Classical SO(n) Spins on Geometrically Frustrated Crystals: A Real-Space Renormalization Group Approach

Abstract

A real-space renormalization group (RG) framework is formulated for classical SO(n) spin models defined on d-dimensional crystal lattices composed of corner-sharing hyper-tetrahedra, a class of geometrically frustrated crystal structures. This includes, as specific instances, the classical Heisenberg model on the kagome and pyrochlore crystals. The approach involves computing the partition function and corresponding order parameters for spin clusters embedded in the crystal, to leading order in symmetry-breaking fields generated by surrounding spins. The crystal geometry plays a central role in determining the scaling relations and the associated critical behavior. To illustrate the efficacy of the method, a reduced manifold of symmetry-allowed ordered states for isotropic nearest-neighbor interactions is analyzed. The RG flow systematically excludes the emergence of a $\overrightarrow{q}=\mathbf{0}$ ordered phase within the antiferromagnetic sector, independently of both the spatial dimensionality of the crystal and the number of spin components. Extensions to incorporate more elaborate crystal-symmetry-induced ordering patterns and fluctuation-driven phenomena—such as order-by-disorder—are also discussed.

1. Introduction

Geometrically frustrated antiferromagnets exhibit a wide range of exotic phases at low temperatures [1,2,3,4], which has sustained considerable interest in the field from both the theoretical and experimental points of view [5,6,7,8,9,10,11,12,13,14]. Recent research has increasingly focused on quantum spin models on frustrated lattices—particularly kagome and pyrochlore geometries—as promising platforms for realizing quantum spin liquids [15,16] and chiral magnetic order [17]. At the same time, classical models continue to provide rich insights, with emergent behaviors such as magnetic monopoles and Coulomb phases in classical spin ice compounds attracting attention in both experiments and theory [18,19]. Advances in renormalization group methods—such as pseudofermion and pseudo-Majorana functional RG [7,16]—have significantly improved our ability to analyze frustrated magnets beyond mean-field approximations. Additionally, analytical and numerical efforts have broadened the scope of classical models, including the development of real-space RG schemes [20] and their application to high-dimensional or large-n limits [21]. These approaches help to elucidate critical behavior and ordering mechanisms in systems with extensive degeneracy, where conventional theories often fail.

Recent developments in the study of frustrated magnetic systems have further emphasized the rich phenomenology arising from competing interactions and lattice topology, particularly in geometrically frustrated networks. Advances in numerical methods, such as large-scale Monte Carlo simulations [22,23], tensor network approaches [24], and machine learning-assisted phase classification [25], have opened new avenues to investigate both classical and quantum models beyond mean-field approximations. These techniques have confirmed and expanded the understanding of exotic phases such as classical and quantum spin liquids, Coulombic phases, and unconventional orders. Of particular relevance is the continued interest in understanding the role of thermal and quantum fluctuations in systems with extensive ground-state degeneracy, as well as the mechanisms behind order-by-disorder selection in various lattice geometries [26]. Moreover, renewed efforts to construct real-space renormalization group frameworks in higher dimensions [27] align closely with the goals of this work, reinforcing the importance of developing analytical tools applicable to generic $𝒮𝒪\left(n\right)$ spins on lattices of arbitrary dimensionality and coordination.

The origin of the exotic behavior observed in geometrically frustrated lattices lies in their intrinsic topology, which gives rise to a macroscopically degenerate ground state at the classical level when only nearest-neighbor (NN) exchange interactions are considered. In such systems, the exchange constant no longer sets the relevant energy scale, and the system remains in a spin-liquid-like state at all finite temperatures. Consequently, even weak perturbations can lift the degeneracy and induce (unconventional) ordered states. From a theoretical standpoint, handling this macroscopic degeneracy is a nontrivial challenge. The geometric structure of these lattices further complicates the problem. For instance, it is well known that standard Weiss mean-field theory fails to distinguish between a pyrochlore lattice and a simple cubic one, incorrectly predicting a second-order transition at a temperature on the order of the exchange coupling constant.

Improved mean-field theories, such as the foundational approach proposed by Reimers et al. [28], correctly capture the distinct features of the pyrochlore lattice and provide criteria for the emergence of ordered states, including the so-called $\overrightarrow{q}=\mathbf{0}$ order and more complex incommensurate structures. In recent years, further refinement has come from mean-field techniques based on cluster or tree expansions [29,30,31,32], which have proven effective in both two and three dimensions and have enabled the identification of a range of novel ordered states. Despite the significant progress achieved through these methods, they all share a common limitation: they are ultimately grounded in mean-field theory. As a result, regardless of their sophistication, they yield classical mean-field critical exponents and are typically tailored to specific cases (e.g., Ising spins on the pyrochlore lattice in spin ice models, or Heisenberg and XY spins in other contexts). A general theoretical framework—based on renormalization group principles—that can address arbitrary spin dimensionalities and lattice geometries remains largely undeveloped.

