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

We construct modified spin-wave thermodynamics for frustrated noncollinear antiferromagnets for the first time. The well-known modified spin-wave theory for collinear antiferromagnets diagonalizes a bosonic Hamiltonian subject to the constraint that the total staggered magnetization be zero. Applying this scheme as it is to frustrated noncollinear antiferromagnets ends in a poor thermodynamics, missing the optimal ground state and breaking the local U(1) rotational symmetry. We find a new double-constraint modification scheme to overcome this difficulty, which is tuned especially to frustrated spiral magnets but spontaneously goes back to the standard single-constraint condition at the onset of a collinear N\'eel-ordered classical ground state. We apply this new scheme to triangular-based polyhedral and planar antiferromagnets with particular interest in a possible contrast between edge- versus corner-sharing geometries. Under such circumstances that very few methods are available to calculate finite-temperature properties of frustrated noncollinear quantum magnets in the thermodynamic limit, our newly developed modified spin-wave theory predicts that the specific heat of the kagome-lattice Heisenberg antiferromagnet in the corner-sharing geometry remains having both mid-temperature broad maximum and low-temperature narrow peak in the thermodynamic limit, while the specific heat of the triangular-lattice Heisenberg antiferromagnet in the edge-sharing geometry retains a low-temperature sharp peak followed by a mid-temperature weak anormaly in the thermodynamic limit.

