A companion paper (Range Versus Entanglement) built a graph Laplacian from the ground state mutual information matrix of the antiferromagnetic Heisenberg (XXZ) spin chain and found that AICc model selection on the lowest eight nonzero eigenvalues consistently prefers a linear dispersion over the quadratic dispersion predicted by a continuum Laplace-Beltrami operator. This paper traces that tension and finds it is substantially, though not entirely, a methodological artifact. The exact open chain dispersion relation has a curvature inflection at k=N/2, proved analytically and confirmed numerically; a fixed eight-mode fitting window straddles or approaches this inflection whenever N is 16 or smaller, biasing AICc toward the linear model for any locally supported adjacency matrix on an open chain, whether or not it is derived from entanglement. This is demonstrated directly on a content-free, unweighted nearest-neighbor chain, which reproduces the same crossover with no physics at all, and on two-body concurrence, which shows the same crossover as mutual information. Once the fitting window is defined adaptively to stay below the proven inflection, AICc prefers the quadratic model at every informative system size. A genuine residual survives this correction: the fitted curvature of mutual information's low-k spectrum has the opposite sign from concurrence's or the ideal chain's, indicating a real, still-unexplained feature specific to spin-spin correlation derived adjacency matrices.
A companion paper (Range Versus Entanglement) built a graph Laplacian from the ground state mutual information matrix of the antiferromagnetic Heisenberg (XXZ) spin chain and found that AICc model selection on the lowest eight nonzero eigenvalues consistently prefers a linear dispersion over the quadratic dispersion predicted by a continuum Laplace-Beltrami operator. This paper traces that tension and finds it is substantially, though not entirely, a methodological artifact. The exact open chain dispersion relation has a curvature inflection at k=N/2, proved analytically and confirmed numerically; a fixed eight-mode fitting window straddles or approaches this inflection whenever N is 16 or smaller, biasing AICc toward the linear model for any locally supported adjacency matrix on an open chain, whether or not it is derived from entanglement. This is demonstrated directly on a content-free, unweighted nearest-neighbor chain, which reproduces the same crossover with no physics at all, and on two-body concurrence, which shows the same crossover as mutual information. Once the fitting window is defined adaptively to stay below the proven inflection, AICc prefers the quadratic model at every informative system size. A genuine residual survives this correction: the fitted curvature of mutual information's low-k spectrum has the opposite sign from concurrence's or the ideal chain's, indicating a real, still-unexplained feature specific to spin-spin correlation derived adjacency matrices.
This paper builds a graph Laplacian from the ground state mutual information matrix of the isotropic antiferromagnetic Heisenberg (XXZ) spin chain and uses it as a case study in what this class of construction does and does not establish. A first-pass analysis reproduces several textbook-looking signatures, including a statistically significant nonzero finite-size intercept in the lowest nonzero eigenvalue that persists and grows under periodic boundaries, naively suggestive of an entanglement-induced mass scale. This reading does not survive scrutiny: the intercept's magnitude varies by an order of magnitude under different adjacency normalizations and can change sign under an AICc-preferred finite-size model with logarithmic corrections. Repeating the construction with the classical, non-entanglement, connected spin correlator reproduces the same intercept phenomenon more cleanly; toggling only quantum coherence on and off via discord leaves it almost unchanged; and restricting to an unambiguous entanglement-only measure, two-body concurrence, collapses the intercept to a value statistically indistinguishable from zero. Across every construction tested, the intercept tracks whether the underlying matrix has long-range support, not whether it is quantum in origin.
A companion paper, The Gravitational Baseline Principle, proposes a minimal pre-geometric quantum substrate and identifies two precise technical conditions that established theorems require before continuum geometry can be said to emerge from it: linear, Lorentz-consistent dispersion, and a mutual-information matrix that encodes a genuine metric distance. This paper tests both directly by exact diagonalization on the critical spin-1/2 antiferromagnetic Heisenberg chain, with a gapped or dimerized chain as a control. The dispersion condition is confirmed to high precision, matching the exact des Cloizeaux-Pearson result and decisively excluding a quadratic alternative. The metric-encoding condition is falsified in its originally hoped-for Euclidean form but not featurelessly: the specific functional form points toward a hyperbolic alternative, confirmed directly via the standard mathematical definition of hyperbolicity, the Gromov four-point delta, rather than a curve fit. An attempt to extend this to a 2D substrate is reported honestly as inconclusive at reachable system sizes, along with a genuine methodological finding: geometrically frustrated lattice shapes can have degenerate ground states that make naive mutual-information calculations non-reproducible unless explicitly checked. A final test sweeps several small dimerization strengths, checking whether the crossover away from hyperbolicity tracks the correlation length's own predicted field-theory scaling. The correlation length itself confirms the predicted scaling cleanly, but whether the geometric diagnostic's own breakdown tracks the same scale is only weakly supported and reported as an open result, not a positive one.