In this work, we develop a real-space renormalization group approach—specifically, a generalization of the Effective Field Renormalization Group (EFRG) technique—to study classical $𝒮𝒪\left(n\right)$ vector spin models on d-dimensional lattices of corner-sharing d-simplexes. These lattices generalize well-known frustrated geometries such as the kagome and pyrochlore networks and are characterized by a non-Bravais structure with a basis of $d+1$ sublattices. The method we present allows for the analysis of critical behavior across arbitrary dimensions and spin component numbers, providing a unified framework for studying ordered and disordered phases in geometrically frustrated systems.

This work expands upon and refines the theoretical framework introduced in earlier studies by the author. Notably, Ref. [33] presented an initial application of the EFRG technique to classical $𝒮𝒪\left(n\right)$ antiferromagnets on d-dimensional corner-sharing hyper-tetrahedral lattices. Although that study concentrated primarily on identifying phenomenological features of the critical behavior, the present manuscript provides a more detailed and rigorous analytical foundation for the method, including many detailed calculations in various appendixes. Additionally, this work incorporates insights from subsequent investigations involving quantum spin systems on similar lattice geometries [34], thereby furnishing a potential way to bridge the gap between classical and quantum descriptions of geometrically frustrated magnets. In particular, we derive explicit expressions for the reduced partition functions and systematically construct the scaling equations required to identify possible ordered phases. As such, this contribution establishes a broader and more general formalism capable of addressing classical vector spin models in arbitrary dimensions and with varying spin component numbers.

The remainder of this paper is organized as follows. In Section 2, we introduce the classical $𝒮𝒪\left(n\right)$ spin model defined on d-dimensional lattices constructed from corner-sharing d-simplexes and discuss its geometric properties. Section 3 presents the real-space renormalization group framework we employ, namely the EFRG method, and derives the associated scaling relations. In Section 4, we compute the reduced partition functions of spin clusters of different sizes and extract the corresponding order parameters. Section 5 analyzes the critical behavior of the model and the conditions under which ordered phases, including the $\overrightarrow{q}=\mathbf{0}$ state, may arise. Finally, in Section 6, we summarize the main results and outline directions for future work.

2. The Model

As stated above, the aim of this work is to develop a real-space renormalization group framework to investigate the critical behavior of classical $𝒮𝒪\left(n\right)$ spins—classical vector spins with n components that transform under the $𝒮𝒪\left(n\right)$ group—on d-dimensional lattices composed of corner-sharing hyper-tetrahedra. Geometrically, a d-dimensional hyper-tetrahedron, also known as the regular d-simplex, is the convex hull of $d+1$ equidistant points. In the present context, this means that each of the $d+1$ vertices is a nearest neighbor to every other vertex within the same simplex.

Well-known examples of $𝒮𝒪\left(n\right)$ spin systems include the classical XY model ($n=2$) and the Heisenberg model ($n=3$). Corresponding examples of d-simplexes are the 1-simplex (a line segment) in one dimension, the 2-simplex (an equilateral triangle) in two dimensions, and the 3-simplex (a regular tetrahedron) in three dimensions. When arranged in corner-sharing configurations, these simplexes form the one-dimensional chain, kagome lattice ($d=2$), and pyrochlore lattice ($d=3$).

Due to their non-Bravais character, these lattices possess a basis of $d+1$ spins per unit cell, corresponding to the vertices of a single simplex. Consequently, the full lattice can be constructed from $d+1$ interpenetrating face-centered cubic (FCC) Bravais sublattices, which are mutually shifted by all possible permutations of the components of the d-dimensional vector $(\frac{1}{4},\frac{1}{4},0,0,\dots ,0)$. Each spin in the system can therefore be uniquely labeled by a pair of indices: one for the simplex (i) and one for the sublattice ($\alpha$) to which it belongs.

Specifically, a spin ${\overrightarrow{s}}_{i\alpha }=(s_{i\alpha }^{\left(1\right)},s_{i\alpha }^{\left(2\right)},\dots ,s_{i\alpha }^{\left(n\right)})$ is a unit vector ($|{\overrightarrow{s}}_{i\alpha }|=1$) located at sublattice site $\alpha =1,2,\dots ,d+1$ within the i-th simplex ($i=1,\dots ,N$), where $N(d+1)$ is the total number of spins in the lattice.

