Amplitude mode in the planar triangular antiferromagnet Na0.9MnO2

Amplitude modes arising from symmetry breaking in materials are of broad interest in condensed matter physics. These modes reflect an oscillation in the amplitude of a complex order parameter, yet are typically unstable and decay into oscillations of the order parameter’s phase. This renders stable amplitude modes rare, and exotic effects in quantum antiferromagnets have historically provided a realm for their detection. Here we report an alternate route to realizing amplitude modes in magnetic materials by demonstrating that an antiferromagnet on a two-dimensional anisotropic triangular lattice (α-Na0.9MnO2) exhibits a long-lived, coherent oscillation of its staggered magnetization field. Our results show that geometric frustration of Heisenberg spins with uniaxial single-ion anisotropy can renormalize the interactions of a dense two-dimensional network of moments into largely decoupled, one-dimensional chains that manifest a longitudinally polarized-bound state. This bound state is driven by the Ising-like anisotropy inherent to the Mn3+ ions of this compound.

Transforming this H 2 term into momentum space gives This has the form with ω k = 2[J 1 + D + J 2 cos(k · e 3 )] and λ k = 2[J 1 cos(k · e 1 ) + J 2 cos(k · e 2 )]. This can be simplified by applying the Bogoliubov transformation, The parameter θ k is determined by inserting Eq. (9) into Eq. (8) and requiring that the Using the general expression, we obtain two conditions: The solution to Eq. (11) is assuming ω k > 0, which is satisfied here. This gives the single magnon spin energy The expression in the text is obtained by using the explicit forms for ω k and λ k given above, and rewriting them in the reciprocal lattice units defined in the main text: k · e 1 = 2πK, k · e 2 = π(H − K), and k · e 3 = −π(H + K).

Spin wave theory and path integral
In this section, we expand about the 0 K antiferromagnetic ground state and ignore the coupling J 2 between chains for simplicity, since it is very small and does not signficantly influence the magnon binding in this limit. This reduces the problem to a single Heisenberg chain with single-ion Ising anisotropy: with J = J 1 , D > 0, and we assume D J. As in the prior subsection, we apply spin wave theory using the Holstein-Primakoff relations in Eq. (2), (3). In the departure from the prervious subsection, we, however, do not, at this point, truncate to quadratic order in the boson operators. The Holstein-Primakoff representation inserted into H gives a complicated expression for H[a † , a, b † , b], which we do not write explicitly. Now we write the path integral representation of this boson problem, which has the action It is very convenient to make the transformation b ↔b. This leads to The benefit of this form is that the U(1) symmetry of S z conservation is manifest, and the resulting action has no anomalous terms. It additionally is convenient for a field-theoretical analysis, which allows a simplified treatment of the low energy physics at small momentum and low energy. This is advantageous because going beyond quadratic order otherwise introduces a very algebraically complicated problem.

Continuum field theory
Now we proceed to convert this into a continuum field theory. We first make the transformation and perform a gradient expansion, i.e. take a continuum limit. We let x = 2n and n → ∫ dx/(2). The action can be expanded to quartic order as S = S 0 + S 2 + S 4 , with Here we see that the η field has a large mass of order JS, and hence dropped a second spatial derivative term of η, which is sub-dominant. The contribution of order DS to the mass of η may also be neglected. Now we can integrate out η to obtain an action for φ alone: It is convenient to make the following rescalings: The transformation on τ is equivalent to measuring energy in units of 2JS. This brings the quadratic action into canonical form: with m 2 = 2D/J. With these rescalings, the Green's function is just Now, carrying through all the same rescalings, the quartic part of the action reads Here we set the massive field η → 0, and kept only leading terms in gradients, using the dimension counting m ∼ ∂ x .

