Thermal features of Heisenberg antiferromagnets on edge- versus corner-sharing triangular-based lattices: a message from spin waves

In order to express the Heisenberg Hamiltonian (1) in terms of Holstein-Primakoff bosons, we first transform it into the rotating frame with its z axis pointing along each local spin direction in the classical ground state. We denote the local coordinate system by $(\tilde{x},\tilde{y},\tilde{z})$ distinguishably from the laboratory frame (x, y, z). For coplanar antiferromagnets with their spins lying in the z-x plane, for instance, the spin components in the laboratory and rotating frames are related with each other as

$$\begin{eqnarray}&&\begin{array}{c}\left[\begin{array}{c}{S}_{{{\bf{r}}}_{l}}^{z}\\ {S}_{{{\bf{r}}}_{l}}^{x}\\ {S}_{{{\bf{r}}}_{l}}^{y}\end{array}\right]=\left[\begin{array}{ccc}\cos {\phi }_{{{\bf{r}}}_{l}} & -\sin {\phi }_{{{\bf{r}}}_{l}} & 0\\ \sin {\phi }_{{{\bf{r}}}_{l}} & \cos {\phi }_{{{\bf{r}}}_{l}} & 0\\ 0 & 0 & 1\end{array}\right]\,\left[\begin{array}{c}{S}_{{{\bf{r}}}_{l}}^{\tilde{z}}\\ {S}_{{{\bf{r}}}_{l}}^{\tilde{x}}\\ {S}_{{{\bf{r}}}_{l}}^{\tilde{y}}\end{array}\right],\end{array}\end{eqnarray} \tag{ 2 }$$

where ${\phi }_{{{\bf{r}}}_{l}}$ is the angle formed by the axes z and $\tilde{z}$ at r _l . The local coordinate notation is applicable to collinear antiferromagnets as well with substantial advantage especially in our comparative study. In the local coordinate system, (1) reads

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal H } & = & J\displaystyle \sum _{\langle l,l^{\prime} \rangle }\left[\left({S}_{{{\bf{r}}}_{l}}^{\tilde{z}}{S}_{{{\bf{r}}}_{l^{\prime} }}^{\tilde{z}}+{S}_{{{\bf{r}}}_{l}}^{\tilde{x}}{S}_{{{\bf{r}}}_{l^{\prime} }}^{\tilde{x}}\right)\cos ({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{l^{\prime} }})+\left({S}_{{{\bf{r}}}_{l}}^{\tilde{z}}{S}_{{{\bf{r}}}_{l^{\prime} }}^{\tilde{x}}-{S}_{{{\bf{r}}}_{l}}^{\tilde{x}}{S}_{{{\bf{r}}}_{l^{\prime} }}^{\tilde{z}}\right)\sin ({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{l^{\prime} }})\right.\\ & & \left.+{S}_{{{\bf{r}}}_{l}}^{\tilde{y}}{S}_{{{\bf{r}}}_{l^{\prime} }}^{\tilde{y}}\right].\end{array}\end{eqnarray} \tag{ 3 }$$

When we employ the Holstein-Primakoff bosons

$$\begin{eqnarray}&&{S}_{{{\bf{r}}}_{l}}^{\tilde{z}}=S-{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}},{S}_{{{\bf{r}}}_{l}}^{\tilde{x}}-{{iS}}_{{{\bf{r}}}_{l}}^{\tilde{y}}\equiv {S}_{{{\bf{r}}}_{l}}^{\tilde{+}\dagger }\equiv {S}_{{{\bf{r}}}_{l}}^{\tilde{-}}={a}_{{{\bf{r}}}_{l}}^{\dagger }\sqrt{2S-{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}}\equiv \sqrt{2S}{a}_{{{\bf{r}}}_{l}}^{\dagger }{{ \mathcal R }}_{{{\bf{r}}}_{l}}(S)\end{eqnarray} \tag{ 4 }$$

and expand the square root ${{ \mathcal R }}_{{{\bf{r}}}_{l}}(S)$ in descending powers of S,

$$\begin{eqnarray}&&{{ \mathcal R }}_{{{\bf{r}}}_{l}}(S)=1-\displaystyle \sum _{l=1}^{\infty }\displaystyle \frac{(2l-3)!!}{l!}{\left(\displaystyle \frac{{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}}{4S}\right)}^{l},\end{eqnarray} \tag{ 5 }$$

the Hamiltonian (3) is expanded into the series

$$\begin{eqnarray}&&{ \mathcal H }=\displaystyle \sum _{m=0}^{4}{{ \mathcal H }}^{\left(\tfrac{m}{2}\right)}+O\left({S}^{-\tfrac{1}{2}}\right),\end{eqnarray} \tag{ 6 }$$

where ${{ \mathcal H }}^{\left(\tfrac{m}{2}\right)}$, on the order of ${S}^{\tfrac{m}{2}}$, read

$$\begin{eqnarray}&&{{ \mathcal H }}^{(2)}={{JS}}^{2}\displaystyle \sum _{\langle l,l^{\prime} \rangle }\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{l^{\prime} }}\right),\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{ \mathcal H }}^{\left(\tfrac{3}{2}\right)}=-J\sqrt{\displaystyle \frac{{S}^{3}}{2}}\displaystyle \sum _{\langle l,l^{\prime} \rangle }\left({a}_{{{\bf{r}}}_{l}}^{\dagger }-{a}_{{{\bf{r}}}_{l}}-{a}_{{{\bf{r}}}_{l^{\prime} }}^{\dagger }+{a}_{{{\bf{r}}}_{l^{\prime} }}\right)\sin \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{l^{\prime} }}\right)=0,\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal H }}^{\left(1\right)} & = & -{JS}\displaystyle \sum _{\left\langle l,{l}^{{\rm{{\prime} }}}\right\rangle }\left[<mml:mpadded width="0"/>\left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}+{a}_{{{\bf{r}}}_{{l}^{{\rm{{\prime} }}}}}^{\dagger }{a}_{{{\bf{r}}}_{{l}^{{\rm{{\prime} }}}}}\right)\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{{l}^{{\rm{{\prime} }}}}}\right)-\left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{{l}^{{\rm{{\prime} }}}}}^{\dagger }+{a}_{{{\bf{r}}}_{l}}{a}_{{{\bf{r}}}_{{l}^{{\rm{{\prime} }}}}}\right)\right.\\ & & \times \displaystyle \frac{\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{{l}^{{\rm{{\prime} }}}}}\right)-1}{2}\left.-\left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{{l}^{{\rm{{\prime} }}}}}+{a}_{{{\bf{r}}}_{{l}^{{\rm{{\prime} }}}}}^{\dagger }{a}_{{{\bf{r}}}_{l}}\right)\displaystyle \frac{\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{{l}^{{\rm{{\prime} }}}}}\right)+1}{2}\right],\end{array}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}=J\sqrt{\displaystyle \frac{S}{2}}\displaystyle \sum _{\langle l,l^{\prime} \rangle }\left[\left({a}_{{{\bf{r}}}_{l}}^{\dagger }+{a}_{{{\bf{r}}}_{l}}\right){a}_{{{\bf{r}}}_{l^{\prime} }}^{\dagger }{a}_{{{\bf{r}}}_{l^{\prime} }}-{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}\left({a}_{{{\bf{r}}}_{l^{\prime} }}^{\dagger }+{a}_{{{\bf{r}}}_{l^{\prime} }}\right)\right]\sin \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{l^{\prime} }}\right),\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal H }}^{\left(0\right)} & = & J\displaystyle \sum _{\left\langle l,{l}^{{\prime} }\right\rangle }\left\{<mml:mpadded width="0"/>{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}^{\dagger }{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{{l}^{{\prime} }}}\right)-\left[{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}^{\dagger }\left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}+{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}^{\dagger }{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}\right)\right.\right.\\ & & \left.+\left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}+{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}^{\dagger }{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}\right){a}_{{{\bf{r}}}_{l}}{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}\right]\displaystyle \frac{\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{{l}^{{\prime} }}}\right)-1}{8}-\left[{a}_{{{\bf{r}}}_{l}}^{\dagger }\left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}+{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}^{\dagger }{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}\right){a}_{{{\bf{r}}}_{{l}^{{\prime} }}}\right.\\ & & \left.+{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}^{\dagger }\left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}+{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}^{\dagger }{a}_{{{\bf{r}}}_{{l}^{{\prime} }}}\right){a}_{{{\bf{r}}}_{l}}\right]\left.\displaystyle \frac{\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{{l}^{{\prime} }}}\right)+1}{8}\right\},\end{array}\end{eqnarray} \tag{ 11 }$$

considering that the sum of the relative rotation angles over the nearest neighbors of an arbitrary site ${\sum }_{\kappa =1}^{z}\left({\phi }_{{{\bf{r}}}_{l}+{{\bf{\delta }}}_{l:\kappa }}-{\phi }_{{{\bf{r}}}_{l}}\right)$ equals zero or a multiple of π according as z is even or odd in the classical ground states in question (figure 1).

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The unit cells of the square and triangular lattices both can be reduced to a single site, while those of the honeycomb and kagome lattices contain at least two and three sites, respectively. In any case, we label lattice sites (l = 1, ⋯ ,L) with their belonging unit cells (n = 1, ⋯ ,N) and further internal degrees of freedom (σ = 1, ⋯ ,L/N ≡ p) as l ≡ p(n − 1) + σ. We denote the primitive translation vectors of the square and triangular lattices by

$$\begin{eqnarray}&&\mathrm{square}:\ {{\bf{a}}}_{z}=a\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right),{{\bf{a}}}_{x}=a\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\mathrm{triangular}:\ {{\bf{a}}}_{1}=a\left(\begin{array}{c}-\displaystyle \frac{1}{2}\\ 0\\ -\displaystyle \frac{\sqrt{3}}{2}\end{array}\right),{{\bf{a}}}_{2}=a\left(\begin{array}{c}-\displaystyle \frac{1}{2}\\ 0\\ \displaystyle \frac{\sqrt{3}}{2}\end{array}\right),{{\bf{a}}}_{0}\equiv -{{\bf{a}}}_{1}-{{\bf{a}}}_{2}=a\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\end{eqnarray} \tag{ 13 }$$

and then those of the honeycomb and kagome lattices are given by a ₀ − a ₁, a ₁ − a ₂, a ₂ − a ₀, and 2 a ₁, 2 a ₂, 2 a ₀, respectively (cf figure 1). The nearest-neighbor vectors of the lth site are given by

$$\begin{eqnarray}&&\mathrm{square}:\ {{\bf{\delta }}}_{l:\kappa }=\pm {{\bf{a}}}_{z},\pm {{\bf{a}}}_{x},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\mathrm{honeycomb}:\ {{\bf{\delta }}}_{l:\kappa }={\left(-\right)}^{l}{{\bf{a}}}_{1},{\left(-\right)}^{l}{{\bf{a}}}_{2},{\left(-\right)}^{l}{{\bf{a}}}_{0},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\mathrm{triangular}:\ {{\bf{\delta }}}_{l:\kappa }=\pm {{\bf{a}}}_{1},\pm {{\bf{a}}}_{2},\pm {{\bf{a}}}_{0},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\mathrm{kagome}:\ {{\bf{\delta }}}_{l:\kappa }=\pm {{\bf{a}}}_{(l+1\ \mathrm{mod}\ 3)},\pm {{\bf{a}}}_{(l+2\ \mathrm{mod}\ 3)},\end{eqnarray} \tag{ 17 }$$

while the center of the nth unit cell is expressed as

$$\begin{eqnarray}&&\mathrm{square}:\ {{\bf{R}}}_{n}={n}_{1}{{\bf{a}}}_{z}+{n}_{2}{{\bf{a}}}_{x}={a}^{{\rm{t}}}\left({n}_{2},0,{n}_{1}\right),\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\mathrm{honeycomb}:\ {{\bf{R}}}_{n}={n}_{1}({{\bf{a}}}_{2}-{{\bf{a}}}_{0})+{n}_{2}({{\bf{a}}}_{0}-{{\bf{a}}}_{1})=\sqrt{3}{a}^{{\rm{t}}}\left[\displaystyle \frac{\sqrt{3}({n}_{2}-{n}_{1})}{2},0,\displaystyle \frac{{n}_{2}+{n}_{1}}{2}\right],\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\mathrm{triangular}:\ {{\bf{R}}}_{n}={n}_{1}{{\bf{a}}}_{2}+{n}_{2}{{\bf{a}}}_{0}={a}^{{\rm{t}}}\left({n}_{2}-\displaystyle \frac{{n}_{1}}{2},0,\displaystyle \frac{\sqrt{3}{n}_{1}}{2}\right),\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\mathrm{kagome}:\ {{\bf{R}}}_{n}=2{n}_{1}{{\bf{a}}}_{2}+2{n}_{2}{{\bf{a}}}_{0}=2{a}^{{\rm{t}}}\left({n}_{2}-\displaystyle \frac{{n}_{1}}{2},0,\displaystyle \frac{\sqrt{3}{n}_{1}}{2}\right)\end{eqnarray} \tag{ 21 }$$

(cf figure 1) with n₁ and n₂ being the unique set of integers to represent R _n . When we express each site position r _l ≡ r _p(n−1)+σ as

$$\begin{eqnarray}&&\mathrm{square}:\ {{\bf{r}}}_{1(n-1)+\sigma }={{\bf{R}}}_{n},\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\mathrm{honeycomb}:\ {{\bf{r}}}_{2(n-1)+\sigma }={{\bf{R}}}_{n}+{\left(-\right)}^{\sigma }{{\bf{a}}}_{0},\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&\mathrm{triangular}:\ {{\bf{r}}}_{1(n-1)+\sigma }={{\bf{R}}}_{n},\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&\mathrm{kagome}:\ {{\bf{r}}}_{3(n-1)+\sigma }={{\bf{R}}}_{n}+{{\bf{a}}}_{\sigma -1},\end{eqnarray} \tag{ 25 }$$

and define the 'interunit' and/or 'intraunit' ordering wave vectors

$$\begin{eqnarray}&&\mathrm{square}:\ {\bf{Q}}=\displaystyle \frac{1}{a}\left(\pi ,0,\pi \right),{\bf{q}}={\bf{0}}\qquad \qquad (180^\circ \ {\rm{N}}\acute{{\rm{e}}}\mathrm{el}),\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&\mathrm{honeycomb}:\ {\bf{Q}}={\bf{0}},{\bf{q}}=\displaystyle \frac{1}{a}\left(\pi ,0,0\right)\qquad \qquad (180^\circ \ {\rm{N}}\acute{{\rm{e}}}\mathrm{el}),\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&\mathrm{triangular}:\ {\bf{Q}}=\displaystyle \frac{1}{a}\left(\displaystyle \frac{4\pi }{3},0,0\right),{\bf{q}}={\bf{0}}\qquad \qquad (120^\circ \ {\rm{N}}\acute{{\rm{e}}}\mathrm{el}),\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{\rm{k}}{\rm{a}}{\rm{g}}{\rm{o}}{\rm{m}}{\rm{e}}:{\bf{Q}}={\bf{0}},{\bf{q}}=\displaystyle \frac{1}{\sqrt{3}a}\left(0,0,\displaystyle \frac{8\pi }{3}\right)\,\,({\bf{Q}}={\bf{0}}),\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{\rm{k}}{\rm{a}}{\rm{g}}{\rm{o}}{\rm{m}}{\rm{e}}:\,{\bf{Q}}=\displaystyle \frac{1}{2a}\left(\displaystyle \frac{8\pi }{3},0,0\right),{\bf{q}}={\bf{0}}\,\,(\sqrt{3}\times \sqrt{3}),\end{eqnarray} \tag{ 30 }$$

the rotation angle ${\phi }_{{{\bf{r}}}_{l}}\equiv {\phi }_{{{\bf{r}}}_{p(n-1)+\sigma }}$ can be given by

$$\begin{eqnarray}&&\mathrm{square}:\ {\phi }_{{{\bf{r}}}_{1(n-1)+\sigma }}={\bf{Q}}\cdot {{\bf{R}}}_{n},\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&\mathrm{honeycomb}:\ {\phi }_{{{\bf{r}}}_{2(n-1)+\sigma }}={\bf{Q}}\cdot {{\bf{R}}}_{n}+\sigma {\bf{q}}\cdot {{\bf{a}}}_{0},\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&\mathrm{triangular}:\ {\phi }_{{{\bf{r}}}_{1(n-1)+\sigma }}={\bf{Q}}\cdot {{\bf{R}}}_{n},\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&\mathrm{kagome}:\ {\phi }_{{{\bf{r}}}_{3(n-1)+\sigma }}={\bf{Q}}\cdot {{\bf{R}}}_{n}+{\bf{q}}\cdot {{\bf{a}}}_{\sigma -1}.\end{eqnarray} \tag{ 34 }$$

Intending to calculate thermal quantities of the high-temperature-superconductor-parent material La₂CuO₄ in terms of SWs, Takahashi [103, 104] considered diagonalizing a bosonic Hamiltonian of the square-lattice antiferromagnet subject to the constraint that the staggered-magnetization expectation value be zero at every temperature T,

$$\begin{eqnarray}&&\langle {{ \mathcal M }}^{\tilde{z}}{\rangle }_{T}\equiv \displaystyle \sum _{l=1}^{L}\langle {S}_{{{\bf{r}}}_{l}}^{\tilde{z}}{\rangle }_{T}={LS}-\displaystyle \sum _{l=1}^{L}\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}{\rangle }_{T}=0.\end{eqnarray} \tag{ 35 }$$

Hirsch and Tang [79] also initiated such an idea by truncating the bosonic Hamiltonian at the harmonic part, ${ \mathcal H }\simeq {\sum }_{m=1}^{2}{{ \mathcal H }}^{(m)}\equiv {{ \mathcal H }}_{\mathrm{harm}},$ and diagonalizing

$$\begin{eqnarray}&&{\widetilde{{ \mathcal H }}}_{\mathrm{harm}}\equiv {{ \mathcal H }}_{\mathrm{harm}}+\mu \displaystyle \sum _{l=1}^{L}\left(S-{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}\right)\ \end{eqnarray} \tag{ 36 }$$

with such μ as to satisfy the constraint condition (35). Several authors [78, 80–82, 103–105] sophisticated the thus-modified LSWs (MLSWs) into modified ISWs (MISWs) taking account of the quartic interaction (11) within a mean-field approximation (cf (41)). As long as we try such a variational construction of MISWs for noncollinear antiferromagnets [106–108], the cubic interaction (10), which conserves neither the total magnetization nor the number of magnons, remains ineffective. The interactions on the order of S to a fractional power, ${{ \mathcal H }}^{\left(\tfrac{m}{2}\right)}$ (m = 1, − 1, ⋯ ), are characteristic of noncollinear antiferromagnets. Without them, the Q = 0 and $\sqrt{3}\times \sqrt{3}$ ground states of the kagome-lattice antiferromagnet remain degenerate in energy with each other.