In response to stimulative experimental [49,50,51,52,53] and/or numerical [54,55] findings, various SL phases have been nominated for a possible ground state, including an algebraic or U(1) Dirac SL [9,10,56,57,58], whose low-energy physics is governed by four flavors of massless two-component Dirac fermions coupled to a U(1) gauge filed, a gapped Z 2 SL [15,16,14], which exhibits gapped vortex excitations of an emergent Z 2 gauge field as well as a flat spinon band at low energies, and a gapped chiral SL [59,60], which simultaneously and spontaneously breaks space reflection and time reversal symmetries. Since one or more gapped Z 2 SLs may be emergent from their "parent" gapless U(1) SL by small pairings of fermionic spinons, such neighboring SLs are so close in energy and thus hard to distinguish numerically. Even state-of-theart simulations based on tensor-network wavefunctions lead to a debate over the U(1) Dirac SL [61,62], Z 2 gapped SL [63], and VBC with a 36-site unit cell [64]. Under such circumstances, a symmetry-based analysis of SL phases provides an otherwise unavailable insight. Wen [65] employed PSGs to characterize hundreds of symmetric SLs with SU(2), U(1), or Z 2 gauge structures in the context of fermionic mean-field theories. Wang and Vishwanath [15] pushed his exploration further for Schwingerboson mean-field states, focusing on the Ising gauge group but accessing both SL and conventional magnetically ordered states. Besides classifying topological orders (gauge configurations) in SLs, the language of projective symmetry can identify itinerant spinon excitation modes [66] and further describe their Raman response [67,68].
The triangular and kagome lattices can be viewed as edge-and corner-sharing triangles in two dimensions, respectively, and their Heisenberg antiferromagnets turn out to have distinct low-energy structures [69,70]. An antiferromagnetic order established in the thermodynamic limit manifests itself in finite-size clusters, whose low-energy spectra can be described in terms of a "quantum top" [22,23,24]. In the case of the spin- 1 2 antiferromagnetic Heisenberg Hamiltonian on the equilateral triangular lattice of L sites, the three identical ferromagnetic sublattices each have a collective spin of length L/6 and they couple to reproduce rotationally invariant states, which approximate the lowest-lying eigenstates in different subspaces labeled by total spin S, E L (S) − E L (S min ) = S(S + 1)/(2I L ) (S > S min ), where S min is either 0 or 1 2 according as L is even or odd and the inertia of the top I L is an extensive quantity, proportional to the perpendicular susceptibility [24] and therefore to L [22,23]. Hence, plotting the lowest energy level as a function of S(S + 1) yields a "Pisa tower" [28,69,70,23] with a slope ∝ 1/L. The Pisa tower is well separated from the softest magnons that converge to the ground state as 1/ √ L [24]. Thus and thus, an extensive set of low-lying levels, including both magnetic and nonmagnetic ones, collapse onto the ground state in the thermodynamic limit. In the spin- 1 2 regular kagome-lattice Heisenberg antiferromagnet, on the contrary, there is absolutely no such low-energy structure. The candidate levels to form a tower of states neither scale as S(S +1) nor separate from the above continuum of excitations. The magnetic (triplet) excitations are separated from the ground state by a gap and this gap is filled with a continuum of nonmagnetic (singlet) excitations adjacent to the ground state [71,72,73,74]. Such differences in level structure should manifest themselves in thermodynamics at low temperatures. Especially in the kagomelattice antiferromagnet, a large amount of low-temperature residual entropy may cause some additional structure, whether a peak [75] or shoulder [76], to the main Schottky maximum in the temperature profile of the specific heat. While the spin susceptibility is thermally activated due to the singlet-triplet gap [54,55], the specific heat does not decrease exponentially in the gap with a possibility of defining an extra energy scale set by low-lying singlets.
We are thus motivated to perform a comparative study of Heisenberg antiferromagnets in the triangular-based edge-versus corner-sharing geometry. We calculate the specific-heat curves of Heisenberg antiferromagnets not only on the twodimensional regular triangular and kagome lattices but also on "zero-dimensional" analogs, i.e., Platonic and Archimedean polyhedra consisting of edge-and corner-sharing triangles, respectively. We aim to verify whether they have generic thermal features, and if any, further clarify why they behave so. In order to explore finite-temperature properties of infinite systems, we modify the conventional antiferromagnetic spin-wave (SW) thermodynamics [77] applicably to low-dimensional spin spirals. We make drastic and far-reaching reforms in the traditional modified SW (MSW) theory designed for low-dimensional collinear antiferromagnets [78,79,80,81,82].
It is very hard to calculate thermodynamic quantities of quantum frustrated antiferromagnets on an infinite lattice in two or more dimensions. There are limited numerical approaches available, and what is worse, their findings are not necessarily consistent with each other. Among others, is the long-standing debate on the lowtemperature profile of the magnetic specific heat of the spin-1 2 kagome-lattice Heisenberg antiferromagnet [75,76], originating from the milikelvin heat-capacity measurements of 3 He adsorbed on graphite [83]. Early calculations of small clusters [84,85] argued that the specific heat of the nearest-neighbor pair-exchange-coupled spin- 1 2 Heisenberg antiferromagnet on the regular kagome lattice is very likely to have a low-temperature peak in addition to the main maximum at k B T /J ≃ 2/3 with J being the antiferromagnetic exchange. The Lanczos diagonalization technique combined with random sampling [86,87] and the thermal-pure-quantum-states formulation of statistical mechanics [88] enabled us to calculate somewhat larger medium-sized clusters, but their findings are not yet decisive of whether the low-temperature structure remains a peak [89] or reduces to a shoulder [88,76] in the thermodynamic limit. Ingeniously devised quantum Monte Carlo algorithms [90,91] revealed that much larger clusters still exhibit a double-peak temperature profile in their specific-heat curves. Recently tensor-network-based methods have come into use in exploring thermodynamics, but their findings [92] are quite otherwise in favor of the conclusion that the low-temperature structure of the specific heat is a shoulderlike hump rather than a true maximum. While high-temperature series expansion may also be employed to calculate the specific heat in the thermodynamic limit, its reliability has been under debate. When analyzed through standard Padé approximants, the specific-heat hightemperature series expansion extrapolated down to absolute zero has a large missing entropy [85]. In an attempt to improve the convergence of standard Padé approximations at low temperatures much below J/k B , Bernu and Misguich [93] introduced such biased approximants as to satisfy the energy and entropy sum rules obeyed by the specific heat. In the same context, Rigol, Bryant, and Singh [94,95] proposed a numerical linkedcluster algorithm, intending to capture advantages of both high-temperature expansion and exact diagonalization, together with sequence extrapolation techniques to accelerate the convergence of linked clusters. Both approaches [94,95,96,97] concluded that the missing entropy should be compensated by the appearance of an additional lowtemperature peak below the major maximum.
Thus, even via the use of such a wide variety of tools, yet the issue remains to be settled. That is the very reason why we develop a new "language" of our own. Our newly developed MSW theory is not exact, to be sure, but it is widely applicable to frustrated antiferromagnets in all dimensions in the thermodynamic limit to capture their thermal features very well, serving as a mirror of their ground states in the context of whether they are classically ordered or quantum disordered. We investigate temperature profiles of the specific heat for the nearest-neighbor antiferromagnetic Heisenberg model where J is assumed to be positive, S r l are the vector spin operators of magnitude S attached to the site at r l , and δ l:κ are the vectors connecting the site at r l with its z nearest neighbors. We start with some bipartite lattices and then proceed to a variety of triangular-based polyhedral and planar lattices. We suppose that they each consist of L sites with the unique coordination number z.

