Entanglement of the quantum Dyson hierarchical Ising chain beyond its critical area law: the true long-range ferromagnetic chain, tensor-network structure, disjoint blocks, excited states, and quench dynamics

The real-space renormalization group (RG) solution of the quantum Dyson hierarchical Ising chain - a non-translationally-invariant, tree-structured approximation to the one-dimensional ferromagnetic Ising model with couplings decaying as $r^{-1-\sigma}$ - yields a ground state that is a tree tensor network with rank-two isometries, whose block entanglement entropy obeys an area law at criticality even though correlations decay algebraically (Pappalardi, Calabrese and Parisi, 2019). The problem is the five linked questions that the source paper's conclusions leave open, unified by one objective: to determine which entanglement features of the hierarchical RG description are artifacts of the tree structure and which reflect genuine long-range quantum criticality. The components are: (1) the scaling of the ground-state block entanglement entropy at the quantum critical point of the true, translationally invariant long-range ferromagnetic Ising chain, for which the source authors expect an area-law violation and call for direct numerical simulation; (2) a tensor-network-theoretic characterization of the RG state and the generality of its finite-rank/area-law property among tree tensor networks; (3) the mutual information and entanglement negativity between two disjoint blocks of the RG state; (4) the entanglement of the low-lying RG excited states; and (5) the real-time entanglement dynamics of the hierarchical model after quenches. No resolution of any of these questions is known in the literature to date.

Background

Entanglement entropy and area laws. For a quantum many-body system in a pure state, the entanglement entropy $S_A$ of a subregion $A$ is the von Neumann entropy of the reduced density matrix $\rho_A$ obtained by tracing out the complement of $A$. In one dimension with short-range interactions, gapped ground states obey an area law - $S_A$ for a block of $\ell$ contiguous sites stays bounded as $\ell$ grows - while at a continuous quantum critical point $S_A$ grows like $(c/3)\ln\ell$, with $c$ the central charge of the conformal field theory (CFT) describing the criticality. This logarithmic law is tied to conformal invariance, which itself rests on Lorentz (Euclidean rotational) symmetry of the scaling limit.

Long-range interactions. One-dimensional spin chains with ferromagnetic couplings decaying algebraically as $J(r)\sim r^{-(1+\sigma)}$ are realizable with trapped ions (ions confined in electromagnetic traps, whose collective vibrations mediate tunable power-law couplings between their internal spin states). For $0<\sigma<1$ (the Dyson regime) the interactions are genuinely long-range and the CFT arguments break down: the dynamical critical exponent $z$ (relating time and length scaling) is no longer 1, conformal invariance is lost, and the critical entanglement scaling is not fixed by the standard theory. Numerical work on antiferromagnetic long-range Ising chains (Koffel, Lewenstein and Tagliacozzo, 2012) found exotic area-law violations even in gapped phases, while the fully-connected (mean-field) ferromagnetic limit has a critical entropy growing logarithmically with block size. Beyond mean field, the entanglement scaling for ferromagnetic long-range interactions remained an open issue even in one dimension. For gapped, non-critical one-dimensional long-range systems, Kuwahara and Saito (2020) proved that an area law holds rigorously, so the open issue is specifically criticality.

Dyson hierarchical model and its RG. Dyson (1969) introduced a hierarchical approximation to the long-range ferromagnetic Ising chain: on a chain of $L=2^n$ spins, blocks of $2^k$ spins interact pairwise with coupling $J_k = J\,2^{-k(1+\sigma)}$, so that all interactions sit on a binary tree (spins at sites $2j$ and $2j+1$ never interact directly; the model is not translationally invariant). The classical hierarchical model reproduces the critical behavior of the true long-range Ising model well. Monthus (2015) introduced the quantum version - hierarchical Ising couplings in the $x$ spin direction plus a transverse field $h\sum_i \sigma_i^z$ - and solved it by a real-space renormalization group (RG): at each step every block Hamiltonian is projected onto its two lowest states, halving the number of spins. Iterating gives the ground state and a flow for the effective coupling ratio $K_k = J_k\mu_k^2/h_k$, with a critical fixed point $K_c(\sigma)$ separating ferromagnetic and paramagnetic phases and algebraically decaying critical correlations.

