The large-m limit, and spin liquid correlations in kagome-like spin models

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Introduction
Exact insights into collective behaviors are rare even for the simplest systems of many interacting constituents. For instance, no exact solutions of short-range classic-spin models on periodic lattices are known above spatial dimension d = 2 [1]. An exception is critical behaviors above upper critical dimensions, due to methods of the renormalization group [2]. Properties of d = 3 spin models are deduced from approximations. A classic approximation is a limit of the large spin-space dimension m, and expansions approximation. The paper suggests that, potentially with the exception of the T interval where orderby-disorder mechanisms [5] are relevant, the structure of the equilibrium pair correlation function does not support the division into two distinct regimes of a paramagnet and a collective paramagnet. The kagome-like models thus may be "transparent" to the conventional paramagnetic treatment deep below the mean-field critical temperature Θ c that is usually interpreted as signaling the onset of a collective paramagnetic regime.

Main result
The paper is based on extending, connecting and interpreting two known observations (1) and (2) below. Observation (1) is essentially due to [6,7]; observation (2) stems from [8][9][10][11][12]. Consider a lattice consisting of equivalent corner sharing triangles. This can be a d = 2 kagome lattice, or a d = 3 kagomelike lattice, shown in figure 1. Other lattices, for which the argument of the paper holds, can be seen e.g., in figure 2 (b), (c) of [13], or in figure 1 of [12]. Kagome-like lattices describe magnetic materials. For example, in gadolinium gallium garnet, magnetic Gd 3+ ions occupy sites of two inter-penetrating species of the lattice of figure 1 (a), which are separated by a distance larger than the n.n. distance in each species. See, e.g., [14] for a list of positions of Gd 3+ in the cubic unit cell.
Define an isotropic m-vector model on the lattice of figure 1 (a) by the Hamiltonian: where i , j span N sites of the lattice, each site i carries an m-dimensional isotropic O(m) vector spin µ i of length m, the spins are coupled via a dot product, and entries of the symmetric interaction matrix J (i , j ) are 0, except for the nearest neighbor (n.n.) sites, when they are half the n.n. coupling J = 1. Let 〈· · · 〉 (β) denote a Gibbs ensemble average defined by (2.1). For instance, a spin-spin correlation matrix χ µν reads:

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The large-m limit, and kagome spin models 2) and the following text]. The difference ∆ between the eigenvalues number 1 and 1/3 N + 1 is exactly zero in two limits, the Ising limit m = 1 and the spherical limit m → ∞. High-T expansion [15] suggests non-zero ∆ for other values of m. The magnitude of ∆ can be thought of as a measure of deviations of correlations from their m → ∞ form. Small deviations can explain the heuristics of applicability of the m → ∞ limit, and the variational mean-field theory, to study correlations deep in the collective paramagnetic regimes [9][10][11][12] of the finite-m models. (b) The behavior of n.n. classical m-vector models on regular frustrated lattices, such as the d = 2 kagome lattice, can be divided into four regimes. As temperature T is lowered, a model can firstly cross-over from a high-T paramagnetic phase to a correlated collective paramagnetic phase at about the mean-field critical temperature Θ c . Then, the model can undergo (a sequence of) cross-overs or phase transitions due to order-by-disorder mechanisms [5] that are activated at temperatures Θ obd Θ c . Strictly at T ≡ 0, there can be a discontinuity, that separates the T → 0 phase from the microcanonical ground states phase. The paper suggests that the juxtaposition of T ≡ 0, collective paramagnetic and paramagnetic phases as separated entities may be not supported by the structure of the equilibrium correlation function. The exception might be the regime 0 < T Θ obd where the order-by-disorder is important. The diagram is qualitative and does not show exact energy scales.
where µ, ν enumerate the components of spins, β = 1/T is the inverse temperature, and Tr µ means integration over all degrees of freedom. We assumeχ µν =χ × δ µν with δ µν being Kronecker delta. The assumption would hold for phases that preserve the global spin rotational symmetry of the Hamiltonian, for example in the paramagnetic, collective paramagnetic and E = 0 phases [cf. figure 2 (b)]. In this way, anyχ µν is fully characterized by the matrixχ, whose dimensions are N × N independently of m. Below, we are interested in the properties ofχ as a function of m.
For any finite model (2.1) of L 3 cubic unit cells with periodic boundary conditions, the following two statements aboutχ are correct.
(1) For the Ising case m = 1, µ i = ±1, the macroscopic number 1/3 N + 1 out of N eigenvalues ofχ coincide (are degenerate) at any β. At β > 0, they are the largest eigenvalues in the spectrum ofχ. Here, N = 12L 3 is the number of spins in the model.
The eigenspace L − of the degenerate eigenvalues ofχ is solely determined by the interaction matrixĴ of (2.1) and is independent of β. Specifically, L − coincides with the eigenspace ofĴ of dimension 1/3N + 1 corresponding to its degenerate minimal eigenvalue −1. Informally: (2) For the case m → ∞ at β > 0, the macroscopic number 1/3 N + 1 of the largest eigenvalues ofχ are again degenerate and describe the same L − (2.3).

