Renormalized Classical Theory of Quantum Magnets

The development of classical and semiclassical approximations to treat quantum systems has played an important role in physics since the early days of quantum mechanics [1]. While semiclassical theories become exact when a “control parameter” $\alpha$ of the quantum mechanical Hamiltonian $\hat{\cal{H}}(\alpha)$ is sent to infinity, ${\cal H}_{\rm cl}=\lim_{\alpha\to\infty}\hat{\cal H}(\alpha)$, a vast amount of experimental evidence accumulated over many decades indicates that semiclassical approximations often remain accurate for small values of $\alpha$. This is particularly evident in quantum magnetism, where $\hat{\cal H}$ is a spin Hamiltonian and $\alpha$ is typically associated with the spin $S$ of the magnetic ions: ${\cal H}_{\rm cl}=\lim_{S\to\infty}\hat{\cal H}(S)$. Although $S$ is of order one for most quantum magnets, semiclassical approaches are the standard tool to describe these materials because of their extraordinary success in reproducing experimental observations, such as the collective modes of magnetic materials. Indeed, a very active area of quantum magnetism is the search for materials that exhibit strong deviations from semiclassical theories. While deviation is the rule for most quasi-1D magnets, finding examples of quasi-2D and 3D magnets that fall into this category is challenging ¹¹1Large deviations from semiclassical behavior in 2D or 3D materials generally arise for low-$S$ materials, and near points where the classical spin Hamiltonian exhibits extensive ground state degeneracy.

Given the importance of classical methods, we must observe that the classical limit (CL) of a spin Hamiltonian $\hat{\cal H}$ is not unique, and the correct choice of CL for a given system is crucial. In particular, when $\hat{\cal H}$ contains nonlinear terms in the spin operators of a given site, the traditional large-$S$ CL introduces systematic errors that can be avoided by the use of an alternative limit. This has wide-ranging consequences. For example, Landau-Lifshitz (LL) dynamics and spin wave theory (SWT), both based on the large-$S$ limit, are used extensively in the inverse modeling problem of extracting $\hat{\cal H}$ from scattering data. These approaches underestimate the magnitudes of nonlinear contributions, such as single-ion (SI) anisotropies or biquadratic interactions, by a substantial amount when $S$ is small. Additionally, many magnetic phenomena of current interest, such as skyrmions, are essentially classical in nature, and classical methods are widely used to produce phase diagrams [3]. We will show that these phase diagrams exhibit an $S$-dependence that is lost in the large-$S$ CL. Finally, the choice of adequate CL is essential as the starting point for developing semiclassical approaches, such as SWT and its generalizations.

Here we present an alternative CL appropriate for a wide range of Hamiltonians, namely those for which bilinear (e.g., exchange) interactions are dominant, but which also possess comparatively weak SI anisotropies or other terms that are nonlinear in the spin operators. This alternative limit does not take $S$ as the control parameter but instead uses $\lambda_{1}$, which labels the degenerate irreps of SU$(N)$. Correspondingly, quantum corrections are organized in powers of $1/\lambda_{1}$ instead of $1/S$, and the classical limit is obtained by sending $\lambda_{1}$ to infinity instead of $S$. The resulting Renormalized Classical Spin (RCS) Hamiltonian is identical to the traditional classical Hamiltonion except that nonlinear terms have been renormalized by coefficients expressed in powers of $1/S$. These $1/S$ factors do not emerge from the introduction of higher-order quantum corrections; instead, they are the consequence of group theoretical considerations when comparing two different classical limits. Application of these renormalizations factors is straightforward and yields a classical theory that precisely recovers the quantum expectation value of nonlinear terms with respect to any dipolar spin state.

