Finite-temperature phase transitions in $S=\frac{1}{2}$ three-dimensional Heisenberg magnets from high-temperature series expansions

Many frustrated spin models on three-dimensional (3D) lattices are currently being investigated, both experimentally and theoretically, and develop new types of long-range orders in their respective phase diagrams. They present finite-temperature phase transitions, most likely in the Heisenberg 3D universality class. However, the combination between the 3D character and frustration makes them hard to study. We present here several methods derived from high-temperature series expansions (HTSEs), which give exact coefficients directly in the thermodynamic limit up to a certain order; for several 3D lattices, supplementary orders than in previous literature are reported for the HTSEs. We introduce an interpolation method able to describe thermodynamic quantities at $T>T_c$, which we use here to reconstruct the magnetic susceptibility and the specific heat and to extract universal and nonuniversal quantities (for example, critical exponents, temperature, energy, entropy, and other parameters related to the phase transition). While the susceptibility associated with the order parameter is not usually known for more exotic long-range orders, the specific heat is indicative of a phase transition for any kind of symmetry breaking. We present examples of applications on ferromagnetic and antiferromagnetic models on various 3D lattices and benchmark our results whenever possible.

Figure 1

Density of poles as defined in Eq. (14) from the Dlog Padé method on $\overline{\chi }\left(\beta \right)$ for the ferromagnetic cases on the fcc, bcc, $\mathit{sc}$, ssc, and pyrochlore lattices as a function of $\beta -{\beta }_c$. For each curve, the poles from the four highest orders are taken.

Figure 2

Density of residues ($\gamma$) from the Dlog Padé method on $\overline{\chi }\left(\beta \right)$ for the ferromagnetic case on the fcc, bcc, $\mathit{sc}$, ssc, and pyrochlore lattices as a function of $\gamma$. For each lattice, we use the four highest orders. The dashed line indicates the result from field theory renormalization group [14, 15].

Figure 3

Poles and residues from the Dlog Padé method on $\overline{\chi }\left(\beta \right)$ for the ferromagnetic case on the fcc, bcc, $\mathit{sc}$, ssc, and pyrochlore lattices as a function of $(\beta -{\beta }_c)/{\beta }_c$. The inset shows the poles and residues for the pyrochlore lattice.

Figure 4

Density of poles from the Dlog Padé method on $\overline{\chi }\left(e\right)$ for the ferromagnetic model. Results are shown as a function of $e-e_c$, where $e_c$ is the value at which there is a peak. For each lattice, poles from HTSEs at orders from $n-3$ to $n$ are used, where $n$ is given in Table 1.

Figure 5

Critical temperature $T_c$ (top panel) and the difference between the ground-state and critical energies $e_0-e_c$ (bottom panel) as a function of the coordination number $Z$ for the ferromagnetic model on the fcc, bcc, $\mathit{sc}$, ssc, and pyrochlore lattices obtained from the Dlog Padé method on $\overline{\chi }$. We also show the interpolation between the $\mathit{sc}$ and ssc lattices by using an effective $Z$ (see main text).

Figure 6

Top panel: Values of the singularity parameters $A$ and $B$ as a function of $\alpha$ for the ferromagnetic case on the fcc lattice obtained from the two interpolation methods IM1 and IM2. We show the results for the three highest orders of the HTSE of $c_v$. The pink dots are the results obtained with the Dlog Padé method on $c_v\left(\beta \right)-B$. Bottom panel: The quality $Q$ [see Eq. (9)].

Figure 7

Values of the nonuniversal parameters $A$ and $B$ as a function of $\alpha$ for the ferromagnetic model on the fcc, bcc, and $\mathit{sc}$ lattices; using only the highest order in the HTSEs. Results obtained with IM1 and IM2 are shown with full and dashed lines, while symbols correspond to the Dlog Padé method on $c_v\left(\beta \right)-B$.

Figure 8

Reconstructed $c_v$ from the two interpolation methods for the ferromagnetic case on the fcc, bcc, $\mathit{sc}$, and pyrochlore lattices. We show the results for two different values of $\alpha$. The inset shows a zoom for the fcc lattice close to the critical temperature.

Figure 9

Interpolation method results for $c_v\left(\beta \right)$ for the antiferromagnetic model on the $\mathit{sc}$ (left) and bcc (right) lattices. The quality of results $Q$ is shown in color scale in a region of the parameter space defined by $T_c$ and $A$ or $B$. The critical exponent $\alpha$ is set to $-0.12$. The quality $Q$ as a function of $T_c$ is shown in the top panels, in blue and orange for IM1 and IM2, respectively. Dashed black lines indicate QMC results [43, 44].

Figure 10

Interpolation method IM2 results for $\overline{\chi }\left(\beta \right)$ for the ferromagnetic case on the fcc, bcc, $\mathit{sc}$, and pyrochlore lattices. The quality of results $Q$ is shown in color scale in a region of the parameter space $\{{\beta }_c,\gamma \}$ close to the best values. White circles indicate the poles and residues from the Dlog Padé method.