Thermal mutual information of order beta to the one-quarter power in one dimension
A bounded-local one-dimensional Hamiltonian whose full Gibbs state has middle-cut mutual information at least c beta^(1/4), including all charge and spin sectors.
Yuxuan Zhang · 13 November 2026 · Version 1.0
How quickly can correlations across a single cut grow as a one-dimensional quantum system is cooled? This construction gives an explicit family whose thermal mutual information grows at least as the one-quarter power of inverse temperature:
\[I(L:R)_{\rho_{\beta_N}}\ge c\,\beta_N^{1/4}, \qquad \rho_{\beta_N}=\frac{e^{-\beta_NH_N}}{\operatorname{Tr}e^{-\beta_NH_N}}.\]The Hamiltonian has fixed local dimension, nearest-neighbor interactions and uniformly bounded local terms. The bound concerns the full Gibbs state, including every particle-number and spin sector. This is a complete proof candidate with sequential internal analytical review; independent expert review and comprehensive priority assessment remain outstanding.
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The precise result
For every \(N\ge2^{30}\) divisible by four, the construction gives an open chain of \(N\) sites of dimension nine. At
\[\beta_N=2^{44}N^4,\qquad I(L:R)_{\rho_{\beta_N}}\ge\frac{\log 2}{8}N.\]Logarithms here are natural. Thus one can take \(c=(\log2)/(8\cdot2^{11})\). The small local potential depends on the chain length; its coefficients remain uniformly bounded. The theorem is an asymptotic existence result along this sequence, with conservative constants. It does not claim practical temperatures, translation invariance or an efficient preparation algorithm.
Why a finite density of holes helps
Each of two rings carries holes and spin-one-half particles. Neighboring particles can exchange places with holes, while a boundary interaction lets a hole cross the endpoint only when both rings do so together. A contact term favors a singlet between the first particles on the two rings. Folding each ring produces the two halves of a one-dimensional chain.
Synchronized motion aligns the ordered spins into singlet pairs. The new boundary rule requires a genuine particle-hole exchange on each ring, which keeps that alignment intact when there are many holes. Endpoint transport and an overlapping-singlet constraint separate all unwanted spin states by an energy of order \(N^{-3}\).
A weak local potential favors a central density band. Its effect is handled with a pointwise scalar ground-state transform, so it does not overwhelm the small spin separation. The proof then controls every charge sector. At the stated temperature, the full thermal weight outside the useful subspace is less than \(e^{-13N}\). Local operations extract \(N/4\) nearly perfect singlet pairs, giving the mutual-information bound by data processing.
Relation to earlier work
Gottesman and Hastings introduced the two-ring architecture, singlet contact and folding. Their ground-state entanglement-gap relation has a one-quarter exponent. That exponent alone does not give the same temperature dependence for a thermal state.
Kuwahara, Alhambra and Anshu, Section III.2, obtain a thermal lower bound of order \((\beta/\log\beta)^{1/5}\). The proposed advance here is the finite-density construction and its full-sector thermal estimate, which replace that exponent by \(1/4\). The path-exclusion spectral gap is inherited from Caputo, Liggett and Richthammer.
The source package compares the exact statements and also checks the recent Gibbs-mixture representation result of Bakshi, Choi and Pilatowsky-Cameo. The distinction from these inspected papers is supported by the comparison; it is not a claim that every concurrent or equivalent result has been ruled out.
Evidence and review
The original checker passes 22,113 exact comparisons between the physical Hamiltonian and its charge-spin representation, together with 50 charge means and four deliberately incorrect operator controls. New exact code checks the ground-state transform with nonconstant rational weights and detects an incorrect edge factor. Numerical endpoint and spin checks are included as diagnostics.
The all-size proof rests on the written argument, especially the endpoint transport estimate, unequal-count spin inequality and uniform comparison across sectors. The review examined those steps rather than relying only on the finite tests. The source package distinguishes the proof, exact controls, numerical diagnostics and review limits.
The author directed the research and workflow. AI systems assisted with derivation, source comparison, computation and drafting. This release records an internally reviewed candidate; it is separate from external refereeing or acceptance by a problem catalogue.