Let us take a look at the variational MISW thermodynamics for noncollinear versus collinear antiferromagnets before we take an alternative step forward. In order to handle variational MISWs, we introduce the multivalued double-angle-bracket notation applicable for various approximation schemes [81, 82]

$$\begin{eqnarray} &&\langle\langle { \mathcal A } \rangle\rangle \equiv \displaystyle \frac{1}{2} \langle\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}+{a}_{{{\bf{r}}}_{l}+{{\bf{\delta }}}_{l:\kappa }}^{\dagger }{a}_{{{\bf{r}}}_{l}+{{\bf{\delta }}}_{l:\kappa }} \rangle\rangle = \langle\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}} \rangle\rangle ,\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray} &&\langle\langle { \mathcal B } \rangle\rangle \equiv \displaystyle \frac{1}{2} \langle\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}+{{\bf{\delta }}}_{l:\kappa }}^{\dagger }+{\rm{H}}.{\rm{c}}. \rangle\rangle = \langle\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}+{{\bf{\delta }}}_{l:\kappa }}^{\dagger } \rangle\rangle ,\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray} &&\langle\langle { \mathcal C } \rangle\rangle \equiv \displaystyle \frac{1}{4} \langle\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}^{\dagger }+{a}_{{{\bf{r}}}_{l}+{{\bf{\delta }}}_{l:\kappa }}^{\dagger }{a}_{{{\bf{r}}}_{l}+{{\bf{\delta }}}_{l:\kappa }}^{\dagger }+{\rm{H}}.{\rm{c}}. \rangle\rangle =\displaystyle \frac{1}{2} \langle\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}^{\dagger }+{\rm{H}}.{\rm{c}}. \rangle\rangle = \langle\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}^{\dagger } \rangle\rangle ,\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray} &&\langle\langle { \mathcal D } \rangle\rangle \equiv \displaystyle \frac{1}{2} \langle\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}+{{\bf{\delta }}}_{l:\kappa }}+{\rm{H}}.{\rm{c}}. \rangle\rangle = \langle\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}+{{\bf{\delta }}}_{l:\kappa }} \rangle\rangle ,\end{eqnarray} \tag{ 40 }$$

which we shall read as the quantum average in the Holstein-Primakoff-boson vacuum $\langle \,{\rangle }_{0}^{\prime}$ for the LSW formalism, the quantum average in the Bogoliubov-boson (magnon) vacuum 〈 〉₀ for the Wick-decomposition-based ISW (WDISW) formalism, or the temperature-T thermal average 〈 〉_T for the Hartree–Fock-decomposition-based ISW (HFISW) formalism. Note that all the averages (37)–(40) are independent of the site indices r _l and δ _l:κ by virtue of translation and rotation symmetries. We decompose the O(S⁰) quartic Hamiltonian (11) into quadratic terms

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal H }}^{(0)}\simeq J\displaystyle \sum _{\langle l,l^{\prime} \rangle }\left\{\left[\left( \langle\langle { \mathcal A } \rangle\rangle -\displaystyle \frac{ \langle\langle { \mathcal B } \rangle\rangle + \langle\langle { \mathcal D } \rangle\rangle }{2}\right)\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{l^{\prime} }}\right)+\displaystyle \frac{ \langle\langle { \mathcal B } \rangle\rangle - \langle\langle { \mathcal D } \rangle\rangle }{2}\right]\right.\\ \quad \times \left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}+{a}_{{{\bf{r}}}_{l^{\prime} }}^{\dagger }{a}_{{{\bf{r}}}_{l^{\prime} }}\right)+\left[\left( \langle\langle { \mathcal B } \rangle\rangle -\displaystyle \frac{2 \langle\langle { \mathcal A } \rangle\rangle + \langle\langle { \mathcal C } \rangle\rangle }{4}\right)\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{l^{\prime} }}\right)+\displaystyle \frac{2 \langle\langle { \mathcal A } \rangle\rangle - \langle\langle { \mathcal C } \rangle\rangle }{4}\right]\\ \quad \times \left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l^{\prime} }}^{\dagger }+{a}_{{{\bf{r}}}_{l}}{a}_{{{\bf{r}}}_{l^{\prime} }}\right)+\left[\left( \langle\langle { \mathcal D } \rangle\rangle -\displaystyle \frac{2 \langle\langle { \mathcal A } \rangle\rangle + \langle\langle { \mathcal C } \rangle\rangle }{4}\right)\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{l^{\prime} }}\right)-\displaystyle \frac{2 \langle\langle { \mathcal A } \rangle\rangle - \langle\langle { \mathcal C } \rangle\rangle }{4}\right]\\ \quad \times \left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l^{\prime} }}+{a}_{{{\bf{r}}}_{l^{\prime} }}^{\dagger }{a}_{{{\bf{r}}}_{l}}\right)-\left[\displaystyle \frac{ \langle\langle { \mathcal B } \rangle\rangle + \langle\langle { \mathcal D } \rangle\rangle }{8}\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{l^{\prime} }}\right)+\displaystyle \frac{ \langle\langle { \mathcal B } \rangle\rangle - \langle\langle { \mathcal D } \rangle\rangle }{8}\right]\\ \quad \times \left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}^{\dagger }+{a}_{{{\bf{r}}}_{l^{\prime} }}^{\dagger }{a}_{{{\bf{r}}}_{l^{\prime} }}^{\dagger }+{a}_{{{\bf{r}}}_{l}}{a}_{{{\bf{r}}}_{l}}+{a}_{{{\bf{r}}}_{l^{\prime} }}{a}_{{{\bf{r}}}_{l^{\prime} }}\right)-\left[\Space{0ex}{1.0ex}{0ex}\Space{0ex}{3.0ex}{0ex} \langle\langle { \mathcal A }{ \rangle\rangle }^{2}+ \langle\langle { \mathcal B }{ \rangle\rangle }^{2}+ \langle\langle { \mathcal D }{ \rangle\rangle }^{2}-\left(2 \langle\langle { \mathcal A } \rangle\rangle + \langle\langle { \mathcal C } \rangle\rangle \right)\right.\\ \quad \times \left.\displaystyle \frac{ \langle\langle { \mathcal B } \rangle\rangle + \langle\langle { \mathcal D } \rangle\rangle }{2}\right]\cos \left({\phi }_{{{\bf{r}}}_{l}}-{\phi }_{{{\bf{r}}}_{l^{\prime} }}\right)\left.-\left(2 \langle\langle { \mathcal A } \rangle\rangle - \langle\langle { \mathcal C } \rangle\rangle \right)\displaystyle \frac{ \langle\langle { \mathcal B } \rangle\rangle - \langle\langle { \mathcal D } \rangle\rangle }{2}\right\}\equiv {{ \mathcal H }}_{\mathrm{quad}}^{(0)}\end{array}\end{eqnarray} \tag{ 41 }$$

to have the tractable SW Hamiltonian ${ \mathcal H }\simeq {{ \mathcal H }}^{(2)}+{{ \mathcal H }}^{(1)}+{{ \mathcal H }}_{\mathrm{quad}}^{(0)}\equiv {{ \mathcal H }}_{\mathrm{quad}}$ and define its MSW extension

$$\begin{eqnarray}&&{\widetilde{{ \mathcal H }}}_{\mathrm{quad}}\equiv {{ \mathcal H }}_{\mathrm{quad}}+\mu \displaystyle \sum _{l=1}^{L}\left(S-{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}\right).\end{eqnarray} \tag{ 42 }$$

For collinear antiferromagnets, every local rotation angle ${\phi }_{{{\bf{r}}}_{l}}$ reads either 0 or π and every relative rotation angle ${\phi }_{{{\bf{r}}}_{l}+{{\bf{\delta }}}_{l:\kappa }}-{\phi }_{{{\bf{r}}}_{l}}$ becomes π (cf (31) and (32)), resulting that their bosonic Hamiltonians ${ \mathcal H }={\sum }_{m=-2}^{\infty }{{ \mathcal H }}^{\left(-m\right)}$ commute with the total uniform magnetization

$$\begin{eqnarray}&&{{ \mathcal M }}^{z}\equiv \displaystyle \sum _{l=1}^{L}{S}_{{{\bf{r}}}_{l}}^{z}=\displaystyle \sum _{l=1}^{L}\left({S}_{{{\bf{r}}}_{l}}^{\tilde{z}}\cos {\phi }_{{{\bf{r}}}_{l}}-{S}_{{{\bf{r}}}_{l}}^{\tilde{x}}\sin {\phi }_{{{\bf{r}}}_{l}}\right)\end{eqnarray} \tag{ 43 }$$

in each order and (41) reduces to

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal H }}_{\mathrm{quad}}^{(0)} & = & J\displaystyle \sum _{\langle l,l^{\prime} \rangle }\left[{\left( \langle\langle { \mathcal A } \rangle\rangle - \langle\langle { \mathcal B } \rangle\rangle \right)}^{2}-\left( \langle\langle { \mathcal A } \rangle\rangle - \langle\langle { \mathcal B } \rangle\rangle \right)\left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}+{a}_{{{\bf{r}}}_{l^{\prime} }}^{\dagger }{a}_{{{\bf{r}}}_{l^{\prime} }}-{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l^{\prime} }}^{\dagger }-{a}_{{{\bf{r}}}_{l}}{a}_{{{\bf{r}}}_{l^{\prime} }}\right)\right]\\ & \equiv & {{ \mathcal H }}_{\mathrm{BL}}^{(0)}\end{array}\end{eqnarray} \tag{ 44 }$$

without any contribution from ${ \mathcal C }$ and ${ \mathcal D }$ to cause a net change in the magnetization (43). If we define an effective Hamiltonian of the bilinear version,

$$\begin{eqnarray}&&{\widetilde{{ \mathcal H }}}_{\mathrm{BL}}\equiv {{ \mathcal H }}_{\mathrm{BL}}+\mu \displaystyle \sum _{l=1}^{L}\left(S-{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}\right);{{ \mathcal H }}_{\mathrm{BL}}\equiv {{ \mathcal H }}^{(2)}+{{ \mathcal H }}^{(1)}+{{ \mathcal H }}_{\mathrm{BL}}^{(0)},\end{eqnarray} \tag{ 45 }$$

and demand that $\langle\langle { \mathcal A } \rangle\rangle$ and $\langle\langle { \mathcal B } \rangle\rangle$ be the self-consistent Hartree–Fock fields to diagonalize (45) subject to the constraint condition (35), we obtain Takahashi's MSW thermodynamics of a square-lattice antiferromagnet [78]. What will happen if we apply this Takahashi scheme and its some variations to noncollinear antiferromagnets?

Let us define the geometric functions

$$\begin{eqnarray}&&{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\equiv \displaystyle \frac{1}{z}\displaystyle \sum _{\kappa =1}^{z}{e}^{i{{\bf{k}}}_{\nu }\cdot {{\bf{\delta }}}_{l:\kappa }}=\displaystyle \frac{1}{z}\displaystyle \sum _{\kappa =1}^{z}{e}^{i{{\bf{k}}}_{\nu }\cdot {{\bf{\delta }}}_{p(n-1)+\sigma :\kappa }}\ (\sigma =1,\cdots ,p)\end{eqnarray} \tag{ 46 }$$

for the square (z = 4, p = 1), honeycomb (z = 3, p = 2), and triangular (z = 6, p = 1) lattices and

$$\begin{eqnarray}&&{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\equiv \displaystyle \sum _{\tau ,\tau ^{\prime} =1}^{p}\displaystyle \sum _{\sigma ^{\prime} =1}^{3}\displaystyle \frac{| {\epsilon }_{\tau \tau ^{\prime} \sigma ^{\prime} }| }{2}{u}_{{{\bf{k}}}_{\nu }:\tau \sigma }{u}_{{{\bf{k}}}_{\nu }:\tau ^{\prime} \sigma }\cos {{\bf{k}}}_{\nu }\cdot {{\bf{a}}}_{\sigma ^{\prime} -1}\ (\sigma =1,\cdots ,p)\end{eqnarray} \tag{ 47 }$$

with ${u}_{{{\bf{k}}}_{\nu }:\sigma ^{\prime} \sigma }$ satisfying the eigenvalue equations

$$\begin{eqnarray}\begin{array}{c}\left[\begin{array}{ccc}-2{\gamma }_{{{\bf{k}}}_{\nu }:\sigma } & \cos {{\bf{k}}}_{\nu }\cdot {{\bf{a}}}_{2} & \cos {{\bf{k}}}_{\nu }\cdot {{\bf{a}}}_{1}\\ \cos {{\bf{k}}}_{\nu }\cdot {{\bf{a}}}_{2} & -2{\gamma }_{{{\bf{k}}}_{\nu }:\sigma } & \cos {{\bf{k}}}_{\nu }\cdot {{\bf{a}}}_{0}\\ \cos {{\bf{k}}}_{\nu }\cdot {{\bf{a}}}_{1} & \cos {{\bf{k}}}_{\nu }\cdot {{\bf{a}}}_{0} & -2{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\end{array}\right]\,\left[\begin{array}{c}{u}_{{{\bf{k}}}_{\nu }:1\sigma }\\ {u}_{{{\bf{k}}}_{\nu }:2\sigma }\\ {u}_{{{\bf{k}}}_{\nu }:3\sigma }\end{array}\right]=0,\displaystyle \sum _{{\sigma }^{{\prime} }=1}^{3}{u}_{{{\bf{k}}}_{\nu }:{\sigma }^{{\prime} }\sigma }^{2}\equiv 1(\sigma =1,\cdots ,p)\end{array}\end{eqnarray} \tag{ 48 }$$

for the kagome (z = 4, p = 3) lattice. We number the three eigenmodes for the kagome lattice in ascending order,

$$\begin{eqnarray}\begin{array}{rcl}{\gamma }_{{{\bf{k}}}_{\nu }:1} & \equiv & \displaystyle \frac{1}{4}\left(1-\sqrt{8\displaystyle \prod _{\tau =1}^{3}\cos {{\bf{k}}}_{\nu }\cdot {{\bf{a}}}_{\tau -1}+1}\right),{\gamma }_{{{\bf{k}}}_{\nu }:2}\equiv -\displaystyle \frac{1}{2},\\ {\gamma }_{{{\bf{k}}}_{\nu }:3} & \equiv & \displaystyle \frac{1}{4}\left(1+\sqrt{8\displaystyle \prod _{\tau =1}^{3}\cos {{\bf{k}}}_{\nu }\cdot {{\bf{a}}}_{\tau -1}+1}\right).\end{array}\end{eqnarray} \tag{ 49 }$$