The Hamiltonian governing the system reads as follows:

$$\tilde{\mathcal{H}}=J_1\sum _{〈i\alpha ,j\beta 〉}{\overrightarrow{s}}_{i\alpha }·{\overrightarrow{s}}_{j\beta }+J_2\sum _{〈〈i\alpha ,j\beta 〉〉}{\overrightarrow{s}}_{i\alpha }·{\overrightarrow{s}}_{j\beta }-\sum _{i,\alpha }{\overrightarrow{s}}_{i\alpha }·{\overrightarrow{h}}_0,$$

(1)

where $J_1$ and $J_2$ are the nearest-neighbor and next-nearest-neighbor (NNN) exchange couplings, respectively; $J_1>0$ corresponds to antiferromagnetic interactions, and $J_1<0$ to ferromagnetic ones; ${\overrightarrow{h}}_0$ is the external magnetic field; $\langle ·\rangle$ denotes summation over NN pairs, and $\langle \langle ·\rangle \rangle$ over NNN pairs.

It is customary to define a dimensionless Hamiltonian by dividing $\tilde{\mathcal{H}}$ by $k_{\mathrm{B}}T$, where T denotes the temperature:

$$\mathcal{H}=K_1\sum _{〈i\alpha ,j\beta 〉}{\overrightarrow{s}}_{i\alpha }·{\overrightarrow{s}}_{j\beta }+K_2\sum _{〈〈i\alpha ,j\beta 〉〉}{\overrightarrow{s}}_{i\alpha }·{\overrightarrow{s}}_{j\beta }-\sum _{i,\alpha }{\overrightarrow{s}}_{i\alpha }·{\overrightarrow{H}}_0,$$

(2)

where $K_1=J_1/k_{\mathrm{B}}T$ and $K_2=J_2/k_{\mathrm{B}}T$ are the dimensionless NN and NNN exchange couplings, respectively; $K_1>0$ corresponds to antiferromagnetic interactions, and $K_1<0$ to ferromagnetic ones; ${\overrightarrow{H}}_0={\overrightarrow{h}}_0/k_{\mathrm{B}}T$ is the dimensionless external magnetic field.

An important feature of these lattices is that both NN and NNN of a given spin ${\overrightarrow{s}}_{i\alpha }$ always belong to different sublattices. This is a direct consequence of the simplex geometry. However, this property does not extend to higher-order neighbors: for instance, third-nearest neighbors of a spin in sublattice $\alpha$ may also belong to sublattice $\alpha$.

3. Phenomenological Real-Space Renormalization Group Methods: The EFRG Framework

The idea behind phenomenological renormalization group methods [35] is grounded in the finite-size scaling hypothesis [36]. To illustrate this concept, let us consider a magnetic system governed by a Hamiltonian of the form $\mathcal{H}=\mathcal{H}(K,H)$, where K is a coupling parameter and H denotes a uniform external magnetic field. Near a second-order phase transition, the singular part of the free energy is expected to scale according to

$$f_s(ϵ,H)=l^{-d}{f}_s(l^{1/\nu }ϵ,l^{y_H}H),$$

(3)

where $ϵ=K-K_c$ measures the deviation from the critical coupling $K_c$, l is an arbitrary scaling factor, d represents the spatial dimensionality of the system, $y_H$ is the magnetic critical exponent, and $\nu$ is the exponent associated with the correlation length.

Any thermodynamic observable P derived from the singular component of the free energy will also exhibit scaling behavior near the critical point, typically obeying a power law of the form

$$P\sim {\left|ϵ\right|}^{-\sigma },$$

(4)

where $\sigma$ is the corresponding critical exponent. For example, the magnetization scales as

$$m(ϵ,H)=l^{-d+y_H}m(l^{1/\nu }ϵ,l^{y_H}H).$$

(5)

Incorporating finite-size effects, the generalized scaling form of a thermodynamic quantity P becomes [36]

$$P(ϵ,H,L)=l^{\varphi }P(l^{1/\nu }ϵ,l^{y_H}H,l^{-1}L),$$

(6)

where $\varphi =\sigma /\nu$ defines the anomalous dimension of P, and L is the characteristic system size. By introducing a smaller system of size $L^{\prime }$ and choosing the scaling factor as $l=L/L^{\prime }$, along with the transformed parameters ${ϵ}^{\prime }=l^{1/\nu }ϵ=K^{\prime }-K_c$ and $H^{\prime }=l^{y_H}H$, Equation (6) simplifies to

$${\displaystyle \frac{P_{L^{\prime }}(K^{\prime },H^{\prime })}{{L^{\prime }}^{\varphi }}}={\displaystyle \frac{P_L(K,H)}{L^{\varphi }}}.$$

(7)

This expression establishes a transformation $(K,H)\to (K^{\prime },H^{\prime })$, which can be exploited to extract critical points and associated critical exponents.

