Structure of Essential Spectra and Discrete Spectrum of the Energy Operator of Two-Magnon Systems with Four-Spin Exchange Hamiltonian

We consider the energy operator of two-magnon systems with four-spin exchange
Hamiltonian and investigated the structure of essential spectra and discrete spectrum of the
system. It is known that the continuous spectrum of the energy operator of two-magnon systems
for an arbitrary spin value in the Heisenberg model is consists of the segment [m_min, M_max],
and the discrete spectrum of the system is consists of no more than 2ν eigenvalues, lying in the
outside of the continuous spectrum of the system. In fact, when the spin value of the system
is more than half, then it is necessary to take into account the multipole exchange interactions
between the atoms of the nearest neighbors. In this case, the corresponding Hamiltonian is called
a non-Heisenberg one. The spectrum of the energy operator of two-magnon systems in the nonHeisenberg case consists of a continuous spectrum and no more than a eigenvalues. In this case, an
additional eigenvalue appears in the system or some existing eigenvalues disappear, compared to
the Heisenberg case. It should be noted that in the non-Heisenberg case, the continuous spectrum
of the system depends on the value of the spin of the system. Also note that the spectrum of
the system for the integer value and the half-integer value of the spin differ from each other.

All these distinctive features of the spectrum of the energy operator of two-magnon systems of
the non-Heisenberg Hamiltonian from the Heisenberg Hamiltonian. We show that the essential
spectrum of the energy operator of two-magnon systems with a four-spin exchange Hamiltonian
consists of a single point {0}, and the discrete spectrum of the system consists of no more than
6ν eigenvalues for arbitrary values of ν, here ν is the lattice dimensionality of Z^ν.
These results show that the spectrum of two-magnon systems with a four-spin exchange
Hamiltonian is very different from the spectra of a system with a Heisenberg Hamiltonian, as
well as with a non-Heisenberg Hamiltonian.

Keywords: Two-magnon system, Heisenberg model, Non-Heisenberg model, Spectr, Continuous
spectrum, Essential Spectrum, Discrete Spectrum, Eigenvalues, Few-spin Exchange Hamiltonian.