Diagrammatic analysis
We first consider the longitudinal spin fluctuations and magnon bound state in a diagrammatic analysis. The correlation function of the longitudinal spin fluctuations, is the leading term in the fluctuations of the spin along the z axis, c.f. Eq. (2). Its Fourier transform is diagrammatically If we neglect the final diagram in Eq. (26), we see the first three terms in a ladder series. We will sum this series to obtain what is hopefully the leading approximation of longitudinal correlation function. From Eq. (24) we can read off the vertex Note that the entire vertex is of order 1/S, as expected from the semi-classical nature of the spin wave expansion. It consists of two terms. The first is parametrically small in the anisotropy-induced mass, while the second represents magnon interactions that would be present in the isotropic Heisenberg limit. The latter interactions have larger amplitude but vanish in the small momentum limit.
Next consider the special case of zero momentum, i.e. at the Bragg peak position, k = 0. In this case a representative term in the ladder sum has internal momenta as indicated here: Here we have not shown the internal frequency variables explicitly, to keep the diagram uncluttered. Comparing with the vertex in Eq. (27), one can can see that ladder diagrams with this structure enjoy an important simplification. The momentum integration over q corresponding to the left-most loop in Eq. (28) contains a denominator (the product of two Green's functions) which is even in q, but the momentum-dependent vertex factors from the second term in Eq. (27) is odd in q, and thus vanish under integration. This holds true by induction, working from left to right in the diagram, for every vertex. Therefore, within the ladder sum, we may replace the vertex in Eq. (27) by the constant term −m 2 /S. One obtains then the usual RPA type form for the ladder sum, with the polarization bubble Performing the Ω integral and simplifying, we obtain with (k) = √ k 2 + m 2 . One can see from Eq. (31) that the polarization bubble is analytic and real for |ω| < 2m, i.e. below the threshold for 2-magnon creation. We are seeking a bound state, so are interested in this regime. The integral is fully convergent at large k, so we can take the limits to be infinite, and rescale k → mk and take iω n → ω to obtain The condition for a bound state is the vanishing denominator of the RPA expression in Eq. (29), i.e.
We see that for large S (which we are assuming), the polarization must be large to achieve this condition. This implies the bound state occurs close to the edge of the continuum, i.e. 2m − ω m, so that the integral in Eq. (32) is large. Let with δ 1. Then the integral in Eq. (32) is dominated by momenta of order δ, so we can neglect the factor of k 2 in the square root in the denominator of the first term in the integrand. Then the integral can be carried out and one obtains the solution for δ, Note that from Eq. (34), δ is defined as the binding energy divided by the two-magnon gap. So if we define the one-magnon gap ∆, then we have

Quantum mechanics approach
Now the simplicity of the result, and especially the dropping out of the complicated vertex factors in the ladder diagrams, suggests there is a simpler explanation. Intuitively, the binding energy is small even compared to the small gap m (by the 1/S 2 factor). This suggests the bound state is composed of wavevectors k b which are small compared to the mass m. In that case, a non-relativistic approximation holds. The condition k b m would justify the neglect of the gradient terms in the interaction, Eq. (24). Then one ought to reduce the problem to one of just two particles interaction via a delta function potential of strength 1/S.
To see how this works in practice, we introduce the mode expansion for a complex scalar field, standard in field theory textbooks (or from the analysis of acoustic phonons): where the a k , b k are two independent sets of canonical fermion annihilation operators. They represent magnons with spin S z = +1 and S z = −1, respectively. The quadratic Hamiltonian corresponding to S 2 is then The longitudinal bound state should be considered a state with one of each type of fermion. Hence we insert the mode expansion into the Hamiltonian corresponding to S 4 (which is just the same as S 4 without the τ integral), and keep only terms which describe a-b interactions: If only states with momenta |k | m are involved, the second term in the square brackets above is negligible, and moreover we may approximate k ≈ m. Then we have We recognize an attractive delta-function interaction between a and b particles, with strength u = −1/S 1. In the same limit |k | m, we may approximate k ≈ m+k 2 /2m in H 2 in Eq. (38). The problem is now a fully non-relativistic one. For the case of one a and one b particle, we can write the first quantized Hamiltonian The ground state of this Hamiltonian has zero total momentum and the wavefunction with λ = mu/2 and the energy E = 2m − E b , with binding energy E b = mu 2 /4. Using this result and u = 1/S, and writing, in these units, the single magnon gap ∆ = m, we obtain This is the same as we found in the diagrammatic treatment.
What have we gained? Well we don't need the diagrams! But we also gain a bit more understanding. We see that that λ/m = u/2 = 1/2S 1, which justifies the assumption k m. Furthermore, due to the Galilean invariance of Eq. (41), the general solution for the ground state of Eq. (41) with non-zero momentum k is just that of a particle with mass 2m, i.e.
So the bound state has a higher mass than the single spin wave, and appears consequently to have a flatter dispersion. This is, however, only valid for k m. More generally, the dispersion for larger k will appear more relativistic.