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Bulletin of the American Physical Society

Ground state problem of frustrated spin systems due to quantum annealing using the transverse interaction

Yôhei Saika, Jun‐ichi Inoue

Published Mar 15, 2010• 0 citations
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We investigated quantum annealing (QA) via the transverse interaction with XY-type anisotropy for a ground state problem for a small composed of 4 S=1/2 quantum spins interacting with anti-ferromagnetic interactions with each other. By solving the Schrodinger equation for the QA system, we found that a preferable solution can be derived by tuning the XY-type anisotropy of the kinetic term among multiple candidates of the QA system. Similar behavior was suggested from the static property obtained by the spin wave theory established in statistical physics. In addition, we clarified that the ground state of the target system can be obtained by the QA starting from an initial state including excited states of the kinetic term, if the interval of time of the QA is set to be large to some extent.
We suggest using the method of quantum annealing for computing the ground state of the Heisenberg spin chains. Our initial Hamiltonian describes a spin system in a highly non-uniform magnetic field. The initial Hamiltonian gradually transforms into the Heisenberg Hamiltonian, which describes the exchange interaction and the Zeeman interaction with the uniform magnetic field. We demonstrate quantum annealing for 2-, 3-, and 9-spin systems. In particular, we consider the alternate (ferro- and antiferromagnetic) exchange integral. By changing the sign of the exchange integrals we switch from the frustrated to the corresponding non-frustrated spin system. We discuss the similarity and difference between the frustrated and non-frustrated spin systems.
In this paper we show quantum fluctuation effect of fully frustrated Ising spin systems. Quantum annealing has been expected to be an efficient method to find ground state of optimization problems. However it is not clear when to use the quantum annealing. In order to clarify when the quantum annealing works well, we have to study microscopic properties of quantum annealing. In fully frustrated Ising spin systems, there are macroscopically degenerated ground states. When we apply quantum annealing to fully frustrated systems, we cannot obtain each ground state with the same probability. This nature is consistent with "order by disorder" which is well-known mechanism in frustrated systems.