Applying Equation (7) to different observables leads to distinct RG schemes. For instance, when applied to the correlation length, it yields

$${\displaystyle \frac{{\xi }^{\prime }(K^{\prime },H^{\prime })}{L^{\prime }}}={\displaystyle \frac{\xi (K,H)}{L}},$$

(8)

indicating that the anomalous dimension of the correlation length is unity. Among the phenomenological RG approaches, the Effective Field Renormalization Group method has proven especially effective for spin systems on lattices. In this formulation, Equation (7) is applied to the order parameters of two spin clusters of different sizes—characterized by the number of constituent spins—and calculated via the Callen–Suzuki identity [37,38]:

$$〈O_p〉={〈{\displaystyle \frac{{Tr}_p\left[O_p{e}^{-{\mathcal{H}}_p}\right]}{{Tr}_p\left[e^{-{\mathcal{H}}_p}\right]}}〉}_{\mathcal{H}},$$

(9)

where the partial trace ${Tr}_p$ is taken over the p spin variables within the finite-size cluster defined by the Hamiltonian ${\mathcal{H}}_p$, $O_p$ is the corresponding order parameter, and ${\langle ·\rangle }_{\mathcal{H}}$ denotes the canonical thermal average over the full system governed by $\mathcal{H}$.

The influence of spins external to the cluster is incorporated through an unknown symmetry-breaking field (SBF), $h_p$, which mimics the collective effect of the environment. Standard mean-field theory postulates $h_p=\langle O_p\rangle$, though this approximation fails in the context of geometrically frustrated lattices, as discussed later. In contrast, within the EFRG framework, $h_p$ is treated as an effective order parameter that also obeys the scaling relation in Equation (5), resulting in a more accurate depiction of the critical properties in such systems [20,39].

Near the critical point, it suffices to evaluate the order parameter in the limit $h_p\ll 1$ and $H\ll 1$, yielding the linear response expression:

$$m_p(K,h_p,H)=\langle O_p\rangle =f_p\left(K\right)h_p+g_p\left(K\right)H.$$

(10)

Performing an analogous calculation for a second cluster of size $p^{\prime }$ and invoking the scaling relation (5) leads to

$$f_{p^{\prime }}\left(K^{\prime }\right)h_{p^{\prime }}+g_{p^{\prime }}\left(K^{\prime }\right)H^{\prime }={\left({\displaystyle \frac{p}{p^{\prime }}}\right)}^{1-y_H/d}\left[f_p\left(K\right)h_p+g_p\left(K\right)H\right].$$

(11)

Assuming the internal symmetry-breaking fields also follow the same scaling behavior, one arrives at

$$f_{p^{\prime }}\left(K^{\prime }\right)=f_p\left(K\right),$$

(12)

which determines the fixed point of the RG transformation, i.e., $K^{\prime }=K=K_c$. Additionally,

$$g_{p^{\prime }}\left(K^{\prime }\right)={\left({\displaystyle \frac{p}{p^{\prime }}}\right)}^{1-2y_H}{g}_p\left(K\right),$$

(13)

from which the magnetic critical exponent $y_H$ can be inferred. In the absence of an external field, Equation (11) describes the emergence of spontaneous ordering below $K_c$.

Linearizing the recursion relation near criticality yields the thermal RG eigenvalue:

$${\lambda }_T={\left.{\displaystyle \frac{\partial f_p/\partial K}{\partial f_{p^{\prime }}/\partial K^{\prime }}}\right|}_{K=K_c}.$$

(14)

From this, the correlation length exponent $\nu$ follows using the identity ${\lambda }_T=l^{y_T}=l^{1/\nu }$ and Equation (8):

$$\nu ={\displaystyle \frac{1}{y_T}}={\displaystyle \frac{1}{d}}{\displaystyle \frac{ln(p/p^{\prime })}{ln{\lambda }_T}}.$$

(15)

The magnetic exponent $y_H$ can be extracted from Equation (13):

$$y_H={\displaystyle \frac{d}{2}}\left(1+{\displaystyle \frac{1}{ln(p/p^{\prime })}}ln{\displaystyle \frac{g_p\left(K_c\right)}{g_{p^{\prime }}\left(K_c\right)}}\right).$$

(16)

Once $\nu$ and $y_H$ are known, the remaining critical exponents of the model governed by the Hamiltonian $\mathcal{H}$ can be derived from standard scaling relations [36].