The 2019 result. Pappalardi, Calabrese and Parisi (2019) computed the entanglement entropy of the RG ground state at criticality for three block partitions. Because each RG step is a two-state (rank-two isometric) projection, the RG ground state is a tree tensor network (TTN), i.e. a wavefunction built by contracting small isometric tensors on a tree graph; equivalently, its block reduced density matrices have finite rank (at most 16, and at most 8 for the natural maximum-rank partition). Consequently, at the critical point the block entropy saturates to a finite value - an area law - even though the same state reproduces algebraically decaying correlations with the correct RG exponents. For $\sigma\gg1$ a pre-asymptotic regime appears in which the entropy grows logarithmically in the block size with a very small prefactor $\sim K^{-2}$, up to block sizes $\ell\sim 2^{K^2}$, before saturating (the limits $K\to\infty$ and $\ell\to\infty$ do not commute); in the small-$K$ regime the saturation value scales as $K^2\ln K$. Comparison with exact diagonalization (numerically diagonalizing the full Hamiltonian matrix of a small system) shows that the eight largest eigenvalues of the reduced density matrix (the entanglement spectrum) already reproduce the exact entropy to relative error of order $10^{-5}$. This critical area law is anomalous: it contrasts with the logarithmic scaling of both short-range critical chains and fully-connected ferromagnets, and with the area-law violations of antiferromagnetic long-range chains. Finite-entanglement critical points (critical points that persist only because the variational state caps the entanglement it can hold) of a related kind are known in the language of matrix product states - wavefunctions built from a chain of contracted small tensors, the standard variational class for one-dimensional ground states (Wolf, Ortiz, Verstraete and Cirac, 2006) - and the source paper invokes this connection when proposing a deeper TTN characterization of the RG state; the same conclusions point to negativity between disjoint blocks (Calabrese, Cardy and Tonni, 2012) and to the entanglement of low-energy excitations of critical chains (Alcaraz, Ibáñez Berganza and Sierra, 2011) as the short-range benchmarks for the questions below.

Problem Statement

The problem is the set of open questions raised by the critical area-law result of Pappalardi, Calabrese and Parisi (2019) for the renormalized ground state of the quantum Dyson hierarchical Ising chain. Its five components serve one common objective: to determine which entanglement features of the hierarchical RG description of the long-range ferromagnetic Ising chain are artifacts of its tree structure and which reflect genuine long-range quantum criticality. Components (a) and (e) test the RG description against the true chain and against real time; components (b)-(d) complete the entanglement characterization of the hierarchical model beyond single-interval ground-state entropies. Its five components are: (a) the critical block-entropy scaling of the true chain determined as a function of $\sigma$; (b) a definite genericity verdict for the stated tree-tensor-network class; (c) the mutual information and entanglement negativity between two disjoint blocks determined at criticality; (d) the entanglement entropies of the lowest projected excited states together with their scaling; and (e) the post-quench entanglement-growth law of the hierarchical model.

The two models involved are: (i) the quantum Dyson hierarchical Ising chain, with Hamiltonian built from hierarchical couplings $J_k = J\,2^{-k(1+\sigma)}$ between blocks of $2^k$ spins (arranged on a binary tree over $L=2^n$ sites, interactions along the $x$ spin direction, transverse field $h$), solved by the two-state real-space RG projection; and (ii) the true, translationally invariant ferromagnetic long-range Ising chain, $\hat{H} = -\sum_{i<j} J |i-j|^{-(1+\sigma)}\,\sigma^x_i \sigma^x_j - h \sum_i \sigma^z_i$, with power-law exponent $1+\sigma$; the interesting regime is the long-range (Dyson) regime $0<\sigma<1$, with the large-$\sigma$ crossover retained as in the source analysis.

