Strict majorization flow of the half-chain entanglement spectrum under bond healing in the open XX chain: An exact finite-size theorem with mode-resolved monotonicity
We prove an exact, finite-size theorem for the open XX spin chain of length N = 4m whose central bond has a tunable strength t ∈ (0,1]: as the bond heals (t increases), the half-chain ground-state entanglement spectrum becomes strictly more uniform in the majorization order. Mode by mode, the 2m single-particle entanglement (correlation-matrix) eigenvalues pair as 1/2 ± s_k(t), and every s_k(t) is strictly decreasing on (0,1] — a Sign Lemma whose proof is the mathematical core of the paper. As consequences: every strictly Schur-concave entanglement functional (von Neumann entropy, all Rényi entropies of order α > 0) is strictly increasing along the bond-healing path; the largest Schmidt coefficient is strictly decreasing; and the Nielsen LOCC order strictly increases. The proof is entirely elementary and closed at finite size: a two-leg reflection folding reduces the problem to a rank-one-perturbed Jacobi matrix family J(t); the mode velocities dr_k/dt satisfy an exact master identity expressing them as a convex combination of Stieltjes-type sums W_a with weights Ω_a > 0 certified by a zero-counting argument on a Cauchy-type oscillation pencil; and a uniform endpoint bound W_a(t) ≥ W_a(1) > 1/π + 1/(2K) (K = 2M+1) closes the chain. All constants are explicit, every analytic step is checkable by hand, and the finite rational-interval certificates are mechanically re-derivable by exact arithmetic from the stated rules; numerical verifications (double precision plus 60-digit micro-mode layer) accompany every proved statement as non-load-bearing corroboration. We state the interacting case (XXZ, Δ > 0) as a precise conjecture supported by exact-diagonalization numerics, locate exactly where the free-fermion proof breaks, discuss the relation to the entropic g-theorem (complementary, not overlapping), and propose quantum-simulator and photonic-ring tests. The phenomenon and its mode-resolved monotonicity appear to be new; the failure of such monotonicity in nearby settings (bulk RG, strong-coupling Kondo defects) is recalled to delimit the scope. Status labelling. Every claim in this paper carries one of the tags [proved] (complete hand-checkable proof in the text), [established] (proof structure complete with mechanical expansions delegated to an appendix or supplement, all load-bearing steps proved), [conjecture], or [numerical]. Theorem M and the Sign Lemma are [proved].
We prove an exact, finite-size theorem for the open XX spin chain of length N = 4m whose central bond has a tunable strength t ∈ (0,1]: as the bond heals (t increases), the half-chain ground-state entanglement spectrum becomes strictly more uniform in the majorization order. Mode by mode, the 2m single-particle entanglement (correlation-matrix) eigenvalues pair as 1/2 ± s_k(t), and every s_k(t) is strictly decreasing on (0,1] — a Sign Lemma whose proof is the mathematical core of the paper. As consequences: every strictly Schur-concave entanglement functional (von Neumann entropy, all Rényi entropies of order α > 0) is strictly increasing along the bond-healing path; the largest Schmidt coefficient is strictly decreasing; and the Nielsen LOCC order strictly increases. The proof is entirely elementary and closed at finite size: a two-leg reflection folding reduces the problem to a rank-one-perturbed Jacobi matrix family J(t); the mode velocities dr_k/dt satisfy an exact master identity expressing them as a convex combination of Stieltjes-type sums W_a with weights Ω_a > 0 certified by a zero-counting argument on a Cauchy-type oscillation pencil; and a uniform endpoint bound W_a(t) ≥ W_a(1) > 1/π + 1/(2K) (K = 2M+1) closes the chain. All constants are explicit, every analytic step is checkable by hand, and the finite rational-interval certificates are mechanically re-derivable by exact arithmetic from the stated rules; numerical verifications (double precision plus 60-digit micro-mode layer) accompany every proved statement as non-load-bearing corroboration. We state the interacting case (XXZ, Δ > 0) as a precise conjecture supported by exact-diagonalization numerics, locate exactly where the free-fermion proof breaks, discuss the relation to the entropic g-theorem (complementary, not overlapping), and propose quantum-simulator and photonic-ring tests. The phenomenon and its mode-resolved monotonicity appear to be new; the failure of such monotonicity in nearby settings (bulk RG, strong-coupling Kondo defects) is recalled to delimit the scope. Status labelling. Every claim in this paper carries one of the tags [proved] (complete hand-checkable proof in the text), [established] (proof structure complete with mechanical expansions delegated to an appendix or supplement, all load-bearing steps proved), [conjecture], or [numerical]. Theorem M and the Sign Lemma are [proved].