Intending to diagonalize the effective Hamiltonian (42), we define Fourier transforms of the Holstein-Primakoff boson operators as

$$\begin{eqnarray}&&{a}_{{{\bf{k}}}_{\nu }:\sigma }=\displaystyle \frac{1}{\sqrt{N}}\displaystyle \sum _{n=1}^{N}{e}^{-i{{\bf{k}}}_{\nu }\cdot {{\bf{r}}}_{p(n-1)+\sigma }}{a}_{{{\bf{r}}}_{p(n-1)+\sigma }}\ (\nu =1,\cdots ,N;\sigma =1,\cdots ,p),\end{eqnarray} \tag{ 50 }$$

where the wavevector k _ν runs over the full paramagnetic Brillouin zone, as is shown in figure 1. While their Bogoliubov transforms, i.e. the ideal MSW creation and annihilation operators, are defined according to their belonging lattice, we eventually obtain a diagonal Hamiltonian in the unified form

$$\begin{eqnarray}&&{\widetilde{{ \mathcal H }}}_{\mathrm{quad}}=\displaystyle \sum _{l=0}^{2}{E}^{(l)}+\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma =1}^{p}{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }^{\dagger }{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }+\mu {LS},\end{eqnarray} \tag{ 51 }$$

where E⁽²⁾ is the classical ground-state energy, E⁽¹⁾ and E⁽⁰⁾ are its O(S¹) quantum and O(S⁰) variational [106, 108] corrections, respectively, and ${\alpha }_{{{\bf{k}}}_{\nu }:\sigma }^{\dagger }$ creates a magnon of the σ species with wavevector k _ν at a cost of energy ${\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }$. Further details are given by each lattice. The square (z = 4, p = 1) and honeycomb (z = 3, p = 2) lattices have the expressions

$$\begin{eqnarray}\begin{array}{rcl}{E}^{(2)} & = & -\displaystyle \frac{z}{2}{{LJS}}^{2},{E}^{(1)}=-\displaystyle \frac{z}{2}L\left({JS}-\displaystyle \frac{\mu }{z}\right)+\displaystyle \frac{1}{2}\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma =1}^{p}{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma },\ \\ {E}^{(0)} & = & \displaystyle \frac{z}{2}{LJ}\left[{\left( \langle\langle { \mathcal A } \rangle\rangle - \langle\langle { \mathcal B } \rangle\rangle \right)}^{2}+ \langle\langle { \mathcal A } \rangle\rangle - \langle\langle { \mathcal B } \rangle\rangle \right],\end{array}\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&{\omega }_{{{\bf{k}}}_{\nu }:\sigma }\equiv \displaystyle \frac{{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }}{{zJ}\left(S- \langle\langle { \mathcal A } \rangle\rangle + \langle\langle { \mathcal B } \rangle\rangle \right)}\equiv \sqrt{{\left[1-\displaystyle \frac{\mu }{{zJ}\left(S- \langle\langle { \mathcal A } \rangle\rangle + \langle\langle { \mathcal B } \rangle\rangle \right)}\right]}^{2}-{\left|{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\right|}^{2}},\end{eqnarray} \tag{ 53 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }_{{{\bf{k}}}_{\nu }:\sigma } & = & {a}_{{{\bf{k}}}_{\nu }:p+1-\sigma }\cosh {\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }+{a}_{-{{\bf{k}}}_{\nu }:\sigma }^{\dagger }\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{| {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }| }\sinh {\vartheta }_{{{\bf{k}}}_{\nu }:\sigma },\ \\ {\alpha }_{-{{\bf{k}}}_{\nu }:\sigma }^{\dagger } & = & {a}_{{{\bf{k}}}_{\nu }:\sigma }\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{| {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }| }\sinh {\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }+{a}_{-{{\bf{k}}}_{\nu }:p+1-\sigma }^{\dagger }\cosh {\vartheta }_{{{\bf{k}}}_{\nu }:\sigma },\\ \cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma } & = & \displaystyle \frac{1}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left[1-\displaystyle \frac{\mu }{{zJ}\left(S- \langle\langle { \mathcal A } \rangle\rangle + \langle\langle { \mathcal B } \rangle\rangle \right)}\right],\sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }=\displaystyle \frac{| {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }| }{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\end{array}\end{eqnarray} \tag{ 54 }$$

in common. The triangular (z = 6, p = 1) and kagome (z = 4, p = 3) lattices have the expressions

$$\begin{eqnarray}\begin{array}{rcl}{E}^{(2)} & = & -\displaystyle \frac{z}{2}L\displaystyle \frac{J}{2}{S}^{2},{E}^{(1)}=-\displaystyle \frac{z}{2}L\left(\displaystyle \frac{J}{2}S-\displaystyle \frac{\mu }{z}\right)+\displaystyle \frac{1}{2}\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma =1}^{p}{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma },\\ {E}^{(0)} & = & \displaystyle \frac{z}{2}L\displaystyle \frac{J}{2}\left[{\left( \langle\langle { \mathcal A } \rangle\rangle -\displaystyle \frac{3 \langle\langle { \mathcal B } \rangle\rangle }{2}+\displaystyle \frac{ \langle\langle { \mathcal D } \rangle\rangle }{2}\right)}^{2}-\displaystyle \frac{3{\left( \langle\langle { \mathcal B } \rangle\rangle - \langle\langle { \mathcal D } \rangle\rangle \right)}^{2}}{4}\right.\\ & & \left.-\displaystyle \frac{ \langle\langle { \mathcal B } \rangle\rangle \left( \langle\langle { \mathcal B } \rangle\rangle - \langle\langle { \mathcal C } \rangle\rangle \right)}{2}+\displaystyle \frac{3 \langle\langle { \mathcal D } \rangle\rangle \left( \langle\langle { \mathcal D } \rangle\rangle - \langle\langle { \mathcal C } \rangle\rangle \right)}{2}+ \langle\langle { \mathcal A } \rangle\rangle -\displaystyle \frac{3 \langle\langle { \mathcal B } \rangle\rangle }{2}+\displaystyle \frac{ \langle\langle { \mathcal D } \rangle\rangle }{2}\right],\end{array}\end{eqnarray} \tag{ 55 }$$

$$\begin{eqnarray}&&{\omega }_{{{\bf{k}}}_{\nu }:\sigma }\equiv \displaystyle \frac{{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }}{{zJS}}\equiv \sqrt{{\left(\displaystyle \frac{1}{2}+\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+F+F^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }-\displaystyle \frac{\mu }{{zJS}}\right)}^{2}-{\left(\displaystyle \frac{3{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+G+G^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\right)}^{2}},\end{eqnarray} \tag{ 56 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }_{{{\bf{k}}}_{\nu }:\sigma } & = & \displaystyle \sum _{\tau =1}^{3}{u}_{{{\bf{k}}}_{\nu }:\tau \sigma }\left({a}_{{{\bf{k}}}_{\nu }:\tau }\cosh {\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }+{a}_{-{{\bf{k}}}_{\nu }:\tau }^{\dagger }\sinh {\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }\right),\\ {\alpha }_{-{{\bf{k}}}_{\nu }:\sigma }^{\dagger } & = & \displaystyle \sum _{\tau =1}^{3}{u}_{{{\bf{k}}}_{\nu }:\tau \sigma }\left({a}_{{{\bf{k}}}_{\nu }:\tau }\sinh {\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }+{a}_{-{{\bf{k}}}_{\nu }:\tau }^{\dagger }\cosh {\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }\right),\\ \cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma } & = & \displaystyle \frac{1}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+F+F^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }-\displaystyle \frac{\mu }{{zJS}}\right),\\ \sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma } & = & \displaystyle \frac{1}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{3{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+G+G^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\right)\end{array}\end{eqnarray} \tag{ 57 }$$

in common with the abbreviations

$$\begin{eqnarray}\begin{array}{rcl}F & \equiv & -\displaystyle \frac{ \langle\langle { \mathcal A } \rangle\rangle }{2S}+\displaystyle \frac{3 \langle\langle { \mathcal B } \rangle\rangle }{4S}-\displaystyle \frac{ \langle\langle { \mathcal D } \rangle\rangle }{4S},F^{\prime} \equiv -\displaystyle \frac{ \langle\langle { \mathcal A } \rangle\rangle }{4S}+\displaystyle \frac{3 \langle\langle { \mathcal C } \rangle\rangle }{8S}-\displaystyle \frac{ \langle\langle { \mathcal D } \rangle\rangle }{2S},\\ G & \equiv & \displaystyle \frac{ \langle\langle { \mathcal B } \rangle\rangle }{8S}-\displaystyle \frac{3 \langle\langle { \mathcal D } \rangle\rangle }{8S},G^{\prime} \equiv \displaystyle \frac{3 \langle\langle { \mathcal A } \rangle\rangle }{4S}-\displaystyle \frac{ \langle\langle { \mathcal B } \rangle\rangle }{2S}-\displaystyle \frac{ \langle\langle { \mathcal C } \rangle\rangle }{8S}\end{array}\end{eqnarray} \tag{ 58 }$$

and the understanding that ${u}_{{{\bf{k}}}_{\nu }:11}$ for the triangular lattice be unity.

Variational MISW—modified WDISW (MWDISW) and HFISW (MHFISW)—thermodynamics can be formulated in terms of the self-consistent fields $\langle\langle { \mathcal A } \rangle\rangle$ to $\langle\langle { \mathcal D } \rangle\rangle$, which depend on how the SWs are interacting. The MSW thermal distribution function reads

$$\begin{eqnarray}&&\langle {\alpha }_{{{\bf{k}}}_{\nu }:\sigma }^{\dagger }{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }{\rangle }_{T}=\displaystyle \frac{\mathrm{Tr}\left[{e}^{-{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }^{\dagger }{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }/{k}_{{\rm{B}}}T}{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }^{\dagger }{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }\right]}{\mathrm{Tr}\left[{e}^{-{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }^{\dagger }{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }/{k}_{{\rm{B}}}T}\right]}=\displaystyle \frac{1}{{e}^{{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }/{k}_{{\rm{B}}}T}-1}\equiv {\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }\end{eqnarray} \tag{ 59 }$$

with ${\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }$ containing part or all of $\langle\langle { \mathcal A } \rangle\rangle$ to $\langle\langle { \mathcal D } \rangle\rangle$. Every time we encounter the double angle brackets $\langle\langle { \mathcal A } \rangle\rangle$ to $\langle\langle { \mathcal D } \rangle\rangle$, we read them according to the scheme of the time,

$$\begin{eqnarray}&&\langle { \mathcal A }{\rangle }_{0}^{\prime} =\langle { \mathcal B }{\rangle }_{0}^{\prime} =0\qquad \qquad (\mathrm{MLSW});\end{eqnarray} \tag{ 60 }$$

$$\begin{eqnarray}\begin{array}{rcl}\langle { \mathcal A }{\rangle }_{0} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\displaystyle \frac{1}{2{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left[1-\displaystyle \frac{\mu }{{zJ}\left(S- \langle\langle { \mathcal A } \rangle\rangle + \langle\langle { \mathcal B } \rangle\rangle \right)}\right]-\displaystyle \frac{1}{2},\\ \langle { \mathcal B }{\rangle }_{0} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\displaystyle \frac{| {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }{| }^{2}}{2{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\qquad \qquad (\mathrm{MWDISW});\end{array}\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}\begin{array}{rcl}\langle { \mathcal A }{\rangle }_{T} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\displaystyle \frac{1}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left[1-\displaystyle \frac{\mu }{{zJ}\left(S- \langle\langle { \mathcal A } \rangle\rangle + \langle\langle { \mathcal B } \rangle\rangle \right)}\right]\left({\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }+\displaystyle \frac{1}{2}\right)-\displaystyle \frac{1}{2},\\ \langle { \mathcal B }{\rangle }_{T} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\displaystyle \frac{| {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }{| }^{2}}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left({\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }+\displaystyle \frac{1}{2}\right)\qquad \qquad (\mathrm{MHFISW})\end{array}\end{eqnarray} \tag{ 62 }$$

for the square (z = 4, p = 1) and honeycomb (z = 3, p = 2) lattices and

$$\begin{eqnarray}&&\langle { \mathcal A }{\rangle }_{0}^{\prime} =\langle { \mathcal B }{\rangle }_{0}^{\prime} =\langle { \mathcal C }{\rangle }_{0}^{\prime} =\langle { \mathcal D }{\rangle }_{0}^{\prime} =0\qquad \qquad (\mathrm{MLSW});\end{eqnarray} \tag{ 63 }$$

$$\begin{eqnarray}\begin{array}{rcl}\langle { \mathcal A }{\rangle }_{0} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\displaystyle \frac{1}{2{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+F+F^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }-\displaystyle \frac{\mu }{{zJS}}\right)-\displaystyle \frac{1}{2},\\ \langle { \mathcal B }{\rangle }_{0} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{2{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{3{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+G+G^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\right),\\ \langle { \mathcal C }{\rangle }_{0} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\displaystyle \frac{1}{2{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{3{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+G+G^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\right),\\ \langle { \mathcal D }{\rangle }_{0} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\left[\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{2{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+F+F^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }-\displaystyle \frac{\mu }{{zJS}}\right)-\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{2}\right]\qquad \qquad (\mathrm{MWDISW});\end{array}\end{eqnarray} \tag{ 64 }$$

$$\begin{eqnarray}\begin{array}{rcl}\langle { \mathcal A }{\rangle }_{T} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\displaystyle \frac{1}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+F+F^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }-\displaystyle \frac{\mu }{{zJS}}\right)\left({\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }+\displaystyle \frac{1}{2}\right)-\displaystyle \frac{1}{2},\\ \langle { \mathcal B }{\rangle }_{T} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{3{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+G+G^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\right)\left({\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }+\displaystyle \frac{1}{2}\right),\\ \langle { \mathcal C }{\rangle }_{T} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\displaystyle \frac{1}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{3{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+G+G^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\right)\left({\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }+\displaystyle \frac{1}{2}\right),\\ \langle { \mathcal D }{\rangle }_{T} & = & \displaystyle \frac{1}{N}\displaystyle \sum _{\nu =1}^{N}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma =1}^{p}\left[\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}+F+F^{\prime} {\gamma }_{{{\bf{k}}}_{\nu }:\sigma }-\displaystyle \frac{\mu }{{zJS}}\right)\right.\\ & & \left.\times \left({\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }+\displaystyle \frac{1}{2}\right)-\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{2}\right]\qquad \qquad (\mathrm{MHFISW})\end{array}\end{eqnarray} \tag{ 65 }$$

for the triangular (z = 6, p = 1) and kagome (z = 4, p = 3) lattices, where they each still contain $\langle\langle { \mathcal A } \rangle\rangle$ and $\langle\langle { \mathcal B } \rangle\rangle$ (collinear antiferromagnets) or $\langle\langle { \mathcal A } \rangle\rangle$ to $\langle\langle { \mathcal D } \rangle\rangle$ (noncollinear antiferromagnets) to be self-consistently determined. The constraint condition is then written as

$$\begin{eqnarray}&&\langle {{ \mathcal M }}^{\tilde{z}}{\rangle }_{T}=L\left(S-\langle { \mathcal A }{\rangle }_{T}\right)=0.\end{eqnarray} \tag{ 66 }$$

In the MHFISW scheme, we solve the simultaneous equations (62) plus (66) for $\langle { \mathcal A }{\rangle }_{T}$, $\langle { \mathcal B }{\rangle }_{T}$, and μ and (65) plus (66) for $\langle { \mathcal A }{\rangle }_{T}$ to $\langle { \mathcal D }{\rangle }_{T}$ and μ. In the MWDISW scheme, we solve the simultaneous equations (61) plus (66) for $\langle { \mathcal A }{\rangle }_{0}$, $\langle { \mathcal B }{\rangle }_{0}$, and μ and (64) plus (66) for $\langle { \mathcal A }{\rangle }_{0}$ to $\langle { \mathcal D }{\rangle }_{0}$ and μ. In the MLSW scheme, with (60) or (63) in mind, we have only to solve (66) for μ. We define the internal energy E as ${\sum }_{m=2}^{4}\langle {{ \mathcal H }}^{\left(\tfrac{m}{2}\right)}{\rangle }_{T}$ and ${\sum }_{m=0}^{4}\langle {{ \mathcal H }}^{\left(\tfrac{m}{2}\right)}{\rangle }_{T}$ in the MLSW and MISW schemes, respectively, where the thermal average of the quartic Hamiltonian is evaluated through the use of the Bloch-De Dominicis theorem [109]. The thermal average of ${{ \mathcal H }}^{\left(\tfrac{m}{2}\right)}$ with an odd m generally vanishes, while that of ${{ \mathcal H }}^{\left(\tfrac{m}{2}\right)}$ with an even m reads

$$\begin{eqnarray}\begin{array}{rcl}\langle {{ \mathcal H }}^{(2)}{\rangle }_{T} & = & -\displaystyle \frac{z}{2}{{LJS}}^{2},\langle {{ \mathcal H }}^{(1)}{\rangle }_{T}={zLJS}\left(\langle { \mathcal A }{\rangle }_{T}-\langle { \mathcal B }{\rangle }_{T}\right),\\ \langle {{ \mathcal H }}^{(0)}{\rangle }_{T} & = & -\displaystyle \frac{z}{2}{LJ}{\left(\langle { \mathcal A }{\rangle }_{T}-\langle { \mathcal B }{\rangle }_{T}\right)}^{2}\end{array}\end{eqnarray} \tag{ 67 }$$

for the square and honeycomb lattices and

$$\begin{eqnarray}\begin{array}{rcl}\langle {{ \mathcal H }}^{(2)}{\rangle }_{T} & = & -\displaystyle \frac{z}{2}L\displaystyle \frac{J}{2}{S}^{2},\langle {{ \mathcal H }}^{(1)}{\rangle }_{T}={zL}\displaystyle \frac{J}{2}S\left(\langle { \mathcal A }{\rangle }_{T}-\displaystyle \frac{3\langle { \mathcal B }{\rangle }_{T}}{2}+\displaystyle \frac{\langle { \mathcal D }{\rangle }_{T}}{2}\right),\\ \langle {{ \mathcal H }}^{(0)}{\rangle }_{T} & = & -\displaystyle \frac{z}{2}L\displaystyle \frac{J}{2}\left[{\left(\langle { \mathcal A }{\rangle }_{T}-\displaystyle \frac{3\langle { \mathcal B }{\rangle }_{T}}{2}+\displaystyle \frac{\langle { \mathcal D }{\rangle }_{T}}{2}\right)}^{2}-\displaystyle \frac{3{\left(\langle { \mathcal B }{\rangle }_{T}-\langle { \mathcal D }{\rangle }_{T}\right)}^{2}}{4}\right.\\ & & \left.-\displaystyle \frac{ \langle\langle { \mathcal B } \rangle\rangle \left( \langle\langle { \mathcal B } \rangle\rangle - \langle\langle { \mathcal C } \rangle\rangle \right)}{2}+\displaystyle \frac{3 \langle\langle { \mathcal D } \rangle\rangle \left( \langle\langle { \mathcal D } \rangle\rangle - \langle\langle { \mathcal C } \rangle\rangle \right)}{2}\right]\end{array}\end{eqnarray} \tag{ 68 }$$

for the triangular and kagome lattices.

