arXiv:2608.30761

Resonance statistics, Fock-space branching, and long-range pair networks in slowly varying interacting chains

Yogeshwar Prasad

Published Aug 31, 2026• 0 citations

Abstract

In a slowly varying aperiodic potential $h_i=h\cos(2πβi^n+ϕ)$ with random power-law density interactions $V_{ij}/|i-j|^α$, the resonant-object ensemble changes across $α=2n$, wherever Hartree fragmentation is operative, from isolated two-level bonds to a mixture of fragmented multi-site clusters and isolated wing bonds that survive at finite density, while the pair-resonance scaling $x_{\rm pair}^{4-2n-α}\ln x_{\rm pair}\propto h^2/(Vt_0)$ is unchanged~\cite{letter}. Here we develop the microscopic resonance theory underlying these results, together with its domain of validity. We derive the exact phase-averaged resonance statistics --- the supply $N_0\propto L^{2-n}/h$, the correlated common-phase comb, and the closed-form Hartree variance --- proving that the interaction leaves the leading supply law unchanged. Exact construction of the resonant Fock-space graph at $L\le18$ shows that the order-one forward-branching scale $h_{\rm FB}\propto L^{2-n}$ carries no giant component: one-step branching and connectivity are inequivalent. We separate the fixed-pair matching law from the shell-averaged $q\ln(1/q)$ law of the bond ensemble and develop the comb into a mesoscopic shell theory; we map the fragmentation domain, with its support threshold $V_*(α)$ and the boundary-healing recursion; and we treat the marginal case $α=2$, where shell and matching marginalities compound into a double logarithm. Three long-range thresholds emerge with distinct meanings --- $α=1/2$ (the exact variance threshold of the random Fock-space energy and the square-summability boundary of the leading LIOM-dressing estimate), $α=2n$ (change of the local resonant objects), and $α=2$ (marginality of the long-range shell sum) --- none of which is, by itself, a localization transition.

DOI: 10.48550/arxiv.2608.30761

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arXiv:2608.30761

Resonance statistics, Fock-space branching, and long-range pair networks in slowly varying interacting chains

Yogeshwar Prasad

Published Aug 31, 2026• 0 citations

arXiv HTML PDF

Abstract

In a slowly varying aperiodic potential

h_i=h\cos(2πβi^n+ϕ)

with random power-law density interactions

V_{ij}/|i-j|^α

, the resonant-object ensemble changes across

α=2n

, wherever Hartree fragmentation is operative, from isolated two-level bonds to a mixture of fragmented multi-site clusters and isolated wing bonds that survive at finite density, while the pair-resonance scaling

x_{\rm pair}^{4-2n-α}\ln x_{\rm pair}\propto h^2/(Vt_0)

is unchanged~\cite{letter}. Here we develop the microscopic resonance theory underlying these results, together with its domain of validity. We derive the exact phase-averaged resonance statistics --- the supply

N_0\propto L^{2-n}/h

, the correlated common-phase comb, and the closed-form Hartree variance --- proving that the interaction leaves the leading supply law unchanged. Exact construction of the resonant Fock-space graph at

L\le18

shows that the order-one forward-branching scale

h_{\rm FB}\propto L^{2-n}

carries no giant component: one-step branching and connectivity are inequivalent. We separate the fixed-pair matching law from the shell-averaged

q\ln(1/q)

law of the bond ensemble and develop the comb into a mesoscopic shell theory; we map the fragmentation domain, with its support threshold

V_*(α)

and the boundary-healing recursion; and we treat the marginal case

α=2

, where shell and matching marginalities compound into a double logarithm. Three long-range thresholds emerge with distinct meanings ---

α=1/2

(the exact variance threshold of the random Fock-space energy and the square-summability boundary of the leading LIOM-dressing estimate),

α=2n

(change of the local resonant objects), and

α=2

(marginality of the long-range shell sum) --- none of which is, by itself, a localization transition.