(a) Critical entanglement of the true chain. Determine the scaling of the ground-state block entanglement entropy with block size at the quantum critical point $h = h_c(\sigma)$ of the true chain, as a function of $\sigma$: does it obey an area law, as the hierarchical RG projection suggests, or a stronger scaling, as both limiting cases (short-range critical chains and the fully-connected ferromagnet) display? This entails locating $h_c(\sigma)$, controlling finite-size and crossover effects (the hierarchical solution already shows decades-long pre-asymptotic logarithmic regimes), and quantifying the agreement or disagreement with the hierarchical prediction.

(b) Tensor-network characterization of the RG state. Characterize the hierarchical RG ground state within tensor-network theory: how generic is the observed finite-rank/area-law phenomenon within the class of binary-tree tensor networks over $L=2^n$ sites with fixed bond dimension (the class containing the RG state) - that is, whether the finite-rank/area-law property holds for the whole class or only for a typical subset of it under a stated ensemble of isometries - and which of its properties depend crucially on the hierarchical (non-translation-invariant) tree structure, including its relation to known finite-entanglement critical states.

(c) Disjoint-block entanglement. Compute the entanglement entropies, the mutual information (the correlation measure $I(A:B)=S_A+S_B-S_{A\cup B}$, which detects both classical and quantum correlations between the blocks), and the entanglement negativity (the partial-transpose-based mixed-state entanglement measure) between two disjoint blocks of the RG ground state at criticality, establishing whether the single-interval area law coexists with long-distance entanglement between separated blocks.

(d) Excited-state entanglement. Characterize the entanglement of the low-lying excited states accessible from the RG projection, and determine whether such excitations encode universal features of the long-range critical point, analogously to the entanglement of low-energy excitations in short-range critical chains.

(e) Real-time entanglement dynamics. Study the entanglement dynamics of the quantum hierarchical model after global or local quenches - in particular near the critical point - and determine to what extent the RG framework predicts the sub-linear, logarithmic-in-time entanglement growth that is observed and analytically understood in true long-range chains.