We show in figure 2 the thus-obtained MSW findings for the specific heat C ≡ ∂E/∂T, together with those obtained by Auerbach-Arovas' Schwinger-boson mean-field theory [105, 110–112], in comparison with modern numerical approaches capable of touching bulk properties. Without any frustration, we can calculate the internal energy for sufficiently large planes by a quantum Monte Carlo method. What Bernu and Misguich [93, 96] call the entropy method is a stable specific-heat interpolation scheme between low and high temperatures intending to improve the convergence of the high-temperature series expansion with the help of two sum rules on the energy and entropy. This technique allows us to compute accurately the specific heat in the thermodynamic limit possibly down to absolute zero, which is never the case with a direct Padé analysis of the series. Various state-of-the-art renormalization group techniques based on tensor-network states [113] are also a potentially powerful tool applicable to thermodynamics of frustrated spin models without suffering from a negative-sign problem. Linearized thermal tensor renormalization, whether on the conventional Trotter-Suzuki-discretized linear quasicontinuous grid in inverse-temperature β ≡ 1/k_B T [114], on some interleaved β grids [115], or in a Trotter-error-free manner based on the numerically exact Taylor series expansion of the whole density operator ${e}^{-\beta { \mathcal H }}$ [116], indeed reveals the thermal quantities of quantum magnets on the linear-chain, square, honeycomb, and triangular lattices of finite length and/or width with pronounced precision and high efficiency. However, elimination of the boundary effects is tricky in two dimensions [117] and any linearization of the kagome lattice in this context seems to be difficult. Some algorithms address an infinite system directly without dealing with boundary effects or finite-size corrections [118]. By mapping a two-dimensional quantum lattice model into a three-dimensional closed tensor network and contracting the three-dimensional brick-wall tensor network with the imaginary time length corresponding to temperature in question [119], finite-temperature properties of even an infinite two-dimensional kagome antiferromagnet can be calculated [92]. However, the full update—contraction of the whole tensor network via the infinite time-evolving block decimation algorithm [120]—is awfully time-consuming and has to be reduced to a cluster update [92, 121] in most cases, through the use of the Bethe approximation [92, 121, 122] for instance. Selective discard of some or all of the loops in the original lattice and inevitable truncation of the bond dimensions in contracting tensors may yield significant errors near a critical point [119] and/or at low temperatures [117].

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With all these in mind, let us observe the traditional, here in the sense of constraining the staggered magnetization to be zero, SC-MSW thermodynamics of collinear antiferromagnets, figures 2(a) and (b), first. While the lowest-order calculations, MLSW findings, succeed in designing antiferromagnetic peaks of C, they are far from precise at low temperatures [82]. The low-temperature quantitativity is significantly improved by taking account of the SW interaction ${{ \mathcal H }}^{(0)}$. The MHFISW findings are highly precise at sufficiently low temperatures [82], while they completely fail to reproduce the overall temperature dependences. The worst of them is an artificial phase transition of the first order to the trivial paramagnetic solution at a certain finite temperature. The specific heat jumps down to zero when k_B T/J reaches 0.9108 and 0.9568 for the square and honeycomb lattices, respectively, where the bond order parameter $\langle { \mathcal B }{\rangle }_{T}$ vanishes, satisfying

$$\begin{eqnarray}&&\left(1-\displaystyle \frac{\mu }{{zJ}\langle { \mathcal B }{\rangle }_{T}}\right)\langle { \mathcal B }{\rangle }_{T}=\displaystyle \frac{{k}_{{\rm{B}}}T}{{zJ}}\mathrm{ln}\left(1+\displaystyle \frac{1}{S}\right).\end{eqnarray} \tag{ 69 }$$

The SB MF formulation [110–112] coincides with the MHFISW thermodynamics [78, 82, 103, 105] except for an overall numerical factor in each thermal quantity. By correcting what Arovas and Auerbach call 'the overcounting of the number of independent boson degrees of freedom' [111, 112], the two approaches yield exactly the same specific heat as a function of temperature. The two schemes are no longer degenerate with each other in noncollinear antiferromagnets, as is demonstrated in figures 2(c) and (d).

In order to retain the MHFISW precise low-temperature findings by all means and connect them naturally with the correct high-temperature asymptotics, we bring SWs into interaction in a different manner from the Hartree–Fock approximation. A new treatment of the O(S⁰) quartic Hamiltonian ${{ \mathcal H }}^{(0)}$ consists of applying the Wick theorem [123] based on the magnon operators ${\alpha }_{{{\bf{k}}}_{\nu }:\sigma }^{\dagger }$ and ${\alpha }_{{{\bf{k}}}_{\nu }:\sigma }$ to it and neglecting the residual normal-ordered interaction $:{{ \mathcal H }}^{(0)}:$. Then we have the bilinear Hamiltonian (44) with $\langle\langle { \mathcal A } \rangle\rangle$ and $\langle\langle { \mathcal B } \rangle\rangle$ read as the SW ground-state expectation values $\langle { \mathcal A }{\rangle }_{0}$ and $\langle { \mathcal B }{\rangle }_{0}$ given by equation (61) for the collinear antiferromagnets and the quadratic Hamiltonian (41) with $\langle\langle { \mathcal A } \rangle\rangle$ to $\langle\langle { \mathcal D } \rangle\rangle$ read as $\langle { \mathcal A }{\rangle }_{0}$ to $\langle { \mathcal D }{\rangle }_{0}$ given by equation (64). We continue to observe figures 2(a) and (b). Unlike MHFISWs, MWDISWs are free from thermal breakdown and succeed in reproducing the Schottky-like peak of C in a pretty good manner, becoming degenerate with MHFISWs at sufficiently low temperatures and giving correct high-temperature asymptotics much better than MLSWs. The MWDISW thermodynamics is precise at both low and high temperatures and free from any thermal breakdown.

Next we observe the SC-MSW thermodynamics of noncollinear antiferromagnets, figures 2(c) and (d). While the numerical findings to compare are not necessarily conclusive at low temperatures, the present MSW findings are all far from consistent with them on the whole. In both cases, the MWDISW temperature profiles are neither reminiscent of the major broad maximum consequent from the exchange coupling constant J nor any reference for the high-temperature asymptotics. Especially for the kagome-lattice antiferromagnet, MWDISWs completely fail to guess a low-temperature shoulder or additional peak below the main maximum. As far as we bring SWs into interaction variationally, any scattering involving an odd number of magnons does not play any role in designing thermodynamics. ${{ \mathcal H }}^{\left(\tfrac{m}{2}\right)}$, or Hamiltonians on the order of S to a fractional power in general, must play a key role in creating thermal features peculiar to frustrated noncollinear antiferromagnets, as will be demonstrated later, and therefore, we consider an alternative approach to SW interactions intending to make them effective in addition.

The first requirement originates from the traditional constraint condition (35) to keep the number of Holstein-Primakoff bosons constant but gives due consideration to general noncollinear antiferromagnets as well,

$$\begin{eqnarray}&&S-\langle { \mathcal A }{\rangle }_{T}=S-\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}{\rangle }_{T}=\delta ,\end{eqnarray} \tag{ 71 }$$

where δ is a temperature-independent nonnegative parameter to regulate the number of Holstein-Primakoff bosons and therefore quasiparticle antiferromagnons. With vanishing δ, the generalized constraint condition (71) reduces to the original one (35) designed for collinear antiferromagnets. With this in mind, we require δ to minimize the ground-state energy. When δ moves away from 0 to S, the ground-state energy, whether within or beyond the harmonic approximation, monotonically increases in any collinear antiferromagnet but takes one and only minimum in any noncollinear antiferromagnet. Therefore, δ reasonably remains zero in collinear antiferromagnets, while tuning δ to an optimal value further stabilizes SWs in noncollinear antiferromagnets. Such determined δ dramatically improves the MSW description of the uniform magnetic susceptibility as well as specific heat.

Before detailing the specific-heat calculations, let us take a look at how hard it is for SWs to describe magnetic susceptibilities of noncollinear antiferromagnets especially in lower than three dimensions and how effectively our DC modification scheme works on them in this context. In case the total magnetization in question ${{ \mathcal M }}^{\lambda }\equiv {\sum }_{l=1}^{L}{S}_{{{\bf{r}}}_{l}}^{\lambda }$ is not commutable with the Hamiltonian ${\widetilde{{ \mathcal H }}}_{\mathrm{harm}}$, we define the canonical correlation function [128] for general operators ${ \mathcal P }$ and ${ \mathcal Q }$ as

$$\begin{eqnarray}&&\langle { \mathcal P };{ \mathcal Q }{\rangle }_{\beta }\equiv \displaystyle \frac{1}{\beta {\hslash }}{\int }_{0}^{\beta {\hslash }}\displaystyle \frac{\mathrm{Tr}[{e}^{-\beta {\widetilde{{ \mathcal H }}}_{\mathrm{harm}}}{ \mathcal P }(\tau ){ \mathcal Q }]}{\mathrm{Tr}[{e}^{-\beta {\widetilde{{ \mathcal H }}}_{\mathrm{harm}}}]}d\tau ,\end{eqnarray} \tag{ 72 }$$

where we denote (Euclidean) time evolution by ${ \mathcal P }(\tau )\equiv {e}^{\tau {\widetilde{{ \mathcal H }}}_{\mathrm{harm}}/{\hslash }}{ \mathcal P }{e}^{-\tau {\widetilde{{ \mathcal H }}}_{\mathrm{harm}}/{\hslash }}$. While any canonical correlation is symmetric, $\langle { \mathcal P };{ \mathcal Q }{\rangle }_{\beta }=\langle { \mathcal Q };{ \mathcal P }{\rangle }_{\beta }$, it is not necessarily the case with a simple canonical-ensemble average,

$$\begin{eqnarray}&&\langle { \mathcal P }{ \mathcal Q }{\rangle }_{1/\beta {k}_{{\rm{B}}}}\equiv \displaystyle \frac{\mathrm{Tr}[{e}^{-\beta {\widetilde{{ \mathcal H }}}_{\mathrm{harm}}}{ \mathcal P }{ \mathcal Q }]}{\mathrm{Tr}[{e}^{-\beta {\widetilde{{ \mathcal H }}}_{\mathrm{harm}}}]}.\end{eqnarray} \tag{ 73 }$$

We further introduce the spin fluctuation operator $\delta {S}_{{{\bf{r}}}_{l}}^{\lambda }\equiv {S}_{{{\bf{r}}}_{l}}^{\lambda }-\langle {S}_{{{\bf{r}}}_{l}}^{\lambda }{\rangle }_{T}$ and define its Fourier transform as

$$\begin{eqnarray}&&\delta {S}_{{{\bf{k}}}_{\nu }:\sigma }^{\lambda }=\displaystyle \frac{1}{\sqrt{N}}\displaystyle \sum _{n=1}^{N}{e}^{-i{{\bf{k}}}_{\nu }\cdot {{\bf{r}}}_{p(n-1)+\sigma }}\delta {S}_{{{\bf{r}}}_{p(n-1)+\sigma }}^{\lambda },\end{eqnarray} \tag{ 74 }$$

just like (50). Calculating canonical correlations between the zero-wave-vector Fourier components of spin fluctuations with β set equal to the inverse temperature 1/k_B T gives the temperature-T uniform magnetic susceptibility

$$\begin{eqnarray}&&{\chi }^{\lambda \lambda }=\displaystyle \frac{{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}{{k}_{{\rm{B}}}T}\displaystyle \frac{L}{p}\displaystyle \sum _{\sigma ,\sigma ^{\prime} =1}^{p}\langle \delta {S}_{{\bf{0}}:\sigma }^{\lambda \,\dagger };\delta {S}_{{\bf{0}}:\sigma ^{\prime} }^{\lambda }{\rangle }_{1/{k}_{{\rm{B}}}T}.\end{eqnarray} \tag{ 75 }$$

When ${\sum }_{\sigma =1}^{p}[{S}_{{{\bf{k}}}_{\nu }:\sigma }^{\lambda },{\widetilde{{ \mathcal H }}}_{\mathrm{harm}}]=0$, the relevant canonical correlation reduces to a static structure factor,

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{p}\displaystyle \sum _{\sigma ,\sigma ^{\prime} =1}^{p}\langle \delta {S}_{{{\bf{k}}}_{\nu }:\sigma }^{\lambda \,\dagger };\delta {S}_{{{\bf{k}}}_{\nu }:\sigma ^{\prime} }^{\lambda }{\rangle }_{1/{k}_{{\rm{B}}}T}\\ \quad =\displaystyle \frac{1}{L}\displaystyle \sum _{n,n^{\prime} =1}^{N}\displaystyle \sum _{\sigma ,\sigma ^{\prime} =1}^{p}{e}^{i{{\bf{k}}}_{\nu }\cdot \left({{\bf{r}}}_{p(n-1)+\sigma }-{{\bf{r}}}_{p(n^{\prime} -1)+\sigma ^{\prime} }\right)}\langle \delta {S}_{{{\bf{r}}}_{p(n-1)+\sigma }}^{\lambda }\delta {S}_{{{\bf{r}}}_{p(n^{\prime} -1)+\sigma ^{\prime} }}^{\lambda }{\rangle }_{T}\equiv {S}^{\lambda \lambda }({{\bf{k}}}_{\nu }),\end{array}\end{eqnarray} \tag{ 76 }$$

and therefore, the λ λ diagonal element of the uniform susceptibility tensor times temperature becomes the mean-square fluctuation of the uniform magnetization along the λ axis,

$$\begin{eqnarray}&&\displaystyle \frac{{k}_{{\rm{B}}}T}{{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}{\chi }^{\lambda \lambda }=\langle {\left({{ \mathcal M }}^{\lambda }\right)}^{2}{\rangle }_{T}-\langle {{ \mathcal M }}^{\lambda }{\rangle }_{T}^{2}={{LS}}^{\lambda \lambda }({\bf{0}}).\end{eqnarray} \tag{ 77 }$$

This is fundamentally the case with the SU(2) Heisenberg model, but its SW Hamiltonian is no longer invariant under SU(2) rotations. For the square- and honeycomb-lattice antiferromagnets, ${\widetilde{{ \mathcal H }}}_{\mathrm{harm}}$ (36) commutes with ${{ \mathcal M }}^{z}$ and (75) reads in the bosonic language

$$\begin{eqnarray}&&\displaystyle \frac{{\chi }^{{zz}}}{{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}=\displaystyle \frac{1}{{k}_{{\rm{B}}}T}\displaystyle \sum _{\rho =1}^{p}\displaystyle \sum _{\nu =1}^{N}{\bar{n}}_{{{\bf{k}}}_{\nu }:\rho }\left({\bar{n}}_{{{\bf{k}}}_{\nu }:\rho }+1\right),\end{eqnarray} \tag{ 78 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\chi }^{{xx}}}{{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}=\displaystyle \frac{{\chi }^{{yy}}}{{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}=\displaystyle \frac{N}{{k}_{{\rm{B}}}T}\sum _{\rho =1}^{p}\left(S-\langle { \mathcal A }{\rangle }_{T}\right)\left(\cosh 2{\vartheta }_{{\bf{0}}:\rho }-\sinh 2{\vartheta }_{{\bf{0}}:\rho }\right)\left({\bar{n}}_{{\bf{0}}:\rho }+\displaystyle \frac{1}{2}\right).\end{eqnarray} \tag{ 79 }$$

Note that (78) satisfies (77). The SC-MWDISW thermodynamics of the square-lattice antiferromagnet is highly successful [82] for the susceptibility as well as the specific heat, giving precise low-temperature analytics, avoiding the artificial discontinuous transition to the paramagnetic phase at any finite temperature, and reproducing the correct high-temperature asymptotics. For the triangular- and kagome-lattice antiferromagnets, ${\widetilde{{ \mathcal H }}}_{\mathrm{harm}}$ [cf (90)] no longer commutes with any of ${{ \mathcal M }}^{x}$, ${{ \mathcal M }}^{y}$, and ${{ \mathcal M }}^{z}$, and (75) reads in the bosonic language