Renormalized Classical Theory. Classical and semiclassical approximations are based on coherent states, which link the quantum and classical worlds. The coherent states of a Lie group are obtained by applying the group elements to a reference state known as the highest weight state, and the resulting manifold of coherent states constitutes the phase space of the resulting classical theory [4, 5]. Quantum spin systems admit more than one CL since there is freedom to choose different Lie groups, each of which generates a different set of coherent states. A natural choice is the group of spin rotations, SU$(2)$, which leads to the traditional dipole-only CL of quantum spin systems. The resulting phase space of coherent states $|\Omega_{j}\rangle$ ($j$ is a lattice site index) is the 2D sphere, $S^{2}\cong$ CP¹, which represents different possible orientations of dipole moments $\vec{\Omega}_{j}=\langle\Omega_{j}|\hat{\bm{S}}_{j}|\Omega_{j}\rangle$ ($|\vec{\Omega}_{j}|=S$). The CL of the Hamiltonian $\hat{\cal{H}}$ is obtained by evaluating the expectation value of $\hat{\mathcal{H}}$ with respect to an arbitrary product of SU($2$) coherent states in the large-$S$ limit, i.e., by sending the spin irreps of SU$(2)$ to infinity,

$${\cal H}_{\rm cl}[\vec{\Omega}]=\lim_{S\to\infty}\langle\vec{\Omega}|\hat{\cal H}(S)|\vec{\Omega}\rangle$$

(1)

with $|\vec{\Omega}\rangle\equiv\bigotimes_{j}|\vec{\Omega}_{j}\rangle$. The dynamics associated with the classical Hamiltonian are obtained by considering the Heisenberg equations of motion in the same limit, yielding

$$\dot{\vec{\Omega}}_{j}=\frac{\partial\cal{H}_{\rm cl}}{\partial\vec{\Omega}_{j}}\times{\vec{\Omega}}_{j}.$$

(2)

This is the well-known LL equation [6, 7]. Gilbert extended this dynamics with the introduction of damping in 1954 [8], and the resulting LL-Gilbert (LLG) equation is now a fundamental tool in applied magnetism [9, 10, 11].

An alternative CL [12, 13, 14, 15, 16], relevant when $S>1/2$, is obtained by considering coherent states of SU($N$) with $N=2S+1$: $|\Psi_{j}\rangle$  ²²2 Note the use of SU$\left(N\right)$ coherent states does not require an SU$(N)$-invariant Hamiltonian. Instead, SU$(N)$ appears as the group of possible time-evolution operators of an approximate local Hamiltonian. See SM.. Just as an SU(2) coherent state may be uniquely associated with a 3-vector of dipole moments, an SU($N$) coherent state may be uniquely associated with a $(N^{2}-1)$-vector of dipole and multipole moments, $\Psi_{j}^{\alpha}=\langle\Psi_{j}|\hat{T}^{\alpha}|\Psi_{j}\rangle$, where $\hat{T}^{\alpha}$ are generators of SU($N$). These generators may be selected such that $\hat{T}^{\alpha}=\hat{S}^{\alpha}$ for $\alpha=1\ldots 3$. To take the classical limit, the expectation value of $\hat{\mathcal{H}}$ is evaluated with respect to an arbitrary SU($N$) coherent state in the $\lambda_{1}\to\infty$ limit, where $\lambda_{1}$ labels irreps of SU($N$) ³³3$\lambda_{1}$ is the only finite eigenvalue when the maximal weight state is applied to the Cartan subalgebra. As with $S$ in the case of SU$(2)$, $\lambda_{1}$ is uniquely associated with the dimension of an irrep of SU$(N)$. See [29, 12] for details.,

$${\cal H}_{\rm cl}^{{\rm SU}(N)}[\vec{\Psi}]=\lim_{\lambda_{1}\to\infty}\langle\vec{\Psi}|\hat{\cal H}(\lambda_{1})|\vec{\Psi}\rangle=\langle\vec{\Psi}|\hat{\cal H}|\vec{\Psi}\rangle$$

(3)

with $|\vec{\Psi}\rangle\equiv\bigotimes_{j}|\vec{\Psi}_{j}\rangle$ . The final equality holds because the expectation value in the $\lambda_{1}\to\infty$ limit is the same as the expectation value in the fundamental represention [19]. In other words, the classical Hamiltonian is the exact expectation value with respect to an SU($N$) coherent state. Note that $\hat{\mathcal{H}}$ retains its dependence on $S$, but $S$ is not used as the control parameter in the limiting procedure. The local phase space of this classical theory is CP^2S rather than CP¹.

