Reaction-diffusion processes described by three-state quantum chains and integrability

Abstract

The master equation of one-dimensional three-species reaction-diffusion processes is mapped onto an imaginary-time Schrödinger equation. In many cases the Hamiltonian obtained is that of an integrable quantum chain. Within this approach we search for all 3-state integrable quantum chains whose spectra are known and which are related to diffusive-reactive systems. Two integrable models are found to appear naturally in this context: the U q SU (2)-invariant model with external fields and the 3-state U q SU (P/M )-invariant Perk-Schultz models with external fields. A nonlocal similarity transformation which brings the Hamiltonian governing the chemical processes to the known standard forms is described, leading in the case of periodic boundary conditions to a generalization of the Dzialoshinsky-Moriya interaction.

Classical and quantum sl(1‖2) superalgebras, Casimir operators and quantum chain Hamiltonians

“…The Hamiltonian (6.15) and the slightly more general Hamiltonian indeed brings the Hamiltonian H(q, q 12 , q 13 , q 23 ) to the Hamiltonian H(q, q ′ 12 = 1, q ′ 13 = 1, q ′ 23 = 1). An analogous transformation was also (and earlier) found in [2] in the context of Reaction-Diffusion processes. The same similarity transformation applied to (6.19) with q 12 = −q 13 = q 23 = s transforms all its non diagonal terms to 1, leading to the Perk-Schultz Hamiltonian H P,M with P = 2, M = 1 [12].…”

“…Notice the use of a (still) different symbol for the inner product. The ones of the previous section relate, via equations (4.17), (4.27), products in U qs with graded coproducts in A qs while the one appearing in (5.10) conforms to the (standard) Hopf algebra duality requirement xy, a = x, a (1) y, a (2) , ∆a ≡ a (1) ⊗ a (2) . (5.11)…”

Reaction–Diffusion Processes from Equivalent Integrable Quantum Chains

“…The recent interest in this setup comes from the integrability of quantum Hamiltonians in one dimension [19,21,23,24,25,26,27]…”

“…While large classes of reaction-diffusion problems are now recognised as being integrable, see [23,25,26] for lists including systems with more than one kind of particles 7 , it is still far from obvious how to exploit the algebraic structure hidden beneath it in order to get explicit expressions for the desired particle number correlators for ∆ = 0. A very interesting alternative was proposed in [48], where subsets of observables were identified for which closed equation of motion can be obtained, thus leading to partially integrable systems.…”

“…This arguments also goes through when particle reactions are added [47]. In the case of more than one species of particles, the relation between H and H ′ becomes even more involved, as explained in [26].…”