Symmetries in quantum mechanics and cor epresentation theory
Lead Research Organisation:
University of Bristol
Department Name: Mathematics
Abstract
This research will consider the role of discrete symmetries in quantum mechanical systems,
with an emphasis on systems whose classical dynamics is chaotic. Mathematically,
symmetries are described by operators that are either unitary or antiunitary and either
commute or anticommute with the Hamiltonian of the system. Considering the case without
unitary symmetries that commute with the Hamiltonian, this already allows to classify
quantum systems into Altland's and Zirnbauer's ten universality classes. If the classical
dynamics of the system is chaotic, its spectral statistics and in some cases also the average
level density agrees with predictions from ensembles of random matrices chosen according
to the symmetry class. This classification generalises to systems that also have unitary
symmetries commuting with the Hamiltonian, however in this case the spectral
characteristics to be considered are those of certain subspectra associated to different
behaviour under the unitary symmetries.
Previous research has shown that this can lead to new physical phenomena. For example, it
can be used to obtain random matrix statistics of the Gaussian Symplectic Ensemble (GSE)
in systems without spin, and this led recently to the first experimental observation of GSE
statistics in a physical system. Charlie will develop this type of approach more
systematically.
A general description of the interplay between unitary and antiunitary symmetries requires to
view symmetries in the context of corepresentation theory. A first step in this direction will be
to classify the behaviour of the simplest symmetry groups with commuting and
nticommuting operators in terms of corepresentations.
with an emphasis on systems whose classical dynamics is chaotic. Mathematically,
symmetries are described by operators that are either unitary or antiunitary and either
commute or anticommute with the Hamiltonian of the system. Considering the case without
unitary symmetries that commute with the Hamiltonian, this already allows to classify
quantum systems into Altland's and Zirnbauer's ten universality classes. If the classical
dynamics of the system is chaotic, its spectral statistics and in some cases also the average
level density agrees with predictions from ensembles of random matrices chosen according
to the symmetry class. This classification generalises to systems that also have unitary
symmetries commuting with the Hamiltonian, however in this case the spectral
characteristics to be considered are those of certain subspectra associated to different
behaviour under the unitary symmetries.
Previous research has shown that this can lead to new physical phenomena. For example, it
can be used to obtain random matrix statistics of the Gaussian Symplectic Ensemble (GSE)
in systems without spin, and this led recently to the first experimental observation of GSE
statistics in a physical system. Charlie will develop this type of approach more
systematically.
A general description of the interplay between unitary and antiunitary symmetries requires to
view symmetries in the context of corepresentation theory. A first step in this direction will be
to classify the behaviour of the simplest symmetry groups with commuting and
nticommuting operators in terms of corepresentations.
Organisations
People |
ORCID iD |
| Francesco Mezzadri (Primary Supervisor) | |
| Charlotte Johnson (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509619/1 | 01/10/2016 | 30/09/2021 | |||
| 1793787 | Studentship | EP/N509619/1 | 01/10/2016 | 31/05/2020 | Charlotte Johnson |
| Description | Statistics of chaotic quantum systems can be categorised as one of ten types (A, AI, AII, AIII, BDI, C, CI, CII, D, DIII) according to how the system behaves under time-reversal and electron-to-positron swapping. Normally different classes need different forms of time-reversal and electron-to-positron swapping; however we have shown that it is possible to pick one type and then add geometric properties to the system to move between the different classes. This realises all ten classes using a consistent setup. This involved investigating how Dirac Graphs transform under electron-positron swapping, and modifying traditional semi-classical methods to deal with quotient graphs. The definition of symmetric Schrodinger and Dirac graphs has been expanded to cover all possible cases geometric, geometric-time-reversal and geometric-charge-conjugation symmetries. Algorithms to efficiently initialise highly symmetric Dirac Graphs for numerics have been written, as well as solving methods. Algorithms to take advantage of symmetry in doing semi-classics have been designed. |
| Exploitation Route | The algorithm (Joyner, Sieber, Muller, EPL 107.5, 2014) this work was based on showed that the class AII can be realised on bosonic systems rather than fermions has been tested experimentally using microwave systems (Rehemanjiang et al. Phys Rev Let 177.6, 2016). It is possible that a modification of this setup could be used to verify the new results in the lab. |
| Sectors | Other |
| Title | Classification of Small Z2 Groups by Their Possible Wigner-Dyson Statistics |
| Description | Classification of all the possible subspectra shown by groups of the form G=U+A according to the Dyson tenfold way. Current classification covers all groups of order |G|<= 100 (|U|<=50). |
| Type Of Material | Database/Collection of data |
| Year Produced | 2017 |
| Provided To Others? | No |
| Impact | Collection allows a desired Wigner-Dyson class to be chosen, then example groups looked up. This has allowed minimal examples of the classes to be identified on quantum graphs. |
| Title | Classification of Small Z2 x Z2 Groups by Their Possible Altland-Zirnbauer Statistics |
| Description | Classification of all the possible subspectra shown by groups of the form G=U+A+C+P according to the Altland-Zirnbauer tenfold way. Current classification covers all groups of order |G|<= 40 (|U|<=10). |
| Type Of Material | Database/Collection of data |
| Year Produced | 2017 |
| Provided To Others? | No |
| Impact | Use has allowed the identification of minimal examples of the Altland-Zirnbauer classes on Dirac Graphs. The classification allows a desired class to be chosen and then groups with this class to be looked up. |