The spin-$\frac{1}{2}$ Heisenberg ferromagnet on the pyrochlore lattice: A Green's function study

We consider the pyrochlore-lattice quantum Heisenberg ferromagnet and discuss the properties of this spin model at arbitrary temperatures. To this end, we use the Green's function technique within the random-phase (or Tyablikov) approximation as well as the linear spin-wave theory and quantum Monte Carlo simulations. We compare our results to the ones obtained recently by other methods to corroborate our findings. Finally, we contrast our results with the ones for the simple-cubic-lattice case: both lattices are identical at the mean-field level. We demonstrate that thermal fluctuations are more efficient in the pyrochlore case (finite-temperature frustration effects). Our results may be of use for interpreting experimental data for ferromagnetic pyrochlore materials.


Introduction
One of the standard three-dimensional models for a study of geometrically frustrated spin systems is the pyrochlore Heisenberg antiferromagnet [15,16]. It is known that the nearest-neighbor exchange interactions, because of the lattice geometry (a three-dimensional network of corner-sharing tetrahedra), cannot establish magnetic ordering even at zero temperature T = 0. Besides the purely theoretical interest in the spin-system physics of this celebrated model, there are various real materials which can be described with the help of the antiferromagnetic Heisenberg model on the pyrochlore lattice [17,18].
Less attention has been paid to the pyrochlore-lattice Heisenberg ferromagnet. Clearly, the set of the eigenvalues of the Heisenberg Hamiltonian (energy levels) does not depend on the sign of the exchange interaction. However, the ordering of these energy levels is different for the antiferromagnetic and ferromagnetic signs of the exchange interaction. As a result, the low-energy levels for the pyrochlore-lattice Heisenberg antiferromagnet are the high-energy levels for the pyrochlore-lattice Heisenberg ferromagnet and they may manifest themselves only in finite-temperature thermodynamics, when due to thermal fluctuations they can come into play. Several systematic theoretical studies [19,20] demonstrated that this is really the case. For example, it was found that the Curie temperature for the pyrochlore lattice is lower than for the simple-cubic lattice [19][20][21], although both lattices have the same number of nearest neighbors and thus are identical at the mean-field level.
In what follows, we first introduce the model and present the corresponding linear spin-wave theory (section 2). In section 3, we describe the Green's function approach to the model and in section 4 we discuss the low-and intermediate-temperature properties of the model making connections to previous studies. In section 5, we close the paper with a short summary pointing out some related problems which can be discussed by the same approach.

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The spin-1 2 Heisenberg ferromagnet on the pyrochlore lattice We begin with a brief description of the linear spin-wave theory which should be valid in the low-temperature limit. Using the Holstein-Primakoff transformation we introduce the bosonic operators a m;α (and a † m;α ) with α = 1, 2, 3, 4 and m = 1, . . . , N . Furthermore, we introduce the bosonic operators where q · R m = q 1 m 1 + q 2 m 2 + q 3 m 3 and q a = 2πz a /L a , z a = 1, . . . , L a (a = 1, 2, 3). [Note that q 1 = 1 2 (q y +q z ), q 2 = 1 2 (q x +q z ), and q 3 = 1 2 (q x +q y ).] Then, the spin Hamiltonian given in equation (2.1) can be approximately rewritten in the following form: After diagonalizing the bilinear Bose form in equation (2.5) we arrive at The energies of the linear spin waves (magnons) −2sJω γ;q = |J|ω γ;q are determined by the eigenvalues of the matrix F ω 1;q = ω 2;q = 4, ω 3;q = 2 + D q , ω 4;q = 2 − D q , The calculated spin-wave dispersion coincides with the result reported in [26] [see figure 1 (c) of that paper]. Now, from equation (2.7) we can obtain the internal (intrinsic) energy per site q ω γ;q n q;γ (2.9) and the magnetization per site Here, n q;γ = 1/(e |J |ω γ;q /T − 1) denotes the Bose-Einstein distribution function (we set k B = 1 for brevity) and s = 1 2 . Below we use the linear spin-wave theory predictions (2.9) and (2.10) for comparison with the Green's function approach results.