References

1. S. Pappalardi, P. Calabrese, G. Parisi, Entanglement entropy of the long-range Dyson hierarchical model, J. Stat. Mech. (2019) 073102, DOI 10.1088/1742-5468/ab2903 (arXiv:1904.00015), https://doi.org/10.1088/1742-5468/ab2903
2. F. J. Dyson, Existence of a phase-transition in a one-dimensional Ising ferromagnet, Commun. Math. Phys. 12, 91 (1969), DOI 10.1007/BF01645907, https://doi.org/10.1007/bf01645907
3. C. Monthus, Dyson hierarchical quantum ferromagnetic Ising chain with pure or random transverse fields, J. Stat. Mech. (2015) P05026, DOI 10.1088/1742-5468/2015/05/P05026 (arXiv:1503.03727), https://doi.org/10.1088/1742-5468/2015/05/P05026
4. T. Koffel, M. Lewenstein, L. Tagliacozzo, Entanglement entropy for the long-range Ising chain in a transverse field, Phys. Rev. Lett. 109, 267203 (2012), DOI 10.1103/PhysRevLett.109.267203 (arXiv:1207.3957), https://doi.org/10.1103/PhysRevLett.109.267203
5. M. M. Wolf, G. Ortiz, F. Verstraete, J. I. Cirac, Quantum Phase Transitions in Matrix Product Systems, Phys. Rev. Lett. 97, 110403 (2006), DOI 10.1103/PhysRevLett.97.110403, https://doi.org/10.1103/PhysRevLett.97.110403
6. P. Calabrese, J. Cardy, E. Tonni, Entanglement Negativity in Quantum Field Theory, Phys. Rev. Lett. 109, 130502 (2012), DOI 10.1103/PhysRevLett.109.130502, https://doi.org/10.1103/PhysRevLett.109.130502
7. F. C. Alcaraz, M. Ibáñez Berganza, G. Sierra, Entanglement of Low-Energy Excitations in Conformal Field Theory, Phys. Rev. Lett. 106, 201601 (2011), DOI 10.1103/PhysRevLett.106.201601, https://doi.org/10.1103/PhysRevLett.106.201601
8. J. Schachenmayer, B. P. Lanyon, C. F. Roos, A. J. Daley, Entanglement Growth in Quench Dynamics with Variable Range Interactions, Phys. Rev. X 3, 031015 (2013), DOI 10.1103/PhysRevX.3.031015, https://doi.org/10.1103/PhysRevX.3.031015
9. A. S. Buyskikh, M. Fagotti, J. Schachenmayer, F. Essler, A. J. Daley, Entanglement growth and correlation spreading with variable-range interactions in spin and fermionic tunneling models, Phys. Rev. A 93, 053620 (2016), DOI 10.1103/PhysRevA.93.053620, https://doi.org/10.1103/PhysRevA.93.053620
10. A. Lerose, S. Pappalardi, Origin of the slow growth of entanglement entropy in long-range interacting spin systems, Phys. Rev. Research 2, 012041 (2020), DOI 10.1103/PhysRevResearch.2.012041 (arXiv:1811.05505), https://doi.org/10.1103/PhysRevResearch.2.012041
11. T. Kuwahara, K. Saito, Area law of noncritical ground states in 1D long-range interacting systems, Nat. Commun. 11, 4478 (2020), DOI 10.1038/s41467-020-18055-x (arXiv:1908.11547), https://doi.org/10.1038/s41467-020-18055-x
12. C. Monthus, Properties of the simplest inhomogeneous and homogeneous Tree-Tensor-States for long-ranged quantum spin chains with or without disorder, Physica A 576, 126040 (2021), DOI 10.1016/j.physa.2021.126040 (arXiv:2001.10731), https://doi.org/10.1016/j.physa.2021.126040
13. L. Capizzi, G. Giachetti, A. Santini, M. Collura, Spreading of a local excitation in a quantum hierarchical model, Phys. Rev. B 106, 134210 (2022), DOI 10.1103/PhysRevB.106.134210 (arXiv:2207.06790), https://doi.org/10.1103/PhysRevB.106.134210
14. X. Turkeshi, P. Ruggiero, P. Calabrese, Negativity spectrum in the random singlet phase, Phys. Rev. B 101, 064207 (2020), DOI 10.1103/PhysRevB.101.064207 (arXiv:1910.09571), https://doi.org/10.1103/PhysRevB.101.064207
15. L. Lepori, S. Paganelli, F. Franchini, A. Trombettoni, Mutual information for fermionic systems, Phys. Rev. Research 4, 033212 (2022), DOI 10.1103/PhysRevResearch.4.033212 (arXiv:2012.00045), https://doi.org/10.1103/PhysRevResearch.4.033212
16. A. Solfanelli, S. Ruffo, S. Succi, N. Defenu, Logarithmic, fractal and volume-law entanglement in a Kitaev chain with long-range hopping and pairing, JHEP 05 (2023) 066, DOI 10.1007/JHEP05(2023)066 (arXiv:2301.13231), https://doi.org/10.1007/JHEP05(2023)066
17. N. Defenu, A. Lerose, S. Pappalardi, Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions, Phys. Rep. 1074, 1-92 (2024), DOI 10.1016/j.physrep.2024.04.005 (arXiv:2307.04802), https://doi.org/10.1016/j.physrep.2024.04.005
18. V. Pagni, G. Giachetti, A. Trombettoni, N. Defenu, One-dimensional long-range Ising model: two (almost) equivalent approximations (2025), arXiv:2510.02458, https://arxiv.org/abs/2510.02458
19. Harikrishnan K J, D. Sadhukhan, A. K. Pal, Scaling of entanglement entropy and correlations in the variable-range extended Ising model (2025), arXiv:2504.01846, https://arxiv.org/abs/2504.01846
20. G. Wang, Y.-Z. Wu, X. Hao, D. Wu, J. Ren, Quantum Fisher information and multipartite entanglement in the long-range transverse-field Ising model, Commun. Theor. Phys. 78, 055101 (2026), DOI 10.1088/1572-9494/ae3236, https://doi.org/10.1088/1572-9494/ae3236