Longitudinal modes of spin fluctuations in iron-based superconductors

TL;DR

This paper investigates the longitudinal spin fluctuation modes in iron-based superconductors using an effective magnetic model, revealing insights into their magnetic behavior near phase boundaries.

Contribution

It provides a unified analysis of longitudinal spin modes in iron-based superconductors, especially near phase boundaries, enhancing understanding of their magnetic fluctuations.

Findings

Longitudinal modes vary near phase boundaries.

Results aid in identifying magnetic fluctuation nature.

Analysis applies to collinear antiferromagnetic phase.

Abstract

Iron-based superconductors can exhibit different magnetic ground states and are in a critical magnetic region where frustrated magnetic interactions strongly compete with each other. Here we investigate the longitudinal modes of spin fluctuations in an unified effective magnetic model for iron-based superconductors. We focus on the collinear antiferromagnetic phase (CAF) and calculate the behavior of the longitudinal modes when different phase boundaries are approached. The results can help to determine the nature of the magnetic fluctuations in iron-based superconductors.

Figures

Equations

$$H$$

$$\displaystyle\begin{array}[]{ll}S^{z}_{i}\rightarrow-S^{z}_{i},&S^{\pm}_{i}\rightarrow-S^{\mp}_{i},\\ S^{z}_{i}=s-a^{\dagger}_{i}a_{i},&S^{+}_{i}(S^{-}_{i})=\chi a_{i}(a^{\dagger}_{i}\chi),\end{array}$$

$$H\approx \sum _k\frac{1}{2}s{\Psi }_k^{†}{H}_k{\Psi }_{-k}-\sum _k\frac{1}{2}sTr(H_k)+E_0,$$

$$H_{k}=\left(\begin{array}[]{cc}\omega_{k}&\gamma_{k}\\ \gamma^{\dagger}_{k}&\omega^{T}_{-k}\end{array}\right),\qquad\sigma_{zn}=\left(\begin{array}[]{cc}I_{n}&0\\ 0&-I_{n}\end{array}\right)$$

$$|L_q⟩\equiv \frac{1}{\sqrt{N}}\sum _i{e}^{i\vec{q}\cdot {\vec{r}}_i}{S}_i^z|0⟩=\frac{1}{\sqrt{N}}\sum _k{a}_k^{†}{a}_{k+q}|0⟩.$$

$$E(q)=\frac{N(q)}{S(q)},$$

$$N(q)=\sum _m{N}_m(q)+\sum _m{N}_m^K(q)$$

$$N_m(q)=$$

$$N_m^K(q)=$$

$${\Pi }_{m+}=$$

$${\Pi }_{m-}=$$

$${\Pi }_{m+}^K=$$

$${\Pi }_{m-}^K=$$

$$\displaystyle S(q)=\frac{1}{n}Tr\Bigl{[}\rho(0)+\frac{1}{N^{\prime}}\sum_{k}(\rho_{k}\rho_{k+q}+\Delta_{k}\Delta_{k+q})\Bigr{]}.$$

$$\displaystyle\begin{array}[]{l}\epsilon_{k}=s\sqrt{\omega^{2}_{k}-\gamma^{2}_{k}},\\ \rho_{k}=\bigl{(}\sqrt{\omega_{k}^{2}/(\omega_{k}^{2}-\gamma_{k}^{2})}-1\bigr{)}/2,\\ \Delta_{k}=-\gamma_{k}/(2\sqrt{\omega_{k}^{2}-\gamma_{k}^{2}}).\end{array}$$

$${\omega }_k=$$

$${\gamma }_k=$$

$${\Psi }_k{\Psi }_q^{†}-(({\Psi }_q^{†}{)}^T{\Psi }_k^T{)}^T={\delta }_{kq}{\sigma }_{zn}.$$