Modified spin-wave theory of collinear antiferromagnets
Takahashi [78] and Hirsch et al. [79,80] had an idea of so modifying the conventional SW (CSW) theory as to describe collinear antiferromagnets in lower than three dimensions at finite temperatures, where Holstein-Primakoff [98] or Dyson-Maleev [99,100] bosons are constrained to keep the total staggered magnetization zero via a Bogoliubov transformation dependent on temperature. If we apply this pioneering but naive singleconstraint (SC)-modification scheme to frustrated noncollinear antiferromagnets as it is, what will happen? Let us begin by calculating the prototypical antiferromagnetic MSW thermodynamics for the triangular and kagome lattices in comparison with those for the bipartite square and honeycomb lattices, intending to ascertain what is the problem in the prototypical MSW as well as CSW formalisms when applied to spiral magnets. Considering that we treat antiferromagnetic spin spirals in various geometries, we employ Holstein-Primakoff, rather than Dyson-Maleev, bosons. Within the linear SW (LSW) treatment, they are no different from each other. When we go beyond it to obtain interacting SWs (ISWs), the Holstein-Primakoff and Dyson-Maleev representations of any spin Hamiltonian generally differ from each other, the former remaining Hermitian but the latter becoming non-Hermitian. As long as the spins align antiparallel among neighbors, however, the Wick or Hartree-Fock decompositions of the Holstein-Primakoff and Dyson-Maleev bosonic Hamiltonians coincide with each other up to O(S 0 ) [81,82]. This is no longer the case with any noncollinear alignment among the spins, whose bosonic Hamiltonian reads a series in descending powers of √ S rather than S and generally contains terms consisting of an odd number of boson operators. A usual treatment [101,102] of such bosonic Hamiltonians consists of diagonalizing the harmonic part exactly and then taking account of higher-order interactions as perturbations, where Holstein-Primakoff bosons are much more tractable than Dyson-Maleev bosons.

Bosonic Hamiltonian
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 (x,ỹ,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  where φ r l is the angle formed by the axes z andz 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 When we employ the Holstein-Primakoff bosons and expand the square root R r l (S) in descending powers of S, the Hamiltonian (3) is expanded into the series where H ( m 2 ) , on the order of S m 2 , read considering that the sum of the relative rotation angles over the nearest neighbors of an arbitrary site z κ=1 (φ r l +δ l:κ − φ r l ) equals zero or a multiple of π according as z is even or odd in the classical ground states in question (Fig. 1).  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 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 2a 1 , 2a 2 , 2a 0 , respectively (cf. Fig. 1). The nearest-neighbor vectors of the lth site are given by square : honeycomb : kagome : while the center of the nth unit cell is expressed as square : R n = n 1 a z + n 2 a x = a t (n 2 , 0, n 1 ) , honeycomb : kagome : R n = 2n 1 a 2 + 2n 2 a 0 = 2a (cf. Fig. 1) with n 1 and n 2 being the unique set of integers to represent R n . When we express each site position r l ≡ r p(n−1)+σ as square : r 1(n−1)+σ = R n , honeycomb : r 2(n−1)+σ = R n + (−) σ a 0 , kagome : r 3(n−1)+σ = R n + a σ−1 , and define the "interunit" and/or "intraunit" ordering wave vectors honeycomb : the rotation angle φ r l ≡ φ r p(n−1)+σ can be given by

Traditional single-constraint condition
Intending to calculate thermal quantities of the high-temperature-superconductorparent material La 2 CuO 4 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 , Hirsch and Tang [79] also initiated such an idea by truncating the bosonic Hamiltonian at the harmonic part, H ≃ 2 m=1 H (m) ≡ H harm , and diagonalizing with such µ as to satisfy the constraint condition (35). Several authors [78,80,81,82,103,104,105] 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 [106,107,108], 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, H ( m 2 ) (m = 1, −1, · · · ), are characteristic of noncollinear antiferromagnets. Without them, the Q = 0 and √ 3 × √ 3 ground states of the kagome-lattice antiferromagnet remain degenerate in energy with each other.

