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Theory of Coexisting Transverse Spin Freezing and Long-Ranged Antiferromagnetic Order in Lightly Doped La_{2-x}Sr_x CuO_4

We provide an explanation of the spin-freezing transition recently observed by Chou et al. (Phys. Rev. Lett. 71, 2323 (1993)) in La_{2-x}Sr_x CuO_4 for x<= 0.02. We propose that topological excitations of the 2D Heisenberg quantum antiferromagnet having non-coplanar transverse components have a pair-interaction energy that qualitatively and quantitatively agrees with the observed values of spin-freezing temperature as a function of doping.

To begin, let us summarize the experimental results. Chou et al. [3] have measured the 139 La NQR for different dopings and temperatures in lightly doped La 2−x Sr x CuO 4 for low temperatures. They measured the 139 La NQR frequency spectrum, and the relaxation times for the system's magnetization to return to equilibrium. The splitting of the 2ν Q NQR peaks due to the internal field associated with the AFM ordering of the spins at the Néel temperature led to a doping-dependent ordering temperature T N (x) -this determination of T N (x) is in agreement with other measurements, such as (i) their observations of the spin-lattice relaxation rate, and (ii) that found using static susceptibility measurements [5].
Further, a large enhancement of this splitting was found at very low T , viz. at temperatures less than ∼ 30 K, with the enhancement increasing with increasing x. Such behaviour could be caused by an alteration of the underlying spin texture at this temperature. A change in the spin state may also be inferred from measurements of the relaxation rate of the 3ν Q transition -below ∼ 70K there is a large enhancement of this rate. The rate continues to increase until it rapidly drops off at very low temperatures, viz. temperatures around 5 -16 K. Using this low temperature behaviour of the relaxation rate, Chou et al. [3] identified the maximum of the NQR rate with the spin-freezing temperature T f . As a function of doping it was determined to be approximately given by T f (x) ∼ (815K)x, and it is this linear relation that we have sketched in Fig. 1. Above 30K (which may be interpreted to be the charge localization transition temperature [6]) the nuclear magnetization relaxed with a multiexponential decay, while below this temperature a dramatic change was noted in that now the relaxation of the magnetization was found to be described by M(∞) − M(t) ∝ exp[−(t/T * 1 ) 1/2 ]. This stretched exponential decay has been found in many systems studied by NQR (see, e.g. Ref. [4]), and can be associated with a spin-glass ordering. However, the staggered moment of the AFM phase does not disappear, as indicated, e.g., in (i) the zerotemperature extrapolation of the 2ν Q splitting to the undoped value, or (ii) the continuous change of the splitting as T f is traversed (see Fig. 1(c) of [3] for the data corresponding to both of these results), and thus the spin glass ordering is found to coexist with the long-ranged AFM order.
Similar experimental results were found in both other NQR work [7] at x = .012 and .015, as well as in electron paramagnetic resonance (EPR) studies of the same system at x = .009 by Rettori et al. [8] (see their Fig. 12) -his latter study again found an increase in internal field observed in the −7/2 → − 5/2 transition.
Clearly, it would be desirable to explain the above-mentioned behaviour of the spin texture, and associated experimental measurements, and that is the focus of this paper.
However, we wish to stress that this phenomenon is certainly not limited to the weakly doped Bednorz-Müller high T c compound, nor to AFMs. For example, previous experimental work on metallic spin glasses [9], and related computational studies [10] on the moderately frustrated three-dimensional Heisenberg model, aids in the understanding of the high T c system we are studying. The work on the moderately frustrated Heisenberg model leads to the phase diagram shown in Fig. 2 of Ref. [10], and for dopant levels less than 25 % is similar to that of our Fig. 1. In particular, the linear dependence of T xy is the same.
The authors of Ref. [10] identified the coexisting freezing of the transverse spin degrees of freedom (transverse in the sense of being perpendicular to the already present ferromagnetic moment) and long-ranged ferromagnetic order at the spin-freezing temperature T xy ; this latter ordering is not associated with a phase transition, only involving a short-ranged freezing of spins, and does not eliminate the ferromagnetic ordering. Unfortunately, these authors did not attempt to come to an analytic understanding of this similar linear dependence of the spin-freezing temperature on doping. Nor did they attempt to justify the magnitude of the slope of this pseudo-phase boundary. That is precisely what we will present in this paper, albeit for at a doped two-dimensional AFM.