As with any real-space RG formulation, the primary limitation of this method lies in the finite size of the clusters employed. Nevertheless, it has been demonstrated that phenomenological RG techniques yield reliable results even for relatively small systems, and in some instances, they outperform more traditional RG approaches when suitably chosen clusters are used [35].

4. Evaluation of the Partition Functions for Finite-Size Clusters

As discussed in the preceding section, the core objective of the EFRG method is to evaluate the order parameters of two spin clusters of different sizes. This calculation proceeds in two main steps. First, the reduced partition function of a cluster is computed,

$${\tilde{Z}}_p(K,h_p)=\sum _p{\mathrm{e}}^{-{\tilde{\mathcal{H}}}_p/k_{\mathrm{B}}T},$$

(17)

or, in terms of the dimensionless Hamiltonian used throughout the rest of the manuscript,

$${\tilde{Z}}_p(K,h_p)=\sum _p{\mathrm{e}}^{-{\mathcal{H}}_p},$$

(18)

where the summation runs over the p spin degrees of freedom defined by the cluster Hamiltonian ${\mathcal{H}}_p$. In the case of classical $𝒮𝒪\left(n\right)$ spins, this sum becomes an integral over $(n-1)$-dimensional hyperspherical solid angles:

$$\sum _p\cdots =\sum _{{\overrightarrow{s}}_1,\cdots ,{\overrightarrow{s}}_p}\cdots ⟶\int \prod _{\alpha }\mathrm{d}{\Omega }_{i\alpha }\cdots .$$

(19)

For convenience, we work with the normalized partition function:

$$Z_p(K,h_p)={\tilde{Z}}_p{\left(0\right)}^{-1}{\tilde{Z}}_p(K,h_p).$$

(20)

Second, the order parameter is obtained from this normalized function via an alternative form of the Callen–Suzuki identity:

$$m_p={〈{\nabla }_{h_p}ln{Z}_p〉}_{\mathcal{H}}.$$

(21)

The geometry of the frustrated lattices considered in this work suggests that the minimal meaningful clusters are (i) a single-spin cluster and (ii) a full simplex consisting of $p=d+1$ interacting spins. We proceed to evaluate the partition functions for both cases.

4.1. Partition Function of the Single-Spin Cluster

The Hamiltonian of a 1-spin cluster, located at simplex i and sublattice $\alpha$, in the presence of symmetry-breaking fields (SBFs) generated by the surrounding environment and a uniform external magnetic field, is given by

$${\mathcal{H}}_{1,i\alpha }={\overrightarrow{s}}_{i\alpha }·{\overrightarrow{\xi }}_{1,i\alpha },$$

(22)

where the effective field ${\overrightarrow{\xi }}_{1,i\alpha }$ takes the form:

$${\overrightarrow{\xi }}_{1,i\alpha }=-{\overrightarrow{H}}_0+K_1\sum _{\beta \ne \alpha }\sum _{j\in \mathrm{NN}}{\overrightarrow{s}}_{j\beta }+K_2\sum _{\beta \ne \alpha }\sum _{j\in \mathrm{NNN}}{\overrightarrow{s}}_{j\beta }.$$

(23)

Here, the first term accounts for NN, while the second includes NNN, excluding the spin at site $i\alpha$ itself.

The normalized partition function is given by

$$Z_{1n,i\alpha }={\displaystyle \frac{\Gamma (n/2)}{2{\pi }^{n/2}}}{\int }_{\forall }\mathrm{d}{\Omega }_{i\alpha }{\mathrm{e}}^{-{\overrightarrow{s}}_{i\alpha }·{\overrightarrow{\xi }}_{1,i\alpha }}.$$

(24)

Following the method of Parsons [40], the integral can be evaluated in closed form (see Appendix A). The result is

$$Z_{1n,i\alpha }={\mathcal{I}}_{n/2-1}\left({\xi }_{1,i\alpha }\right),$$

(25)

where ${\mathcal{I}}_{\nu }\left(z\right)=2^{\nu }\Gamma (\nu +1)z^{-\nu }{I}_{\nu }\left(z\right)$ is the normalized hyperspherical modified Bessel function of the first kind [41]. Here, $I_{\nu }\left(z\right)$ denotes the standard modified Bessel function of order $\nu$.

For certain values of n, this expression simplifies to well-known forms. For example, in the Heisenberg case ($n=3$), one obtains

$$Z\left({\xi }_{1,i\alpha }\right)={\displaystyle \frac{sinh{\xi }_{1,i\alpha }}{{\xi }_{1,i\alpha }}},$$

(26)

which corresponds to the classical paramagnetic response of a spin under an external field [42].

