Computing the Many Body Density of States of a system of non interacting identical quantum particles

The modeling of many-body (MB) quantum systems undergoing an out-of-equilibrium evolution

requires one to go beyond the low-energy physics and local or few body Densities of States (DoS).

MB localization, thermalization (or lack of) and quantum chaos are phenomena in which states at

various energy scales contribute to the dynamics. Enumerating these states with a many-body DoS

is a crucial step in order to build a statistical description of systems displaying such phenomena.

Surprisingly, such a calculation, which dates back to H. Bethe [1], has proved to be very difficult

even in the non interacting case [2].

We provide in this work an exact analytic method based on a principal component analysis which

characterize universal properties of a fermonic MB spectrum. This is done by decoupling the

combinatoric information inherent to a set of MB configurations, from any particular choice of the

single-body spectrum with given number of states and particles.

The method can account for symmetries and the associated conserved quantities, and can be extended

to bosons. We give applications to simple models such as tight-binding and transverse Ising chains.

Exact and fast numerical computations can be made for large systems since the time-consuming part

is universal and can be re-used for all MB systems of a given size [3].

[1] H. Bethe, Phy. Rev. (1936)

[2] V. Zelevinsky and M. Horoi, Progress in Particle and Nuclear Physics (2018)

[3] R. Lefevre, K. deZawadzki and G. Ithier, arXiv:2208.02236