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{\chi }^{zz}}{{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}=\displaystyle \frac{{\chi }^{xx}}{{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}=\sum _{\rho ,{\rho }^{{\prime} }=1}^{p}\sum _{\kappa =\pm }\displaystyle \frac{{e}^{i\kappa ({\bf{Q}}+{\bf{q}})\cdot ({{\bf{a}}}_{\rho -1}-{{\bf{a}}}_{{\rho }^{{\prime} }-1})}}{16}\hspace{60mm}\\ \,\times \displaystyle \sum _{\sigma ,{\sigma }^{{\prime} }=1}^{p}\displaystyle \sum _{\nu =1}^{N}{u}_{{{\bf{k}}}_{\nu }:\rho \sigma }{u}_{{{\bf{k}}}_{\nu }:{\rho }^{{\prime} }\sigma }{u}_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:\rho {\sigma }^{{\prime} }}{u}_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\rho }^{{\prime} }{\sigma }^{{\prime} }}\left\{{\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }({\bar{n}}_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}+1)\displaystyle \frac{{e}^{\beta ({\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }-{\epsilon }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }})}-1}{{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }-{\epsilon }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}}\right.\\ \,\times \left[(\cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }+1)(\cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}+1)+\sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }\sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}\right]\hspace{35mm}\\ \,-({\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }+1){\bar{n}}_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}\displaystyle \frac{{e}^{-\beta ({\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }-{\epsilon }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }})}-1}{{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }-{\epsilon }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}}\hspace{75mm}\\ \,\times \left[(\cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }-1)(\cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}-1)+\sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }\sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}\right]\hspace{35mm}\\ \,+{\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }{\bar{n}}_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}\displaystyle \frac{{e}^{\beta ({\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }+{\epsilon }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }})}-1}{{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }+{\epsilon }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}}\hspace{88mm}\\ \,\times \left[(\cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }+1)(\cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}-1)+\sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }\sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}\right]\hspace{35mm}\\ \,-({\bar{n}}_{{{\bf{k}}}_{\nu }:\sigma }+1)({\bar{n}}_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}+1)\displaystyle \frac{{e}^{-\beta ({\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }+{\epsilon }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }})}-1}{{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }+{\epsilon }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}}\hspace{65mm}\\ \,\times \left.\left[(\cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }-1)(\cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}+1)+\sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }\sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }+\kappa {\bf{Q}}:{\sigma }^{{\prime} }}\right]\Space{0ex}{3.47ex}{0ex}\right\}\hspace{35mm}\\ \,+N\displaystyle \sum _{\rho ,{\rho }^{{\prime} }=1}^{p}\displaystyle \sum _{\kappa =\pm }\displaystyle \frac{{e}^{i\kappa ({\bf{Q}}+{\bf{q}})\cdot ({{\bf{a}}}_{\rho -1}-{{\bf{a}}}_{{\rho }^{{\prime} }-1})}}{4}\sum _{\sigma =1}^{p}{u}_{\kappa {\bf{Q}}:\rho \sigma }{u}_{\kappa {\bf{Q}}:{\rho }^{{\prime} }\sigma }\left(\displaystyle \frac{S}{2}-\displaystyle \frac{\langle { \mathcal A }{\rangle }_{T}}{2}-\displaystyle \frac{\langle { \mathcal C }{\rangle }_{T}}{4}\right)\hspace{30mm}\\ \,\times (\cosh 2{\vartheta }_{\kappa {\bf{Q}}:\sigma }+\sinh 2{\vartheta }_{\kappa {\bf{Q}}:\sigma })\left[{\bar{n}}_{\kappa {\bf{Q}}:\sigma }\displaystyle \frac{{e}^{\beta {\epsilon }_{\kappa {\bf{Q}}:\sigma }}-1}{{\epsilon }_{\kappa {\bf{Q}}:\sigma }}-({\bar{n}}_{\kappa {\bf{Q}}:\sigma }+1)\displaystyle \frac{{e}^{-\beta {\epsilon }_{\kappa {\bf{Q}}:\sigma }}-1}{{\epsilon }_{\kappa {\bf{Q}}:\sigma }}\right]\,\hspace{25mm}\end{array}\end{eqnarray} \tag{ 80 }$$

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{\chi }^{{yy}}}{{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}=N\sum _{\rho ,{\rho }^{{\prime} }=1}^{p}\sum _{\sigma =1}^{p}{u}_{{\bf{0}}:\rho \sigma }{u}_{{\bf{0}}:{\rho }^{{\prime} }\sigma }\left(\displaystyle \frac{S}{2}-\displaystyle \frac{\langle { \mathcal A }{\rangle }_{T}}{2}+\displaystyle \frac{\langle { \mathcal C }{\rangle }_{T}}{4}\right)(\cosh 2{\vartheta }_{{\bf{0}}:\sigma }-\sinh 2{\vartheta }_{{\bf{0}}:\sigma })\\ \times \left[{\bar{n}}_{{\bf{0}}:\sigma }\displaystyle \frac{{e}^{\beta {\epsilon }_{{\bf{0}}:\sigma }}-1}{{\epsilon }_{{\bf{0}}:\sigma }}-\left({\bar{n}}_{{\bf{0}}:\sigma }+1\right)\displaystyle \frac{{e}^{-\beta {\epsilon }_{{\bf{0}}:\sigma }}-1}{{\epsilon }_{{\bf{0}}:\sigma }}\right].\,\,\,\,\,\,\,\end{array}\end{eqnarray} \tag{ 81 }$$

We evaluate (80) and (81) modifying CLSWs under various constraint conditions and compare them with numerical linked-cluster and tensor-network-based renormalization-group calculations in figure 3. Considering that SC MLSWs never fail to qualitatively reproduce the overall temperature profile of the uniform magnetic susceptibility of any collinear antiferromagnet [82], their findings for the triangular- and kagome-lattice antiferromagnets are far from successful especially at low temperatures. They misread the per-site static uniform susceptibility as divergent and quasidivergent with temperature going down to absolute zero in the triangular- and kagome-lattice antiferromagnets, respectively. The complete divergence of the zero-temperature per-site susceptibility of the triangular-lattice antiferromagnet is because of SC MLSWs softening at k = Q and artificially tuned SC MLSWs or more elaborate DC MLSWs solve this difficulty with their Q modes hardening (cf equation (80) and figures 7(b)–(d)). Introducing the boson number tuning parameter δ into the first constraint (71) dramatically improves the MSW thermodynamics and imposing the second constraint (cf (89) in the following) in addition on such MSWs gives a still better description of the susceptibility at low temperatures. In the high-temperature limit, on the other hand, SWs spontaneously meet the second requirement (89) inducing the relevant chemical potential η to vanish. Therefore, the superiority of DC MLSWs over SC MLSWs at high temperatures owes much to tuning of the first constraint (71). In collinear antiferromagnets, SC MLSWs yield the correct high-temperature asymptotics [82]

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{\chi }{L{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}\equiv \mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{{\sum }_{\lambda =x,y,z}{\chi }^{\lambda \lambda }}{3L{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}=\displaystyle \frac{S(S+1)}{3{k}_{{\rm{B}}}T},\end{eqnarray} \tag{ 82 }$$

while in noncollinear antiferromagnets, SC MSWs constrained to (71), whether with or without δ, no longer hit the exact sum rule in general but say that

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{\chi }{L{\left(g{\mu }_{{\rm{B}}}\right)}^{2}}=\displaystyle \frac{S(S+1)}{3{k}_{{\rm{B}}}T}\left[\displaystyle \frac{(S-\delta )(S-\delta +1)}{S(S+1)}+\displaystyle \frac{1}{(S+1)\mathrm{ln}\left(1+\tfrac{1}{S-\delta }\right)}\right].\end{eqnarray} \tag{ 83 }$$

In the case of $S=\tfrac{1}{2}$, SC MLSWs estimate ${\mathrm{lim}}_{T\to \infty }\chi {k}_{{\rm{B}}}T/L{\left(g{\mu }_{{\rm{B}}}\right)}^{2}$ at 0.4017, compared to the exact value 1/4. However, tuning δ optimally under the DC condition (cf table 1) yields much better estimates, 0.1963 and 0.2769 for the triangular and kagome-lattice antiferromagnets, respectively. We are thus convinced of the necessity of adjusting the traditional constraint condition (35) to noncollinear antiferromagnets and the validity of imposing the extended constraint condition (71) on their Holstein-Primakoff bosons so as to stabilize their ground states.

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We can have another good understanding of equation (71) in a different aspect. Let us recall the Schwinger boson representation

$$\begin{eqnarray}\begin{array}{rcl}{S}_{{{\bf{r}}}_{l}}^{z} & = & \displaystyle \frac{{a}_{{{\bf{r}}}_{l}:\uparrow }^{\dagger }{a}_{{{\bf{r}}}_{l}:\uparrow }-{a}_{{{\bf{r}}}_{l}:\downarrow }^{\dagger }{a}_{{{\bf{r}}}_{l}:\downarrow }}{2}\equiv \displaystyle \frac{{n}_{{{\bf{r}}}_{l}:\uparrow }-{n}_{{{\bf{r}}}_{l}:\downarrow }}{2},{S}_{{{\bf{r}}}_{l}}^{x}-{{iS}}_{{{\bf{r}}}_{l}}^{y}\equiv {S}_{{{\bf{r}}}_{l}}^{+\dagger }\equiv {S}_{{{\bf{r}}}_{l}}^{-}={a}_{{{\bf{r}}}_{l}:\downarrow }^{\dagger }{a}_{{{\bf{r}}}_{l}:\uparrow };\\ {{\bf{S}}}_{{{\bf{r}}}_{l}}^{2} & = & \displaystyle \frac{{n}_{{{\bf{r}}}_{l}:\uparrow }+{n}_{{{\bf{r}}}_{l}:\downarrow }}{2}\left(\displaystyle \frac{{n}_{{{\bf{r}}}_{l}:\uparrow }+{n}_{{{\bf{r}}}_{l}:\downarrow }}{2}+1\right),\end{array}\end{eqnarray} \tag{ 84 }$$

where physical states demand that the bosons should satisfy the constraint condition

$$\begin{eqnarray}&&\displaystyle \sum _{\sigma =\uparrow ,\downarrow }{a}_{{{\bf{r}}}_{l}:\sigma }^{\dagger }{a}_{{{\bf{r}}}_{l}:\sigma }=2S\end{eqnarray} \tag{ 85 }$$

at each lattice site. In what are called Schwinger-boson mean-field theories [110–112, 129–132], the L requirements (85) are relaxed and imposed only on average, i.e. merely a single Lagrange multiplier μ is introduced and the term $\mu {\sum }_{l=1}^{L}\left(2S-{\sum }_{\sigma =\uparrow ,\downarrow }{a}_{{{\bf{r}}}_{l}:\sigma }^{\dagger }{a}_{{{\bf{r}}}_{l}:\sigma }\right)$ is added to the Hamiltonian. The determination condition for μ,

$$\begin{eqnarray}&&\displaystyle \frac{1}{L}\displaystyle \sum _{l=1}^{L}\displaystyle \sum _{\sigma =\uparrow ,\downarrow }\langle {a}_{{{\bf{r}}}_{l}:\sigma }^{\dagger }{a}_{{{\bf{r}}}_{l}:\sigma }{\rangle }_{T}=2S,\end{eqnarray} \tag{ 86 }$$

is reminiscent of the MSW constraint condition (35). In the Holstein-Primakoff representation, each spin is described by a single Bose oscillator together with the nonholonomic constraint $0\leqslant {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}\leqslant 2S$, whereas in the Schwinger representation, each spin is replaced by two bosons together with the holonomic constraint (85). Then, there may be a Schwinger-representation analog of the generalized MSW constraint condition (71). The idea of the number of bosons being tunable is practiced indeed in an extended mean-field Schwinger boson framework developed by Messio et al [36, 37, 59, 60] intending to explore chiral spin liquids [37, 59, 60] in the kagome-lattice antiferromagnet. The Schwinger-boson mean-field solution with its bond order parameters being chosen as real for the $S=\tfrac{1}{2}$ regular kagome-lattice Heisenberg antiferromagnet is a long-range Néel-ordered phase of the $\sqrt{3}\times \sqrt{3}$ type with a gapless spinon spectrum. However, if we allow the left-hand side of (86) to deviate from twice the real spin quantum number, a SL phase characterized by a fully gapped spinon spectrum stabilizes instead [36, 37, 59] on the way of its value going away from unity down to zero, the extreme quantum limit. The analogy between the Holstein-Primakoff and Schwinger representations strongly motivates us to tune δ in the generalized MSW constraint (71).

The second requirement is related to spin rotational symmetry. There is a crucial difference, for all similarities as have been noted above and observed in figure 2, between the Schwinger and Holstein-Primakoff bosons. The Schwinger-boson mean-field formalism retains the SU(2) invariance of the original Hamiltonian (1), whereas every kind of SW theory, whether modified or not, reduces this to U(1) or less. The Schwinger-boson representation of the SU(2) spin variables, (84) constrained to (85), indeed gives an isotropic mean-field expectation value of each spin component,

$$\begin{eqnarray}&&\langle {S}_{{{\bf{r}}}_{l}}^{x}{S}_{{{\bf{r}}}_{l}}^{x}{\rangle }_{T}=\langle {S}_{{{\bf{r}}}_{l}}^{y}{S}_{{{\bf{r}}}_{l}}^{y}{\rangle }_{T}=\langle {S}_{{{\bf{r}}}_{l}}^{z}{S}_{{{\bf{r}}}_{l}}^{z}{\rangle }_{T}=\displaystyle \frac{S(S+1)}{2},\end{eqnarray} \tag{ 87 }$$

but misreads the spin magnitude as 3/2 times as large as the correct value, $\langle {{\bf{S}}}_{{{\bf{r}}}_{l}}^{2}{\rangle }_{T}=3S(S+1)/2$ [110, 111]. The Holstein-Primakoff-boson representation in the local spin reference frame (4) gives the correct spin magnitude but breaks the original spin rotational symmetry,

$$\begin{eqnarray}\begin{array}{rcl}\langle {S}_{{{\bf{r}}}_{l}}^{\tilde{x}}{S}_{{{\bf{r}}}_{l}}^{\tilde{x}}{\rangle }_{T} & = & \left(S-\langle { \mathcal A }{\rangle }_{T}\right)\left(\displaystyle \frac{1}{2}+\langle { \mathcal A }{\rangle }_{T}\right)-\displaystyle \frac{\langle { \mathcal C }{\rangle }_{T}^{2}}{2}+\langle { \mathcal C }{\rangle }_{T}\left(S-\displaystyle \frac{1}{4}-\displaystyle \frac{3\langle { \mathcal A }{\rangle }_{T}}{2}\right)+O({S}^{-1}),\\ \langle {S}_{{{\bf{r}}}_{l}}^{\tilde{y}}{S}_{{{\bf{r}}}_{l}}^{\tilde{y}}{\rangle }_{T} & = & \left(S-\langle { \mathcal A }{\rangle }_{T}\right)\left(\displaystyle \frac{1}{2}+\langle { \mathcal A }{\rangle }_{T}\right)-\displaystyle \frac{\langle { \mathcal C }{\rangle }_{T}^{2}}{2}-\langle { \mathcal C }{\rangle }_{T}\left(S-\displaystyle \frac{1}{4}-\displaystyle \frac{3\langle { \mathcal A }{\rangle }_{T}}{2}\right)+O({S}^{-1}),\\ \langle {S}_{{{\bf{r}}}_{l}}^{\tilde{z}}{S}_{{{\bf{r}}}_{l}}^{\tilde{z}}{\rangle }_{T} & = & {S}^{2}+(1-2S)\langle { \mathcal A }{\rangle }_{T}+2\langle { \mathcal A }{\rangle }_{T}^{2}+\langle { \mathcal C }{\rangle }_{T}^{2};\ \displaystyle \sum _{\lambda =x,y,z}\langle {S}_{{{\bf{r}}}_{l}}^{\tilde{\lambda }}{S}_{{{\bf{r}}}_{l}}^{\tilde{\lambda }}{\rangle }_{T}=S(S+1).\end{array}\end{eqnarray} \tag{ 88 }$$

For collinear antiferromagnets, $\langle { \mathcal C }{\rangle }_{T}$ spontaneously vanishes to retain a U(1) symmetry of each local spin operator in the rotating frame as well as the global U(1) symmetry related to the conservation of ${{ \mathcal M }}^{z}$ (43). For noncollinear antiferromagnets, the bosonic Hamiltonian ${{ \mathcal H }}_{\mathrm{harm}}$ no longer commutes with ${{ \mathcal M }}^{z}$, yet we should expect each spin operator to possess rotational symmetry about its local quantization axis $\tilde{z}$, which is met by demanding that $\langle {S}_{{{\bf{r}}}_{l}}^{\tilde{x}}{S}_{{{\bf{r}}}_{l}}^{\tilde{x}}{\rangle }_{T}=\langle {S}_{{{\bf{r}}}_{l}}^{\tilde{y}}{S}_{{{\bf{r}}}_{l}}^{\tilde{y}}{\rangle }_{T}$, i.e.

$$\begin{eqnarray}&&\langle { \mathcal C }{\rangle }_{T}=\displaystyle \frac{\langle {a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}^{\dagger }+{a}_{{{\bf{r}}}_{l}}{a}_{{{\bf{r}}}_{l}}{\rangle }_{T}}{2}=0.\end{eqnarray} \tag{ 89 }$$