4.2. Partition Function of the p-Spin Cluster

The Hamiltonian of a p-spin cluster (with $p=d+1$) located at simplex i is given by

$${\mathcal{H}}_{p,i}=K_1\sum _{\langle \alpha ,\beta \rangle }{\overrightarrow{s}}_{i\alpha }·{\overrightarrow{s}}_{i\beta }+\sum _{\alpha }{\overrightarrow{s}}_{i\alpha }·{\overrightarrow{\xi }}_{p,i\alpha }.$$

(27)

In this Hamiltonian, the first term represents the interaction between the spins inside the cluster of size p, whereas the second term represents the interaction of those same spins with the symmetry-breaking fields acting on each spin

$${\overrightarrow{\xi }}_{p,i\alpha }=-{\overrightarrow{H}}_0+K_1\sum _{\beta \ne \alpha }\sum _{j\ne i\in \mathrm{NN}}{\overrightarrow{s}}_{j\beta }+K_2\sum _{\beta \ne \alpha }\sum _{j\in \mathrm{NNN}}{\overrightarrow{s}}_{j\beta }.$$

(28)

The summations in the second term includes only spins outside the i-th simplex (i.e., the sum over i goes over all the NN spins except the ones already included in the cluster).

The normalized partition function then reads as follows:

$$Z_{pn,i}={\left({\displaystyle \frac{\Gamma (n/2)}{2{\pi }^{n/2}}}\right)}^p\int \prod _{\alpha =1}^p\mathrm{d}{\Omega }_{i\alpha }{\mathrm{e}}^{-{\mathcal{H}}_{p,i}}.$$

(29)

Exact analytical evaluation of this integral is only feasible for a few specific cases. Nonetheless, the expression can be recast into a more tractable form using the method detailed in Appendix B. This approach involves rewriting the Hamiltonian in terms of the total spin momentum and applying a Hubbard–Stratonovich transformation to decouple the spin interactions via an auxiliary field. For nearest-neighbor ferromagnetic interactions ($K<0$, where the subscript can be dropped) only, one obtains

$$Z_{pn,i}={\displaystyle \frac{{\mathrm{e}}^{pK}}{{(-2\pi K)}^{n/2}}}{\int }_{\forall }{\mathrm{d}}^n\overrightarrow{q}{\mathrm{e}}^{q^2/2K}\prod _{\alpha =1}^p{\mathcal{I}}_{n/2-1}\left(\sqrt{{({\overrightarrow{\xi }}_{p,i\alpha }\mp \overrightarrow{q})}^2}\right).$$

(30)

For antiferromagnetic interactions ($K>0$), the corresponding form becomes

$$Z_{pn,i}={\displaystyle \frac{{\mathrm{e}}^{pK}}{{\left(2\pi K\right)}^{n/2}}}{\int }_{\forall }{\mathrm{d}}^n\overrightarrow{q}{\mathrm{e}}^{-q^2/2K}\prod _{\alpha =1}^p{\mathcal{I}}_{n/2-1}\left(\sqrt{{({\overrightarrow{\xi }}_{p,i\alpha }\mp \mathrm{i}\overrightarrow{q})}^2}\right).$$

(31)

Although these expressions are not analytically solvable in general, near a second-order transition the hyperspherical Bessel functions can be expanded in powers of the SBF, as shown in Appendix C. Truncating the expansion at second order leads to a linear response approximation:

$$Z_{pn,i}=z_{pn,i}^{\left(0\right)}+{\displaystyle \frac{z_{pn}^{\left(0\right)}}{2n}}\sum _{\alpha }{\overrightarrow{\xi }}_{p,i\alpha }·{\overrightarrow{\xi }}_{p,i\alpha }-{\displaystyle \frac{z_{pn}^{\left(2\right)}}{2n}}\sum _{\alpha \ne \beta }{\overrightarrow{\xi }}_{p,i\alpha }·{\overrightarrow{\xi }}_{p,i\beta },$$

(32)

with

$$z_{pn}^{\left(0\right)}={\displaystyle \frac{2e^{pK}}{\Gamma (n/2){\left(4K\right)}^{n/2}}}{\int }_0^{\infty }{q}^{n-1}{\mathcal{Z}}_{n/2-1}^p\left(q\right)e^{-q^2/4K}\mathrm{d}q,$$

(33)

and

$$z_{pn}^{\left(2\right)}={\displaystyle \frac{2\mathrm{sgn}\left(K\right)e^{pK}}{n^2\Gamma (n/2){\left(4K\right)}^{n/2}}}{\int }_0^{\infty }{q}^{n+1}{\mathcal{Z}}_{n/2-1}^{p-2}\left(q\right){\mathcal{Z}}_{n/2}^2\left(q\right)e^{-q^2/4K}\mathrm{d}q.$$

(34)

Here, ${\mathcal{Z}}_{\nu }\left(q\right)$ is defined as ${\mathcal{I}}_{\nu }\left(q\right)$ for ferromagnetic interactions and ${𝒥}_{\nu }\left(q\right)$ for antiferromagnetic ones.