The set of SU$(2)$ coherent states, $|\vec{\Omega_{j}}\rangle$, is a submanifold of the set of SU$(N)$ coherent states, $|\vec{\Psi_{j}}\rangle$. This submanifold is simply the set of states obtained by applying group actions generated by the spin operators to the highest weight state while disregarding the additional generators in the algebra of SU$(N)$. It is therefore natural to define a constrained version of $\mathcal{H}^{{\rm SU}(N)}_{\rm cl}$ by limiting the classical Hamiltonian obtained in the large-$\lambda_{1}$ limit to the coherent states of SU$(2)$:

$${\cal\tilde{H}}_{\rm cl}[\vec{\Omega}]=\lim_{\lambda_{1}\to\infty}\langle\vec{\Omega}|\hat{\cal H}(\lambda_{1})|\vec{\Omega}\rangle=\langle\vec{\Omega}|\hat{\cal H}|\vec{\Omega}\rangle.$$

(4)

This is will be referred to as the renormalized classical spin (RCS) Hamiltonian. By construction, the phase space of ${\cal\tilde{H}}_{\rm cl}$ is the same as that of $\mathcal{H}_{\rm{cl}}$, namely the space of SU($2$) coherent states. Moreover, the associated dynamics will be that given in Eq. (2) [19], with the substitution $\mathcal{H}_{\rm cl}\to\tilde{\mathcal{H}}_{\rm cl}$. ⁴⁴4The full, unconstrained SU$(N)$ theory [12] becomes relevant when anisotropies or other nonlinear terms are strong relative to the linear ones. In this case, the classical spin in the large-$\lambda_{1}$ limit (an SU(N) coherent state) no longer exhibits a rigid dipole and the restricted classical phase space of SU$(2)$ coherent states, the 2-sphere, no longer captures all the relevant degrees of freedom. In other words, both the traditional large-$S$ approach and the RCS theory become bad approximations.

The significance of defining this Hamiltonian becomes clear when one considers the different behavior of the two limits on nonlinear terms. In the $S\to\infty$ limit, an important simplification arises in the factorization of the expectation value of a product of on-site operators into the product of the expectation values of each individual operator:

	$\displaystyle\lim_{S\to\infty}\langle\vec{\Omega}|\hat{S}^{\mu}_{j}\hat{S}^{\nu}_{j}\dots\hat{S}^{\eta}_{j}|\vec{\Omega}\rangle$	$\displaystyle=$	$\displaystyle\langle\vec{\Omega}|\hat{S}^{\mu}_{j}|\vec{\Omega}\rangle\langle\vec{\Omega}|\hat{S}^{\nu}_{j}|\vec{\Omega}\rangle\cdots\langle\vec{\Omega}|\hat{S}^{\eta}_{j}|\vec{\Omega}\rangle$		(5)
		$\displaystyle=$	$\displaystyle\Omega^{\mu}_{j}\Omega^{\nu}_{j}\dots\Omega^{\eta}_{j}.$

However, to describe a quantum mechanical system with a finite value of $S$ that includes nonlinear terms in the components of an on-site spin operator $\hat{\bm{S}}_{j}$, Eq. (5) becomes an extra approximation on top of the one made in Eq. (1). We can obtain a more accurate classical approximation of $\hat{\cal H}(S)$ if we we can avoid this step, which is exact only in the $S\to\infty$ limit.

In contrast, Eq. (4) shows that in the RCS theory the resulting Hamiltonian is exactly the quantum expectation value with respect to a finite-$S$ SU(2) coherent state. The two limits will differ precisely when the $S\to\infty$ limit demands application of the factorization rule

The RCS theory is simply the full SU($N$) theory constrained to the dipole sector. It is applicable when the only collective modes of the Hamiltonian are dipole fluctuations, which is typical when the exchange and Zeeman terms are strong relative to the nonlinear terms. When this condition is satisfied, the full SU($N$) theory will predict extra, non-dipolar modes that contribute little or no intensity to the dynamical spin structure factor. Moreover, these modes are overdamped (via loop corrections to the SU($N$) linear SWT) by the two-magnon continuum. The RCS theory will not produce these spurious modes. Further, the RCS Hamiltonian may be derived by applying a simple renormalization to ${\cal H}_{\rm cl}$.