$$\displaystyle\begin{array}[]{rl}\sigma_{zn}=&T_{k}\sigma_{zn}T^{\dagger}_{k},\\ \Lambda_{k}=&T_{k}^{-1}(\sigma_{zn}H_{k})T_{k},\end{array}$$

$$\displaystyle T_{k}=\left(\begin{array}[]{cc}\mu_{k}&\nu_{k}\\ \nu^{\ast}_{-k}&\mu^{\ast}_{-k}\end{array}\right),$$

$$\displaystyle\langle\Psi_{k}\Psi^{\dagger}_{q}\rangle=\left(\begin{array}[]{cc}\mu_{k}\mu^{\dagger}_{k}&\mu_{k}\nu^{T}_{-k}\\ \nu^{\ast}_{-k}\mu^{\dagger}_{k}&\nu^{\ast}_{-k}\nu^{T}_{-k}\end{array}\right)\delta_{kq}=\left(\begin{array}[]{cc}\rho^{T}_{k}+1&\Delta_{-k}\\ \Delta^{\dagger}_{-k}&\rho_{-k}\end{array}\right)\delta_{kq}.$$

$$\rho (m)$$

$$\Delta (m)$$

$$\displaystyle\begin{array}[]{rl}\langle a^{\dagger}_{l}b_{l+m}\rangle=&\rho_{ab}(m)=\langle a_{l}b^{\dagger}_{l+m}\rangle-\delta_{m0},\\ \langle a_{l}b_{l+m}\rangle=&\Delta_{ab}(m)=\langle a^{\dagger}_{l}b^{\dagger}_{l+m}\rangle.\end{array}$$

$$\displaystyle\begin{array}[]{rl}\langle a_{i}b_{j}a_{i}b_{j}\rangle=&2\Delta_{ab}^{2}+\Delta_{aa}\Delta_{bb},\\ \langle a_{i}b^{\dagger}_{j}a_{i}b^{\dagger}_{j}\rangle=&2\rho_{ab}^{2}+\Delta_{aa}\Delta_{bb},\\ \langle a^{\dagger}_{i}b^{\dagger}_{j}a^{\dagger}_{i}a_{j}\rangle=&2\rho_{aa}\Delta_{ab}+\rho_{ab}\Delta_{aa},\\ \langle a_{i}b^{\dagger}_{j}b^{\dagger}_{j}b_{j}\rangle=&=2\rho_{bb}\rho_{ab}+\Delta_{ab}\Delta_{bb}.\end{array}$$

$$⟨S_i^{†}{S}_j^{-}⟩=$$

$$⟨S_i^{-}{S}_j^{-}⟩=$$

$$\displaystyle\begin{array}[]{rl}\langle S^{-}_{i}S^{-}_{j}S^{-}_{i}S^{-}_{j}\rangle\approx&(2s)^{2}(2\Delta_{ab}^{2}+\Delta_{aa}\Delta_{bb}),\\ \langle S^{\dagger}_{i}S^{-}_{j}S^{\dagger}_{i}S^{-}_{j}\rangle\approx&(2s)^{2}(2\rho_{ab}^{2}+\Delta_{aa}\Delta_{bb}),\\ \langle S^{\dagger}_{i}S^{-}_{j}S^{z}_{i}S^{z}_{j}\rangle\approx&s^{2}\bigl{(}2(s-1)\rho_{ab}-5\rho_{ab}(\rho_{aa}+\rho_{bb})\\ &-5/2\Delta_{ab}(\Delta_{aa}+\Delta_{bb})\bigr{)},\\ \langle S^{\dagger}_{i}S^{\dagger}_{j}S^{z}_{i}S^{z}_{j}\rangle\approx&s^{2}\bigl{(}2(s-2)\Delta_{ab}-5\Delta_{ab}(\rho_{aa}+\rho_{bb})\\ &-5/2\rho_{ab}(\Delta_{aa}+\Delta_{bb})\bigr{)},\\ \end{array}$$

