Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the phase transitions of the five-state clock model on the square lattice. The temperature dependence of the specific heat indicates the system has two phase transitions, as verified clearly in the correlation function. By investigating the magnetic susceptibility, we can only obtained the upper critical temperature as Tc2 = 0.9565 (7). From the fixed-point tensor, we locate the transition temperatures at Tc1 = 0.9029(1) and Tc2 = 0.9520(1), consistent with the MC and the DMRG results.

With an auxiliary weak external magnetic field, we reexamine the fundamental thermodynamic function, Gibbs free energy F (T, h), to study the phase transitions in the classical spin lattice models. A cross derivative, i.e. the second-order partial derivative of F (T, h) with respect to both temperature and field, is calculated to precisely locate the critical temperature, which also reveals the nature of a transition. The strategy is efficient and universal, as exemplified by the 5-state clock model, 2-dimensional (2D) and 3D Ising models, and the XY model, no matter a transition is trivial or exotic with complex excitations. More importantly, other conjugate pairs could also be integrated into a similar cross derivative if necessary, which would greatly enrich our vision and means to investigate phase transitions both theoretically and experimentally.

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This paper presents a theoretical and numerical study of all possible nonlinear electrostatic waves in an unmagnetized and collisionless electron–positron plasma. From a two-component plasma with an arbitrary ion/positron-to-electron mass ratio β and a temperature ratio TR, four possible waves are discussed: Langmuir waves (LWs), electron acoustic waves, ion acoustic waves, and ion-bulk (IBk) waves. Although LWs are ubiquitous, the existence of the other three acoustic waves depends on the values of β and TR. In an electron–positron plasma, only LWs and IBk waves are allowed. The dispersion relation of these waves varies from a “thumb” curve to a “thumb–teardrop” curve when TR < 1. Vlasov simulations are used to verify predicted waves and reveal several characteristics of excited waves, such as sideband instability, harmonic effect, and nonlinear shifts in frequency and wave number. The present results for an electron–positron plasma can be easily applied to plasmas with arbitrary β and TR, especially when the dispersion relations of these waves are close to each other.