Applying the RCS Theory Consider a broad class of spin Hamiltonians containing both interaction and single ion terms,

$$\hat{\mathcal{H}}=\underset{\textrm{Interaction}}{\underbrace{\hat{\mathcal{H}}^{\mathrm{Bil}}+\hat{\mathcal{H}}^{\mathrm{Biq}}}}+\underset{\textrm{SI}}{\underbrace{\hat{\mathcal{H}}^{\mathrm{Z}}+\hat{\mathcal{H}}^{\mathrm{A}}}}.$$

(6)

Each term will be expressed as polynomials of operators $\hat{\bm{S}}_{j}$ that may represent spin degrees of freedom for magnetic ions with weak spin-orbit interaction, or the total angular momentum $\hat{\bm{J}}_{j}$ for magnetic ions with strong spin-orbit coupling.

The dominant interactions are typically bilinear,

$$\hat{\cal{H}}^{\rm Bil}=\frac{1}{2}\sum_{i\neq j}\hat{S}^{\mu}_{i}{\cal J}^{\mu\nu}\hat{S}^{\nu}_{j}.$$

(7)

implying that they are linear in the spin operators of each site. Thus, taking the classical limit – both large-$S$ and RCS – amounts to the substitution of each spin operator by a spin component: $\hat{S}^{\mu}_{j}\rightarrow\langle\Omega_{j}|S^{\mu}_{j}|\Omega_{j}\rangle=\Omega^{\mu}_{j}$. The same applies to the Zeeman term, $\hat{\cal{H}}^{\rm Z},$ which is linear in $\hat{S}^{\mu}_{j}$.

Biquadratic interactions, $\hat{\mathcal{H}}^{\rm Biq}$, and SI anisotropies, $\hat{\mathcal{H}}^{\rm A}$, require special treatment. First consider

$$\hat{\cal{H}}^{\rm A}=\sum_{j,q,k}A_{q}^{(k)}\hat{O}_{q}^{(k)}({\hat{\bm{S}}}_{j}),$$

(8)

which includes only even powers of the spin operators due to time-reversal invariance. It is expressed as a linear combination of Stevens operators, $\hat{O}_{q}^{(k)}$, which span the $k$-irrep of SO(3). The coefficients, $A_{q}^{(k)}$, are the crystal field parameters. The traditional CL of a Stevens operator is $\lim_{S\rightarrow\infty}\langle\Omega|\hat{O}_{q}^{(k)}|\Omega\rangle$, and the RCS limit is $\langle\Omega|\hat{O}_{q}^{(k)}|\Omega\rangle$, where the coherent state is given in the spin-$S$ representation with $S$ finite. Both may be expressed as functions of the two angles that parameterize the SU$(2)$ coherent state $|\Omega\rangle$, and both will transform according to the same irrep. It follows that they are proportional. In general we have

$$\tilde{\cal{H}}^{\rm A}_{\rm cl}=\sum_{j,q,k}\tilde{A}_{q}^{(k)}{\cal O}_{q}^{(k)}(\vec{\Omega}_{j})$$

(9)

with $\tilde{A}_{q}^{(k)}=c^{(k)}A_{q}^{(k)}$. The proportionality constants are [19],

	$\displaystyle c^{\left(2\right)}$	$\displaystyle=$	$\displaystyle 1-\frac{1}{2}S^{-1},\quad c^{\left(4\right)}=1-3S^{-1}+\frac{11}{4}S^{-2}-\frac{3}{4}S^{-3}$
	$\displaystyle c^{\left(6\right)}$	$\displaystyle=$	$\displaystyle 1-\frac{15}{2}S^{-1}+\frac{85}{4}S^{-2}-\frac{225}{8}S^{-3}+\frac{137}{8}S^{-4}-\frac{15}{4}S^{-5}$

As expected, $\lim_{S\to\infty}\tilde{\cal{H}}^{\rm A}_{\rm cl}={\cal{H}}^{\rm A}_{\rm cl}$ because $\lim_{S\to\infty}c^{(k)}=1$, but these renormalization factors become significant for low spin values. For instance,

$$c^{\left(2\right)}(S=1)=1/2,\;c^{\left(4\right)}(S=2)=3/32,\;c^{\left(6\right)}(S=3)=5/324.$$

(10)

Thus, if a spin-1 quadratic SI anisotropy is approximated classically taking the $S\to\infty$ limit, the anisotropy strength will be underestimated by a factor $1/c^{(2)}=2$. Errors become even more severe for higher-order SI anisotropies.