We present exact results on the exactly solvable spin chain of Bravyi et al. [Phys. Rev. Lett. 109, 207202 (2012)]. This model is a spin one chain and has a Hamiltonian that is local and translationally invariant in the bulk. It has a unique (frustration free) ground state with an energy gap that is polynomially small in the system’s size (2n). The half-chain entanglement entropy of the ground state is 12logn+const.[Bravyi et al., Phys. Rev. Lett. 109, 207202 (2012)]. Here we first write the Hamiltonian in the standard spin-basis representation. We prove that at zero temperature, the magnetization is along the z-direction, i.e., ⟨sx⟩=⟨sy⟩=0 (everywhere on the chain). We then analytically calculate ⟨sz⟩ and the two-point correlation functions of sz. By analytically diagonalizing the reduced density matrices, we calculate the Schmidt rank, von Neumann, and Rényi entanglement entropies for the following: 1. Any partition of the chain into two pieces (not necessarily in the middle) and 2. L consecutive spins centered in the middle. Further, we identify entanglement Hamiltonians (Eqs. (49) and (59)). We prove a small lemma (Lemma (1)) on the combinatorics of lattice paths using the reflection principle to relate and calculate the Motzkin walk “height” to spin expected values. We also calculate the, closely related (scaled), correlation functions of Brownian excursions. The known features of this model are summarized in a table in Sec. I.
This record contains the preprint and the complete reproducibility package for “A formal finite-size spectral-map hierarchy for free-fermion entanglement”. The work studies the finite-size deformation of the spectral relation between the discrete prolate (Slepian) eigenvalue and the one-particle entanglement energy in a critical free-fermion chain. Using the exact DPSWF representation, Riccati/Dunham analysis and finite-part period calculus, an all-orders formal hierarchy is derived in which the finite-size coefficients satisfyFj=Pj(D)Φ∞F_j=P_j(D)\Phi_\infty, with PjP_j a rational function of the Euler operator D=t∂tD=t\partial_t. The work establishes universal denominator, degree and zero structures for these rational operators, including additional even-shift factors, and derives their Weber/Bernoulli leading behaviour. Independent checks include exact reconstruction of the half-filling Fisher–Hartwig phase polynomials Q20,Q40,Q60,Q80Q_{20},Q_{40},Q_{60},Q_{80}, exact Legendre reduction at the spectral centre, rational operator reconstruction through j=13j=13, and finite-matrix validation of the first two finite-size coefficients. The archive accompanying the manuscript contains the scripts, exact symbolic calculations, numerical data, regression tests and SHA-256 manifests required to reproduce the principal computational claims. The manuscript explicitly distinguishes the proved formal Dunham hierarchy from the still-open uniform physical connection/remainder theorem.
