Zangara, Bendersky and Pastawski (Phys. Rev. A 91, 042112 (2015)) showed that in closed spin-1/2 systems with all-to-all dipolar interactions (N=12,14,16N=12,14,16), a weak non-secular perturbation generates two- and four-body effective interactions through virtual processes; these proliferate within subspaces of conserved total spin projection and drive Loschmidt-echo decay obeying an effective Fermi golden rule (FGR) once the perturbation exceeds a lower bound that shrinks as NN grows. The open problem is the quantitative theory of this mechanism: evaluate the geometric prefactors entering the effective-FGR rate (the mean connectivity aa, the coefficient bb, and the effective mean level spacing of directly connected states deff≈b Jdipd_{\mathrm{eff}}\approx b\,J_{\mathrm{dip}}, whose inverse deff−1d_{\mathrm{eff}}^{-1} is the effective density of directly connected states), establish the functional scaling of the validity threshold with system size, decide whether taking the thermodynamic limit before the perturbation to zero makes the effective FGR the leading irreversibility mechanism, and determine how the mechanism and its irreversible equilibration survive in short-range or structurally realistic lattices. No work is known in the literature that computes these prefactors or the threshold scaling, so the problem remains open.

Background

An initially local polarization excess injected into a crystal of interacting spin-1/2 nuclei spreads through the sample under spin-spin couplings, a process known in solid-state nuclear magnetic resonance (NMR) as spin diffusion. The Hamiltonian can in principle be inverted by radio-frequency pulse sequences, so an ideal time-reversal procedure should revive the local excitation. The Loschmidt echo (LE) quantifies how much signal is actually recovered after such an imperfect reversal: for forward evolution under a Hamiltonian H^0\hat H_0 followed by backward evolution under H^0+Σ^\hat H_0+\hat\Sigma, it is the locally measured spin autocorrelation whose decay measures the decoherence induced by the perturbation Σ^\hat\Sigma. In the experiments that motivated the source paper, H^0\hat H_0 is the secular (truncated) dipolar Hamiltonian, which conserves the total spin projection and therefore decomposes the 2N2^N-dimensional Hilbert space into subspaces Sν\mathcal S_\nu of definite projection ν\nu, while the unavoidable non-inverted perturbation Σ^\hat\Sigma is a double-quantum term (it changes the projection by Δν=±2\Delta\nu=\pm 2) that couples different subspaces. For a weak perturbation whose matrix elements are below the level spacing of the directly connected states (DCS — eigenstates of H^0\hat H_0 linked by a nonzero matrix element of Σ^\hat\Sigma), the LE decays perturbatively with a rate linear in the perturbation strength. When the perturbation exceeds this spacing, the decay crosses over to a Fermi golden rule (FGR) — the standard perturbative decay-rate formula Γ=(2π/ℏ)∣Σ∣2ρ\Gamma=(2\pi/\hbar)|\Sigma|^2\rho valid once the local density of states (LDOS, the energy distribution the perturbation samples) becomes Lorentzian. Zangara, Bendersky and Pastawski addressed the puzzle that perturbations far too weak to produce such mixing in small clusters nevertheless cause essentially irreversible equilibration in macroscopic samples. Computing the full 2N2^N dynamics of all-to-all dipolar clusters with N=12,14,16N=12,14,16, they found that Σ^\hat\Sigma generates, at second order through virtual excursions into neighboring projection subspaces, effective two- and four-spin interactions V^eff\hat V_{\mathrm{eff}} that mix quasidegenerate states inside each subspace which the bare dipolar Hamiltonian did not couple. Because these effective terms obey fewer selection rules than the bare interaction, they proliferate — they appear where the bare Hamiltonian had null entries — so that the LE decay follows an effective FGR with rate Γeff=(2π/ℏ)⟨Veff2⟩/deff\Gamma_{\mathrm{eff}}=(2\pi/\hbar)\langle V_{\mathrm{eff}}^2\rangle/d_{\mathrm{eff}}, where deffd_{\mathrm{eff}} is the effective mean level spacing among DCS, so that deff−1d_{\mathrm{eff}}^{-1} is the effective density of DCS, once the perturbation exceeds a lower bound that moves to weaker perturbations as NN increases. The prefactors aa (the mean number of states connected to a given state) and bb (with deff≈bJdipd_{\mathrm{eff}}\approx b J_{\mathrm{dip}}, where JdipJ_{\mathrm{dip}} is the characteristic dipolar coupling energy) encode the interplay of NN, lattice coordination and selection rules and were left undetermined. Their analysis assumes a specific order of limits — first N→∞N\to\infty and then ∥Σ^∥→0+\|\hat\Sigma\|\to 0^+ — under which even an arbitrarily slight perturbation would rule the LE decay in the thermodynamic limit (TL), rendering the equilibrated state irreversible. The model was deliberately all-to-all (with 1/N1/\sqrt N-normalized couplings) to reduce statistical fluctuations, at the price of losing the hierarchical, distance-dependent structure of real crystals, where earlier work on spin ladders found persistent decay-rate offsets (Zangara, Dente, Levstein and Pastawski, Phys. Rev. A 86, 012322 (2012)). The same group's NMR experiments on effectively infinite crystals (e.g. polycrystalline adamantane) established a complementary empirical fact: a perturbation-independent LE decay time T3T_3 tied to the reversible interaction time T2T_2 (the 'central hypothesis of irreversibility'), whose connection to the proliferation mechanism remains heuristic. Whether the proliferated-interaction equilibrated state — an infinite-temperature background plus a local excitation that ends effectively 'cooled' to a finite-temperature-like state — constitutes genuine thermalization in the sense of microcanonical or thermal ensembles was explicitly left unelaborated by the authors.