On finite system sizes, monitored one-dimensional free-fermion chains display a broad crossover in entanglement scaling as the measurement rate increases, from sub-extensive behavior at weak monitoring toward an area law at strong monitoring. Analytical field theory and recent large-scale simulations indicate that this apparent change is not a finite-rate transition in the thermodynamic limit. I test, using trajectory-resolved correlation-matrix simulations (chains up to L=96 and up to 128 Born-rule trajectories per parameter point), whether two quantities built from the single-particle entanglement spectrum are useful finite-size diagnostics: the bipartite particle-number fluctuation FA=∑kνk(1−νk) and a participation-style effective number of entangling modes, MA=exp[−∑kwklnwk], with wk∝νk(1−νk). Both quantities track the crossover. However, for trajectory- and time-averaged steady-state values, M is nearly a deterministic function of F: a pooled curve explains 99.7\% of its variance across all rates, so M carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of ∼2 smaller than that of the entropy or F under strong monitoring. Subsystem-scaling comparisons at L=48--96 also show that a pure logarithmic law is not statistically preferred under weak monitoring; a small residual quasi-extensive component and a drifting logarithmic coefficient are instead consistent with a crossover rather than a critical phase. Thus, F remains the natural experimentally motivated companion to the entropy, whereas M is best viewed as a variance-reduced numerical summary of essentially the same information.
This paper studies mutual-information-induced graph Laplacians as a diagnostic of quantum phase structure across multiple models of quantum matter. Using exact diagonalization for system sizes N up to 20, and independent DMRG cross-validation up to N=14, mutual information matrices are computed from the ground states of the XXZ chain, the transverse-field Ising model, a classical spin-glass Hamiltonian, and free fermions, and geometric invariants are extracted from the normalized Laplacian spectrum. A classical spin-glass control produces identically zero mutual information, demonstrating that the pipeline does not manufacture spurious structure when correlations are absent. At the two critical points studied, the exact large-N data reveals a genuine, boundary-condition-robust qualitative distinction between the two universality classes: the XXZ spectral gap is fit far better by a form with a logarithmic finite-size correction characteristic of the SU(2) point's marginally irrelevant operator than by a bare power law, while the transverse-field Ising model gap follows the expected power law cleanly. Every code path is covered by an automated test suite that checks each numerical claim in the paper against its underlying data file.
Mutual information across the commensurate--incommensurate crossover and near the Kosterlitz--Thouless point of the spin-21J1--J2 Heisenberg chain
What quantum mutual information adds to conventional correlation functions is often unclear. We study the spin-1/2 J1-J2 Heisenberg chain, using DMRG on infinite chains and periodic rings and an SU(2) decomposition of the mutual information into correlation channels. For singlet ground states the two-site mutual information is an exact function of the spin correlator, so any additional content must come from blocks. In the dimerized phase, blocks resolve bond-energy correlations in the SU(2)-scalar channel without an independent asymptotic length: singlet and triplet correlation lengths agree to about 1%. The scalar channel dominates block mutual information at large distances in the commensurate regime up to J2/J1=0.40, but falls to a few percent beyond the disorder point, which coincides numerically with the Majumdar-Ghosh point J2/J1=1/2; the structure-factor peak splits only at the Lifshitz point J2/J1=0.520622(2). On the gapless side we compare block mutual information on rings extrapolated to infinite size with the two-interval prediction of SU(2)1 conformal field theory across the Kosterlitz-Thouless point J2/J1=0.241167. There, for block lengths l≤6, the correction decays as l−p with p=1.96-2.07, and any logarithmic term is at least 60 times smaller than in the Heisenberg chain. Expanding in the marginal coupling (from level spectroscopy) the first-order coefficient is consistent with l−2 scaling and the second-order coefficient is positive for x≤1/4, suggesting that the mutual information relative to the field-theory value becomes stationary at the KT point for large blocks. The scalar-channel share approaches 1/4 at small cross ratio x, as expected from SO(4) symmetry; under two stated assumptions its first-order response to the coupling is (3/8)lnx−5/8 at small x, matched by the l=4 data within 1-3% at x≤1/9.
In this dissertation, the momentum-space entanglement properties of several quantum spin chains are investigated. In the second chapter we re- view important results on many-body entanglement. In the third chapter, we numerically study the momentum-space entanglement spectra of bosonic and fermionic formulations of the spin-1/2 XXZ chain [Phys. Rev. Lett. 113, 256404 (2014)]. We investigate the behavior of the entanglement gaps, present in both formulations, across quantum phase transitions in the XXZ chain. In both cases, finite size scaling suggests that the entanglement gap closure does not occur at the physical transition points. For bosons, we find that the entan- glement gap depends on the scaling dimension of the conformal field theory as varied by the X X Z anisotropy. For fermions, the infinite entanglement gap present at the XX point persists well past the phase transition at the Heisenberg point. We elaborate on how this may support the numerical study of phase transitions. In the third chapter, we advocate that in certain critical spin chains a gap in the momentum-space entanglement spectrum separates the universal part of the spectrum, which is determined by the associated conformal field theory, from the non-universal part, which is specific to the model [arXiv:1512.09030]. We provide affirmative evidence from multicritical spin-1 chains with low energy sectors described by the SU(2)2 or the SU(3)1 Wess-Zumino-Witten model. In chapter four, we study momentum- space en- tanglement in quantum spin-half ladders [Phys. Rev. B 93, 125107 (2016)]. When the system is gapped, we analytically find the momentum-space entan- glement Hamiltonian is described by a chiral conformal field theory with a central charge of two. When the system is gapless, the entanglement Hamil- tonian consists of one gapless mode that is linear in subsystem momentum and one mode with a flat dispersion relation. We also analytically include the effect of a certain irrelevant terms (in the renormalization group sense) on the entanglement spectrum. In chapter six, we investigate the momentum- space entanglement spectrum after a quantum quench [arXiv:1603.01997]. We show that the momentum-space entanglement spectrum of the XXZ spin- half chain possesses many universal features both in equilibrium and after a quantum quench.