Full text

I Introduction

Iron-based superconductors have very rich magnetic propertiesDai et al. (2012); Dai (2015). They exhibit many intriguing magnetically ordered ground states, including stripe-like collinear antiferromagnetic (CAF) statede La Cruz et al. (2008), checkerboard-like antiferromagnetic (AFM) stateMarty et al. (2011), bi-collinear antiferromagnetic (BCAF) stateBao et al. (2009), staggered dimmer (DI)(Cao et al., 2015) state and some incomensurate (IC) states. The superconductivity appears to be linked to the magnetism, in particular, the CAF stateMazin et al. (2008). The origin of these magnetic states thus has been one of central focus in this field.

Although both itinerant and local spin theories are reasonably successful in explaining magnetic properties of some certain families of iron-based superconductors, it has been crystal clear that the magnetism is a hybrid with dual characters from both itinerant electrons and local spin moments(Hu et al., 2012; Glasbrenner et al., 2015). However, microscopically, the system can not be simply described by a Kondo lattice type of model because it is very difficult to separate itinerant electrons from localized ones.

A reasonable strategy is to seek an effective magnetic model. With the existence of local magnetic moments, we can still focus on the effective interactions between these local moments by integrating out of itinerant electrons to obtain a minimum magnetic effective model by keeping those Heisenberg-type leading interactions with the shortest distances. This approach has yielded a successful effective model, the $J_{1}-J_{2}-J_{3}-K$ modelHu et al. (2012); Glasbrenner et al. (2015); Fang et al. (2009, 2008); Yaresko et al. (2009); Wysocki et al. (2011); Glasbrenner et al. (2014); Wang et al. (2011). In this model, the nearest neighbor (NN) magnetic interaction $J_{1}$ and the next NN one $J_{2}$ arise mainly through local magnetic direct exchange and magnetic superexchange mechanisms. The third NN interaction $J_{3}$ and the quartic interaction $K$ indicate the existence of the strong couplings to itinerant electrons. The phase diagram of the model has been studied extensivelyHu et al. (2012); Glasbrenner et al. (2015). The mean-field results of the model can account for most magnetic phases and low energy magnetic excitations observed experimentally in iron-based superconductors(Hu et al., 2012) except some recently observed orders in very specific situations such as the double-$\mathbf{Q}$ orders(Giovannetti et al., 2011; Hoyer et al., 2016). The magnetism in the effective model is extremely frustrated due to the strong competition among $J_{1}$, $J_{2}$ and $J_{3}$, which is also consistent with the fact that the long range magnetic order is absent in some iron-based superconductors.

In a model based on local magnetic moments, the spin waves (SW) are the low energy excitations in a given magnetically ordered state. The SW are the transverse modes, namely the magnetic fluctuations perpendicular to the direction of the ordered moment. The longitudinal modes (LM), which are parallel to the ordered moments, are gapped out as Higgs mode. However, if there are several competing magnetic states, the LM can start to appear at low energy even at zero temperature. Therefore, the gaps of the LM (Higgs mass) can provide us important information about the degree of magnetic frustrationHaldane (1983); Wang et al. (2013); Luo et al. (2013).

Recently, several polarized neutron scattering experiments have been carried out in the CAF state of iron-based superconductorsWang et al. (2013); Qureshi et al. (2012); Song et al. (2013); Luo et al. (2013); Li et al. (2017). The gapped LM have been observed. The observed gaps of these LM are much lower than the band width of spin excitations. Thus, these measurements suggest the existence of strong magnetic frustration in the materials. The materials may be close to a quantum critical point or are located close to a spin liquid regionZaliznyak et al. (2015); Zhou et al. (2013).