Thus and thus, we introduce 2L Lagrange multipliers and diagonalize the effective harmonic Hamiltonian

$$\begin{eqnarray}&&{\widetilde{{ \mathcal H }}}_{\mathrm{harm}}\equiv {{ \mathcal H }}_{\mathrm{harm}}+\displaystyle \sum _{l=1}^{L}{\mu }_{l}\left(S-\delta -{a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}\right)+\displaystyle \sum _{l=1}^{L}\displaystyle \frac{{\eta }_{l}}{2}\left({a}_{{{\bf{r}}}_{l}}^{\dagger }{a}_{{{\bf{r}}}_{l}}^{\dagger }+{a}_{{{\bf{r}}}_{l}}{a}_{{{\bf{r}}}_{l}}\right).\end{eqnarray} \tag{ 90 }$$

subject to (71) and (89) at each site. Since the 2L Lagrange multipliers degenerate into merely two in practice as μ₁ = ⋯ = μ_L ≡ μ and η₁ = ⋯ = η_L ≡ η, (90) becomes

$$\begin{eqnarray}&&{\widetilde{{ \mathcal H }}}_{\mathrm{harm}}=\displaystyle \sum _{l=1}^{2}{E}^{(l)}+\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma =1}^{p}{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }^{\dagger }{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }+\mu L(S-\delta )\end{eqnarray} \tag{ 91 }$$

with E^(l) and ${\alpha }_{{{\bf{k}}}_{\nu }:\sigma }^{\dagger }$ having the same meanings as those in (51). For DC MLSWs in the triangular- and kagome-lattice antiferromagnets, the creation energy (56) and the Bogoliubov transformation (57) are rewritten to

$$\begin{eqnarray}&&{\omega }_{{{\bf{k}}}_{\nu }:\sigma }\equiv \displaystyle \frac{{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }}{{zJS}}\equiv \sqrt{{\left(\displaystyle \frac{1}{2}+\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}-\displaystyle \frac{\mu }{{zJS}}\right)}^{2}-{\left(\displaystyle \frac{3{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}-\displaystyle \frac{\eta }{{zJS}}\right)}^{2}},\end{eqnarray} \tag{ 92 }$$

$$\begin{eqnarray}&&\cosh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }=\displaystyle \frac{1}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}-\displaystyle \frac{\mu }{{zJS}}\right),\,\sinh 2{\vartheta }_{{{\bf{k}}}_{\nu }:\sigma }=\displaystyle \frac{1}{{\omega }_{{{\bf{k}}}_{\nu }:\sigma }}\left(\displaystyle \frac{3{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }}{4}-\displaystyle \frac{\eta }{{zJS}}\right).\end{eqnarray} \tag{ 93 }$$

For DC MLSWs in polyhedral-lattice antiferromagnets, we set N and p equal to 1 and L, respectively, omitting their momentum indices as ${\epsilon }_{{{\bf{k}}}_{\nu }:\sigma }\equiv {\epsilon }_{\sigma }$ and

$$\begin{eqnarray}&&{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }\equiv {\alpha }_{\sigma }=\displaystyle \sum _{l=1}^{L}\left({f}_{\sigma l}{a}_{{{\bf{r}}}_{l}}+{g}_{\sigma l}^{* }{a}_{{{\bf{r}}}_{l}}^{\dagger }\right),\end{eqnarray} \tag{ 94 }$$

and make Bogoliubov bosons out of bare Holstein-Primakoff bosons by numerically obtaining the coefficients f_{σ l} and g_{σ l} . Since spins are not necessarily coplanar in the classical ground states of polyhedral-lattice antiferromagnets, we generalize the ground-state energy expressions into three dimensions,

$$\begin{eqnarray}\begin{array}{rcl}{E}^{(1)} & = & {JS}\displaystyle \sum _{\langle l,l^{\prime} \rangle }\left[\sin {\phi }_{{{\bf{r}}}_{l}}\sin {\phi }_{{{\bf{r}}}_{l^{\prime} }}\cos \left({\theta }_{{{\bf{r}}}_{l}}-{\theta }_{{{\bf{r}}}_{l^{\prime} }}\right)+\cos {\phi }_{{{\bf{r}}}_{l}}\cos {\phi }_{{{\bf{r}}}_{l^{\prime} }}\right]+\displaystyle \frac{L}{2}\mu +\displaystyle \frac{1}{2}\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma =1}^{p}{\epsilon }_{{{\bf{k}}}_{\nu }:\sigma },\\ {E}^{(2)} & = & {{JS}}^{2}\displaystyle \sum _{\langle l,l^{\prime} \rangle }\left[\sin {\phi }_{{{\bf{r}}}_{l}}\sin {\phi }_{{{\bf{r}}}_{l^{\prime} }}\cos \left({\theta }_{{{\bf{r}}}_{l}}-{\theta }_{{{\bf{r}}}_{l^{\prime} }}\right)+\cos {\phi }_{{{\bf{r}}}_{l}}\cos {\phi }_{{{\bf{r}}}_{l^{\prime} }}\right].\end{array}\end{eqnarray} \tag{ 95 }$$

In order to investigate effects of the interactions ${ \mathcal V }$ on temperature profiles of the specific heat, we calculate their perturbative corrections to the DC-MLSW free energy. The lth-order correction at temperature T ≡ 1/β k_B is calculated as

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{l}=-\displaystyle \frac{{\left(-1\right)}^{l}}{\beta l!{{\hslash }}^{l}}\int \cdots {\int }_{0}^{\beta {\hslash }}d{\tau }_{1}\cdots d{\tau }_{l}\langle { \mathcal T }\left[{ \mathcal V }({\tau }_{1})\cdots { \mathcal V }({\tau }_{l})\right]{\rangle }_{T},\end{eqnarray} \tag{ 96 }$$

where we denote (Euclidean) time ordering operation by ${ \mathcal T }$. In the context of newly constructed perturbatively corrected (PC)-DC-MSW theory, every thermal-bracket notation 〈 〉_T denotes the temperature-T thermal average with respect to MLSWs (cf (73)) unless otherwise noted. ΔF_l (l = 1, 2, ⋯ ) each make their own contribution on the order of S^m , which we shall denote by ${\rm{\Delta }}{F}_{l}^{(m)}$. When we truncate any correction ΔF_l at the order of S⁻¹, all the remaining corrections are given by

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{1}^{(0)}={\int }_{0}^{\beta {\hslash }}\displaystyle \frac{d{\tau }_{1}}{\beta {\hslash }}{\left\langle {{ \mathcal H }}^{(0)}({\tau }_{1})\right\rangle }_{T},\end{eqnarray} \tag{ 97 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{1}^{(-1)}={\int }_{0}^{\beta {\hslash }}\displaystyle \frac{d{\tau }_{1}}{\beta {\hslash }}{\left\langle {{ \mathcal H }}^{(-1)}({\tau }_{1})\right\rangle }_{T},\end{eqnarray} \tag{ 98 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{2}^{(0)}=-\int {\int }_{0}^{\beta {\hslash }}\displaystyle \frac{d{\tau }_{1}d{\tau }_{2}}{2\beta {{\hslash }}^{2}}{\left\langle { \mathcal T }\left[{{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}({\tau }_{1}){{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}({\tau }_{2})\right]\right\rangle }_{T},\end{eqnarray} \tag{ 99 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{2}^{(-1)}=-\int {\int }_{0}^{\beta {\hslash }}\displaystyle \frac{d{\tau }_{1}d{\tau }_{2}}{2\beta {{\hslash }}^{2}}\left\{{\left\langle { \mathcal T }\left[{{ \mathcal H }}^{(0)}({\tau }_{1}){{ \mathcal H }}^{(0)}({\tau }_{2})\right]\right\rangle }_{T}+{\left\langle { \mathcal T }\left[{{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}({\tau }_{1}){{ \mathcal H }}^{\left(-\tfrac{1}{2}\right)}({\tau }_{2})\right]\right\rangle }_{T}\right\},\end{eqnarray} \tag{ 100 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{3}^{(-1)}=\int \int {\int }_{0}^{\beta {\hslash }}\displaystyle \frac{d{\tau }_{1}d{\tau }_{2}d{\tau }_{3}}{6\beta {{\hslash }}^{3}}{\left\langle { \mathcal T }\left[{{ \mathcal H }}^{(0)}({\tau }_{1}){{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}({\tau }_{2}){{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}({\tau }_{3})\right]\right\rangle }_{T}.\end{eqnarray} \tag{ 101 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{4}^{(-1)}=-\int \int \int {\int }_{0}^{\beta {\hslash }}\displaystyle \frac{d{\tau }_{1}d{\tau }_{2}d{\tau }_{3}d{\tau }_{4}}{24\beta {{\hslash }}^{4}}{\left\langle { \mathcal T }\left[{{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}({\tau }_{1}){{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}({\tau }_{2}){{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}({\tau }_{3}){{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}({\tau }_{4})\right]\right\rangle }_{T}.\end{eqnarray} \tag{ 102 }$$

The O(S⁰) corrections ${\rm{\Delta }}{F}_{1}^{(0)}$ and ${\rm{\Delta }}{F}_{2}^{(0)}$ are both strictly calculated, whereas the O(S⁻¹) corrections are approximately evaluated in an empirical manner. It is interesting to observe temperature profiles of the specific heat for each order of S. We define the up-to-O(S¹) internal energy as

$$\begin{eqnarray}&&E={E}_{\mathrm{harm}}\equiv \langle {{ \mathcal H }}_{\mathrm{harm}}{\rangle }_{T},\end{eqnarray} \tag{ 103 }$$

the up-to-O(S⁰) internal energy as

$$\begin{eqnarray}&&E={E}_{\mathrm{harm}}+\displaystyle \frac{\partial \beta {\rm{\Delta }}{F}_{1}^{(0)}}{\partial \beta }+\displaystyle \frac{\partial \beta {\rm{\Delta }}{F}_{2}^{(0)}}{\partial \beta },\end{eqnarray} \tag{ 104 }$$

and the up-to-O(S⁻¹) internal energy as

$$\begin{eqnarray}&&E\simeq {E}_{\mathrm{harm}}+\displaystyle \frac{\partial \beta {\rm{\Delta }}{F}_{1}^{(0)}}{\partial \beta }+\displaystyle \frac{\partial \beta {\rm{\Delta }}{F}_{2}^{(0)}}{\partial \beta }+\displaystyle \frac{\partial \beta {\rm{\Delta }}{F}_{2}^{(-1)}}{\partial \beta }+\displaystyle \frac{\partial \beta {\rm{\Delta }}{F}_{3}^{(-1)}}{\partial \beta }+\displaystyle \frac{\partial \beta {\rm{\Delta }}{F}_{4}^{(-1)}}{\partial \beta }.\end{eqnarray} \tag{ 105 }$$

In terms of the unperturbed temperature Green functions

$$\begin{eqnarray}\begin{array}{rcl}{G}_{0}({{\bf{k}}}_{\nu }:\sigma \sigma ^{\prime} ;\tau ) & \equiv & -{\left\langle { \mathcal T }\left[{\alpha }_{{{\bf{k}}}_{\nu }:\sigma }(\tau ){\alpha }_{{{\bf{k}}}_{\nu }:\sigma ^{\prime} }^{\dagger }(0)\right]\right\rangle }_{T},\\ {G}_{0}({{\bf{k}}}_{\nu }:\sigma \sigma ^{\prime} ;i{\omega }_{n}) & \equiv & {\int }_{0}^{\beta {\hslash }}d\tau \,{e}^{i{\omega }_{n}\tau }{G}_{0}({{\bf{k}}}_{\nu }:\sigma \sigma ^{\prime} ;\tau )\end{array}\end{eqnarray} \tag{ 106 }$$

with bosonic Matsubara frequencies ω_n ≡ 2n π/β ℏ, the leading corrections on the order of S⁰, ${\rm{\Delta }}{F}_{1}^{(0)}$ and ${\rm{\Delta }}{F}_{2}^{(0)}$, can be written as

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{1}^{(0)}\equiv {E}^{(0)}+\displaystyle \frac{1}{\beta {\hslash }}\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma ,\sigma ^{\prime} =1}^{p}\displaystyle \sum _{n=-\infty }^{\infty }{{\rm{\Sigma }}}_{1}^{(0)}({{\bf{k}}}_{\nu }:\sigma \sigma ^{\prime} ){G}_{0}({{\bf{k}}}_{\nu }:\sigma ^{\prime} \sigma ;i{\omega }_{n}),\end{eqnarray} \tag{ 107 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{2}^{(0)}\equiv \displaystyle \frac{1}{\beta {\hslash }}\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma ,\sigma ^{\prime} =1}^{p}\displaystyle \sum _{n=-\infty }^{\infty }{{\rm{\Sigma }}}_{2}^{(0)}({{\bf{k}}}_{\nu }:\sigma \sigma ^{\prime} ;i{\omega }_{n}){G}_{0}({{\bf{k}}}_{\nu }:\sigma ^{\prime} \sigma ;i{\omega }_{n}),\end{eqnarray} \tag{ 108 }$$

where E⁽⁰⁾ is the O(S⁰) perturbative correction to the DC-MLSW ground-state energy E⁽²⁾+E⁽¹⁾ (91),

$$\begin{eqnarray}\begin{array}{rcl}{E}^{\left(0\right)} & = & -J\displaystyle \sum _{\left\langle l,{l}^{{\prime} }\right\rangle }\left\{\left[\left\langle { \mathcal A }{\rangle }_{0}^{2}+\langle { \mathcal B }{\rangle }_{0}^{2}+\langle { \mathcal D }{\rangle }_{0}^{2}+\langle { \mathcal A }{\rangle }_{0}\right]\left[\sin {\phi }_{{{\bf{r}}}_{l}}\sin {\phi }_{{{\bf{r}}}_{{l}^{{\prime} }}}\cos \left({\theta }_{{{\bf{r}}}_{l}}-{\theta }_{{{\bf{r}}}_{{l}^{{\prime} }}}\right)\right.\right.\right.\\ & & \left.+\cos {\phi }_{{{\bf{r}}}_{l}}\cos {\phi }_{{{\bf{r}}}_{{l}^{{\prime} }}}\right]-\left[\left\langle { \mathcal A }{\rangle }_{0}\langle { \mathcal B }{\rangle }_{0}+\displaystyle \frac{\left\langle { \mathcal D }{\rangle }_{0}\langle { \mathcal C }{\rangle }_{0}+\langle { \mathcal B }{\rangle }_{0}\right.}{2}\right]\left[\left(\cos {\phi }_{{{\bf{r}}}_{l}}\cos {\phi }_{{{\bf{r}}}_{{l}^{{\prime} }}}-1\right)\cos \left({\theta }_{{{\bf{r}}}_{l}}-{\theta }_{{{\bf{r}}}_{{l}^{{\prime} }}}\right)\right.\right.\\ & & \left.+\sin {\phi }_{{{\bf{r}}}_{l}}\sin {\phi }_{{{\bf{r}}}_{{l}^{{\prime} }}}\right]-\left[\left\langle { \mathcal A }{\rangle }_{0}\langle { \mathcal D }{\rangle }_{0}+\displaystyle \frac{\left\langle { \mathcal B }{\rangle }_{0}\langle { \mathcal C }{\rangle }_{0}+\langle { \mathcal D }{\rangle }_{0}\right.}{2}\right]\left[\left(\cos {\phi }_{{{\bf{r}}}_{l}}\cos {\phi }_{{{\bf{r}}}_{{l}^{{\prime} }}}+1\right)\cos \left({\theta }_{{{\bf{r}}}_{l}}-{\theta }_{{{\bf{r}}}_{{l}^{{\prime} }}}\right)\right.\right.\\ & & \left.+\left.\sin {\phi }_{{{\bf{r}}}_{l}}\sin {\phi }_{{{\bf{r}}}_{{l}^{{\prime} }}}\right]\right\}\end{array}\end{eqnarray} \tag{ 109 }$$

with 〈 〉₀ denoting the quantum average in the DC-MLSW vacuum. We refer to ${{\rm{\Sigma }}}_{1}^{(0)}({{\bf{k}}}_{\nu }:\sigma \sigma ^{\prime} )$ [figure 4(a) and equation (B.11)] and ${{\rm{\Sigma }}}_{2}^{(0)}({{\bf{k}}}_{\nu }:\sigma \sigma ^{\prime} ;i{\omega }_{n})$ (figure 4(b) and equation (B.13)) as the primary self-energies [133] and pick up only their iterations in any higher-order corrections (cf appendix B as well),

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{F}_{2}^{(-1)} & \simeq & \displaystyle \frac{1}{\beta {\hslash }}\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma ,\cdots ,\sigma \unicode{x02057}=1}^{p}\displaystyle \sum _{n=-\infty }^{\infty }{{\rm{\Sigma }}}_{1}^{(0)}({{\bf{k}}}_{\nu }:\sigma \sigma ^{\prime} ){G}_{0}({{\bf{k}}}_{\nu }:\sigma ^{\prime} \sigma ^{\prime\prime} ;i{\omega }_{n}){{\rm{\Sigma }}}_{1}^{(0)}({{\bf{k}}}_{\nu }:\sigma ^{\prime\prime} \sigma \prime\prime\prime )\\ & & \times {G}_{0}({{\bf{k}}}_{\nu }:\sigma \prime\prime\prime \sigma \unicode{x02057};i{\omega }_{n}),\end{array}\end{eqnarray} \tag{ 110 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{F}_{3}^{(-1)} & \simeq & \displaystyle \frac{1}{\beta {\hslash }}\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma ,\cdots ,\sigma \unicode{x02057}=1}^{p}\displaystyle \sum _{n=-\infty }^{\infty }{{\rm{\Sigma }}}_{1}^{(0)}({{\bf{k}}}_{\nu }:\sigma \sigma ^{\prime} ){G}_{0}({{\bf{k}}}_{\nu }:\sigma ^{\prime} \sigma ^{\prime\prime} ;i{\omega }_{n}){{\rm{\Sigma }}}_{2}^{(0)}({{\bf{k}}}_{\nu }:\sigma ^{\prime\prime} \sigma \prime\prime\prime ;i{\omega }_{n})\\ & & \times {G}_{0}({{\bf{k}}}_{\nu }:\sigma \prime\prime\prime \sigma \unicode{x02057};i{\omega }_{n}),\end{array}\end{eqnarray} \tag{ 111 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{F}_{4}^{(-1)} & \simeq & \displaystyle \frac{1}{\beta {\hslash }}\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma ,\cdots ,\sigma \unicode{x02057}=1}^{p}\displaystyle \sum _{n=-\infty }^{\infty }{{\rm{\Sigma }}}_{2}^{(0)}({{\bf{k}}}_{\nu }:\sigma \sigma ^{\prime} ;i{\omega }_{n}){G}_{0}({{\bf{k}}}_{\nu }:\sigma ^{\prime} \sigma ^{\prime\prime} ;i{\omega }_{n}){{\rm{\Sigma }}}_{2}^{(0)}({{\bf{k}}}_{\nu }:\sigma ^{\prime\prime} \sigma \prime\prime\prime ;i{\omega }_{n})\\ & & \times {G}_{0}({{\bf{k}}}_{\nu }:\sigma \prime\prime\prime \sigma \unicode{x02057};i{\omega }_{n}).\end{array}\end{eqnarray} \tag{ 112 }$$

Such diagrams are emergent in ${\rm{\Delta }}{F}_{2}^{(-1)}$, ${\rm{\Delta }}{F}_{3}^{(-1)}$, and ${\rm{\Delta }}{F}_{4}^{(-1)}$ but absent in ${\rm{\Delta }}{F}_{1}^{(-1)}$.