Green's function method. Random-phase approximation
Now, we turn to the Green's function approach [3-5, 11, 29-32]. We introduce the following operators and the (retarded) Green's function see, e.g., [29]. The first-order equation of motion reads: Calculating the commutator in the right-hand side of the second equation in equation (3.3) (see appendix) and introducing the random-phase (or Tyablikov) approximation s z A s ± B → s z s ± B [29] (see appendix) we obtain the closed-form equation for the Green's function where −2J s z F is the frequency matrix, G(ω) is the matrix of Green's functions, and M = 2 s z 1 is the moment matrix, see also equation (A.4). The matrix F appeared already in the linear spin-wave calculations, see equation (2.6). We can find the eigenvalues and the eigenvectors of the matrix F, which satisfy β F αβ β|γq = ω γ;q α|γq . The eigenvalues ω γ;q , γ = 1, 2, 3, 4 are given in equation (2.8). The (orthonormal) eigenvectors α|γq , γ = 1, 2, 3, 4 are too lengthy to be presented here.
Finally, using the identity 1 = γ |γq γq|, we can resolve equation (3.4) with respect to G, G = 2 Here, γq| β is the β-th component of the eigenvector γq|. Clearly, the magnetization s z which enters equation (3.5) should be determined self-consistently, see equation (3.7) below. As can be seen from equation (3.5), the excitation energies are given by −2J s z ω γ;q = 2|J| s z ω γ;q . In the limit T → 0, when s z → 1 2 , they coincide with the magnon energies considered in section 2 and, therefore, we call these excitations magnons. Moreover, they coincide with the results of [19] in this limit (see the upper panel in figure 2 of that paper). For finite temperatures T > 0 the dependences on q remain unchanged, and the energies simply decrease due to the factor s z < 1 2 . More sophisticated effects of the temperature on the excitation energies were discussed within the rotation-invariant (or Kondo-Yamaji [33]) Green's function approach in [19].1 Since D q → 2 − 1 16 |q| 2 as q → 0, ω 4;q is the acoustic branch of the spectrum, i.e., ω 4;q → ρ|q| 2 , ρ = 1 8 |J| s z as q → 0. The spin stiffness ρ is proportional to s z ; it shows the same temperature behavior as s z vanishing as T → T c .
1The rotation-invariant Green's function approach considers the equation of motion up to the second order, which obviously involves Green's functions of higher order than the initial ones. Several products of three-spin operators are then simplified by a decoupling like where α AB is a vertex parameter which is introduced to improve the approximation made by decoupling. After all, the correlation functions, the vertex parameter, and the condensation term which is related to magnetic long-range order are determined self-consistently [within the Tyablikov approximation, one faces only one resulting equation for s z , see equation (3.7)]. For further details see [19].

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Knowing the Green's functions, we calculate the equal-time correlation functions via the relation These correlation functions yield important thermodynamic characteristics. In particular, using the iden- α q s − q;α s + q;α , we arrive at the following formula for s z : In fact, this is the equation for determining self-consistently the magnetization s z .
We can obtain the internal energy of the spin model at hand by averaging the Hamiltonian (2.1). Since we have calculated only the Green's function s + |s − rather than s z |s z we should eliminate the correlations s z s z from the consideration in order to treat all correlations in H on equal footing, i.e., within the same (Tyablikov) approximation [31,34]. After some manipulations (see appendix), we arrive at the following expression for the internal energy within the Tyablikov approximation: where φ a and φ ab are defined in equation (2.6). Differentiation of the internal energy with respect to T gives the specific heat. The Green's function (3.5) gives the dynamic susceptibility, , which is related to the dynamic structure factor Furthermore, the Green's function given in equation (3.5) is proportional to s z and at zero field it disappears in the high-temperature paramagnetic phase above the critical (Curie) temperature T c . However, the (initial) susceptibility χ, defined as s z = χh, where h is an infinitesimally small applied magnetic field can be determined as follows. For the spin model with the Hamiltonian H − h m;α s z m;α , the Green's function G αβ (ω) within the random-phase approximation is again given by equation (3.5), although with 2J s z ω γ;q − h instead of 2J s z ω γ;q in the denominator. Therefore, the equation for s z now reads cf. equation (3.7). Following, e.g., [29], for T > T c we set s z = 0 in the left-hand side of equation (3.10) and then expand the exponent and substitute s z = χh to arrive at the equation for 1/ χ above the critical temperature T c : (3.11) All calculations described in this section are, in principle, well known [3-5, 11, 29-32]. However, to our best knowledge, we are not aware of any random-phase Green's function paper for the pyrochlorelattice quantum Heisenberg ferromagnet.