Longitudinal excitations was viewed as support for itinerant magnetism(You et al., 2011; Knolle et al., 2010; Wang et al., 2013). In this paper, we use a new method to analytically calculate the LM from the effective exchange model, in particular, in the parameter region of the CAF state near the phase boundary. We find that the LM become visible at low energy close to the phase boundary and they have different dispersion relations from the transverse SW modes. They disperse very rapidly along the antiferromagnetic direction and have very little dispersion along the FM direction in the CAF phase. This feature is absent in other magnetically ordered states so that it is unique for the CAF state. Therefore, our results suggest that the measurement of the LM in a paramagnetic state that has a finite magnetic correlation length can be used to determine how the system is close to the CAF state.

II The $J_{1}-J_{2}-J_{3}-K$ Model

The $J_{1}-J_{2}-J_{3}-K$ modelHu et al. (2012); Glasbrenner et al. (2015) is described by

[TABLE]

which includes the nearest ($J_{1}$), second ($J_{2}$) and third ($J_{3}$) nearest neighbor Heisenberg interactions and $K$($>0$) quartic termChandra et al. (1990); Yaresko et al. (2009). Various magnetic phases can be classified in Fig.1. We will focus on the LM in the CAF phase and the behavior of approaching the phase boundaries .

This Hamiltonian can be solved within the standard spin wave (Holstein-Primakov) theory. Starting from the above depicted classical ground state, we rotate the down spins and then represent the spins $\vec{S}_{i}$ with magnons $a_{i},a^{\dagger}_{i}$:

[TABLE]

with $\chi=\sqrt{2s-a^{\dagger}_{i}a_{i}}$. When the local environment for each spin is not identical, there are several spins in one magnetic cell and the site $i=\{c,a\}$ with $c=1,\cdots,N^{\prime}$ counting for the cells and $a=1,\cdots,n$ for the magnon types. Up to two-magnon operators (we would omit the vector notation for position and momentum), the approximate Hamiltonian in momentum space is

[TABLE]

with $H_{k}$ the $2n\times 2n$ matrix, $E_{0}$ the classical ground state energy and $\Psi^{\dagger}_{k}=(\mathbf{a}^{\dagger}_{k},\mathbf{a}^{T}_{-k})$, where $\mathbf{a}_{k}$ is a column collection of magnon annihilators and $\mathbf{a}^{T}_{k}$ is the matrix transpose. As derived in the appendix.(A), this Hamiltonian can be diagonalized by Bogoliubov transformationXiao (2009) and the SW spectra $\epsilon_{nk}$ are the eigenvalues of $\sigma_{zn}H_{k}$. Here

[TABLE]

with $I_{n}$ the $n\times n$ identical matrix and $\omega_{k}=\omega^{\dagger}_{k}=\omega^{T}_{-k}$, $\gamma_{k}=\gamma^{T}_{-k}=\gamma^{\dagger}_{k}$. When $n=1$, it is easy to find $\epsilon_{k}=s\sqrt{\omega_{k}^{2}-\gamma_{k}^{2}}$. This SW spectrum is generally gapless at the order momentum as the Goldstone mode. Coupling anisotropy can interpret the observed gap(Wang et al., 2013). We would shift the Heisenberg to XXZ coupling $\vec{S_{i}}\cdot\vec{S_{j}}=S^{x}_{i}S^{x}_{j}+S^{y}_{i}S^{y}_{j}+AS^{z}_{i}S^{z}_{j}$, with $A=1+\delta_{a}$. This positive $\delta_{a}$ will pick up the easy axis and speed up the numerical calculations.