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3.3.1. Triangular-based polyhedral-lattice antiferromagnets

Intending to verify to what extent our PC-DC-MSW scheme can depict the thermal features of Heisenberg antiferromagnets in the triangular-based geometry, we first investigate various polyhedral-lattice antiferromagnets whose thermal features can precisely be calculated by the finite-temperature Lanczos method [86, 134]. In figure 5, we show the DC-MSW findings in comparison with these numerically exact solutions. As for their classical ground states from which Holstein-Primakoff bosons emerge, the spins are not coplanar with a relative angle of 116.6° between nearest neighbors in the icosahedral antiferromagnet [135], whereas all spins are (assumed to be) coplanar with a relative angle of 120° between nearest neighbors in the rest three each [135, 136]. Similar to the kagome-lattice antiferromagnet, the cuboctahedral antiferromagnet has noncoplanar singlet ground states degenerate in energy with the coplanar one [136]. These ground states of the four polyhedral-lattice antiferromagnets are schematically shown in figure 5. As for the boson number tuning parameter δ, we calculate the up-to-O(S^m ) (m = 1, 0, − 1) DC-MSW ground-state energies E∣_T=0 ((103), (104), (105)) as functions of δ given by hand and find the minimum point of each curve. The thus-obtained optimal values of δ are listed in table 1.

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The octahedron and icosahedron consist of edge-sharing triangles, while the cuboctahedron and icosidodecahedron can be viewed as corner-sharing triangles. The former and latter antiferromagnetic spin spirals yield such temperature profiles of the specific heat as to bear remarkable resemblance to those of antiferromagnetic small clusters of periodic triangular [29, 31, 137] and kagome [76, 88, 97] lattices (cf figure 6), marked by a quite sharp low-temperature maximum followed by a long gentle down slope from intermediate to high temperatures [138] and a main round maximum at intermediate temperatures accompanied by an additional low-temperature modest peak [139–141], respectively. The PC-DC-MSW findings for the former reproduce these temperature profiles quite well, while those for the latter are indeed quantitatively less precise but intriguingly suggestive of a distinct double-peak temperature profile. These successful findings are not the case with DC MLSWs. Even DC MLSWs may be better than SC MLSWs [142] at reproducing the overall temperature profiles of thermal quantities, but they neither depict the low-temperature steep peak of the antiferromagnetic specific heat characteristic of the edge-sharing triangular geometry nor hint at the double-peck structure of the antiferromagnetic specific heat peculiar to the corner-sharing triangular geometry. It is not until the fractional-power interactions ${{ \mathcal H }}^{\left(\tfrac{m}{2}\right)}$ become effective that DC MSWs give a fairly good description of the thermal features in various triangular-based geometries.

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3.3.2. Triangular-based planar-lattice antiferromagnets

Now we proceed to the two-dimensional triangular- and kagome-lattice antiferromagnets in the thermodynamic limit of our main interest. In figure 6, we present the PC-DC-MSW findings, whose optimal values of δ are also listed in table 1, in comparison with numerical findings obtained by the high-temperature-series-expansion-based entropy method and tensor-network-based renormalization-group approach. Small clusters of periodic triangular lattice exhibit a quite sharp low-temperature maximum in their temperature profiles of the specific heat in common, while those of periodic kagome lattice exhibit a prominent low-temperature peak besides the main round maximum in common. The entropy method claims that the former and latter features do not and do survive in the thermodynamic limit, respectively, whereas the tensor-network approach claims that the former and latter features do and do not survive in the thermodynamic limit, respectively. The thermodynamic-limit properties of frustrated noncollinear antiferromagnets are thus hard to evaluate even by such sophisticated methods. Under such circumstances, our elaborate PC-DC-MSW theory predicts that the former and latter features both survive in the thermodynamic limit. What a precious message from SWs! This is their first clear report on thermodynamics of frustrated noncollinear quantum magnets. PC DC MSWs succeed in reproducing such distinctive specific-heat curves of triangular-based edge-sharing Platonic and corner-sharing Archimedean polyhedral-lattice antiferromagnets as well. Note that MLSWs are ignorant of any individual fine structure of the specific heat. Not only the elaborate modification scheme but also quantum corrections, especially those caused by the O(S⁰) primary self-energies, are key ingredients in this context, as will be discussed in more detail in section 4.

It is generally very hard to provide a precise thermodynamics of two- or higher-dimensional noncollinear antiferromagnets over the whole temperature range of absolute zero to infinity. Under such circumstances, our PC DC MISWs can give not only a qualitative description but also semiquantitative information of thermal features as follows. The exact high-temperature asymptotics of the specific heat is given by

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{C}{{k}_{{\rm{B}}}}=\displaystyle \frac{z}{6}{\left[\displaystyle \frac{{JS}(S+1)}{{k}_{{\rm{B}}}T}\right]}^{2}\equiv A{\left(\displaystyle \frac{J}{{k}_{{\rm{B}}}T}\right)}^{2}.\end{eqnarray} \tag{ 113 }$$

DC MLSWs say that

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{C}{{k}_{{\rm{B}}}}=\displaystyle \frac{S-\delta +\tfrac{1}{2}}{\mathrm{ln}\left(1+\tfrac{1}{S-\delta }\right)}{\left(\displaystyle \frac{{JS}}{{k}_{{\rm{B}}}T}\right)}^{2}\displaystyle \sum _{\nu =1}^{N}\displaystyle \sum _{\sigma =1}^{p}{\left(\displaystyle \frac{3}{4}z{\gamma }_{{{\bf{k}}}_{\nu }:\sigma }\right)}^{2}\end{eqnarray} \tag{ 114 }$$

and the coefficient A quantitatively ameliorates with perturbative corrections, as is demonstrated in table 2. While PC DC MISWs overestimate and underestimate the specific heat at low and intermediate temperatures, respectively, they satisfy fairly well a sum rule on the entropy,

$$\begin{eqnarray}&&{\int }_{0}^{\infty }\displaystyle \frac{C}{{{Lk}}_{{\rm{B}}}T}{dT}={\int }_{T=0}^{T=\infty }\displaystyle \frac{d{ \mathcal S }}{{{Lk}}_{{\rm{B}}}}=\mathrm{ln}(2S+1),\end{eqnarray} \tag{ 115 }$$

as is demonstrated in table 3. Considering that a direct Padé analysis of high-temperature expansion series of the specific heat generally fails to satisfy this entropy sum rule [93, 96, 141], it is even surprising that our MSWs remain quantitative to this extent at high temperatures. Although they are not necessarily successful in precisely reproducing a thermal quantity at each temperature, yet they never fail to predict the overall thermal features. This is because the DC-MLSW dispersion relations to construct the PC-DC-MSW thermodynamics are equipped with the key ingredients in the low-lying energy spectra of the triangular- and kagome-lattice antiferromagnets, which we shall further explain in the next closing section.

Table 2. DC-MSW estimates of the high-temperature asymptotic specific heat coefficient $A\equiv {\mathrm{lim}}_{T\to \infty }{\left({k}_{{\rm{B}}}T/J\right)}^{2}C/{k}_{{\rm{B}}}$ (cf (113) and (114)) in comparison with the exact values $z{\left[S(S+1)\right]}^{2}/6$ for the $S=\tfrac{1}{2}$ Hamiltonian (1) on the L → ∞ triangular and kagome lattices.

DC MSW up to	O(S1)	O(S0)	O(S−1)	exact
Triangular	0.3674	0.6877	0.6751	0.5625
Kagome	0.3350	0.4134	0.4104	0.3750

Table 3. DC-MSW estimates of the entropy sum rule (115) for the $S=\tfrac{1}{2}$ Hamiltonian (1) on the L → ∞ triangular and kagome lattices, compared to the exact value $\mathrm{ln}2$.

DC MSW up to	O(S1)	O(S0)	O(S−1)
Triangular	$\ \mathrm{ln}1.8230\$	$\ \mathrm{ln}1.7463\$	$\ \mathrm{ln}1.8267\$
Kagome	$\ \mathrm{ln}2.1495\$	$\ \mathrm{ln}2.1030\$	$\ \mathrm{ln}2.1430\$

The low-energy Lanczos spectrum of the $S=\tfrac{1}{2}$ antiferromagnetic Heisenberg model for a Kagome cluster of L = 48 consists of a huge number of singlet states below the lowest-lying triplet state [74]. With further increasing size L, the lowest triplet remains separated from the ground state by a gap [54, 55], whereas the singlets in between presumably develop into a gapless continuum adjacent to the ground state [73, 74, 76]. While the singlet-singlet gap strongly depends on which boundary condition is adopted, toric or cylindric, the fact remains unchanged that it is always smaller than the singlet-triplet gap [54]. With all these in mind, let us observe MSW eigenspectra under various constraint conditions. We show in figure 7 the DC-MLSW dispersion relations and the consequent up-to-O(S⁻¹) PC-DC-MSW specific-heat curves, together with the CLSW and SC-MLSW dispersion relations, for the triangular- and kagome-lattice antiferromagnets, where the DC-MLSW specific heat ∂E_harm/∂T and the perturbative corrections ${\partial }^{2}\beta {\rm{\Delta }}{F}_{l}^{(m)}/\partial T\partial \beta$ to that each are also shown separately. CLSWs of the kagome-lattice Heisenberg antiferromagnets [figure 7(e)] are well known to have a dispersionless zero-energy mode [33, 124], resulting that we can neither calculate any kind of magnetic susceptibility nor evaluate any further perturbative correction to the harmonic energy E_harm even at absolute zero [102]. Modifying these CLSWs under the traditional SC condition (35) pushes up their whole eigenspectrum, lifting the degeneracy between the two dispersive bands [figure 7(f)]. Further modifying them under the DC condition, (71) plus(89), puts one of the dispersive modes back into its original gapless appearance, keeping the flat band apart from the ground state [figure 7(h)]. Note that any artificially tuned SC condition, (71) without (89) but with a nonzero δ, brings no qualitative change into figure 7(f), i.e. induces no eigenstate to emerge below the flat band [figure 7(g)]. The second constraint (89) is necessary to recover a linear Goldstone mode looking up at the floating flat band.

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Such an elaborate DC-MLSW spectrum bears some resemblance to the low-energy Lanczos spectrum in that the dispersionless DC-MLSW mode may correspond to the low-lying triplet eigenstates separated from the ground state by a gap and the linear DC-MLSW mode slipping under the flat band can serve as the singlet eigenstates filling the singlet-triplet spin gap. Although neither the total spin nor the total magnetization is a good quantum number for the DC-MLSW Hamiltonian (90), yet its dispersionless and thus localized excitation mode possibly has close relevance to the singlet-triplet spin gap. The flat band of CLSWs in the kagome-lattice antiferromagnet consists of excitations localized within an arbitrary hexagon of nearest-neighbor spins [33], and so is that of MLSWs, no matter which modification scheme is imposed, SC or DC, and though the excitation energy is no longer zero. Interestingly enough, the VBC with a 36-site unit cell [45], which is one of the most promising candidate ground states for the kagome-lattice antiferromagnetic Heisenberg model, has low-lying triplet excitations localized inside a hexagon of nearest-neighbor spins. It actually has 18 triplet modes in the reduced Brillouin zone and many of them are dispersionless [47, 143]. In a bond operator mean-field theory [143], the second-lowest flat mode, which lies slightly higher than the lowest flat mode defining the spin gap, is completely localized within either of the two 'perfect hexagons' in the 36-site unit cell and therefore doubly degenerate. Series expansions around the decoupled-dimer limit [47] reveal doubly degenerate flat modes of the same nature lying lowest in the spin-triplet channel. They are indeed dispersionless in second-order perturbation theory [47], being completely localized inside a perfect hexagon, but turn dispersive to the extent of about one-hundredth of J to third order in the expansion [48], weakly hopping from one perfect hexagon to the next. Our DC-MLSW flat mode (figure 8(c2)) also becomes weakly dispersive to the same extent [figure 8(d2)] when it is renormalized with the primary self-energies (B.13) [(B.11) have no effect in this context, to be precise]. The primary self-energies (B.11) push the gapped two modes upward [figures 8(d2), 8(d3);(B.13) have no effect in this context, to be precise], retaining the Goldstone mode [figure 8(d1)], so that a prominent low-temperature peak can emerge out of the main round maximum in the temperature profile of the kagome-lattice-antiferromagnet specific heat. It is likely that the four-boson scattering ${{ \mathcal H }}^{(0)}$ is responsible for locating the major broad maximum, while the three-boson scattering ${{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}$ designs the additional low-temperature modest peak. For further details of spectral-function calculations, see appendix B.

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Once again, neither the well-pronounced low-temperature peak nor the mid-temperature broad maximum is obtainable within any SC-MSW thermodynamics. Just like the singlet states filling the spin gap in the Lanczos spectrum, the sufficient density of antiferromagnon eigenstates lying below the flat band properly apart from the ground state enables DC MSWs to reproduce the bimodal temperature profile of the kagome-lattice-antiferromagnet specific heat. It is not the case at all with DC MLSWs [MSW up to O(S¹) in figure 6(b)] but is indeed the case with up-to-O(S⁰) or higher DC MSWs [MSW up to O(S⁰) and O(S⁻¹) in figure 6(b)]. The bimodal temperature profile owes much to the second-order perturbative correction ${\rm{\Delta }}{F}_{2}^{(0)}$ originating from the three-boson scattering ${{ \mathcal H }}^{\left(\tfrac{1}{2}\right)}$ peculiar to noncollinear antiferromagnets. The marked ups and downs of ${\partial }^{2}\beta {\rm{\Delta }}{F}_{2}^{(0)}/\partial T\partial \beta$ [Figure 7(${\rm{h}}^{\prime}$)] are indeed thanks to the flat-band [figure 8(c2)] contribution $\sigma =\sigma ^{\prime} =2$ in (108). We further find that $\partial \beta {\rm{\Delta }}{F}_{1}^{(0)}/\partial \beta \gt 0$, while $\partial \beta {\rm{\Delta }}{F}_{2}^{(0)}/\partial \beta \lt 0$, i.e. the former and latter read repulsive and attractive interactions, respectively. Considering that at sufficiently low temperatures, the latter yields a remarkable amount of entropy, whereas the former slightly cancel this [See k_B T/J ≲ 0.2 in figure 7(${\rm{h}}^{\prime}$)], it is quite possible that bound states of antiferromagnons localized within a hexagon of nearest-neighbor spins play a significant role in reproducing the low-temperature entropy distinctive of the kagome-lattice antiferromagnet. Indeed, Singh and Huse [48] stated that low-lying triplet excitations against the VBC with a 36-site unit cell, including those localized inside a perfect hexagon, attract one another and form many bound states in the spin-singlet channel.

