Exact Properties of Néel Cat States and Constant Entanglement at a Special Ordered Point of the Hierarchical Quantum Ising Model
Garoual, Anass
This preprint studies the hierarchical quantum transverse-field Ising model (HQTFIM) with couplings defined by a 2-adic hierarchy, 𝐽 𝑖 𝑗 ∝ 2 − 𝛼 𝑣 2 ( 𝑗 − 𝑖 ) J ij ∝2 −αv 2 (j−i) . Building on the classical Dyson hierarchical Ising model, we construct an explicit block renormalization scheme for the dimensionless transverse field 𝑥 = ℎ / 𝐽 x=h/J and derive a closed RG map with a non-trivial fixed point 𝑥 \* ( 𝛼 ) x \* (α). The corresponding field ℎ \* ( 𝛼 ) = 𝑥 \* ( 𝛼 ) 𝐽 0 h \* (α)=x \* (α)J 0 is interpreted as an “RG self-similarity field” deep inside the ordered phase. On the analytical side, we prove two exact structural results for the ideal Néel cat state |psi_cat> = (|Néel> + |anti-Néel>)/sqrt(2) on chains of length N = 2^H. First, for any dyadic bipartition A:B, the von Neumann entanglement entropy of the reduced state on A is exactly one bit. Second, we construct an explicit tree tensor network (TTN) representation of |psi_cat> on a depth-H binary tree with bond dimension chi = 2, independent of system size. This makes precise in what sense the Néel cat is an extreme example of an area-law state with a simple tensor-network structure. On the numerical side, we use dense exact diagonalization (Python / NumPy / SciPy) to compare the HQTFIM to the standard one-dimensional nearest-neighbour transverse-field Ising model (TFIM). For alpha = 1.5 and h/J0 = x^*(1.5) ≈ 0.404, the HQTFIM ground state in the even-parity sector exhibits: (i) a fidelity F_N with the ideal Néel cat that increases with N; (ii) a half-chain entanglement entropy S_N pinned extremely close to one bit; and (iii) a staggered order parameter M_stagg^2 very close to one. By contrast, the standard TFIM at the same ratio h/J has essentially zero Néel-cat fidelity and negligible staggered order. We further compare the HQTFIM at h = h^*(alpha) with the standard TFIM at its exact critical point h_c = J. The standard TFIM shows the expected logarithmic growth of entanglement, S_N ~ (c/3) log N with an effective central charge c_eff ≈ 0.7 on the small system sizes we access, and vanishing staggered order. In sharp contrast, at h^*(alpha) the HQTFIM shows entanglement consistent with a strict area law, S_N → 1 bit, and strong Néel order. A numerical scan in h for fixed N identifies a “true” critical field h_c(alpha) for the HQTFIM from the rapid drop of M_stagg^2(h); we find h_c(alpha) >> h^*(alpha), with ratios between ~5 and ~28 over alpha in [1.2, 3.0]. Thus the RG self-similarity field h^*(alpha) is not a quantum critical point but a special ordered-phase point with an unusually clean Néel-cat ground state. Finally, we characterize finite-size corrections at h^*(alpha) via power-law fits 1 − F_N ~ N^{-beta(alpha)} and 1 − M_stagg^2 ~ N^{-gamma(alpha)}, with beta(alpha) ≈ 1 and gamma(alpha) ≈ 1.8–2.1 and a systematic but mild dependence on alpha. We clearly separate mathematically rigorous statements (block spectrum, exact entanglement, TTN construction) from numerical or conjectural claims (properties of the HQTFIM in the thermodynamic limit and the location/universality class of the true critical point). The work suggests a concrete set of open problems on rigorous phase diagrams, entanglement scaling, and excitation spectra for quantum hierarchical models.
•Dec 7, 2025•0 citations