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.
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.
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.
We introduce a geometric scaling relation that characterizes the local scale behavior of correlations using the informational distance dE=K0/I, where I is the mutual information. We define a geometric conversion factor, G≡∂rdE, which quantifies the local scale. We show that G relates directly to I via G∝Iκ. For systems with power-law correlations I(r)∼r−X, the metric scaling exponent is κ=1/X−1/2. A key consequence is that the geometric scale G is uniform (position-independent) if and only if κ=0, which occurs precisely at X=2. This identifies X=2 as the unique condition for a uniform and metric informational distance. We validate this relation using DMRG simulations of the 1D XXZ chain and exact results for the XX model. We demonstrate two falsifiable diagnostics: (i) G(r) is flat in the bulk at criticality (X≈2) but varies strongly when gapped; (ii) a coordinate-agnostic slope test of logG versus logI at the XX benchmark (X=2) yields κ≃0. This approach provides a coordinate-independent method for identifying scaling regimes that helps to reduce ambiguity from non-universal amplitudes and from the fitting choices in standard power-law analyses, and defines a simple post-processing pipeline that can be applied directly to numerical or experimental mutual-information data.
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.
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.
We numerically investigate classical and quantum correlations in one-dimensional quantum critical systems. The infinite matrix product state (iMPS) representation is employed in order to consider an infinite-size spin chain. By using the infinite time-evolving block decimation algorithm, iMPS ground state wave functions are obtained at critical points for the transverse-field spin-1/2 XY model. From the ground state wave functions, we calculate classical and quantum correlations and mutual information. All of the correlations are found to exhibit a power-law decay with the increments of the lattice distance for both the transition lines of the Ising universality class and the Gaussian universality class. Such power-law scaling behaviors of the correlations manifest the existence of diversing correlation lengths, which means scale invariance. The critical features of the correlations can be characterized by introducing a critical exponent of the power-law decaying correlations. Similar to the critical exponent η of the spin-spin correlation for the universality classes in the transverse-field XY model, we calculate the critical exponents of the two-spin classical and quantum correlations as well as that of the corresponding mutual information. All of the correlations have the same critical exponents, i.e., ηI=ηC=ηD at a critical point, where the superscripts I, C, and D stand for mutual information, classical correlation, and quantum correlation, respectively. Furthermore, the critical exponent η of the spin-spin correlation is shown to relate to η=ηα/2 with α∈{I,C,D}.
Abstract Whether the spatial geometry implied by a quantum state can be recovered from its entanglement structure is a central question in the "space from Hilbert space" program. We study the sharpest tractable version of this question: can equal-time mutual information (MI) detect metric inhomogeneity, as opposed to topology and dimensionality alone? Working with inhomogeneous critical free fermions, whose emergent curved-CFT metric is known in closed form, we derive an exact expression for the Weyl-dressed conformal separation between two points, D/r = e^([ϕ(a)+ϕ(b)]/2) ⟨e^(−ϕ)⟩ with ϕ = ln v, and expand it in cumulants of ϕ. The leading term is δ1 = 1/2 [ϕ(a) + ϕ(b)] − ⟨ϕ⟩, which vanishes identically for linear ϕ and is small for many gently varying monotonic profiles; when it vanishes, the leading response degrades to 1/2 Var(ϕ). This explains a practical obstruction: the standard linear-v and linear-ln v ramp experiments are second-order-only probes of the emergent metric, and we verify numerically that it improves predictive variance over plain lattice distance by only ∆R^2 ≈ 4 × 10^(−3). The criterion is constructive. Profiles that place the inhomogeneity in the interval interior while matching endpoint velocities restore first-order sensitivity. Exact free-fermion computations on chains of N = 240 sites confirm the resulting predictions: MI is suppressed across a velocity valley and enhanced across a velocity bump at identical lattice separation, a sign reversal no lattice-distance model can produce; and 42 profiles drawn from four functional families collapse onto a single line ln[I_profile/I_flat] = −X_resp δ1 with X_resp = 1.64 and R^2 = 0.9969, improving to R^2 = 0.9991 when the second-order cumulant is included. We conclude that equal-time MI does encode the emergent conformal geometry, but that identifiability—not signal strength—is the binding constraint, and we give a design rule for experiments that respect it. Repository Contents & Replication Scripts This repository provides the complete, standalone codebase used to execute the exact free-fermion correlation-matrix computations and regenerate all numerical data tables, out-of-sample family cross-validations, and data collapse figures presented in the text. manuscript.pdf: Complete unblinded research article text. free_fermion_sim.py: Simulates the inhomogeneous hopping (XX) chains across N=240 site geometries and computes the exact fermionic mutual information matrices via Peschel's correlation matrix method. cumulant_analysis.py: Extracts the response coordinate coordinates, maps the profile variants (Gaussian, sech², Lorentzian, raised cosine), and handles the multi-family data collapse regression profiles. Environment Requirements: Python 3.8+ with standard scientific computing installations (numpy, scipy, matplotlib). Double-Blind Review & Anonymity Compliance Author names, institutional affiliations, contact emails, and regional funding details are permanently withheld from these files to guarantee total anonymity. Content processing and textual structuring were partially optimized using conversational generative AI models, with all underlying mathematical derivations, code architecture, and numerical assertions independently validated by the author.