Abstract We study the half-chain entanglement entropy in the ground state of the spin-1/2 XX chain across an extended random defect, where the strength of disorder decays with the distance from the interface algebraically as Δ l ∼ l − κ . In the whole regime κ ⩾ 0 , the average entanglement entropy is found to increase logarithmically with the system size L as S L ≃ c e f f ( κ ) 6 ln L + c o n s t , where the effective central charge c e f f ( κ ) depends on κ . In the regime κ < 1 / 2 , where the extended defect is a relevant perturbation, the strong-disorder renormalization group method gives c e f f ( κ ) = ( 1 − 2 κ ) ln 2 , while, in the regime κ ⩾ 1 / 2 , where the extended defect is irrelevant in the bulk, numerical results indicate a non-zero effective central charge, which increases with κ . The variation of c e f f ( κ ) is thus found to be non-monotonic and discontinuous at κ
For a region of a free-fermion ground state the entanglement Hamiltonian K_A = log[(1 - C_A) C_A^-1] splits into a part proportional to the identity and a trace-free remainder. The identity part shifts every level rigidly, so every observable built from level differences is blind to it. This paper is about that part. It is not a bulk quantity. The block carries one ladder per entangling cut, the trace is the difference of the two cut-phase charges, and on the exactly dimerised chain with a single cut carrying the whole charge it has the closed form 2 artanh(v / sqrt(v^2 + t^2)), derived in one line and measured to 10^-11. In exact arithmetic the trace is a definite number: zero to thirty-three digits when the two cut phases agree, and 0.786153199019314523 when they do not. The apparent divergence reported in earlier work was double-precision truncation of a finite sum, and is retired here. Along the two lines through the Rice-Mele plane where the half-line entanglement Hamiltonian is known in closed form, it is proportional to the Bisognano-Wichmann deformation T = (1/2){X, H} with no additive constant, and the ladder spacing and phase are fixed by the literature. Between those lines the commuting operator survives, measured at 10^-13 against controls that fail at 1.5 and 29, but the proportionality does not: the level map acquires an intercept b that is zero to 3 x 10^-9 on both lines and of order 10^-1 in the interior, and is identical whether the region has one entangling cut or two. That intercept is the measured failure of exact Bisognano-Wichmann proportionality. At first order in the staggering v the intercept is derived rather than fitted. The symbol's determinant is det(z - P) = z(z - 1) independently of momentum, so the factorisation is canonical everywhere; its logarithm is affine in an involution R = H/E with R^2 = I; the abelian Szego-type boundary functional is identically even in v and therefore cannot carry an odd intercept; and the first-order piece follows from the chiral zero mode of the dimerised chain, whose amplitudes are (-r)^j C(2j,j) 4^-j in closed form. The result is b/v = 2(E' - K')/(1 - r^2) + K', K' = K(1 - r^2), E' = E(1 - r^2), r = t1/t2, whose expansion in the modulus u is, in exact rationals, h(u)/(pi/8) = 1 - (5/8) u^2 - (9/64) u^4 - (65/1024) u^6 - (595/16384) u^8 - ... So pi/8, -5/8 and -9/64 are consequences rather than recognitions, and the natural elliptic candidate [9E - K]/16 - pi/8 is excluded by 7.7 percent on the first coefficient it does not get to choose. The closed form agrees with direct measurement extrapolated to v -> 0 by two to three orders better than the fitted series it replaces; a previously reported u^6 coefficient of -0.0648 is corrected here to -65/1024 and identified as truncation. The trace-free half is treated in parallel. A Clausius relation with a single temperature holds exactly when that half is proportional to the region's own Hamiltonian. At a ground state it never is, and the measured temperature is zero, which is correct rather than a failure. In a thermal state the same instrument returns 1/beta to five significant figures, and the boundary of the region where it does so collapses onto two dimensionless ratios. Two further results are included. The offset vanishes when the two cuts sit at the same phase in the unit cell, without requiring chirality, time reversal, inversion or cleanliness, and a mirror-symmetrised disordered chain gives a larger offset than its unsymmetrised control, which rules reflection out as the mechanism. And in two dimensions the offset is exactly additive along the boundary for a transverse coupling proportional to the identity in sublattice space, for a reason short enough to state, with a derived crossing past which additivity fails and a sigma_z coupling that breaks it smoothly. Every claim carries one of four grades: [T] proven here, [V] computationally verified with stated controls and a