II.1 The Longitudinal Mode

In quantum antiferromagnet, the longitudinal modes are the amplitude fluctuation of the ordered moments $\langle S^{z}\rangle$, essentially, magnon density. Different from the itinerant approach(Knolle et al., 2010) and nonlinear-$\sigma$ model(Haldane, 1983; You et al., 2011), LM can also be viewed as two magnon resonance(Affleck and Wellman, 1992) or magnon density wave(Xian and Merdan, 2014) in Holstein-Primakov theory. Following Feynman’s approach to the helium superfluidFeynman (1954), the LM can be defined as:

[TABLE]

With respect to the ground state, the LM has the spectrum:

[TABLE]

with $N(q)\equiv\langle L_{q}|H|L_{q}\rangle$ and $S(q)\equiv\langle L_{q}|L_{q}\rangle$ the structure factor. Separating the Hamiltonian by neighborhood ($\vec{m}$) coupling: $H=\sum_{m}H_{m}+\sum_{m}H^{K}_{m}$, we have

[TABLE]

with (dependent on whether the $\vec{m}$ neighbor spins are anti-parallel or parallel (AP/P))

[TABLE]

where $\sum_{a}$ sums over different magnon types and the correlation function $\Pi$’s are (see appendix.A)

[TABLE]

Here $\rho_{ab}(m)\equiv\langle a^{\dagger}_{i}b_{i+m}\rangle,\ \Delta_{ab}(m)\equiv\langle a_{i}b_{i+m}\rangle$ are the correlations of type $a,b$ magnon at site $i,i+m$. We have omitted the neighborhood $(m)$ to simplify the notation. The quartic $K$ term modifies the exchange $J$ by $\pm 2AKs^{2}$ at the first order of large $s$. The $n\times n$ correlation matrices in momentum space $\rho_{k},\ \Delta_{k}$ are the Fourier transformation of $\rho(m)$, $\Delta(m)$. The structure factor is

[TABLE]

In the case of one type magnon, the SW spectrum and the correlators can be solved analytically:

[TABLE]

II.2 The CAF Phase

The CAF phase with ordered momentum $\mathbf{Q}=X=(\pi,0)$ is an example of the exactly solvable case of Eq.(21) with

[TABLE]

The spectra of SW and LM for FeSe at $9\ GPa$(Glasbrenner et al., 2015) are depicted in Fig.2, in the unit of $J_{1}s$ ($51.1$ meV). The Goldstone modes appear at $\Gamma$ and $\mathbf{Q}$ in the SW spectrum Fig.2(a). In our calculation, the anisotropic exchange opens a tiny gap $0.07J_{1}s$ $(\sim\sqrt{\delta_{a}})$. This is often introduced to explain the observed anisotropic gap in the transverse modes(Wang et al., 2013). In totally isotropic case $(\delta_{a}=K=0)$, it would be gapless at all high symmetry points $YMX\Gamma$ as one can see from Eq.(21). Zooming into $\mathbf{Q}$, the SW has oval-shape equal energy line as seen from the 3D view in Fig.2(c) which has been measured experimentally.

The LM spectrum is depicted in Fig.2(b). It has a deep valley structure along the $k_{x}=\pi$ line. The dispersion is flat in $(\pi,\pm\delta_{q})$ direction, but it is steep in $(\pi\pm\delta_{q},0)$ as shown in Fig.2(b,d). The valley structure is manifested in Fig.2(d). It is interesting to point out that at $\mathbf{Q}=(\pi,0)$, the structure factor $S(\mathbf{Q})=0$. This is protected by the symmetry $\gamma_{k+Q}=-\gamma_{k}$, not an issue of approximation. The coupling in $c$ axis can break this symmetry, resulting in a well-defined LM. Indeed, $J_{c}$ is important for the occurrence of long range magnetic order. Yet, for numerical consideration, the longitudinal gap can be obtained from the nearby region since Eq.(8) is a smooth function except at $\mathbf{Q}$.

The dispersion along some high symmetry directions are shown in Fig.2(e). As expected, the transverse mode converges to almost zero and its linear dispersion shows slight anisotropy at $\mathbf{Q}$. The LM at this point is gapped and strongly anisotropic. The dispersion is flat in $q_{y}$ ($XM$) direction and sharp in $q_{x}$ ($X\Gamma$) direction. The calculated longitudinal gap $\Delta=0.87J_{1}s=44.5$ meV, which is not measured yet.