Modified spin-wave interaction-Variational treatment
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] A ≡ 1 2 a † r l a r l + a † r l +δ l:κ a r l +δ l:κ = a † r l a r l , D ≡ 1 2 a † r l a r l +δ l:κ + H.c. = a † r l a r l +δ l:κ , which we shall read as the quantum average in the Holstein-Primakoff-boson vacuum ′ 0 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(S 0 ) quartic Hamiltonian (11) into quadratic terms For collinear antiferromagnets, every local rotation angle φ r l reads either 0 or π and every relative rotation angle φ r l +δ l:κ − φ r l becomes π [cf. (31) and (32)], resulting that their bosonic Hamiltonians H = ∞ m=−2 H −m commute with the total uniform magnetization in each order and (41) reduces to without any contribution from C and D to cause a net change in the magnetization (43). If we define an effective Hamiltonian of the bilinear version, and demand that A and B 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 for the square (z = 4, p = 1), honeycomb (z = 3, p = 2), and triangular (z = 6, p = 1) lattices and with u kν :σ ′ σ satisfying the eigenvalue equations    −2γ kν :σ cos k ν · a 2 cos k ν · a 1 cos k ν · a 2 −2γ kν :σ cos k ν · a 0 cos k ν · a 1 cos k ν · a 0 −2γ kν :σ for the kagome (z = 4, p = 3) lattice. We number the three eigenmodes for the kagome lattice in ascending order, Intending to diagonalize the effective Hamiltonian (42), we define Fourier transforms of the Holstein-Primakoff boson operators as e −ikν ·r p(n−1)+σ a r p(n−1)+σ (ν = 1, · · · , N; σ = 1, · · · , p), where the wavevector k ν runs over the full paramagnetic Brillouin zone, as is shown in Fig. 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 where E (2) is the classical ground-state energy, E (1) and E (0) are its O(S 1 ) quantum and O(S 0 ) variational [106,108] corrections, respectively, and α † kν :σ creates a magnon of the σ species with wavevector k ν at a cost of energy ε kν :σ . Further details are given by each lattice. The square (z = 4, p = 1) and honeycomb (z = 3, p = 2) lattices have the expressions in common. The triangular (z = 6, p = 1) and kagome (z = 4, p = 3) lattices have the expressions u kν:τ σ a kν :τ sinhϑ kν :σ + a † −kν :τ coshϑ kν :σ , in common with the abbreviations and the understanding that u kν :11 for the triangular lattice be unity.

Variational single-constraint modified-spin-wave thermodynamics
Variational MISW-modified WDISW (MWDISW) and HFISW (MHFISW)thermodynamics can be formulated in terms of the self-consistent fields A to D , which depend on how the SWs are interacting. The MSW thermal distribution function reads with ε kν :σ containing part or all of A to D . Every time we encounter the double angle brackets A to D , we read them according to the scheme of the time, for the square (z = 4, p = 1) and honeycomb (z = 3, p = 2) lattices and for the triangular (z = 6, p = 1) and kagome (z = 4, p = 3) lattices, where they each still contain A and B (collinear antiferromagnets) or A to D (noncollinear antiferromagnets) to be self-consistently determined. The constraint condition is then written as In the MHFISW scheme, we solve the simultaneous equations (62)   of Padé approximants to high-temperature series subject to the energy and entropy sum rules (HTS + SRs) [93,96,110], and tensor-network-based renormalization-group calculations (TN) [92,111]. The findings referred to as TN in (d) [92] are obtained on the infinite two-dimensional kagome lattice, while those as TN (6 × ∞) in (c) [111] are extrapolations to the infinite cylindrical triangular lattice with hexagonal ends aslant. Schwinger-boson mean-field calculations (SB MF) are also shown for reference.
for the square and honeycomb lattices and for the triangular and kagome lattices. We show in Fig. 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 [105,112,113,114], 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 hightemperature 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 tensornetwork states [115] 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 [116], on some interleaved β grids [117], or in a Trotter-error-free manner based on the numerically exact Taylor series expansion of the whole density operator e −βH [118], 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 [111] 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 [119]. By mapping a two-dimensional quantum lattice model into a three-dimensional closed tensor network and contracting the three-dimensional brickwall tensor network with the imaginary time length corresponding to temperature in question [120], 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 [121]is awfully time-consuming and has to be reduced to a cluster update [92,122] in most cases, through the use of the Bethe approximation [92,122,123] 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 [120] and/or at low temperatures [111].
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, Figs. 2(a) and 2(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 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 B T vanishes, satisfying The SB MF formulation [112,113,114] 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" [113,114], 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 Figs. 2(c) and 2(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.  (41) with A to D read as A 0 to D 0 given by Eq. (64). We continue to observe Figs. 2(a) and 2(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 hightemperature 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, Figs. 2(c) and 2(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. H ( m 2 ) , 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.