It is noteworthy that $z_{pn}^{\left(0\right)}$ corresponds to the partition function of an isolated spin simplex, whereas $z_{pn}^{\left(2\right)}$ captures inter-cluster correlations responsible for the emergence of long-range order. For the Heisenberg case ($n=3$) with antiferromagnetic interactions, one recovers known expressions such as ${𝒥}_{1/2}\left(z\right)={\displaystyle \frac{sinz}{z}}$ and ${𝒥}_{3/2}\left(z\right)=3\left({\displaystyle \frac{sinz}{z^3}}-{\displaystyle \frac{cosz}{z}}\right)$, consistent with results in Refs. [20,43]. The general formulation provided here, however, extends beyond previous treatments by allowing for arbitrary spin dimensionality and spatial dimension.

5. Critical Behavior and Scaling Analysis

The order parameter for sublattice $\alpha$ can be directly obtained from Equation (32) using Equation (29), yielding

$${\overrightarrow{m}}_{p,i\alpha }={\displaystyle \frac{1}{n}}{〈{\overrightarrow{\xi }}_{p,i\alpha }〉}_{\mathcal{H}}-{\displaystyle \frac{1}{n}}{\displaystyle \frac{z_{pn}^{\left(0\right)}}{z_{pn}^{\left(2\right)}}}\sum _{\beta \ne \alpha }{〈{\overrightarrow{\xi }}_{p,i\beta }〉}_{\mathcal{H}}.$$

(35)

To illustrate how the critical behavior of this system can be analyzed within the Effective Field Renormalization Group framework, we apply the scaling relation in Equation (12) together with Equations (21) and (35). For clarity and analytical tractability, two simplifying assumptions are made. First, we consider only ordered states where all spins belonging to the same sublattice share a common order parameter value. This excludes incommensurate and helical phases, yet retains enough structure to yield meaningful insights into the critical properties of the model. Second, we set the external magnetic field to zero, ${\overrightarrow{h}}_0=0$.

Under these assumptions, the order parameters simplify as follows. For the one-spin cluster,

$${\overrightarrow{m}}_{1\alpha }={\displaystyle \frac{2}{n}}\sum _{\beta \ne \alpha }{\overrightarrow{b}}_{\beta },$$

(36)

and for the d-simplex cluster with $p=d+1$ spins,

$${\overrightarrow{m}}_{p\alpha }={\displaystyle \frac{1}{n}}\left[\sum _{\beta \ne \alpha }{\overrightarrow{b}}_{\beta }^{\prime }+{ϵ}_{pn}\left(K\right)\left({\overrightarrow{b}}_{\alpha }^{\prime }+{\displaystyle \frac{p-2}{p-1}}\sum _{\beta \ne \alpha }{\overrightarrow{b}}_{\beta }^{\prime }\right)\right].$$

(37)

Here, the quantities ${\overrightarrow{b}}_{\alpha }=K{〈{\overrightarrow{s}}_{i\alpha }〉}_{\mathcal{H}}$ and ${\overrightarrow{b}}_{\alpha }^{\prime }$ are the effective internal fields acting on each sublattice due to spins outside the cluster. These internal fields are treated as order parameters and assumed to follow the same scaling behavior as the magnetizations, as prescribed by Equation (5).

Applying this scaling hypothesis to both the order parameters and internal fields leads to a system of linear equations:

$${ϵ}_{pn}\left(K\right){\overrightarrow{b}}_{\alpha }+\left({\displaystyle \frac{p-2}{p-1}}{ϵ}_{pn}\left(K\right)-1\right)\sum _{\beta \ne \alpha }{\overrightarrow{b}}_{\beta }=0.$$

(38)

This set of equations admits three classes of solutions, each corresponding to a distinct phase of the system. The trivial solution ${\overrightarrow{b}}_{\alpha }=0$ for all $\alpha$ represents the paramagnetic phase. In the presence of a nonzero external field ${\overrightarrow{h}}_0$, the magnetic susceptibility can be derived from ${ϵ}_{pn}\left(K\right)$, following the same strategy as in Ref. [20] for Heisenberg spins.