Our paper is organized as follows. In §II we summarize our previous work [11,12] on the spin texture of singly and weakly doped CuO 2 planes; this involves studies of a non-coplanar spin texture where each Sr 2+ dopant produces a spin state reminiscent of a singly charged skyrmion. This section also serves to introduce the effective Hamiltonian that we will use to describe these doped planes. In §III we present an analytic calculation of the two skyrmion interaction energy, and then use this to suggest the dependence of T f (x) on doping x. In §IV we describe numerical work examining the same two skyrmion interaction energy problem, and which reaches the same qualitative conclusion as in §III, viz. an interaction energy ∝ x, as well as providing a quantitative estimate of the slope. Then, we conclude this section by extrapolating the above work to determine the dependence of the spin-freezing temperature on doping via both a comparison to Chou et al. [3], and a simple counting of the degrees of freedom of the two defect system. Finally, in §V we summarize how this and other work on the weakly doped planes leads to a clear description of the novel spin texture of this system.

II. SPIN TEXTURE OF WEAKLY DOPED PLANES AT LOW T :
Our model of the doped state of a CuO 2 plane at low T is based on a collection of partially delocalised holes. It is assumed that at low temperatures the carriers (viz. O holes) are not mobile, but rather are confined to some small region of a plane -this is entirely consistent with the localisation transition seen in resistivity measurements between 50 and 100 K [13], as well as with the localisation transition inferred from the NQR measurements of Chou [6].
Physically, this localisation can be considered to arise from the electrostatic attraction of the holes to a divalent Sr 2+ ion.
A study of the magnetic spin texture of the ground state of the system for one hole was previously discussed by one of us [11], within the framework of the t − J model. There, the hole was only free to move around one square plaquette, and it was found that the distorted AFM spins' state could be described, in a semi-classical sense [14], via a noncoplanar spin state with a rotational twist generated by the motional current of the hole circulating around the plaquette. Since the hole can circulate in either the clockwise or counter-clockwise directions, the ground state is two-fold degenerate. The combination of the above results allows one to infer that the spin texture is analogous to a singly charged skyrmion found in classical chiral field theory [15]. A numerical study [16] found similar results in that the symmetry of the ground state in their work is identical to that of the skyrmion. In order to model many such partially localised carriers an effective Hamiltonian based on only the spin degrees of freedom was recently proposed by two of us [12]. Again, based on the semiclassical representation of the spins, a Hamiltonian thought to mimick the effect of a hole confined to a single plaquette was used: The first term is the usual Heisenberg interaction representing the AFM background. J is the exchange coupling and ij denotes nearest neighbours in the plane; all near-neighbour pairs are summed over. The second term is introduced to generate a spin distortion that is the same as that produced by a hole circulating around the plaquette, the corners of the plaquette being denoted by 1, 2, 3 and 4. It was motivated by the work of Wen, Wilczek, and Zee [17] on chiral spin liquids, and we shall refer to this term as the chiral-plaquette interaction. The prefactor of the interaction, D/S 4 , includes the 1/S 4 factor to ensure that the Heisenberg and one-plaquette interactions both scale as S 2 . In Ref. [12] it was found that the interaction strength D must exceed ∼ 2.2 J to be able to induce a singly-charged skyrmion ground state.
In order to model a plane doped with many O holes, one simply sums the chiral-plaquette interactions over all plaquettes having a Sr 2+ ion above them; we distribute these defects randomly in the plane, and the resulting system is necessarily spatially inhomogeneous. The results of Monte Carlo studies of the spin-spin correlation function [18] for large lattices at temperatures greater than J/2 were consistent with the empirical relation for the spin correlation length ξ(x, T ) found experimentally by Keimer et al. [19]. Note that in order for Eq. (2.2) to be obtained, it was necessary to use D ∼ 3 J. Further, in Ref. [12] we showed that a mean field model of this effective Hamiltonian was also consistent with Raman scattering experiments [20]. Now, in this paper, we extend our study of this effective Hamiltonian and demonstrate that the transverse spin-freezing temperature T f may be quantitatively determined.