Modified spin-wave theory of noncollinear antiferromagnets
What we newly construct is a double-constraint (DC)-MSW theory of antiferromagnetic spin spirals in various geometries, where we perturbatively take account of up-to-O(S −1 ) interactions between DC MSWs. In order to treat noncoplanar antiferromagnets as well, we extend the local spin reference frame (2) to any rotation in three dimensions, rewrite the Hamiltonian (1) into the local coordinate system, and then introduce Holstein-Primakoff bosons in the same way as (4) to expand the Hamiltonian in descending powers of √ S, where H ( m 2 ) , on the order of S m 2 , are explicitly given in Appendix A. We divide the bosonic Hamiltonian (70) is merely a constant and H ( 3 2 ) always vanishes for symmetry reasons, and the interactions 1 m=−2 H ( m 2 ) ≡ V perturbing to the harmonic spin waves. We first modify CLSWs into MLSWs by diagonalizing an effective harmonic Hamiltonian, which we shall denote by H harm , instead of the bare one H harm , and then perturb H harm to fourth order at highest in V within the order of S −1 by calculating relevant one-and two-loop corrections. Such a perturbative treatment of SW interactions is rather orthodox than notable in itself, especially in frustrated noncollinear antiferromagnets [101,102,125,126,127,128]. It is the elaborate constraint condition that distinguishes our MSW theory for frustrated noncollinear antiferromagnets from any other previous related attempts. We design a DC condition for SWs to satisfy, which spontaneously reduces to the well-known traditional SC condition (35) when the spins choose a collinear alignment as their classical ground state.
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 Fig. 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 persite 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. Eq. (80) and Figs. 7(b)-7(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] lim 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 In the case of S = 1 2 , SC MLSWs estimate lim T →∞ χk B T /L(gµ B ) 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.
We can have another good understanding of Eq. (71) in a different aspect. Let us recall the Schwinger boson representation S z r l = a † r l :↑ a r l :↑ − a † r l :↓ a r l :↓ 2 ≡ n r l :↑ − n r l :↓ 2 , S x r l − iS y r l ≡ S + † r l ≡ S − r l = a † r l :↓ a r l :↑ ; S 2 r l = n r l :↑ + n r l :↓ 2 n r l :↑ + n r l : where physical states demand that the bosons should satisfy the constraint condition σ=↑,↓ a † r l :σ a r l :σ = 2S (85) at each lattice site. In what are called Schwinger-boson mean-field theories [112,113,114,130,131,132,133], the L requirements (85) are relaxed and imposed only on average, i.e., merely a single Lagrange multiplier µ is introduced and the term µ L l=1 2S − σ=↑,↓ a † r l :σ a r l :σ is added to the Hamiltonian. The determination condition for µ, 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 ≤ a † r l a r l ≤ 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 = 1 2 regular kagome-lattice Heisenberg antiferromagnet is a long-range Néel-ordered phase of the √ 3 × √ 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 Fig. 2, between the Schwinger and Holstein-Primakoff bosons. The Schwinger-boson meanfield 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, but misreads the spin magnitude as 3/2 times as large as the correct value, S 2 r l T = 3S(S + 1)/2 [112,113]. The Holstein-Primakoff-boson representation in the local spin reference frame (4) gives the correct spin magnitude but breaks the original spin rotational symmetry, λ=x,y,z Sλ r l Sλ r l T = S(S + 1). (88) For collinear antiferromagnets, C 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 M z (43). For noncollinear antiferromagnets, the bosonic Hamiltonian H harm no longer commutes with M z , yet we should expect each spin operator to possess rotational symmetry about its local quantization axisz, which is met by demanding that Sx r l Sx r l T = Sỹ r l Sỹ r l T , i.e., C T = a † r l a † r l + a r l a r l T 2 = 0.
Thus and thus, we introduce 2L Lagrange multipliers and diagonalize the effective harmonic Hamiltonian subject to (71) and (89) at each site. Since the 2L Lagrange multipliers degenerate into merely two in practice as µ 1 = · · · = µ L ≡ µ and η 1 = · · · = η L ≡ η, (90) becomes with E (l) and α † kν :σ 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 For DC MLSWs in polyhedral-lattice antiferromagnets, we set N and p equal to 1 and L, respectively, omitting their momentum indices as ε kν :σ ≡ ε σ and and make Bogoliubov bosons out of bare Holstein-Primakoff bosons by numerically obtaining the coefficients f σl and g σl . Since spins are not necessarily coplanar in the classical ground states of polyhedral-lattice antiferromagnets, we generalize the groundstate energy expressions into three dimensions,