As approaching the phases boundaries, the order parameter would be soften due to the competition among various magnetic orders(Affleck and Wellman, 1992) and the LM would appear in low energy as frustration arises. The longitudinal gap $\Delta$ is depicted in Fig,3 as approaching the phase boundaries. The gap is taken as the limit from the flat $k_{y}$ direction. The gap $\Delta$ drops to zero near the phases boundaries, appearing as low energy excitation in the neutron scattering experiment. We also noticed that the gaps of the three empty markers (K=0.5), as routes approaching BCAF, DI-horizontally and DI-vertically respectively, drop faster to zero in front of the boundaries, as a possible sign of stronger frustration. No big difference is found for the paths approaching the DI phases horizontally (varying $J_{2}$) and vertically (varying $J_{3}$). Thus in this effective model, localized exchange and itinerant coupling equivalently contribute to frustration to hold LM.

Adjust to the models and parameters in Ref.(Wang et al., 2013), a $53$ meV longitudinal gap can be obtained with our calculation, comparing with the observed value $25\sim 30$ meV. This result is very reasonable. As fluctuation effect in the vicinity of critical region is underestimated in our method, the gap in our calculation should be larger than the true gap. For more accurate quantitative result, the higher order correction has to be considered.

II.3 Other Magnetic Phase

Similar work can be done for various magnetic phases represented in Fig.1, following the method generally described in Appendix.(A). The $J_{1}$ dominated AF state with $\mathbf{Q}^{\prime}=M=(\pi,\pi)$ is another example of exactly solvable case. Its SW the LM along the high symmetry points are in Fig.4. Similar to the CAF phase, the SW is gapless and the LM is gapped at $\mathbf{Q}^{\prime}$. The magnetic environment is $C_{4}$ symmetric and both the SW and LM dispersion are isotropic. Approaching to the phases boundaries, the longitudinal gaps also decreases to appear as low excitations due to phases competition. The LM are always more sensitive to the quantum frustration. Near the quantum critical point, the interaction among magnons in LM breaks the ground state.

III Discussion and Summary

Recently, the inelastic neutron scattering experiments have already measured the LM in iron-based superconducting materialsWang et al. (2013); Luo et al. (2013); Waßer et al. (2016); Zhang et al. (2014); Steffens et al. (2013). These results are typically considered as evidence to support pure itinerant magnetism in these materialsWang et al. (2013); Knolle et al. (2010); You et al. (2011). However, as argued in literature(Hu et al., 2012), a pure itinerant magnetism can not explain all the observed magnetic properties in iron-based superconductors. According to our calculation, LM can also emerge from the effective exchange model.

Combining with the experimental results, our results strongly support that iron-based superconductors are strongly frustrated magnetic systems in a vicinity to many different magnetic phases. In particular, the CAF order is very close to quantum critical transitions to other magnetic orders. The nature of the frustration stems from the competitions between short range local magnetic exchange couplings and other effective magnetic interactions through couplings with itinerant electrons.

In summary, we derived the longitudinal excitation for the general $J_{1}-J_{2}-J_{3}-K$ magnetic model. Specifically, the analytic solution in the CAF state is given. The dispersion of the LM in the CAF state near the quantum critical point is very different from the transverse SW modes.

Acknowledgement: the work is supported by the Ministry of Science and Technology of China 973 program(Grant No. 2015CB921300), National Science Foundation of China (Grant No. NSFC-11334012, No. NSFC-11534014), the Strategic Priority Research Program of CAS (Grant No. XDB07000000) and the International Young Scientist Fellowship of Institute of Physics CAS (Grant No.2017002).

