We investigate the phenomenon of disorder-free localisation in a quantum
system with a global permutation symmetry and the exchange symmetry for
identical particles. We start with a systematic construction of many-fermion
Hamiltonians with a global permutation symmetry using the conjugacy classes of
the permutation group SNβ, with N being the total number of fermions. The
resulting Hamiltonians are interpreted as generators of continuous-time quantum
walk of indistinguishable fermions. In this setup we analytically solve the
simplest example and show that for large N all the states are localised
without the introduction of any disorder coefficients. The localisation is also
time-independent and is not the result of any emergent disorder. This seems to
be an important distinction from other mechanisms of disorder-free
localisation. Furthermore, we show that the localisation is stable to
interactions that preserve the global SNβ symmetry making these systems
potential candidates for a quantum memory. By marking sites and reducing the
symmetry to subgroups of SNβ, the localisation can be obtained for any N by
tuning the symmetry-reducing parameters in the Hamiltonian. Finally we show
that similar localisation also occurs for spin systems governed by a
SNβ-symmetric Heisenberg chain and we make a few comments about
SNβ-symmetric bosonic systems. The models we propose feature all-to-all
connectivity and can be realised on superconducting quantum circuits and
trapped ion systems.Comment: 24 pages, 2 figures ; v2- 27 pages, 2 figures, slight modifications
in the abstract and introductio