Semiclassical foundations of single particle and collective phenomena in quantum chaos
Final Report Abstract
For a long time an explanation of the universal spectral properties of quantum systems with classically chaotic dynamics remained the main challenge of the quantum chaos theory. As this goal was basically achieved in the last two decades for few-particle chaotic systems, the focus of the research has been gradually shifting into the direction of the non-universal features of the quantum systems. In this project we studied chaotic systems with dynamically broken chirality. Their spectral statistics turn out to be anomalous, but can still be investigated by standard tools bored from semiclassical and random matrix theories. The situation, however, changes drustically if one attempts to increase the system dimensions. Although many-body systems played a pronounced role in the foundation of quantum chaos its later on development has been mainly restricted to few particle systems. This limitation is seemingly related to one of the key semiclassical tools, the trace formula, which connects traces of quantum evolution operators with periodic orbits of the underlying classical system. For systems with few degrees of freedom this approach is applicable to a very wide range of time scales, including the Heisenberg times, where the phenomenon of spectral universality holds. On the other hand, an increase in the number of particles N leads to an exponential proliferation of periodic orbits on the classical side of the problem and, simultaneously, to an exponential growth of the effective Hilbert space dimension (resp. density of states) on the quantum side. Thus, it becomes apparent that the conventional quantum chaos path should fail, in general, to reproduce correctly the classical-quantum correspondence in a limit where both N and h^−1 grow simultaneously. In this project we developed an alternative idea to study spectral properties of many-body systems in the thermodynamical limit N → ∞. The key ingredient of our approach is the duality relation which connects traces of the unitary evolution U^T to those of the non-unitary operator U˜^N. Crucially, the dimension of U˜ is independent of N and remains small for short evolution times. This drastically reduces the complexity of the problem from the numerical point of view. What is even more important, the duality relation opens up a second path suitable to address large N systems. Instead of treating the spectrum of the original unitary evolution U we can apply semiclassical techniques to its dual counterpart U .
Publications
- Collective vs. single-particle motion in quantum many-body systems: Spreading and its semiclassical interpretation in the perturbative regime, Europhys. Lett. 96, 20007 (2011)
J. Hämmerling, B. Gutkin, T. Guhr
- Spectral problem of block-rectangular hierarchical matrices J. Stat. Phys. 143, 72 (2011)
B. Gutkin, V.Al. Osipov
- Spectral Statistics of “Cellular” Billiards, Nonlinearity 24, 1743 (2011)
B. Gutkin
- Clustering of periodic orbits in chaotic systems, Nonlinearity 26, 177 (2012)
B. Gutkin, V.Al. Osipov
(See online at https://doi.org/10.1088/0951-7715/26/1/177) - Clustering of periodic orbits and ensembles of truncated unitary matrices, J. Stat. Phys. 123, 1049 - 1064 (2013)
B. Gutkin, V.Al. Osipov
(See online at https://doi.org/10.1007/s10955-013-0859-9) - Spectral properties and dynamical tunneling in constant-width billiards, Phys. Rev. E 90, 022903 (2014)
B. Dietz, T. Guhr, B. Gutkin, M. Miski-Oglu and A. Richter
(See online at https://doi.org/10.1103/PhysRevE.90.022903) - Spectral statistics of nearly unidirectional quantum graphs, J. Phys. A, 48 345101 (2015)
M. Akila, B. Gutkin
(See online at https://doi.org/10.1088/1751-8113/48/34/345101) - Universality in spectral statistics of “open” quantum graphs, Phys. Rev. E 91, 060901(R) (2015)
B. Gutkin, V.Al. Osipov
(See online at https://doi.org/10.1103/PhysRevE.91.060901) - Classical foundations of many-particle quantum chaos, Nonlinearity 29 (2), 325 (2016)
B. Gutkin, V.Al. Osipov
(See online at https://doi.org/10.1088/0951-7715/29/2/325) - Semiclassical Identification of Periodic Orbits in a Quantum Many-Body System, Phys. Rev. Lett. 118 164101 (2017)
M. Akila, D. Waltner, B. Gutkin, P. Braun, T. Guhr
(See online at https://doi.org/10.1103/PhysRevLett.118.164101)