Appendix A Derivation

A.1 Spin Wave spectrum

The magnon satisfies the bosonic commute relation:

[TABLE]

With the help of Bogoliubov transformation $\Psi_{k}=T_{k}\Phi_{k}$, where $\Phi^{\dagger}_{k}=(\mathbf{d}^{\dagger}_{k},\mathbf{d}^{T}_{-k})$, the Hamiltonian can be diagonalized as $T^{\dagger}_{k}H_{k}T_{k}=\sigma_{zn}\Lambda_{k}$, when the transformation matrix satisfy

[TABLE]

where $\Lambda_{k}$ is the diagonalized matrix of $\sigma_{zn}H_{k}$ by similarity transformation and $T_{k}$ is the Bogoliubov matrixXiao (2009):

[TABLE]

where $\mu_{k},\nu_{k}$ are the $n\times n$ coherent phase matrices. One can collect the eigenvectors of matrix $\sigma_{zn}H_{k}$ and then normalize them by equation.(26). It is proven that for the eigenvalue $\epsilon_{k}$ with eigenvector $\mathbf{v}_{k}$, there exists a dual eigenvalue $-\epsilon^{\ast}_{-k}$ with eigenvector $\sigma_{xn}\mathbf{v}^{\ast}_{-k}$. Thus by properly order the eigenvalues and adjust the relative phases, this $T_{k}$ is the Bogoliubov matrix to diagonalize the Hamiltonian via congruence transformation. In algorithm, there could be phase freedom for those eigenvectors, making the matrix non-Bogoliubov, i.e. $T\rightarrow Te^{i\Theta}$ with $\Theta=diag\{\theta_{1},\cdots,\theta_{2n}\}$ the phases matrix. It’s easy to remove those phases and, moreover, this phase freedom does not affect the SW spectra and the LM.

In linear SW theory, the Hamiltonian matrix elements are the Fourier transformation of the neighborhood interaction in real space, so $\omega_{k},\gamma_{k},\epsilon_{k},\mu_{k},\nu_{k}$ all satisfy $f^{\ast}(k)=f(-k)$ and the diagonalized Hamiltonian $\sigma_{zn}\Lambda_{k}$ is real and doubly degenerate.

A.2 Longitudinal Spectrum

The essence of calculating the ground state expectation of the commutators $N_{m}(q)$ and $N^{K}_{m}(q)$ in equ.(9) is to calculate the correlation of spins, up to second $\langle SS\rangle$ and forth order $\langle SSSS\rangle$. Define the magnon correlators first:

[TABLE]

It is easy to see $\rho^{\dagger}_{k}=\rho_{k}$ and $\Delta_{k}=\Delta^{T}_{-k}$. In linear spin theory, $f(-k)=f^{\ast}(k)$ also works for $\rho_{k}$ and $\Delta_{k}$. The real space correlations

[TABLE]

satisfy $\rho^{\ast}(m)=\rho^{\dagger}(-m)=\rho(m)$ and $\Delta^{\ast}(m)=\Delta^{\dagger}(-m)=\Delta(m)$. Thus the correlation among different magnons is the elements of the correlation matrices $\rho(m)$ and $\Delta(m)$:

[TABLE]

For higher (even) operators correlation, thereafter, the contraction rules can be concluded $\langle c_{1}c_{2}\cdots c_{2n-1}c_{2n}\rangle$ ($c_{i}=a_{i},a_{i}^{\dagger}$):

Put the $c^{\prime}s$ operators in pairs, all possible combinations; 2. 2.

Refer to the matrix element of $\rho(m)$ and $\Delta(m)$, write all the operator pairs as two-operator correlation functions in real space.

Referring this contraction rule, the following four operators (and their conjugate) correlation is needed:

[TABLE]

We omit the variables $(j-i)$ within $\rho_{ab},\Delta_{ab}$ for convenience. Relevant to the LM up to the four operators correlation, for $N_{m}(q)$, we need to estimate :

[TABLE]

and in the quartic term $N^{K}_{m}(q)$

[TABLE]

With the above correlators derived, the correlation function $\Pi$’s in $N(q)$ can be obtained.