III. SEMICLASSICAL EVALUATION OF T f :
Here we present a discussion that leads to the analytic prediction for the dependence of the spin-freezing temperature on doping. The theory is based on a calculation of the interaction energy between two skyrmions, and is performed utilizing the semiclassical approach of Shraiman and Siggia [14]. Two of us [21] have previously used a similar approach to calculate the interaction energy between two spin polarons generated in the frustrated bond model [22,23], a calculation that was also verified via exact diagonalization numerical work [21].
To begin, we introduce the spin texture associated with one skyrmion, where the spins are described by a continuum field which is chiral. The direction of the staggered magnetization is denoted by a chiral vector fieldΩ satisfyinĝ at every point in the plane. For a singly charged skyrmion (Q=± 1) placed at the origin of the coordinate system in use, and for one choice of phase (see below), where the staggered magnetization is chosen to be in the positive z direction at infinity, the chiral field is found where r = √ x 2 + y 2 is the distance in the two-dimensional plane from the centre of the skyrmion, and λ is the size of the skyrmion. That λ is the size of the skyrmion is seen from the z component, viz. for r < λ the direction of the staggered magnetization is reversed; formally, λ sets the length scale in the continuum problem. Further, upon changing the topological charge Q from 1 to -1, the handedness of the twisting of the staggered magnetization is found to change. This is seen in the x and y components: note that Ω x ± iΩ y ∝ x ± iy, and thus the twisting of the staggered magnetization is described by a vector which circulates in the counter-clockwise (Q = 1) or clockwise (Q = -1) directions. Lastly, note that these two components of the spin field represent the transverse degrees of freedom and in this paper we shall suggest that it is these components that lock together, due to the mutual interactions between skyrmions mediated by the Heisenberg AFM superexchange, leading to the spin freezing observed experimentally. A schematic representation of the spin texture for the Q = 1 skyrmion is shown in Fig. 2.
We place one skyrmion at r = 0, and then a second at r = (ℓ, 0). A charge of Q = 1 is chosen for the first, and the charge of the second will be determined by minimizing the interaction energy. The latter may be obtained from extrapolating the semiclassical theory [14] to describe this system. There, in the classical approximation, Shraiman and Siggia derived that the interaction between a dipolar spin distortion P a , such as that which would be produced by a ferromagnetic bond in an AFM background (here a = x or y labels a direction in the plane, such as the direction of the ferromagnetic bond), and the surrounding spins is given by The interaction strength g 2 is a phenomenological constant [14] which in the t ≫ J limit is of O(J). The ground state for the spin texture is then found by minimizing the sum of this energy and the usual Heisenberg AFM interactions between spins, the latter being most conveniently represented by the nonlinear sigma model. We cannot associate a dipole moment with the spin texture of a skyrmion, although it can be represented as a superposition of dipolar backflow spin distortions induced by mobile holes subsequently localised onto a single plaquette [24]. A more direct way to represent the interaction of the background spin texture with the distortion induced by the skyrmion is to replace where the staggered magnetization in this equation is that of the skyrmion. This is the analogue of the definition of the semiclassical dipole moment given by Shraiman and Siggia [25] for mobile holes. Then, we posit that one can represent the interaction between two skyrmions as where the superscripts 1 and 2 denote each of the two skyrmions. (Here we shall assume that each of the spin fields describing a skyrmion are those produced by systems with only one defect. Thus, no back-reaction effects are included -these will be considered in the next section.) As an estimate of this energy we evaluate E int at either of the points in space where either of the two skyrmions are located. In the limit of large ℓ we obtain when the second skyrmion is chosen to be a Q = -1 spin distortion, as this leads to the minimum energy. We note that the second skyrmion can be placed anywhere in the lattice with the same result (viz. Eq. (3.6)) being obtained as long as the second skyrmion has Q = -1. However, an arbitrary phase of the skyrmion must be introduced (one that is not included in Eq. (3.2) -see, e.g., Ref. [15] for such a spin field), and then fixed via minimizing E int with respect to this angle. (The interaction energy between two skyrmions with the same topological charge scales as 1/ℓ 4 in the large ℓ limit. As we will see below, this would lead to T f ∝ x 2 , which indeed is not observed experimentally.) For holes randomly placed on a two-dimensional lattice one expects that the doping x would be given by x = 1/ℓ 2 . This is consistent with low temperature neutron scattering studies of Keimer et al. [19] where the correlation length ξ(x, T ) was found to be behave approximately as ξ(x, 0) ∼ 1/ √ x. Thus, expressing the two-skyrmion interaction energy as a function of doping, we finally have Now, we suggest that one may associate the spin-freezing temperature T f with this interaction energy; to be specific, k B T f ∝ |E int |. This is consistent with the notion that there is only short-ranged ordering of the transverse spins (similar to the results of Ref. [10]) and the temperature at which the freezing occurs is such that at higher temperatures thermal fluctuations would destabilize a spin texture having these transverse degrees of freedom fixed. Of course, the energy associated with when the transverse degrees of freedom would become frozen into their ground state involves many skyrmions, and thus we can only consider our treatment to be valid in the low doping limit. Noting that the spins are S = 1/2 quantum spins, and thus we must include the renormalization factor Z c accounting for quantum fluctuations, we finally have that where Z c ∼ 1.2 [26]. This relationship demonstrates that if the transverse degrees of freedom which freeze are associated with the many skyrmion spin texture produced by randomly distributed defects, a linear dependence of T f on doping is expected. This is entirely consistent with the experiments of Chou et al. [3].

