The internal energies of Heisenberg magnetic systems

• We give the best expressions of the internal energies for Heisenberg magnetic systems.
• Around transition temperature, the longitudinal correlation energies for magnetic systems are positive due to strong quantum fluctuation.
• A system with smaller spin quantum number has a stronger fluctuation even if there is spontaneous magnetization.
• The strong quantum fluctuation cannot be totally suppressed by an external magnetic field.
• The expressions of the internal energies when the magnetization has three components are given.

The internal energies, including transverse and longitudinal parts, of quantum Heisenberg systems for arbitrary spin S are investigated by the double-time Green's function method. The expressions for ferromagnetic (FM) and antiferromagnetic (AFM) systems are derived when one-component of magnetization is considered with the higher order longitudinal correlation functions being carefully treated. An unexpected result is that around the order–disorder transition points the neighboring spins in a FM (AFM) system are more likely longitudinally antiparallel (parallel) than parallel (antiparallel) to each other for S≤3/2 in spite of the FM (AFM) exchange between the spins. This is attributed to the strong quantum fluctuation of the systems with small S values. We also present the expressions of the internal energies of FM systems when the three-component of magnetizations are considered.