The remaining term in Eq. (6) corresponds to biquadratic or higher-order interactions, which may be significant in Mott insulators that are not deep inside the Mott regime or in $f$-electron magnets where the spin-orbit coupling is comparable to the intra-atomic Coulomb interaction. As a simple illustration, consider the isotropic biquadratic interaction,

$$\hat{\cal{H}}^{\rm Biq}=\frac{1}{2}\sum_{i\neq j}{\cal K}_{ij}(\hat{\bm{S}}_{i}\cdot\hat{\bm{S}}_{j})^{2}.$$

(11)

After evaluating $\hat{\cal{H}}^{\rm Biq}$ in the $S\rightarrow\infty$ limit and comparing the result to its expectation value with respect to a finite-$S$ coherent state [19], the RCS Hamiltonian is found to be

$$\tilde{\cal{H}}^{\rm Biq}_{\rm cl}=\frac{1}{2}\sum_{i\neq j}{\cal K}_{ij}\left[r(\vec{\Omega}_{1}\cdot\vec{\Omega}_{2})^{2}-\frac{1}{2}\vec{\Omega}_{1}\cdot\vec{\Omega}_{2}+S^{3}+\frac{S^{2}}{4}\right]$$

(12)

with $r=\left(1-\frac{1}{S}+\frac{1}{4S^{2}}\right).$ Besides being renormalized, the biquadratic interaction generates a bilinear term that is absent in the large-$S$ limit and is comparable in amplitude to the biquadratic interaction for $S=1$. Moreover, the renormalization factor becomes $r(S=1)=1/4$, implying that the amplitude of the biquadratic term is 4 times smaller than $r(S\to\infty)=1$.

$S$-dependent Phase Diagrams. If ${\cal H}_{\rm cl}\neq\tilde{\cal H}_{\rm cl}$, the RCS Hamiltonian will produce an $S$-dependent thermodynamic phase diagram that coincides with the usual classical phase diagram only in the $S\to\infty$ limit. This observation has many implications for real magnets, which we illustrate with a model that has been extensively studied in the context of magnetic skyrmion crystals [3]. The spin-$S$ Hamiltonian,

$$\hat{\mathcal{H}}=\frac{1}{2}\sum_{i\neq j}J_{ij}\hat{\bm{S}}_{i}\cdot\hat{\bm{S}}_{j}-h\sum_{i}\hat{S}_{i}^{z}-\frac{K}{2}\sum_{i}(\hat{S}_{i}^{z})^{2},$$

(13)

includes ferromagnetic nearest-neighbor exchange interactions and antiferromagnetic next-nearest-neighbor interactions on a triangular lattice: $J_{ij}=J_{1}<0$ ($J_{ij}=J_{2}>0$) for nearest (next-nearest) neighbors spins $i$ and $j$ and $J_{ij}=0$ otherwise. The second and third terms are, respectively, Zeeman coupling to a magnetic field along the $z$-axis and easy-axis ($K>0$) SI anisotropy. Leonov and Mostovoy [3] used a variational scheme to compute the classical phase diagram of this model, shown in Fig. 1, by taking the large-$S$ limit, i.e., by using the classical Hamiltonian ${\cal H}_{\rm cl}$. For sufficiently small magnetic fields, the lowest energy state of $\mathcal{H}_{\rm cl}$ is the “vertical spiral” (VS) phase with a polarization plane parallel to the $z$-axis and a propagation wave vector $\pm{\bm{Q}_{\nu}}$ ($\nu=1,2,3$) parallel to the three possible directions related by 120^∘ rotations about the $z$-axis. Several multi-${\bm{Q}}$ orderings corresponding to a superposition of more than one spiral are induced upon increasing $h$, as described in detail in Ref. [3]. The most interesting phase is the triple-${\bm{Q}}$ skyrmion crystal (SkX) that extends over the interval $0.05\lesssim K/|J_{1}|\lesssim 0.6$ However, if we use the RCS theory,

$$\tilde{\mathcal{H}}_{\rm cl}=\frac{1}{2}\sum_{i\neq j}J_{ij}{\vec{\Omega}}_{i}\cdot{\vec{\Omega}}_{j}-h\sum_{i}{\Omega}_{i}^{z}-\frac{\tilde{K}(S)}{2}\sum_{i}(\hat{\Omega}_{i}^{z})^{2},$$

