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arXiv:2604.14695 · Physical review. B./Physical review. B

Level statistics of the disordered Haldane-Shastry model with $1/r^α$ interaction

Vengatesan Ganapathy, Pranay Patil, Ajit C. Balram

cond-mat.str-elcond-mat.stat-mechquant-phSubmitted Apr 16, 2026• Announced Apr 17, 2026• 0 citations
v1 · Apr 16, 2026v2 · Oct 16, 2026
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Abstract

Understanding how the interaction range and various types of disorder affect the level statistics of many-body quantum systems and lead to the emergence of many-body localization (MBL) is a challenging open frontier. We study the level statistics of a variant of the spin-$1/2$ Haldane-Shastry model with $1/r^α$ interactions, where $α{\geq}0$ parametrizes the range of the interactions, in the presence of position disorder and/or random magnetic fields. We find that neither position disorder nor random magnetic fields alone yields pristine Poisson statistics in this long-range interacting system; however, Poisson statistics emerge in their combined presence, suggesting the emergence of MBL when both types of disorder coexist. Interestingly, once random magnetic fields break the $SU(2)$ symmetry, the strength of the position disorder, $δ$, appears to play an important role, as evidenced by an approximate scaling collapse of the disorder-averaged gap ratios that is parametrized in terms of a single parameter, $αδ$.

Journal reference
Phys. Rev. B 114, 245118 (2026)
DOI
10.1103/zbks-gdsw
Comments
16 pages, 15 figures

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Understanding how the interaction range and various types of disorder affect the level statistics of many-body quantum systems and lead to the emergence of many-body localization (MBL) is a challenging open frontier. We study the level statistics of a variant of the spin-1/21/21/2 Haldane-Shastry model with 1/rα1/r^α1/rα interactions, where α≥0α{\geq}0α≥0 parametrizes the range of the interactions, in the presence of position disorder and/or random magnetic fields. We find that neither position disorder nor random magnetic fields alone yields pristine Poisson statistics in this long-range interacting system; however, Poisson statistics emerge in their combined presence, suggesting the emergence of MBL when both types of disorder coexist. Interestingly, once random magnetic fields break the SU(2)SU(2)SU(2) symmetry, the strength of the position disorder, δδδ, appears to play an important role, as evidenced by an approximate scaling collapse of the disorder-averaged gap ratios that is parametrized in terms of a single parameter, αδαδαδ.
The Haldane-Shastry model is one of the most studied interacting spin systems. The Yangian symmetry makes it exactly solvable, and the model has semionic excitations. We introduce disorder into the Haldane-Shastry model by allowing the spins to sit at random positions on the unit circle and study the properties of the eigenstates. At weak disorder, the spectrum is similar to the spectrum of the clean Haldane-Shastry model. At strong disorder, the long-range interactions in the model do not decay as a simple power law. The eigenstates in the middle of the spectrum follow a volume law, but the coefficient is small, and the entropy is hence much less than for an ergodic system. In addition, the energy level spacing statistics is neither Poissonian nor of the Wigner-Dyson type. The behavior at strong disorder hence serves as an example of a nonergodic phase, which is not of the many-body localized kind, in a model with long-range interactions and SU(2) symmetry.
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In this paper, we report our numerical analysis of energy level spacing statistics for one-dimensional spin-1/21/21/2 XXZ model with random on-site longitudinal magnetic fields BiB_iBi​. We concentrate on the strong disorder limit where the system is expected to be in either a paramagnetic many-body localized (MBL) state or a spin-glass MBL state. By analyzing the energy-level spacing statistics as a function of strength of random magnetic field h/J⊥h/J_{\perp}h/J⊥​ (−h≤Bi≤h-h\leq B_i\leq h−h≤Bi​≤h), interaction strength Jz/J⊥J_z/J_{\perp}Jz​/J⊥​, energy of the many-body state EEE and the number of spin-↑\uparrow↑ particles in the system M=∑i(siz+12)M=\sum_i(s_i^z+{1\over2})M=∑i​(siz​+21​), we show that there exists a small region between the paramagnetic and spin-glass MBL phases where an ergodic phase emerges. The ergodic phase emerges at the middle of the many-body energy spectrum when M∼N2M\sim{N\over2}M∼2N​ and is not adiabatically connected to the ergodic phase that occurs in the weak-disorder, weak-interaction limit.
We numerically study level statistics of disordered interacting quantum many-body systems. A two-parameter plasma model which controls the level repulsion exponent $\ensuremath{\beta}$ and range hhh of interactions between eigenvalues is shown to reproduce accurately features of level statistics across the transition from the ergodic to many-body localized phase. Analysis of higher-order spacing ratios indicates that the considered $\ensuremath{\beta}\ensuremath{-}h$ model accounts even for long-range spectral correlations and allows us to obtain a clear picture of the flow of level statistics across the transition. Comparing the spectral form factors of the $\ensuremath{\beta}\ensuremath{-}h$ model and of a system across the ergodic-many-body-localized transition, we show that the range of effective interactions between eigenvalues hhh is related to the Thouless time which marks the onset of quantum chaotic behavior of the system. Analysis of level statistics of the random quantum circuit which hosts chaotic and localized phases supports the claim that the $\ensuremath{\beta}\ensuremath{-}h$ model grasps universal features of level statistics in transition between ergodic and many-body-localized phases also for systems breaking time-reversal invariance.
We present a finite-size scaling for both interaction and disorder strengths in the critical regime of the many-body localization (MBL) transition for a spin-1/2XXZ2XXZ2XXZ spin chain with a random field by studying level statistics. We show how the dynamical transition from the thermal to MBL phase depends on interaction together with disorder by evaluating the ratio of adjacent level spacings, and thus, extend previous studies in which interaction coupling is fixed. We introduce an extra critical exponent in order to describe the nontrivial interaction dependence of the MBL transition. It is characterized by the ratio of the disorder strength to the power of the interaction coupling with respect to the extra critical exponent and not by the simple ratio between them.
We present a novel finite-size scaling for both interaction and disorder strengths in the critical regime of the many-body localization (MBL) transition for a spin-1/2 XXZ spin chain with random field by studying the level statistics. We show how the dynamical transition from the thermal to MBL phase depends on interaction together with disorder by evaluating the adjacent gap ratio, and thus, extend previous studies in which the interaction coupling is fixed. We introduce an extra critical exponent in order to describe the nontrivial interaction dependence of the MBL transition. It is characterized by the ratio of the disorder strength to the power of the interaction coupling with respect to the extra critical exponent and not by the simple ratio between them.