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 $c_{\mathbf{r}}=⟨S_0^z{S}_{\mathbf{r}}^z⟩$, 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 $\phi$, $J_1=cos(\phi )$, $J_2=sin(\phi )$. Due to low dimension there is no long-range order at $T\ne 0$, but the short-range holds the memory of the parent zero-temperature ordered phase (antiferromagnetic, stripe or ferromagnetic). $E(\phi )$ and $C_V(\phi )$ demonstrate extrema "above" the long-range ordered phases and in the regions of rapid short-range rearranging. Tracts of $c_{\mathbf{r}}(\phi )$ lines have several nodes leading to nonmonotonic $c_{\mathbf{r}}(T)$ dependence. For any fixed $\phi$ the heat capacity $C_V(T)$ always has maximum, tending to zero at $T\to 0$, in the narrow vicinity of $\phi =155^{\circ }$ it exhibits an additional frustration-induced low-temperature maximum. We have also found the nonmonotonic behaviour of the spin gaps at $\phi =270^{\circ }\pm 0$ and exponentially small antiferromagnetic gap up to ($T\lesssim 0.5$) for $\phi \gtrsim 270^{\circ }$.

16 pages, 9 figures

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