Low-and intermediate-temperature results
In the low-temperature limit, the excitations described by G αβ (ω) (3.2), (3.5) are magnons also emerging in the linear spin-wave theory. Consequently, in the low-temperature limit, the Green's function results are expected to coincide with the ones of the linear spin-wave theory, see the low-temperature region in figures 1 and 2.
Let us compute the critical (Curie) temperature T c . At the critical temperature, s z vanishes; we set s z = 0 in the left-hand side of equation (3.7) and expand the exponent in the right-hand side of equation (3.7) to obtain the critical temperature: Here, in the thermodynamic limit 1 The same equation follows from equation (3.11) after setting 1/ χ = 0. This result should be compared to the corresponding ones obtained by other means given in the second column of table 1. The important feature is visible: the critical temperature calculated within the random-phase approximation Green's function method is the highest one in comparison with the results of other approaches (excluding mean-field) indicating some underestimate of the role of thermal fluctuations. Comparing the results for the pyrochlore and simplecubic cases in the same approximation, we again observe a manifestation of finite-temperature frustration effects [19,35]: thermal fluctuations destroy the magnetic order more effectively for the pyrochlore-lattice geometry, and the critical temperature for the pyrochlore ferromagnet is about 15% smaller than for the simple-cubic ferromagnet. Of course, the mean-field result for both lattices is the same: T c = 3 2 |J|. We can solve equation (3.7) with respect to s z for all temperatures below T c (4.1). This temperature dependence of the magnetization is shown in figure 1 along with the results of the rotation-invariant Green's function method [19] and the quantum Monte Carlo simulations using the ALPS package [36,37] (see also [38]). We also report in figure 1 the corresponding results for the simple-cubic case for comparison. Moreover, in figure 1 we show additionally the temperature dependence of 1/ χ in the paramagnetic phase as it follows from equation (3.11) (thick solid), from the rotation-invariant For comparison, we also show the results for the simple-cubic lattice (black). Note that the mean-field data (solid green) are identical for both lattices. Quantum Monte Carlo simulation data for N = 16 · 32 3 sites (pyrochlore, red) and N = 80 3 sites (simple-cubic, black) are shown by open circles (m) and filled circles (1/ χ). Inset: Magnetization at low temperatures. We also show here the linear spin-wave theory predictions (very thin dotted). 2 pyrochlorelattice (red) Heisenberg ferromagnet (J = −1) obtained within Tyablikov approximation (solid) and rotation-invariant Green's function method (dashed). We compare the results for the pyrochlore lattice (red) and the simple-cubic lattice (black) which are identical at the mean-field level (solid green). We provide in addition quantum Monte Carlo data obtained for N = 16 · 16 3 (pyrochlore, red empty circles) and N = 40 3 (simple-cubic, black empty circles) sites. Very thin dotted (double-dashed) lines represent the 13th order high-temperature expansion raw (extrapolated by the [6,7] Padé approximant) results. Inset: Specific heat at low temperatures. We also show here the linear spin-wave theory predictions (very thin dotted). It is in order to mention here, that for the c(T ) curve of the simple-cubic lattice presented in figure 8 of [19], an incorrect data set was used.

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Green's function method [19] (thick dashed), and from the quantum Monte Carlo simulations (filled circles) for the pyrochlore (red) and simple-cubic (black) lattices. Within the mean-field approximation 1/ χ = 4(T c − T), T c = 3 2 |J| for both lattices, see thick solid green curve. The reported results for the temperature dependences of magnetization and susceptibility agree well with each other and illustrate the finite-temperature frustration effects as well as the quality of various approximations.
The specific heat within the Tyablikov approximation is evaluated by finding the derivative of equation (3.8) with respect to temperature T. In figure 2 we compare the temperature dependence of the specific heat obtained by different methods for the pyrochlore lattice (red) and simple-cubic lattice 33705-7 (black). The mean-field result for both lattices is the same (green): the critical temperature T c is overestimated and c(T c ) is finite. The Tyablikov approximation (solid red and solid black) yields already different values for T c , although the specific heat diverges at the Curie temperature. This contradicts the current critical exponent estimates of the three-dimensional Heisenberg model obtained by field theory [42], high-temperature series analyses, and Monte Carlo [43][44][45] methods which predict that the specific heat is not singular for this model. According to the rotation-invariant Green's function method, c(T c ) is finite (dashed red and dashed black). We also report quantum Monte Carlo data obtained using the ALPS package [36,37] (see also [38]); red (black) empty circles refer to the pyrochlore (simple-cubic) lattice of N = 16 · 16 3 (N = 40 3 ) sites. Moreover, we show the 13th order high-temperature expansion results, raw data (very thin dotted) and [6,7] Padé approximants (very thin double-dashed), obtained using the HTE program freely available at http://www.uni-magdeburg.de/jschulen/HTE, see [40,46]. In summary, the presented data show that the Tyablikov approximation is capable of reproducing reasonably well the dependence c(T) in the ferromagnetic phase, except the region close to T c . In the paramagnetic phase, the rotation-invariant Green's function method and, of course, high-temperature expansion method give reasonably good results.
Finally, we turn to the dynamic structure factor S +− q (ω). Within the Tyablikov approximation, it is given by equation (3.9). For further calculations we replace in equation (3.9) δ(x) by the Lorentzian function ǫ/[π(x 2 + ǫ 2 )] with ǫ = 0.01. Inspired by the experimental paper on the spin-1 2 pyrochlore ferromagnet Lu 2 V 2 O 7 [26], where neutron inelastic scattering data were reported (see figure 2 of that paper), we calculate S +− q (ω) (3.9) at T = 0.0425|J| along the path q = (q, q, q) and present the result as a function of the reduced momentum t = 2 − D q , see figure 3. (Experimentalists collected data from more points in the q-space, not only along the path q = (q, q, q), see [26].) Overall, figure 3 resembles the experimental data reported in [26]; the dynamic structure factor is concentrated along the lines ω 4;q = 4−t, ω 3;q = t, however, because of the chosen path q = (q, q, q), not along the line ω 1;q = ω 2;q = 4. While comparing with experiments (and with the rotation-invariant Green's function result [19]), one should remember that S +− q (ω) presents only the transverse part, while the longitudinal part S zz q (ω) is not taken into consideration. We note in passing that comparing theoretical predictions with experimental data one can obtain the exchange interaction constants, and the exploited here random-phase approximation Green's function method can be used for this purpose.