Modified-spin-wave interaction-Perturbative treatment
In order to investigate effects of the interactions V on temperature profiles of the specific heat, we calculate their perturbative corrections to the DC-MLSW free energy. The lthorder correction at temperature T ≡ 1/βk B is calculated as where we denote (Euclidean) time ordering operation by 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. ∆F l (l = 1, 2, · · · ) each make their own contribution on the order of S m , which we shall denote by ∆F (m) l . When we truncate any correction ∆F l at the order of S −1 , all the remaining corrections are given by The O(S 0 ) corrections ∆F the up-to-O(S 0 ) internal energy as and the up-to-O(S −1 ) internal energy as In terms of the unperturbed temperature Green functions with bosonic Matsubara frequencies ω n ≡ 2nπ/β , the leading corrections on the order of S 0 , ∆F (0) 1 and ∆F (0) 2 , can be written as where E (0) is the O(S 0 ) perturbative correction to the DC-MLSW ground-state energy E (2) + E (1) (91), with 0 denoting the quantum average in the DC-MLSW vacuum. We refer to Σ Such diagrams are emergent in ∆F   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 [86,135]. In Fig. 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 [136], whereas all spins are (assumed to be) coplanar with a relative angle of 120 • between nearest neighbors in the rest three each [137,136]. Similar to the kagome-lattice antiferromagnet, the cuboctahedral antiferromagnet has noncoplanar singlet ground states degenerate in energy with the coplanar one [137]. These ground 2 Hamiltonian (1) on the L → ∞ triangular (a) and kagome (b) lattices in comparison with optimal interpolations of Padé approximants to high-temperature series subject to the energy and entropy sum rules (HTS + SRs) [93,96,110] and tensor-network-based renormalization-group calculations (TN) [92,111]. The findings referred to as TN in (b) [92] are obtained on the infinite two-dimensional kagome lattice, while those as TN (6 × ∞) in (a) [111] are extrapolations to the infinite cylindrical triangular lattice with hexagonal ends aslant. Finite-temperature Lanczos calculations (FTL) for small periodic clusters (L = 12, 24, 36) [29,76] of the same Hamiltonian are also shown for reference. We denote the temperatures at which the up-to-O(S 1 ) specific heat C ≡ ∂E harm /∂T (103) and up-to-O(S −1 ) specific heat C ≡ ∂E/∂T (105) reach their maxima by T Q , which measure 0.4904J/k B and 0.3526J/k B , respectively, and the temperature corresponding to the roton gap located at M in the Brillouin zone by T roton , which measures 0.7579J/k B [cf. Fig. 8 states of the four polyhedral-lattice antiferromagnets are schematically shown in  Table 1.
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 Table 2. DC-MSW estimates of the high-temperature asymptotic specific heat coefficient A ≡ lim T →∞ (k B T /J) 2 C/k B [cf. (113) and (114) specific heat as to bear remarkable resemblance to those of antiferromagnetic small clusters of periodic triangular [29,31,138] and kagome [76,88,97] lattices (cf. Fig.  6), marked by a quite sharp low-temperature maximum followed by a long gentle down slope from intermediate to high temperatures [139] and a main round maximum at intermediate temperatures accompanied by an additional low-temperature modest peak [140,141,142], 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 [143] 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 H ( m 2 ) become effective that DC MSWs give a fairly good description of the thermal features in various triangular-based geometries.

