quantum chaos.
   • This parity dependence is generalized in Corollary 1 for all odd $p$.

4. Exact Entanglement Dynamics:
   Leveraging the dual-unitarity of the mapped model, the authors derive exact expressions for the $\alpha$-Rényi entanglement entropy $S^{(\alpha)}_A(t)$ for a class of solvable initial states (Transverse $T$ and Longitudinal $L$ types).

   • For large system sizes, the entanglement entropy grows linearly with time: $\lim_{L\to\infty} S^{(\alpha)}_A(t) = \min(2(p-1)t, N) \ln(2)$.
   • The growth rate is determined by the maximum range of the mapped interactions ($p-1$).
   • Numerical simulations for $p=3$ and $p=4$ confirm these linear growth regimes and saturation values, validating the theoretical mapping.

Significance and Claims
The paper claims to provide a systematic route for constructing and studying $p$-body dual-unitary Floquet models. By demonstrating that specific $p$-body interactions can be exactly reduced to long-range 2-body interactions, the work bridges the gap between complex multi-body systems and the solvable framework of dual-unitarity.

The authors emphasize that this is the first concrete realization of a contiguous $p$-body kicked-Ising model that is dual-unitary. The work does not claim to solve general non-equilibrium dynamics for arbitrary parameters but establishes exact solvability for specific interaction strengths. The significance lies in expanding the class of analytically tractable models to include genuine multi-body interactions, thereby offering new platforms to study quantum chaos, entanglement spreading, and spectral statistics in higher-order interacting systems. The authors note that while the mapping is derived for spin-1/2 systems, it is expected to generalize to $p$-body generalizations of the kicked Potts model, potentially leading to dual-unitary models with local dimensions $d=3, 4$.