TY - JOUR

T1 - Integrable matrix theory

T2 - Level statistics

AU - Scaramazza, Jasen A.

AU - Shastry, B. Sriram

AU - Yuzbashyan, Emil A.

N1 - Funding Information:
This work, especially to the unfolding technique in Appendix. We also thank J. Lebowitz for helpful discussions. Finally, we thank the PRE referees for suggestions and questions that led to a considerably improved draft of this work
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 -