3.3.2.
Triangular-based planar-lattice antiferromagnets. Now we proceed to the twodimensional triangular-and kagome-lattice antiferromagnets in the thermodynamic limit of our main interest. In Fig. 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-networkbased 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 lowtemperature 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 edgesharing 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(S 0 ) primary self-energies, are key ingredients in this context, as will be discussed in more detail in Sect. 4.
It is generally very hard to provide a precise thermodynamics of two-or higherdimensional 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 DC MLSWs say that 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, as is demonstrated in Table 3. Considering that a direct Padé analysis of hightemperature expansion series of the specific heat generally fails to satisfy this entropy sum rule [93,96,142], 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 lowlying energy spectra of the triangular-and kagome-lattice antiferromagnets, which we shall further explain in the next closing section.

Kagome-lattice Heisenberg antiferromagnet
The low-energy Lanczos spectrum of the S =    /∂T ∂β. In (d ′ ), we denote the temperature at which the specific heat C ≡ ∂E/∂T (105) reaches its maximum by T Q , which measures 0.3526J/k B , and the temperature corresponding to the roton gap located at M in the Brillouin zone by T roton , which measures 0.7579J/k B [cf. Fig. 8(b)].
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 Fig. 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 ∂ 2 β∆F (m) l /∂T ∂β to that each are also shown separately. CLSWs of the kagome-lattice Heisenberg antiferromagnets [ Fig. 7(e)] are well known to have a dispersionless zero-energy mode [33,125], 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 [ Fig. 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 [ Fig. 7(h)]. Note that any artificially tuned SC condition, (71) without (89) but with a nonzero δ, brings no qualitative change into Fig. 7(f), i.e., induces no eigenstate to emerge below the flat band [ Fig. 7(g)]. The second constraint (89) is necessary to recover a linear Goldstone mode looking up at the floating flat band.
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 [47,144]. In a bond operator mean-field theory [144], 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 [ Fig. 8(c2)] also becomes weakly dispersive to the same extent [ Fig. 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 [Figs. 8(d2), 8(d3); (B.13) have no effect in this context, to be precise], retaining the Goldstone mode [ Fig. 8(d1)], so that a prominent lowtemperature 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 H (0) is responsible for locating the major broad maximum, while the three-boson scattering H ( 1 2 ) designs the additional low-temperature modest peak. For further details of spectral-function calculations, see Appendix B.
Once again, neither the well-pronounced low-temperature peak nor the midtemperature 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 kagomelattice-antiferromagnet specific heat. It is not the case at all with DC MLSWs [MSW up to O(S 1 ) in Fig. 6(b)] but is indeed the case with up-to-O(S 0 ) or higher DC MSWs [MSW up to O(S 0 ) and O(S −1 ) in Fig. 6(b)]. The bimodal temperature profile owes much to the second-order perturbative correction ∆F  (108). We further find that ∂β∆F (0) 1 /∂β > 0, while ∂β∆F (0) 2 /∂β < 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 Fig. 7(h ′ )], it is quite possible that bound states of antiferromagnons localized within a hexagon of nearest-neighbor spins play a significant role in reproducing the lowtemperature 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.