stated failure mode, [C] imported from the cited literature, [H] interpretation. The two exact lines, the commuting deformation, the boundary-locality of the entanglement spectrum, and the first law of entanglement entropy are not this work's and are used as given. Sixteen retired claims, including two withdrawn from earlier versions of this manuscript, are listed with the measurement that killed each. Files in this deposit: the_offset_belongs_to_the_boundary_v8_submission.pdf — 35 pages, gate output verbatim, script listings supplied separately. the_offset_belongs_to_the_boundary_v8_complete.pdf — 59 pages, identical body with all scripts printed inline. offset_v8_scripts.zip — 23 files. Every number quoted in the paper is printed by one of them; the two gate files reproduce their stored output byte for byte. KEYWORDS entanglement Hamiltonian; entanglement spectrum; modular Hamiltonian; Bisognano-Wichmann; Rice-Mele model; SSH model; free fermions; boundary charge; cut-phase charge; corner transfer matrix; Toeplitz operators; Wiener-Hopf factorisation; Szego-Widom asymptotics; algebraic Heun operator; complete elliptic integrals; entanglement first law; area law; Landau-Widom counting function ADDITIONAL NOTES Supersedes earlier versions of this manuscript. Two claims present in versions 1 to 3 are withdrawn and recorded in the kill list: that the incommensurate offset has no value, and that the offset is one quantum. Both were double-precision truncation of a finite sum. The u^6 coefficient of the shape function reported in the companion results ledger is likewise superseded by the derived value. Computational verification, gating and drafting assistance by an AI collaborator (Claude, Anthropic). Every gate is independently re-runnable from the ancillary scripts.
The von Neumann entanglement entropy of exact valence-bond ground states is studied in two frustrated one-dimensional spin chains: the spin-1/2 Majumdar-Ghosh (MG) model and the spin-3/2 J1-J2-J3 chain in its fully dimerized (FD) and partially dimerized (PD) phases. Using matrix-product-state representations, the entropy is computed as a function of system size for three complementary bipartitions - half-chain, single-site, and pairwise - under both open and periodic boundary conditions. In all cases, the entropy saturates to a finite constant in the thermodynamic limit, confirming area-law behavior. The saturation values, extracted via finite-size scaling, are directly related to the underlying virtual-spin bond structure. The MG model and FD phase exhibit similar entanglement behavior, differing primarily in saturation magnitude determined by spin value and bond multiplicity, and both display even-odd oscillations and exponential convergence with system size. In contrast, the PD phase shows qualitatively distinct signatures, including multiple half-chain saturation values depending on the bond type at the cut, asymmetric edge contributions in the single-site entropy, and a multi-band structure in the pairwise entropy reflecting the coexistence of single- and double-singlet bonds. These results establish entanglement entropy as a robust signature of frustrated bond architecture, enabling clear distinction among dimerized phases with different spin magnitude, bond multiplicity, and dimerization patterns.
Cubitt, Harrow, Leung, Montanaro, and Winter (CHLMW) exhibited an explicit pair of quantum channels whose minimum output Rényi entropy is nonadditive at order zero, and reported numerical violations at positive orders close to zero. Their paper states that a semidefinite-programming argument yields a rigorous positive-order interval for the same example, without printing the endpoint or a verifiable certificate; a recent paper by Leung, Lovitz, and Wu records that for this pair "no rigorous endpoint was obtained". We close that gap for the trace-preserving normalization of the printed pair fixed in Section 2. Small rational witness matrices prove that every output of either channel has all eigenvalues between 301/100000 and 2/3; a single explicit entangled input has an exact rational joint output spectrum of rank eight; and two independent elementary interval arguments turn these three facts into a proof of strict additivity violation, Spmin(NR⊗NSˉ)<Spmin(NR)+Spmin(NSˉ), for every real order 0<p≤1/22. Every step of the verification reduces to comparisons of integers, and the complete certificate is a few small rational matrices that a reader can check with a short program -- or, for any single order, by hand. To our knowledge, consistent with the assessment of Leung, Lovitz, and Wu, this is the first printed, computer-verifiable certified positive-order endpoint for this explicit pair. We claim no novelty for the phenomenon or for the eigenvalue-floor mechanism, both due to CHLMW, and no optimality of the endpoint.