DOI: 10.48550/arxiv.2608.30761

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Sabyasachi Nag, Arti Garg

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arXiv:2609.15323Sep 14, 20260 citations

In a slowly varying aperiodic potential

h_i=h\cos(2πβi^n+ϕ)

with random power-law density interactions

V_{ij}/|i-j|^α

, the resonant-object ensemble changes across

α=2n

x_{\rm pair}^{4-2n-α}\ln x_{\rm pair}\propto h^2/(Vt_0)

N_0\propto L^{2-n}/h

L\le18

shows that the order-one forward-branching scale

h_{\rm FB}\propto L^{2-n}

carries no giant component: one-step branching and connectivity are inequivalent. We separate the fixed-pair matching law from the shell-averaged

q\ln(1/q)

law of the bond ensemble and develop the comb into a mesoscopic shell theory; we map the fragmentation domain, with its support threshold

V_*(α)

and the boundary-healing recursion; and we treat the marginal case

α=2

, where shell and matching marginalities compound into a double logarithm. Three long-range thresholds emerge with distinct meanings ---

α=1/2

(the exact variance threshold of the random Fock-space energy and the square-summability boundary of the leading LIOM-dressing estimate),

α=2n

(change of the local resonant objects), and

α=2

(marginality of the long-range shell sum) --- none of which is, by itself, a localization transition.

{S}_{\mathrm{cdw}}(q)

---including the size dependence of

{S}_{\mathrm{cdw}}({q}_{min})

, the extrapolation of

{S}_{\mathrm{cdw}}({q}_{min})/L

versus

1/L

, the evolution of

{K}_{\mathrm{eff}}(L)=\frac{1}{2}L\phantom{\rule{0.16em}{0ex}}{S}_{\mathrm{cdw}}({q}_{min})

, and the power-law behavior of

{S}_{\mathrm{cdw}}(2{k}_{F})

---disentangles phase separation (PS), Luttinger liquids with pairing-dominated correlations (

K>1/2

), Luttinger liquids dominated by

2{k}_{F}

bond-order-wave (BOW) correlations (

K<1/2

), and a long-range

2{k}_{F}

charge-density-wave (CDW) phase. The phase structure, controlled by the interaction

V/t

and dimerization $\ensuremath{\delta}t/t$, is as follows: PS for $V/t\ensuremath{\lesssim}\ensuremath{-}2$; a pairing-dominated Luttinger liquid for $\ensuremath{-}2<V/t<0$; for

V/t>0

and moderate $|\ensuremath{\delta}t/t|$ the system realizes a

2{k}_{F}

-BOW-dominated Luttinger liquid; and for approximately $V/t\ensuremath{\gtrsim}1.5$ together with $|\ensuremath{\delta}t/t|\ensuremath{\lesssim}0.9$ it enters a long-range

2{k}_{F}

CDW, whose domain expands toward smaller $|\ensuremath{\delta}t/t|$ with increasing

V/t

. Quantitative agreement between ED and DMRG across system sizes and boundary treatments underscores the robustness of this phase map and provides a benchmark for the competition between interactions and dimerization in one-dimensional dimerized chains.

{W}_{c}

, which separates localized (

W>{W}_{c}

) and chaotic (

W<{W}_{c}

) phases. Using the analysis of the instability of localized states with respect to resonant interactions complemented by numerical finite-size scaling, we show that the critical disordering scales with the number of spins,

N

, as ${W}_{c}\ensuremath{\approx}[1.37{J}_{0}/(4/3\ensuremath{-}\ensuremath{\alpha})]{N}^{4/3\ensuremath{-}\ensuremath{\alpha}}lnN$ for $0<\ensuremath{\alpha}\ensuremath{\le}1$, and as ${W}_{c}\ensuremath{\approx}[{J}_{0}/(1\ensuremath{-}2\ensuremath{\alpha}/3)]{N}^{1\ensuremath{-}2\ensuremath{\alpha}/3}{ln}^{2/3}N$ for $1<\ensuremath{\alpha}<3/2$, while the transition width scales as ${\ensuremath{\sigma}}_{W}\ensuremath{\propto}{W}_{c}/N$. We use this result to predict the spin long-term evolution for a very large number of spins (

N=50

), inaccessible for exact diagonalization, and to suggest the rescaling of hopping interaction with the system size to attain the localization transition at finite disordering in the thermodynamic limit of infinite number of spins.