Thermodynamic properties of the 2D frustrated Heisenberg model for the entire $J_{1}-J_{2}$ circle

Abstract

Using the spherically symmetric self-consistent Green's function method, we consider thermodynamic properties of the $S=1/2$ $J_1$-$J_2$ Heisenberg model on the 2D square lattice. We calculate the temperature dependence of the spin-spin correlation functions $c_{\mathbf{r}}=\langle S_{\mathbf{0}}^{z}S_{\mathbf{r}}^{z}\rangle $, the gaps in the spin excitation spectrum, the energy $E$ and the heat capacity $C_{V}$ for the whole $J_{1}$--$J_{2}$-circle, i.e. for arbitrary $\varphi$, $J_1=cos(\varphi)$, $J_2=sin(\varphi)$. Due to low dimension there is no long-range order at $T\neq 0$, but the short-range holds the memory of the parent zero-temperature ordered phase (antiferromagnetic, stripe or ferromagnetic). $E(\varphi)$ and $C_{V}(\varphi)$ demonstrate extrema "above" the long-range ordered phases and in the regions of rapid short-range rearranging. Tracts of $c_{\mathbf{r}}(\varphi)$ lines have several nodes leading to nonmonotonic $c_{\mathbf{r}}(T)$ dependence. For any fixed $\varphi$ the heat capacity $C_{V}(T)$ always has maximum, tending to zero at $T\rightarrow 0$, in the narrow vicinity of $\varphi = 155^{\circ}$ it exhibits an additional frustration-induced low-temperature maximum. We have also found the nonmonotonic behaviour of the spin gaps at $\varphi=270^{\circ}\pm 0$ and exponentially small antiferromagnetic gap up to ($T\lesssim 0.5$) for $\varphi \gtrsim 270^{\circ}$.