The first nontrivial solution corresponds to the ferromagnetic phase, characterized by uniform order parameters across sublattices, ${\overrightarrow{m}}_{\alpha }={\overrightarrow{m}}_{\beta }$ for all $\alpha \ne \beta$. The critical point for this phase is obtained from the condition:

$${ϵ}_{pn}\left(K\right)=1.$$

(39)

The second nontrivial ordered state is the so-called $\overrightarrow{q}=\mathbf{0}$ phase, in which the total magnetization of the simplex vanishes:

$$\sum _{\alpha }{\overrightarrow{m}}_{\alpha }=0.$$

(40)

This occurs when

$${ϵ}_{pn}\left(K\right)=-(p-1).$$

(41)

A natural question is whether this $\overrightarrow{q}=\mathbf{0}$ ordering can emerge at finite temperature. It is well known that for the kagome and pyrochlore lattices with classical antiferromagnetic Heisenberg spins, such ordering does not occur at any finite temperature when only NN interactions are considered. However, less is known about the behavior of these systems in higher dimensions or for spins with large component numbers.

Exploring this regime is not merely academic: models with large n or d form the basis of perturbative RG techniques, such as the $\epsilon$-expansion and $1/n$ expansions [36]. To examine the onset of $\overrightarrow{q}=\mathbf{0}$ ordering, it is instructive to plot the function ${ϵ}_{pn}\left(K\right)$ for different values of p and n in the antiferromagnetic regime, as shown in Figure 1. This function is smooth and lacks extrema. As shown in Ref. [34], ${ϵ}_{pn}\left(K\right)\to -1$ in the limit $K\to +\infty$ ($T\to 0$), meaning that condition (41) is satisfied at zero temperature for the 1D chain—a well-established result [44].

However, for higher-dimensional lattices, regardless of the number of spin components, condition (41) is never met. Thus, no $\overrightarrow{q}=\mathbf{0}$ ordering occurs at any temperature when only NN interactions are present. The system remains in a classical spin liquid state at all temperatures. Furthermore, the discrepancy from the $\overrightarrow{q}=\mathbf{0}$ criterion grows with increasing dimensionality. This observation is consistent with the Maxwellian counting arguments presented by Moessner and Chalker [45], which show that the degeneracy of the ground state increases extensively with lattice dimension in frustrated Heisenberg models.

Nonetheless, some exceptions to this behavior have been reported. For instance, XY spins on the pyrochlore lattice can exhibit collinear ordering via the order-by-disorder mechanism [45]. Similarly, Heisenberg spins on the kagome lattice may develop coplanar $\sqrt{3}\times \sqrt{3}$ ordering [46,47]. These observations appear to contradict the present conclusions. However, it must be emphasized that our analysis explicitly excludes such forms of order, as we have assumed that all spins within the same sublattice share a common order parameter. As a result, only non-collinear states commensurate with a $d+1$-sublattice decomposition are considered. This is a limitation of the present implementation, not of the EFRG method itself. More general states can be addressed by relaxing the constraint on sublattice uniformity and employing larger cluster sizes. Furthermore, investigating the possibility of order-by-disorder within the EFRG formalism requires inclusion of higher-order terms—particularly quartic terms—in the expansion of the partition function in Equation (32).

6. Conclusions

In this work, we have developed a real-space renormalization group approach for classical $𝒮𝒪\left(n\right)$ vector spin models defined on d-dimensional geometrically frustrated lattices composed of corner-sharing d-simplexes. This framework generalizes the Effective Field Renormalization Group method to systems with arbitrary spin dimensionality and lattice dimension, encompassing as particular cases the classical XY and Heisenberg models on kagome and pyrochlore lattices.

By computing the reduced partition functions of spin clusters of different sizes and constructing appropriate scaling relations, we have obtained analytical criteria for the emergence of various ordered phases. In particular, we have identified the conditions under which the system exhibits a transition to a $\overrightarrow{q}=\mathbf{0}$ state. Our results demonstrate that, in the presence of only nearest-neighbor interactions, this type of ordering is absent for all $d\ge 2$ and arbitrary n, indicating that these systems remain in a classical spin liquid phase down to zero temperature. This finding is in agreement with previous studies based on counting arguments and effective mean-field theories.

Moreover, we have discussed how more complex ordering phenomena, such as order-by-disorder or incommensurate phases, could in principle be captured within the same EFRG framework by lifting the assumption of sublattice-uniform order parameters and by including higher-order terms in the expansion of the partition function. The generality of the method presented here makes it a promising analytical tool for studying a wide class of frustrated systems beyond the limitations of standard mean-field approximations.

Future work may include the extension of this method to include quantum corrections, anisotropic interactions, or coupling to lattice degrees of freedom, as well as comparisons with numerical simulations and experiments in real materials exhibiting geometric frustration.