Problem Statement

Determine, with quantitative and preferably analytic control, how the proliferation of effective interactions makes weak perturbations irreversible in closed many-spin systems, as posed by Zangara, Bendersky and Pastawski (Phys. Rev. A 91, 042112 (2015)) for the secular dipolar Hamiltonian with a double-quantum perturbation and a locally measured Loschmidt echo. The problem has three jointly supporting deliverables whose common objective is to convert the numerically observed proliferation mechanism into a predictive theory of the onset of irreversibility with system size: (1) Evaluate the geometric prefactors of the effective Fermi golden rule — the mean connectivity aa, the coefficient bb, and the effective mean level spacing of directly connected states deff≈b Jdipd_{\mathrm{eff}}\approx b\,J_{\mathrm{dip}} (whose inverse deff−1d_{\mathrm{eff}}^{-1} is the effective density of DCS entering the rate Γeff\Gamma_{\mathrm{eff}}) — as functions of system size NN, lattice coordination and geometry, and interaction range, for the all-to-all model and beyond it. (2) Establish the functional scaling law for the lower perturbation bound below which the LE decay is only perturbative and above which the effective FGR holds, as a function of NN; and decide whether the ordered limits — N→∞N\to\infty taken before ∥Σ^∥→0+\|\hat\Sigma\|\to 0^+ — rigorously make the effective FGR the leading LE-decay mechanism in the thermodynamic limit, as the finite-NN numerics (N=12,14,16N=12,14,16) suggest but do not prove. (3) Assess how the proliferation mechanism and the onset of the effective FGR change in short-range or lattice-specific models with distance-dependent, signed couplings — where the randomization that suppresses destructive interference between virtual paths in the all-to-all model is absent — including whether finite-size rate offsets of the kind observed in hierarchical spin lattices persist. All three deliverables are required: (1) and (2) may be resolved analytically (exact or controlled) or numerically with a controlled finite-size analysis, while a refutation under (3) showing that the mechanism fails in realistic geometries settles only the realistic-geometry question and leaves (1) and (2) open; for a given perturbation strength and lattice geometry, the completed theory should state whether a closed spin system equilibrates irreversibly.

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