Rotational-permutational dual-pairing and long-lived spin order

Abstract

Quantum systems in contact with a thermal environment experience coherent and incoherent dynamics. These drive the system back toward thermal equilibrium after an initial perturbation. The relaxation process involves the reorganization of spin state populations and the decay of spin state coherences. In general, individual populations and coherences may exhibit different relaxation time constants. Particular spin configurations may exhibit exceptionally long relaxation time constants. Such spin configurations are known as long-lived spin order. The existence of long-lived spin order is a direct consequence of the symmetries of the system. For nuclear spin systems, rotational and permutational symmetries are of fundamental importance. Based on the Schur–Weyl duality theorem, we describe a theoretical framework for the study of rotational and permutational dual-symmetries in the context of long-lived spin order. Making use of the proposed formalism, we derive refined bounds on the number on long-lived spin populations and coherences for systems exhibiting rotational-permutational dual-symmetries.

Centralizer theory for long-lived spin states

“…where ζ j is purely real and non-negative. Eigenoperators Ψ 0 j with eigenvalue ζ j = 0 form the null space, null(Γ), of the relaxation superoperator 43,48,53…”

“…Analytic results for the centralizer of generic relaxation algebra are challenging. But it is well known that longlived spin operators predominantly arise whenever the coherent and fluctuating contributions display some type of internal symmetry [43][44][45][46][47][48][49][50][51][52][53][54] . The relaxation algebra then consists of a set spin operators invariant under some symmetry group G. For such cases it is possible to give an explicit characterisation of the centralizer.…”

“…The majority of long-lived spin operators may be found in systems with internal symmetry [43][44][45][46][47][48][49][50][51][52][53][54] . Typical examples include symmetric arrangements of the spin interaction network with respect spin permutations.…”

“…The characterisation of long-lived spin operators may then be based on group theoretical techniques [55][56][57] . Symmetry projection operators have been particularly useful in the characterisation of long-lived spin operators [43][44][45][46][47][48][50][51][52][53][54] . Nonetheless, some small caveats remain.…”

“…Most notably for rapidly rotating methyl groups 31,32,52 . For such systems group theoretical arguments need to be complemented by taking rotational and permutation dual symmetries into account 53 .…”