Derivation
(1) Consider the Ising version m = 1 of (2.1). A star-triangle (Y − ∆) transformation, said to be due to Onsager, relates exactly the zero-field partition function of n.n. Ising models on d = 2 kagome, hexagonal and triangular lattices [16]. A perhaps less known its application is a relationship between the n-spin correlation functions of the three models [17]. In particular, [6,7] showed that the largest eigenvalues of the correlation matrix of the d = 2 kagome Ising model are degenerate at all T . The argument uses a local lattice topology and works for Ising models on lattices at any d , as soon as they consist of corner sharing triangles. This paper adopts the argument [6,7,17] to three-dimensional lattices, such as the kagome-like lattice of figure 1 (a).
Here, β d and A are known functions of β. Introduction of σ decouples µ 1 , µ 2 , µ 3 . We use new variables {σ} to decouple all triangle-coupled kagome spins {µ}, and then sum {µ} out (applying the "decorationiteration" transformation [17]). Graphically, the remaining variables {σ} can be thought of as forming a n.n. Ising model on a lattice of the centers of the original corner-sharing triangles: the hexagonal lattice at d = 2 and a hexagonal-like lattice at d = 3, see figure 1 (a), (b), with the spin number N h = 2/3N . Analytically, the partition functions Z h and Z of any pair of the hexagonal-like and kagome-like Ising models become related by the same, exact formula of [16].
To recast the kagome-like model spin correlations in terms of the hexagonal-like model spin correlations, we again decouple {µ} in the Boltzmann factors by using {σ}. However, when summing {µ} out, we include a product of the chosen µ j µ j in (2.2). Same procedure works for multi-spin correlations. Note that every spin µ i of the kagome-like lattice neighbors a unique pair of spins, say σ i , σ i , that are nearest neighbors on the corresponding hexagonal-like lattice, cf. figure 1 (c). Denotingμ i = σ i + σ i , we obtain [6,7]: Here, 〈· · · 〉 h (β h ) means thermal average in the hexagonal-like model with n.n. coupling 1 at the inverse temperature β h , and M 2 (β h ) = 1/4 (e 4β h − 1) > 0 . Consider a sign-alternating linear combination τ of kagome spins lying on a closed path. For the kagome-like lattice of figure 1 (a), the mode τ can be formed of ten spins µ 1 , . . . , µ 10 of the loop in figure 1 (a): τ = c 1 µ 1 + · · · + c 10 µ 10 , where c 1 = 1, c 2 = −1, . . . , c 10 = −1. Note that (3.2) is a difference of an identity matrix times a constant, and a positive semi-definite matrix. In (3.2), the special choice of τ zeros the second term contribution to the mode susceptibility 〈τ 2 〉, thus maximizing it. This makes any c = (· · · , c k , · · · ), whose entries are non-zero only if they coincide with the sign-alternating coefficients of a loop, the eigenvector corresponding to the largest eigenvalue ofχ.
Observe that (2.1) can be written as a sum of squares minus a constant. It is clear that every c zeros the squares, and thus is the eigenvector ofĴ corresponding to the smallest eigenvalue −1 ofĴ . The linear span of all c forms L − , whose dimension is N minus the dimension of the triangles constraints in the sum of squares, which is N h − 1. We have: dim L − = 1/3 N + 1.
(2) Examine (2.1) at arbitrary m. [18,19] showed that an m-vector lattice model is exactly solvable in the limit m → ∞, where it coincides with the spherical model of Berlin and Kac [20]. In particular, the spherical limit of the correlation matrix (2.2) of model (2.1) reads [19]: where parameter r 0 is fixed by normalization andĴ is the interaction matrix in (2.1). The variational mean-field theory [21], see e.g., [9] for its application in the context of frustrated magnetism, gives the dependence ofχ onĴ in the same form of a [0/1] Padé approximant. Sinceχ is an (analytic) function ofĴ ,χ has a set of degenerate eigenvalues corresponding to L − . Sinceχ is a monotonously decreasing function ofĴ , β > 0, the smallest degenerate eigenvalues in the spectrum ofĴ are the largest in the spectrum ofχ .