(14)

with $\tilde{K}=K(1-\frac{1}{2S})$, the range of stability of the SkX phase is $S$-dependent. In particular, for $S=1$ the SkX phase is stable over a range of $K$ values that is twice as large as the range obtained for $S\to\infty$, i.e., lowering the spin of the magnetic ions becomes a guiding principle to find more robust SkXs in centrosymmetric materials. It is also noteworthy that exactly the same phase diagram is obtained for $S=1$ if we use direct products of coherent states of SU(3) as a variational space [12]. Finally, note that the only difference between $\tilde{\mathcal{H}}_{\rm cl}$ and $\mathcal{H}_{\rm cl}$ is a renormalization of the SI anisotropy. More drastic differences between the phase diagrams of the two classical models, such as the presence of different phases, can be expected if more than one Hamiltonian parameter is renormalized.

More Accurate Dynamics. Besides producing different phase diagrams, $\tilde{\mathcal{H}}_{\rm cl}$ and $\mathcal{H}_{\rm cl}$ will in general lead to different dispersion relations of the normal modes (spin-waves) of a given phase. The linear SWT (LSWT) is obtained by quantizing the harmonic oscillators of each normal mode: the spin-waves of the CL become magnons of the quantum mechanical theory. Since Hamiltonian parameters are typically extracted by fitting the magnons measured with INS, using the renormalized classical Hamiltonian is critical to extract correct values. Consider the fully polarized phase of the Hamiltonian Eq. (14). This has the exact single-magnon dispersion

$$\omega_{\bm{k}}=S[{\cal J}({\bm{k}})-{\cal J}({\bm{0}})]+h+\tilde{K}S,$$

(15)

where ${\cal J}({\bm{k}})=\sum_{j}e^{i{\bm{k}}\cdot{\bm{r}}_{ij}}J_{ij}$. This coincides with the dispersion relation that is obtained from $\tilde{\mathcal{H}}_{\rm cl}$, but differs from the result $\omega_{\bm{k}}=S[{\cal J}({\bm{k}})-{\cal J}({\bm{0}})]+h+KS$ that is obtained in the large-$S$ limit. In other words, the value of the actual SI anisotropy is $(1-1/2S)^{-1}$ times bigger than the value obtained by fitting the magnon dispersion with the unrenormalized LSWT, giving a relative correction of 100$\%$ for $S=1$ and still $17\%$ for $S=7/2$. Moreover, from Eq. (10), in the presence of quartic ($q=4$) and sixth ($q=6$) order SI anisotropy terms, the unrenormalized LSWT predicts amplitudes that are of order $10$ and $100$ times bigger than the actual values for the lowest values of $S$ compatible with these anisotropies.

Progress in quantum magnetism relies heavily on inelastic neutron scattering (INS) data. The recent development of performant LSWT codes (e.g., SpinW) has been instrumental in opening the bottleneck between observables and model Hamiltonians, providing crucial information closer to the beginning of the materials life cycle [21]. Nearly 20% of reports from the last year that make reference to SpinW include Hamiltonians with nonlinear terms in site spin operators. More accurate values may be obtained using either the renormalization factors $c^{(k)}$ or the recipe for generation of non-tabulated terms. Looking forward, estimates should be made either using the appropriate SU($N$) LSWT or the renormalized SU(2) LSWT to exctact more accurate models. The open-source code Sunny provides an implementation of the RCS theory ⁵⁵5https://github.com/SunnySuite/Sunny.jl.

