Periodic orbit theory of Bethe-integrable quantum systems: an
$N$-particle Berry-Tabor trace formula
- URL: http://arxiv.org/abs/2401.17891v1
- Date: Wed, 31 Jan 2024 14:56:17 GMT
- Title: Periodic orbit theory of Bethe-integrable quantum systems: an
$N$-particle Berry-Tabor trace formula
- Authors: Juan Diego Urbina, Michael Kelly, Klaus Richter
- Abstract summary: We show how to extend the Berry-Tabor trace formula into the domain of quantum many-body systems displaying integrability.
Our work paves the way towards the treatment of the important class of integrable many-body systems.
- Score: 0.4143603294943439
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: One of the fundamental results of semiclassical theory is the existence of
trace formulae showing how spectra of quantum mechanical systems emerge from
massive interference among amplitudes related with time-periodic structures of
the corresponding classical limit. If it displays the properties of Hamiltonian
integrability, this connection is given by the celebrated Berry-Tabor trace
formula, and the periodic structures it is built on are KAM tori supporting
closed trajectories in phase space. Here we show how to extend this connection
into the domain of quantum many-body systems displaying integrability in the
sense of the Bethe ansatz, where a classical limit cannot be rigorously defined
due to the presence of singular potentials. Formally following the original
derivation of Berry and Tabor [1, 2], but applied to the Bethe equations
without underlying classical structure, we obtain a many-particle trace formula
for the density of states of N interacting bosons on a ring, the Lieb-Liniger
model. Our semiclassical expressions are in excellent agreement with quantum
mechanical results for $N$ = 2, 3 and 4 particles. For N = 2 we relate our
results to the quantization of billiards with mixed boundary conditions. Our
work paves the way towards the treatment of the important class of integrable
many-body systems by means of semiclassical trace formulae pioneered by Michael
Berry in the single-particle context.
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