Interpretation
It was previously observed, for instance in [10,12], that the m → ∞ formulae provide an excellent fit to the collective paramagnetic correlations of finite-m n.n. m-vector models on the d = 3 pyrochlore

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and a kagome-like frustrated lattice. The variational mean-field theory was observed in e.g., [9,11,22] to quantitatively correctly describe the role of perturbations in lifting the degeneracy in the collective paramagnetic regimes. This paper may shed light as to why these theories are well applicable into collective paramagnetic regimes for the case of kagome-like lattices. 1 We showed above that the upper 1/3 N +1 eigenvalues of the correlation matrixχ (2.2) of model (2.1) are exactly degenerate for m = 1 and m → ∞ at all T , and the corresponding eigenspace L − is independent of T . We can quite naturally conjecture that for the intermediate values 1 m < ∞, the eigenvalues of L − become at all T only weakly dispersed. The upper eigenvalues can remain quasi-degenerate also for m < 1, for instance in the polymer limit m → 0. Observe that at T ≡ 0, i.e., strictly in the phase of the microcanonical ground states, and at m 2, these are the spin states belonging to L − only that contribute toχ. Thus, at T ≡ 0,χ is determined by its approximate form of the m → ∞ projector on the linear space L − .
We can consider the (relative) dispersion ∆ of the quasi-degenerate eigenvalues ofχ as a measure of deviations of correlations from the m → ∞ projector form. The dependence of ∆ on m can have a shape of figure 2 (a). The location m 0 of a maximum might depend on d and on the choice of the kagomelike lattice, but might not exceed 3. For instance, no order-by-disorder phenomenology was observed for larger m on the d = 2 kagome lattice [6,7], pointing that such a model is in the m → ∞ regime, where no order-by-disorder is observed either.
We can next speculate that the m → ∞ projector form ∆ ≈ 0 ofχ is valid in the collective paramagnetic phase at a finite T , which by definition mainly consists of the states from the extensively degenerate manifold L − . The projector form would hold several orders of magnitude in T below Θ c , the mean-field critical temperature, but potentially above Θ obd , the order-by-disorder temperatures, where thermal fluctuations can select subset(s) of L − with the greatest number of low-energy excitations [23,24], cf. figure 2 (b). As L − is known exactly, the interesting question about the structure of correlations at low T may be not the projector form ofχ per se, but the nature of the (small) deviations from it. Above about Θ c , the m → ∞ form (3.3) can be expected to apply naturally. Therefore, the correlations in model (2.1) can be well reproducible by their m → ∞ form for m 2 and at all T , with the potential exception of the phases dictated by the order-by-disorder.
Consider any other Hamiltonian H on a kagome-like lattice, which preserves the symmetry of the lattice. Let, for m 2, H admit the same microcanonical T ≡ 0 degenerate ground states as the original H (2.1). For instance, H can be obtained from (2.1) by using another interaction matrixĴ , for which (2.3) is true, but which is not necessarily the nearest neighbor. The coincidence of ground states for distinct H and H was a dubbed projective equivalence in [25] in the context of spin models on the pyrochlore lattice. As T ≡ 0 states of H and H coincide, the upper eigenvalues of the correlation matrix are again quasi-degenerate for all m 2. The quasi-degeneracy, and the m → ∞ form of correlations for H should be again correct for m 2 and all T , potentially excluding the window 0 < T Θ obd .
Consider another Hamiltonian H different from H by small perturbations such that (2.3) is not valid. The perturbations can force the model H to undergo a phase transition at T > Θ obd , the regime where equation (3.3) is applicable. We can thus use equation (3.3) to study, for instance, the selection of the ordering wave vectors dictated by perturbations. In essence, frustration may be unimportant for applicability of the m → ∞ approximation, while the variational mean-field theory may be used for the study of the role of perturbations deep below the mean-field critical temperature Θ c [9][10][11][12]. If we regard the applicability of these approaches as defining the nature of correlations, there may be no difference between a collective and regular paramagnet. Correspondingly, the m → ∞, and the related variational mean-field approaches may claim back their status as simple, powerful and standard tools for the study of perturbations at low T in kagome-like and other frustrated systems, as was heuristically observed for instance in [9,11]. 1 Pyrochlore lattice case will be presented elsewhere.