Summary
In the present paper we have discussed the thermodynamic properties of the pyrochlore-lattice quantum Heisenberg ferromagnet. Our main results (Curie temperature, temperature dependences of the magnetization and the specific heat, dynamic structure factor) have been obtained within the random-33705-8 phase approximation. We compare them to the linear spin-wave theory at low temperatures as well as to the rotation-invariant Green's function method [19], quantum Monte Carlo simulation, and hightemperature expansion [19,40] results. Although the random-phase approximation goes beyond the mean-field treatment, thus leading to different results for the pyrochlore-lattice and the simple-cubiclattice case, it takes into account not all fluctuations, and ferromagnetic ordering is slightly overestimated in comparison with the second-order rotation-invariant Green's function method results. Nevertheless, the random-phase approximation Green's function method together with the high-temperature expansion provide a simple route to describe quantum Heisenberg ferromagnets for all temperatures. Of course, the quantum Monte Carlo simulations or the rotation-invariant Green's function method are also applicable, though these are more complicated techniques.
The Green's function method which uses the Tyablikov approximation can be applied to some other spin models related to the pyrochlore lattice. For example, one can in a similar way consider the ferromagnet in the presence of the breathing anisotropy or the second-nearest neighbor interactions. However, the most interesting problem within this context is the study of the properties of the pyrochlore-lattice quantum Heisenberg antiferromagnet. In zero magnetic field, the partition functions of the Heisenberg ferro-and antiferromagnet are connected by the obvious relation: Z FM (−T) = Z AFM (T). This implies that the specific heat of the antiferromagnet at temperature T is given by the specific heat of the ferromagnet taken at temperature −T. However, in the thermodynamic limit, because of spontaneous symmetry breaking, this relation is not valid. Clearly, the rotation-invariant Green's function approach, which implies s z = 0, seems to be an appropriate approximation for examining that case. The work in this direction is in progress [47].

A. First-order equation of motion, Tyablikov approximation, and some other calculations
To obtain the first-order equation of motion we have to calculate the commutator in the second equation in equation (3.3). We begin with the case α = 1 in equation (3.3). Then, 1 J s + m;1 , H = n s + m;1 , 1 2 s − n;1 s + n;2 + s + n;3 + s + n;4 + s z n;1 s z n;2 + s z n;3 + s z n;4 + 1 2 s − n;1 s + n 1 −1,n 2 ,n 3 ;2 + s + n 1 ,n 2 −1,n 3 ;3 + s + n 1 ,n 2 ,n 3 −1;4 + s z n;1 s z n 1 −1,n 2 ,n 3 ;2 + s z n 1 ,n 2 −1,n 3 ;3 + s z Within the Tyablikov approximation s z s ± → s z s ± (obviously, two spin operators here are attached to different sites and s z is already site independent) and, therefore, for the definition of φ a see equation (2.6). As a result, according to equation (3.3), we obtain Setting α = 2, α = 3, and α = 4 and repeating such calculations thrice, we obtain three more equations. After all, we may combine them in the matrix form where the matrix F is defined in equation (2.6), cf. equation (3.4).

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for definition of φ a and φ ab see equation (2.6  This is equivalent to equation (3.8). Note also that the expression for the internal energy (3.8) agrees with the formula given in equation (336) of [31].