Previous works have used Bose operator expansions to renormalize SU($2$) LSWT for spin Hamiltonians with SI anisotropy terms. In 1961, Oguchi and Honma performed a $1/S$ expansion of the square root that appears in the Holstein-Primakoff transformation to order $1/S^{0}$ and in this way derived the $(1-1/2S)$ renormalizalization factor of the quadratic SI anisotropy [23]. In 1976, Kowalska and Lindgård found the same renormalization using a generalized crystal-field Hamiltonian with an expansion parameter of the crystal field strength divided by the exchange field strength [24]. Later efforts have echoed these results, whether using Holstein-Primakoff [25] or Dyson-Maleev [26] transformations. However, since these works focused on the linearized dynamics (LSWT), the renormalization of the underlying classical theory and the corresponding non-linear LL equations were not discussed. What remained hidden is that the renormalization of the linearized LSWT Hamiltonian arises from a more fundamental renormalization of the classical Hamiltonian. The failure to recognize this fact presents a danger when adding only some $1/S$ corrections to LSWT calculations. Specifically, the renormalization presented here can change the classical ground state that is used as the starting point for a spin-wave calculation. If $1/S$ corrections are added to the LSWT calculation only, and not to the calculation of the classical ground state, LSWT dispersion calculations may fail, either by gapping out Goldstone modes or by predicting unphysical, imaginary frequencies.

Recall that $\tilde{\cal{H}}_{\rm{cl}}$ is obtained by restricting the generalized dynamics of SU$(2S+1)$ coherent states (phase space CP^2S) to the submanifold of dipoles (SU(2) coherent states in CP¹) Since CP^2S contains $2S$ pairs of conjugate coordinates and momenta, the number of normal modes per spin is $2S$. The quantization of these normal modes is implemented by introducing Holstein-Primakoff bosons $b_{j\nu}$ with $2S$ flavors ($1\leq\nu\leq 2S$). The restriction of the linearized dynamics to the plane tangent to the the CP¹ phase space of SU(2) coherent states is equivalent to projecting out the $2S-1$ bosons representing fluctuations that are orthogonal to this plane.

Quantum corrections to the generalized spin-wave Hamiltonian are implemented via a loop expansion for the propagators of the $2S$ bosons (one for each flavor). This loop expansion is actually an expansion in $1/\lambda_{1}$, where $\lambda_{1}$ labels degenerate irreps of SU($2S+1$) (generalization of the $1/S$ expansion) [27, 12]. The restricted dynamics is obtained by keeping only the bosonic propagator for the magnon modes and integrating out the remaining modes. ⁶⁶6The internal lines of the diagrammatic expansion of this propagator can still have any flavor

These observations reveal that the RCS Hamiltonian and the corresponding renormalized SWT (including quantum corrections) is a projection of a generalized SWT, the “control parameter” of which is $\lambda_{1}$ instead of $S$. Thus, it is not surprising that the renormalization factors contain higher order corrections in $1/S$. The most important corollary is that the renormalized SWT with the above-mentioned quantum corrections still preserves the Goldstone modes of theories that spontaneously break continuous symmetries because these symmetries are preserved to each order of the $1/\lambda_{1}$ expansion.

In summary, we have provided a general renormalization procedure which is essential for deriving accurate classical and semiclassical approximations of spin Hamiltonians with nonlinear terms. While the procedure was illustrated with the most common SI anisotropy terms and isotropic biquadratic interactions, more general cases, such as anisotropic four-spin interactions, can naturally appear in materials with strong spin-orbit coupling, such as $4f$-electron systems. Since semi-classical methods are the most common approximation used for solving the inverse scattering problem, the renormalization procedure presented here is necessary for extracting correct Hamiltonian parameters from INS data. In particular, the traditional method of fitting the measured INS data with LSWT (large-$S$ limit) can lead to serious inconsistencies between the neutron-derived parameters and those derived from other measurements.

Acknowledgements We thank Martin Mourigal, Alan Tennant and Xiaojian Bai for helpful discussions. This work was funded by the U.S. Department of Energy, Office of Science, 631 Office of Basic Energy Sciences, under Award No. DE-SC-DE-SC-0018660. K. B. and H. Z. acknowledge support from the LANL LDRD program. Z. L. (phase diagram of Fig. 1) acknowledges support from U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Science and Engineering Division. D. P. acknowledges support by the DOE Office of Science (Office of Basic Energy Sciences). This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This manuscript has been authored by UT-Batelle, LLC, under contract DEAC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doepublic-access-plan).