TY - JOUR
T1 - Integrable matrix theory
T2 - Level statistics
AU - Scaramazza, Jasen A.
AU - Shastry, B. Sriram
AU - Yuzbashyan, Emil A.
N1 - Publisher Copyright:
© 2016 American Physical Society.
PY - 2016/9/2
Y1 - 2016/9/2
N2 - We study level statistics in ensembles of integrable N×N matrices linear in a real parameter x. The matrix H(x) is considered integrable if it has a prescribed number n>1 of linearly independent commuting partners Hi(x) (integrals of motion) H(x),Hi(x)=0, Hi(x),Hj(x)=0, for all x. In a recent work [Phys. Rev. E 93, 052114 (2016)2470-004510.1103/PhysRevE.93.052114], we developed a basis-independent construction of H(x) for any n from which we derived the probability density function, thereby determining how to choose a typical integrable matrix from the ensemble. Here, we find that typical integrable matrices have Poisson statistics in the N→ limit provided n scales at least as logN; otherwise, they exhibit level repulsion. Exceptions to the Poisson case occur at isolated coupling values x=x0 or when correlations are introduced between typically independent matrix parameters. However, level statistics cross over to Poisson at O(N-0.5) deviations from these exceptions, indicating that non-Poissonian statistics characterize only subsets of measure zero in the parameter space. Furthermore, we present strong numerical evidence that ensembles of integrable matrices are stationary and ergodic with respect to nearest-neighbor level statistics.
AB - We study level statistics in ensembles of integrable N×N matrices linear in a real parameter x. The matrix H(x) is considered integrable if it has a prescribed number n>1 of linearly independent commuting partners Hi(x) (integrals of motion) H(x),Hi(x)=0, Hi(x),Hj(x)=0, for all x. In a recent work [Phys. Rev. E 93, 052114 (2016)2470-004510.1103/PhysRevE.93.052114], we developed a basis-independent construction of H(x) for any n from which we derived the probability density function, thereby determining how to choose a typical integrable matrix from the ensemble. Here, we find that typical integrable matrices have Poisson statistics in the N→ limit provided n scales at least as logN; otherwise, they exhibit level repulsion. Exceptions to the Poisson case occur at isolated coupling values x=x0 or when correlations are introduced between typically independent matrix parameters. However, level statistics cross over to Poisson at O(N-0.5) deviations from these exceptions, indicating that non-Poissonian statistics characterize only subsets of measure zero in the parameter space. Furthermore, we present strong numerical evidence that ensembles of integrable matrices are stationary and ergodic with respect to nearest-neighbor level statistics.
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U2 - 10.1103/PhysRevE.94.032106
DO - 10.1103/PhysRevE.94.032106
M3 - Article
AN - SCOPUS:84989866941
SN - 2470-0045
VL - 94
JO - Physical Review E
JF - Physical Review E
IS - 3
M1 - 032106
ER -