Triangular-lattice Heisenberg antiferromagnet
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 [ Figs. 7(a) and 7(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 [ Fig. 3(a)], unlike those in the square-lattice antiferromagnet [82]. For the square-lattice antiferromagnet, the MLSW Hamiltonian H harm (36) commutes with the total magnetization M z , so that lim T →0 [ (M z ) 2 T − M z 2 T ] becomes zero and lim T →0 χ 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), lim T →0 χ zz /L and lim T →0 χ xx /L both diverge if lim T →0 ε ±Q:1 = 0, while in (81), lim T →0 χ yy /L vanishes even if lim T →0 ε 0:1 = 0 by virtue of the vanishing factor lim T →0 (cosh 2θ 0:1 − sinh 2θ 0:1 ). Only the DC-MLSW dispersion relation Fig. 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 [111] 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 [145] 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. [111] 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 [146,147] that would be expected from a relevant semiclassical nonlinear sigma model, whereas T high originates in the quadratic roton-like excitation band [148,149].
Chakravarty, Halperin, and Nelson [146,147] 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 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 [146,147] at low temperatures with correlation length ultimately diverging exponentially as T → 0 and quantum critical behavior [146,147,150] 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-timederivative of the three-component vector field from the beginning, respectivelynonlinear sigma models suited to frustrated planar-lattice antiferromagnets assuming a noncollinearly ordered ground state, Azaria, Delamotte, and Mouhanna [151] 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 [151] still diverges as (116), where (117) withρ ⊥ s andρ s 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 [151,152] also diverges as (116), where with ρ ⊥ s and ρ s being the bare stiffnesses at T = 0 instead. Their prediction of the correlation length for the quantum model in the renormalized classical regime [151] is precisely available from a large-N expansion based on symplectic symmetry including fluctuations to order 1/N [153] as well, while that for the classical model [151,152] is well consistent with Monte Carlo simulations [154]. The bare spin stiffnesses are readily available by twisting Néel-ordered classical spins by an infinitesimal angle per lattice constant along the relevant direction [155] or indirectly obtainable through [156,157] where χ ⊥ and χ are the bare transverse and longitudinal susceptibilities at T = 0, respectively, while c Q and c 0 are the LSW velocity at the ordering momenta k = ±Q and that at k = 0, respectively, JSa .
In any case we have and then (118) reads which was indeed demonstrated by the Monte Carlo calculations on the classical Heisenberg model [154].
In order to evaluate the actual quantum mechanical correlation length, Elstner, Singh, and Young [158] calculated series for up to order 13 in powers of the inverse temperature 1/k B T and found that the quantity (k B T /J)ln(k B T ξ 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. [111] identified their T low with. However, as the authors themselves [158] pointed out, there may be a crossover to the renormalized classical behavior (116) with (117) and thus an upturn [149,154,158,159] of (k B T /J)ln(k B T ξ 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 [149,151,152,158,159,160] is much smaller than that on the square lattice [146,147,155,159] which is known to be no smaller than 0.7. It must be the consequence of such strong quantum fluctuations in the triangularlattice antiferromagnet that the higher-order quantum corrections we take account of, the lower peak temperature T Q we have [ Fig. 6(a)]. The up-to-O(S m )-MSW estimate of the peak temperature, T  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 [ Fig. 2(a)].
To end, we point out that the other temperature scale T high [111] argued in relation to roton-like excitations [101,148,149,161,162] is also available by our PC DC MSWs. The DC-MLSW and PC-DC-MSW temperature profiles of the triangularlattice-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 M ≡ 1 a π, 0, π √ 3 in the DC-MLSW excitation energy surface ε k:1 (92) into a local minimum surrounded by flat parts as well [163,164], may yield thermodynamic anomalies [101,148,149]. MSWs around the M point form in a quadratic band of rotonlike excitations above the ground state by 0.7579J ≡ k B T roton with stronger intensity than otherwise. We indicate this temperature T roton in Figs. 6(a) and 7(d ′ ). 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(S 0 ) second-order perturbative correction ∆F 2 /∂T ∂β. 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.

Further possible applications
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 specificheat 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- 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- 1 2 Heisenberg antiferromagnet on the regular kagome lattice into an ordered ground state [35,36,37,38,39]. 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 specificheat 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-oxidebased molecular spheres of the Keplerate type with spin-higher-than- 1 2 magnetic centers such as {Mo 72 [143] must be insufficient to capture its thermal features and there may be something beyond Monte Carlo calculations of its classical analog [171].

Acknowledgments
One of the authors (S. Y.) is grateful to B. Schmidt for useful pieces of information on his finite-temperature Lanczos calculations. This work is supported by JSPS KAKENHI Grant Number 22K03502.

Data availability statement
Any data that support the findings of this study are included within the article. cos φ r l − sin φ r l 0 sin φ r l cos θ r l cos φ r l cos θ r l − sin θ r l sin φ r l sin θ r l cos φ r l sin θ r l cos θ r l 1 − e (i ωn−ε k:σ )β i ω n − ε k:σ (n k:σ + 1) δ σσ ′ = δ σσ ′ i ω n − ε kν :σ .