IV. NUMERICAL EVALUATION OF T f :
We have just seen that the freezing of the transverse degrees of freedom of two skyrmions separated by a distance 1/ √ x would lead to a spin-freezing temperature linear in doping x. In this section we present a numerical procedure to obtain the quantitative dependence of T f on the doping again in the low temperature and lightly doped regime. This work is required since the relationship found in the previous section, viz. Eq. (3.8), did not allow for (i) the back-reaction, and subsequent alteration, of a skyrmion's core, and thus must be an overestimate of the interaction energy, and (ii) relies on an assumption for g 2 , the interaction strength. Here we wish to eliminate these limitations and determine a very accurate numerical relationship for T f on x.
To establish a relationship between E int and x we will consider a lattice of classical spins with two skyrmions placed somewhere in the lattice a distance ℓ lattice spacings apart.
We calculate the interaction energy for such configurations by evaluating the ground state using Eq. (2.1) as the Hamiltonian (we have done the calculations for S = 1, and properly convert to S = 1 2 below). In Ref. [12] it was found that the constant D was required to be 2.5 < D/J < 3.3 in order to explain both the Raman scattering data [20]

The method employed in the evaluation of E int is straightforward and involves studying
L×L lattices with open boundary conditions with two chiral-plaquette interactions included on two plaquettes, and then extrapolating to L → ∞; for concreteness, we explain the method by considering Fig. 3. In this figure we have shown a 6 × 6 lattice and have labeled two plaquettes S 1 and S 2 , where the separation between the centres of these plaquettes correspond to ℓ = 2. Firstly, we find the single skyrmion energy for finite L by eliminating the chiral-plaquette interaction on S 2 , and then find the ground state spin configuration and associated energy (with open boundary conditions) -denote this energy by E 1 (L, ℓ), where the inclusion of ℓ in E 1 indicates that the skyrmion is off centre on the lattice. Then, the chiral-plaquette interaction on S 2 is reintroduced and the two-skyrmion energy may be found once the two-skyrmion ground state energy, denoted by E 2 (L, ℓ), is known (again evaluated for open boundary conditions). All zero-force spin configurations were evaluated using a conjugate-force method similar to that employed in Ref. [21]. Consistent with the above semiclassical work, we always find that the ground state for two defects corresponds to opposite topological charges on the plaquettes. Then, in analogy to a two-particle binding energy, one defines where E 0 (L) is the energy of a pure Heisenberg AFM, with open boundary conditions, for an L × L lattice. Finally, the large L limit is taken to evaluate the interaction energy.
An example of this extrapolation is shown in Fig. 4 for D/J = 3 and ℓ = 8 √ 2, the latter corresponding to x = .0078.
In Fig. 5 we have plotted |E int |/J vs x, utilizing ℓ = 1/ √ x, for all three ratios of D/J. We obtain 5.14x ± 0.08, 5.00x ± .08, and 4.74x To determine the quantitative behaviour of T f (x) we must (i) transform the above energies to represent those of S = 1 2 quantum spins, and (ii) determine the ratio of |E int | to k B T f . The first point may be incorporated by multiplying the interaction energies by S(S + 1) Z c .