Though CLSWs do not have a dispersionless zero-energy mode in the triangular-lattice antiferromagnet [21], the fact remains that they cannot describe any magnetic properties as functions of temperature at all. Although SC MLSWs at T = 0 have the same dispersion relation as CLSWs [figures 7(a) and (b)], yet SC MLSWs are no longer useless in thermodynamics with their chemical potential μ effective at every finite temperature, similar to those in the square-lattice antiferromagnet [82]. However, they misread the per-site static uniform susceptibility χ/L as divergent with T → 0 [figure 3(a)], unlike those in the square-lattice antiferromagnet [82]. For the square-lattice antiferromagnet, the MLSW Hamiltonian ${\widetilde{{ \mathcal H }}}_{\mathrm{harm}}$ (36) commutes with the total magnetization ${{ \mathcal M }}^{z}$, so that ${\mathrm{lim}}_{T\to 0}\left[\langle {\left({{ \mathcal M }}^{z}\right)}^{2}{\rangle }_{T}-\langle {{ \mathcal M }}^{z}{\rangle }_{T}^{2}\right]$ becomes zero and ${\mathrm{lim}}_{T\to 0}{\chi }^{{zz}}/L$ stays finite [cf (77)], while its constraint condition (35) makes both χ^xx and χ^yy vanish [cf (79)]. SC MLSWs well sketch the square-lattice-antiferromagnet thermodynamics over the whole temperature range of absolute zero to infinity, indeed [82]. Although SC MLSWs soften at k = ± Q and k = 0 with T → 0 in both the square- and triangular-lattice antiferromagnets, they yield convergent and divergent χ/L with T → 0 in the former and latter, respectively. The difficulty in the latter lies in the expression (80). In (80), ${\mathrm{lim}}_{T\to 0}{\chi }^{{zz}}/L$ and ${\mathrm{lim}}_{T\to 0}{\chi }^{{xx}}/L$ both diverge if ${\mathrm{lim}}_{T\to 0}{\epsilon }_{\pm {\bf{Q}}:1}=0$, while in (81), ${\mathrm{lim}}_{T\to 0}{\chi }^{{yy}}/L$ vanishes even if ${\mathrm{lim}}_{T\to 0}{\epsilon }_{{\bf{0}}:1}=0$ by virtue of the vanishing factor ${{\rm{lim}}}_{T\to 0}(\cosh 2{\vartheta }_{{\bf{0}}:1}-\sinh 2{\vartheta }_{{\bf{0}}:1})$. Only the DC-MLSW dispersion relation figure 7(d) reasonably—in the sense of minimizing the ground-state energy and keeping the excitation spectrum gapless—suppresses the divergence of χ/L at T = 0. We can bring DC MLSWs into interaction with the use of the primary self-energies to give a still better description of the susceptibility. The tensor-network-based renormalization-group calculations [117] are still limited to particular geometries and subject to their boundaries at low temperatures of increasing interest. Whether we use d-log Padé and integrated differential approximants [144] or employ particular sequence extrapolation techniques to accelerate the convergence [95], the extrapolations of high-temperature series work well down to the peak temperature but begin to deviate from each other below the peak. Under such circumstances, we may expect many useful pieces of information of the DC-MSW thermodynamics reachable to both infinite-lattice and low-temperature limits.

Developing an exponential tensor renormalization-group method on cylinder- and strip-shaped triangular lattices, Chen et al [117] calculated their thermal properties and especially revealed two generic temperature scales, 0.2 ≲ k_B T_low/J ≲ 0.28 and k_B T_high/J ≃ 0.55, at which the specific heat reaches its local maximum or exhibit a shoulder-like anormaly. They claimed that T_low corresponds with the onset of the 'renormalized classical' behavior [145, 146] that would be expected from a relevant semiclassical nonlinear sigma model, whereas T_high originates in the quadratic roton-like excitation band [147, 148].

Chakravarty, Halperin, and Nelson [145, 146] claimed that when a planar-lattice quantum Heisenberg antiferromagnet has an ordered ground state, its long-wavelength behavior at certain finite temperatures, which they designate as the renormalized classical regime, can be described by an effective classical nonlinear sigma model in two spatial dimensions obtainable from the pertinent quantum model in two spatial plus one temporal dimensions. In this regime the correlation length reads

$$\begin{eqnarray}&&\displaystyle \frac{\xi }{a}=A{\left(\displaystyle \frac{{k}_{{\rm{B}}}T}{B}\right)}^{x}\exp \left(\displaystyle \frac{B}{{k}_{{\rm{B}}}T}\right)\left[1+O\left(\displaystyle \frac{{k}_{{\rm{B}}}T}{J}\right)\right],\end{eqnarray} \tag{ 116 }$$

where A and B depend on the spin-wave velocities and stiffness coefficients for twisting the spins, both being renormalized by the quantum fluctuations at T = 0. The constant A and power exponent x are nonuniversal numbers in the sense that they depend on our modeling, such as whether the Hamiltonian is a nearest-neighbor exchange model or contains next-nearest-neighbor interactions in addition. Still further, they depend on to which order we calculate, such as whether within the one-loop approximation or up to two-loop corrections. On the other hand, B unambiguously sets the temperature scale for the correlations. Vanishing B signifies such strong quantum fluctuations as to destroy the long-range magnetic order, but otherwise we can identify two distinct regions separated by the crossover temperature k_B T = B, i.e. the renormalized classical behavior [145, 146] at low temperatures with correlation length ultimately diverging exponentially as T → 0 and quantum critical behavior [145, 146, 149] at intermediate temperatures with inverse correlation length given by a certain power function of T. Studying both quantum and classical—here in the sense of integrating out all quantum fluctuations to obtain an effective classical model and neglecting any imaginary-time-derivative of the three-component vector field from the beginning, respectively—nonlinear sigma models suited to frustrated planar-lattice antiferromagnets assuming a noncollinearly ordered ground state, Azaria, Delamotte, and Mouhanna [150] calculated the two-loop [in the context of calculating the (effective) classical model (obtainable via the one-loop renormalization of the coupling constants)] correlation length intending to describe quantum and classical Heisenberg antiferromagnets on the triangular lattice at low temperatures. They argued that the correlation length in the quantum model [150] still diverges as (116), where

$$\begin{eqnarray}&&x=-\displaystyle \frac{1}{2},B=\pi {\tilde{\rho }}_{{\rm{s}}}^{\perp }G(\tilde{\alpha })\equiv {B}_{\mathrm{quant}};G(\tilde{\alpha })\equiv 2+2(1+\tilde{\alpha })\displaystyle \frac{{\tan }^{-1}\sqrt{\tilde{\alpha }}}{\sqrt{\tilde{\alpha }}},\tilde{\alpha }\equiv \displaystyle \frac{{\tilde{\rho }}_{{\rm{s}}}^{\parallel }}{{\tilde{\rho }}_{{\rm{s}}}^{\perp }}-1\end{eqnarray} \tag{ 117 }$$

with ${\tilde{\rho }}_{{\rm{s}}}^{\perp }$ and ${\tilde{\rho }}_{{\rm{s}}}^{\parallel }$ being the transverse (in-plane) and longitudinal spin stiffnesses renormalized by the quantum fluctuations at T = 0, respectively, while the correlation length in the corresponding classical model [150, 151] also diverges as (116), where

$$\begin{eqnarray}&&x=\displaystyle \frac{1}{2},B=\pi {\rho }_{{\rm{s}}}^{\perp }G(\alpha )\equiv {B}_{\mathrm{class}};\alpha \equiv \displaystyle \frac{{\rho }_{{\rm{s}}}^{\parallel }}{{\rho }_{{\rm{s}}}^{\perp }}-1\end{eqnarray} \tag{ 118 }$$

with ${\rho }_{{\rm{s}}}^{\perp }$ and ${\rho }_{{\rm{s}}}^{\parallel }$ being the bare stiffnesses at T = 0 instead. Their prediction of the correlation length for the quantum model in the renormalized classical regime [150] is precisely available from a large-N expansion based on symplectic symmetry including fluctuations to order 1/N [152] as well, while that for the classical model [150, 151] is well consistent with Monte Carlo simulations [153]. The bare spin stiffnesses are readily available by twisting Néel-ordered classical spins by an infinitesimal angle per lattice constant along the relevant direction [154] or indirectly obtainable through [155, 156]

$$\begin{eqnarray}&&{\rho }_{{\rm{s}}}^{\perp }={\chi }^{\perp }{c}_{{\bf{Q}}}^{2},{\rho }_{{\rm{s}}}^{\parallel }={\chi }^{\parallel }{c}_{{\bf{0}}}^{2},\end{eqnarray} \tag{ 119 }$$

where χ^⊥ and χ^∥ are the bare transverse and longitudinal susceptibilities at T = 0, respectively,

$$\begin{eqnarray}&&{\chi }^{\perp }={\chi }^{\parallel }=\displaystyle \frac{2{{\hslash }}^{2}}{9\sqrt{3}{{Ja}}^{2}},\end{eqnarray} \tag{ 120 }$$

while c _Q and c₀ are the LSW velocity at the ordering momenta k = ± Q and that at k = 0, respectively,

$$\begin{eqnarray}&&{c}_{{\bf{Q}}}=\displaystyle \frac{3\sqrt{3}}{2\sqrt{2}}\displaystyle \frac{{JSa}}{{\hslash }},{c}_{{\bf{0}}}=\displaystyle \frac{3\sqrt{3}}{2}\displaystyle \frac{{JSa}}{{\hslash }}.\end{eqnarray} \tag{ 121 }$$

In any case we have

$$\begin{eqnarray}&&{\rho }_{{\rm{s}}}^{\perp }\equiv \displaystyle \frac{\sqrt{3}}{4}{{JS}}^{2},{\rho }_{{\rm{s}}}^{\parallel }\equiv \displaystyle \frac{\sqrt{3}}{2}{{JS}}^{2},\end{eqnarray} \tag{ 122 }$$

and then (118) reads

$$\begin{eqnarray}&&{B}_{\mathrm{class}}=\displaystyle \frac{\sqrt{3}}{4}\pi (2+\pi ){{JS}}^{2}=1.7486J,\end{eqnarray} \tag{ 123 }$$

which was indeed demonstrated by the Monte Carlo calculations on the classical Heisenberg model [153].

In order to evaluate the actual quantum mechanical correlation length, Elstner, Singh, and Young [157] calculated series for

$$\begin{eqnarray}&&{\xi }^{2}=\displaystyle \frac{1}{L}\displaystyle \sum _{l,l^{\prime} =1}^{L}\displaystyle \frac{{\left({{\bf{r}}}_{l}-{{\bf{r}}}_{l^{\prime} }\right)}^{2}}{4{S}^{{zz}}({\bf{Q}})}\langle {S}_{{{\bf{r}}}_{l}}^{z}{S}_{{{\bf{r}}}_{l^{\prime} }}^{z}{\rangle }_{T}{e}^{i{\bf{Q}}\cdot ({{\bf{r}}}_{l}-{{\bf{r}}}_{l^{\prime} })}\end{eqnarray} \tag{ 124 }$$

up to order 13 in powers of the inverse temperature 1/k_B T and found that the quantity $({k}_{{\rm{B}}}T/J)\mathrm{ln}({k}_{{\rm{B}}}T{\xi }^{2}/{{Ja}}^{2})$ extrapolate to about 0.2 ≡ 2B_quant/J as T → 0, which is apparently nonzero but merely about 6% of the corresponding classical value 2B_class = 3.4972J. It is this finding 2B_quant/k_B that Chen et al [117] identified their T_low with. However, as the authors themselves [157] pointed out, there may be a crossover to the renormalized classical behavior (116) with (117) and thus an upturn [148, 153, 157, 158] of $({k}_{{\rm{B}}}T/J)\mathrm{ln}({k}_{{\rm{B}}}T{\xi }^{2}/{{Ja}}^{2})$ with decreasing T at some temperature much lower than the high-temperature series expansion approach can reach, resulting in a larger value of B_quant. Then, it is quite possible that the PC-DC-MSW specific-heat peak temperatures T _Q are reasonable indeed. In any case, the fact remains unchanged that the ratio of B_quant to B_class on the triangular lattice [148, 150, 151, 157–159] is much smaller than that on the square lattice [145, 146, 154, 158] which is known to be no smaller than 0.7. It must be the consequence of such strong quantum fluctuations in the triangular-lattice antiferromagnet that the higher-order quantum corrections we take account of, the lower peak temperature T _Q we have [figure 6(a)]. The up-to-O(S^m )-MSW estimate of the peak temperature, ${T}_{{\bf{Q}}}^{(m)}$, monotonically and significantly decreases with descending powers of S as ${k}_{{\rm{B}}}{T}_{{\bf{Q}}}^{(m)}/J\,=\,0.4904(m=1),0.3894(m=0),0.3526(m=-1)$. On the other hand, it may be the consequence of much weaker quantum fluctuations in the square-lattice antiferromagnet that the MLSW and MWDISW estimates of the peak temperature are almost the same (figure 2(a)).

To end, we point out that the other temperature scale T_high [117] argued in relation to roton-like excitations [101, 147, 148, 160, 161] is also available by our PC DC MSWs. The DC-MLSW and PC-DC-MSW temperature profiles of the triangular-lattice-antiferromagnet specific heat look as though they have qualitatively different excitation mechanisms on their down slopes from intermediate to high temperatures. Quantum renormalization converts the saddle point at ${\rm{M}}\equiv \tfrac{1}{a}\left(\pi ,0,\tfrac{\pi }{\sqrt{3}}\right)$ in the DC-MLSW excitation energy surface _{k :1} (92) into a local minimum surrounded by flat parts (figures 8(a) and (b)). Such local extrema, available on the ways from ${\rm{\Gamma }}\equiv \tfrac{1}{a}\left(0,0,0\right)$ to ${\rm{K}}\equiv \tfrac{1}{a}\left(\tfrac{4\pi }{3},0,0\right)$ and ${\rm{K}}^{\prime} \equiv \tfrac{1}{a}\left(\tfrac{2\pi }{3},0,\tfrac{2\pi }{\sqrt{3}}\right)$ as well [162, 163], may yield thermodynamic anomalies [101, 147, 148]. MSWs around the M point form in a quadratic band of roton-like excitations above the ground state by 0.7579J ≡ k_B T_roton with stronger intensity than otherwise. We indicate this temperature T_roton in figures 6(a) and 7(${\rm{d}}^{\prime}$). We cannot find any anormaly around T_roton in the DC-MLSW specific-heat curve, while we find a steep-to-mild crossover as temperature increases across T_roton in the PC-DC-MSW specific-heat curves, where the O(S⁰) second-order perturbative correction ${\rm{\Delta }}{F}_{2}^{(0)}$ plays a prominent role, just like it does in yielding the bimodal temperature profile of the kagome-lattice-antiferromagnet specific heat (figures 7(${\rm{d}}^{\prime}$) and (${\rm{h}}^{\prime}$)). There is a certain resemblance between the dispersionless DC-MLSW spectrum _{k :1} around the M point in the triangular-lattice antiferromagnet and the wholly flat DC-MLSW spectrum _{k :2} in the kagome-lattice antiferromagnet. Without them, we do not have such marked ups and downs of ${\partial }^{2}\beta {\rm{\Delta }}{F}_{2}^{(0)}/\partial T\partial \beta$. In the kagome-lattice antiferromagnet, the wholly flat band gives such a prominent peak in the density of states to yield the mid-temperature broad maximum, whereas in the triangular-lattice antiferromagnet, only a flat part of the whole spectrum is not sufficient to do this and does no more than soften the down slope.

The ground states of the triangular- and kagome-lattice antiferromagnets are so different from each other as to be ordered and disordered, respectively, and their specific-heat curves are still of different aspect such as a single-peak temperature profile containing two temperature scales and an explicitly bimodal temperature profile, respectively. Nevertheless, MSW excitations behind them have some similarities as well as differences. This is understandable if their ground states sit close to each other sandwiching a quantum critical point. Additional ring-exchange interactions stabilize the nearest-neighbor pair-exchange-coupled spin-$\tfrac{1}{2}$ Heisenberg antiferromagnet on the equilateral triangular lattice into a disordered ground state [28, 29], while additional Dzyaloshinskii-Moriya interactions stabilize the nearest-neighbor pair-exchange-coupled spin-$\tfrac{1}{2}$ Heisenberg antiferromagnet on the regular kagome lattice into an ordered ground state [35–39]. If we monitor the triangular-lattice antiferromagnet with increasing ring-exchange interactions and the kagome-lattice antiferromagnet with increasing Dzyaloshinskii-Moriya interactions through the use of PC DC MSWs, we can indeed see single-to-double-peak and double-to-single-peak crossovers of their specific-heat curves as an evidence of their locating in close vicinity of a quantum critical point.

In the framework of SW languages, frustrated noncollinear antiferromagnets are much less tractable than collinear antiferromagnets, and still less are they when their quantum ground states are disordered. We have challenged this difficulty and just obtained a robust and eloquent PC-DC-MSW thermodynamics, which is designed especially for noncollinear antiferromagnets but is not inconsistent with the traditional SC-MSW thermodynamics for collinear antiferromagnets [78, 103]. Note that the present DC condition degenerates into the traditional SC condition for any collinear ground state.

Our PC-DC-MSW scheme is widely applicable to various frustrated noncollinear quantum magnets. We may take further interest in describing giant molybdenum-oxide-based molecular spheres of the Keplerate type with spin-higher-than-$\tfrac{1}{2}$ magnetic centers such as {Mo₇₂Cr₃₀} [164, 165] and {Mo₇₂Fe₃₀} [165–168] containing 30 Cr³⁺ ($S=\tfrac{3}{2}$) and Fe³⁺ ($S=\tfrac{5}{2}$) ions, respectively, where neither exact diagonalization nor quantum Monte Carlo simulation is feasible. Considering that LSWs, whether modified or not, only give a poor description of the excitation spectrum of the icosahedral keplerate cluster {Mo₇₂Fe₃₀} [169], SC-MLSW thermodynamics [142] must be insufficient to capture its thermal features and there may be something beyond Monte Carlo calculations of its classical analog [170].