The second is a nontrivial exercise, and to resolve it one can appeal to experiment. Chou et al. [3] investigated how best to describe the low-temperature maxima they observed in the NQR rate, and found that if one attempted to describe the maxima by a non-equilibrium freezing of the spins, with a correlation time τ associated with an Arrhenius law, viz.
then the relaxation rate peaked at T f with a related exponent Fig. 3 of Ref. [3]). Clearly, the energy E in the exponent in Eq. (4.3) must be associated with an interaction energy, and thus we incorporate |E int | into this equation.
A theoretical explanation for this relationship follows from application of the equipartition theorem. Apart from continuous symmetries, there are three differing low-energy states of the single defect problem: Q = ± 1, as well as an excitation with zero topological charge (see Ref. [11] for a discussion of this important low-energy excitation). Thus, there are 3 2 static degrees of freedom for the two defect system. However, since the spins are quantum spins, one should include a kinetic or dynamical component for the quantum fluctuations associated with each of these degrees of freedom. Then, counting 1 2 k B T for each such degree of freedom one expects Then, we immediately have estimates for the spin-freezing temperature: where we have used J ∼ 1550 K for the exchange constant. We anticipate error bars of at least 5% on the above values, the major uncertainties coming from Z c , and the exchange constant J.
The experimental result, viz. T f ∼ (815 K) x, shows that for D being close to 3 this two-skyrmion interaction model provides a very credible estimate of the spin-freezing temperature. We stress that the temperature and doping dependence of the spin correlation length was found to agree with Ref. [19], and the Raman scattering spectra of Ref. [20], when the interaction strength D ∼ 3 was used. It is gratifying that quantitative aspects of all three experiments seem to be consistent with the spin texture found when the low-temperature doped planes are described by Eq. (2.1).

V. DISCUSSION:
We have considered a simple model of the ground state spin texture produced by a Sr 2+ defect and a single hole localised in its region [11], and then utilized a model that allows for the description of many such defects [12], viz. Eq. (2.1), where the chiralplaquette interaction is summed over all plaquettes having a divalent Sr above them. We have used a semiclassical theory, and a numerical simulation technique, to determine the interaction energy between two such skyrmion states. We find that the ground state of such a system corresponds to two skyrmions having opposite topological charges, and that then the interaction energy, analogous to a two-particle binding energy, is proportional to 1/ℓ 2 , as in Eq. (3.6), where ℓ is the separation between defects. Then, we associate the doping x with this average distance via ℓ = 1/ √ x, consistent with the low-temperature spin correlation length determined by neutron scattering studies. The interaction energy is then found to scale linearly with doping, and if we associate the spin-freezing temperature with this interaction energy we can reproduce the same linear dependence that is observed experimentally. We have quantified this relationship by using the experimental value for the Arrhenius exponent given in Eq. (4.3). We can also present theoretical arguments that lead to values for the ratio of E to k B T f that give the same numerical value as in Eq. (4.3) (e.g., a simple equipartition theorem argument also gives a ratio of 9).
Alternatively, the frustrated bond model of Aharony et al. [22] predicts an interaction energy which scales linearly as x (see Ref. [21]), but the spin texture of the ground state depends on the location of the frustrated bonds in the plane (e.g., see Ref. [21]). Also, the ratio of Eq. (4.4) would be 4 for the frustrated bond model, and not ∼ 9, as found experimentally. However, it is not clear whether or not the experimental results of Chou et al. [3] depend on probe frequency, and thus it is not clear if this particular kind of experiment could distinguish between these two models of the spin texture of the weakly doped planes. Since the important physics of both these models is that defects produce longranged spin distortions, and there is some evidence that such distortions are also produced by mobile holes at moderate doping levels [12,27], it is encouraging that both models rely on spin deviations coupling the perturbing effects of the holes to the long-wavelength spin waves, thus further justifying our hydrodynamic approach.
We have showed that for our model, with the interaction strength D/J ∼ 3 in Eq. (2.1), the dependence of T f on x is reproduced quite accurately. Further, since our model with this same ratio of D/J also reproduces the neutron scattering measurements of the spin correlation length [19,28] and the Raman scattering spectra [20], it seems likely that these novel topological excitations of the S = 1 2 quantum AFM do indeed play a role in forming the spin textures of the low-temperature, inhomogeneously doped CuO 2 planes.
We wish to thank Fangcheng Chou, Ferdinand Borsa, and David Johnston for valuable discussions. This work was supported by the NSERC of Canada.