### Syntax

spectra = spinwave(obj,Q)

spectra = spinwave(___,Name,Value)

### Description

spinwave(obj,Q,Name,Value) calculates spin wave dispersion and spin-spin correlation function at the reciprocal space points $$Q$$. The function can solve any single-k magnetic structure exactly and any multi-k magnetic structure appoximately and quadratic spinw-spin interactions as well as single ion anisotropy and magnetic field. Biquadratic exchange interactions are also implemented, however only for $$k_m=0$$ magnetic structures.

If the magnetic ordering wavevector is non-integer, the dispersion is calculated using a coordinate system rotating from unit cell to unit cell. In this case the spin Hamiltonian has to fulfill this extra rotational symmetry which is not checked programatically.

Some of the code of the function can run faster if mex files are used. To switch on mex files, use the swpref.setpref('usemex',true) command. For details see the sw_mex and swpref.setpref functions.

### Examples

To calculate and plot the spin wave dispersion of the triangular lattice antiferromagnet ($$S=1$$, $$J=1$$) along the $$(h,h,0)$$ direction in reciprocal space we create the built in triangular lattice model using sw_model.

tri = sw_model('triAF',1)
spec = tri.spinwave({[0 0 0] [1 1 0]})
sw_plotspec(spec)


### Input Arguments

obj
spinw object.
Q
Defines the $$Q$$ points where the spectra is calculated, in reciprocal lattice units, size is $$[3\times n_{Q}]$$. $$Q$$ can be also defined by several linear scan in reciprocal space. In this case Q is cell type, where each element of the cell defines a point in $$Q$$ space. Linear scans are assumed between consecutive points. Also the number of $$Q$$ points can be specified as a last element, it is 100 by defaults.

For example to define a scan along $$(h,0,0)$$ from $$h=0$$ to $$h=1$$ using 200 $$Q$$ points the following input should be used:

Q = {[0 0 0] [1 0 0]  200}


For symbolic calculation at a general reciprocal space point use sym type input.

For example to calculate the spectrum along $$(h,0,0)$$ use:

Q = [sym('h') 0 0]


To calculate spectrum at a specific $$Q$$ point symbolically, e.g. at $$(0,1,0)$$ use:

Q = sym([0 1 0])


### Name-Value Pair Arguments

'formfact'
If true, the magnetic form factor is included in the spin-spin correlation function calculation. The form factor coefficients are stored in obj.unit_cell.ff(1,:,atomIndex). Default value is false.
'formfactfun'
Function that calculates the magnetic form factor for given $$Q$$ value. value. Default value is @sw_mff, that uses a tabulated coefficients for the form factor calculation. For anisotropic form factors a user defined function can be written that has the following header:
F = formfactfun(atomLabel,Q)


where the parameters are:

• F row vector containing the form factor for every input $$Q$$ value
• atomLabel string, label of the selected magnetic atom
• Q matrix with dimensions of $$[3\times n_Q]$$, where each column contains a $$Q$$ vector in $$Å^{-1}$$ units.
'gtensor'
If true, the g-tensor will be included in the spin-spin correlation function. Including anisotropic g-tensor or different g-tensor for different ions is only possible here. Including a simple isotropic g-tensor is possible afterwards using the sw_instrument function.
'fitmode'
If true, function is optimized for multiple consecutive calls (e.g. the output spectrum won’t contain the copy of obj), default is false.
'notwin'
If true, the spectra of the twins won’t be calculated. Default is false.
'sortMode'
If true, the spin wave modes will be sorted by continuity. Default is true.
'optmem'
Parameter to optimise memory usage. The list of Q values will be cut into optmem number of pieces and will be calculated piece by piece to decrease peak memory usage. Default value is 0, when the number of slices are determined automatically from the available free memory.
'tol'
Tolerance of the incommensurability of the magnetic ordering wavevector. Deviations from integer values of the ordering wavevector smaller than the tolerance are considered to be commensurate. Default value is $$10^{-4}$$.
'omega_tol'
Tolerance on the energy difference of degenerate modes when diagonalising the quadratic form, default value is $$10^{-5}$$.
'hermit'
Method for matrix diagonalization with the following logical values:
• true using Colpa’s method (for details see J.H.P. Colpa, Physica 93A (1978) 327), the dynamical matrix is converted into another Hermitian matrix, that will give the real eigenvalues.
• false using the standard method (for details see R.M. White, PR 139 (1965) A450) the non-Hermitian $$\mathcal{g}\times \mathcal{H}$$ matrix will be diagonalised, which is computationally less efficient. Default value is true.
'saveH'
If true, the quadratic form of the Hamiltonian is also saved in the output. Be carefull, it can take up lots of memory. Default value is false.
'saveV'
If true, the matrices that transform the normal magnon modes into the magnon modes localized on the spins are also saved into the output. Be carefull, it can take up lots of memory. Default value is false.
'saveSabp'
If true, the dynamical structure factor in the rotating frame $$S'(k,\omega)$$ is saved. Default value is false.
'title'
Gives a title string to the simulation that is saved in the output.
'fid'
Defines whether to provide text output. The default value is determined by the fid preference stored in swpref. The possible values are:
• 0 No text output is generated.
• 1 Text output in the MATLAB Command Window.
• fid File ID provided by the fopen command, the output is written into the opened file stream.
'tid'
Determines if the elapsed and required time for the calculation is displayed. The default value is determined by the tid preference stored in swpref. The following values are allowed (for more details see sw_timeit):
• 0 No timing is executed.
• 1 Display the timing in the Command Window.
• 2 Show the timing in a separat pup-up window.

### Output Arguments

spectra
structure, with the following fields:
• omega Calculated spin wave dispersion with dimensions of $$[n_{mode}\times n_{Q}]$$.
• Sab Dynamical structure factor with dimensins of $$[3\times 3\times n_{mode}\times n_{Q}]$$. Each (:,:,i,j) submatrix contains the 9 correlation functions $$S^{xx}$$, $$S^{xy}$$, $$S^{xz}$$, etc. If given, magnetic form factor is included. Intensity is in ħ units, normalized to the crystallographic unit cell.
• H Quadratic form of the Hamiltonian. Only saved if saveH is true.
• V Transformation matrix from the normal magnon modes to the magnons localized on spins using the following: $$x_i = \sum_j V_{ij} \times x_j'$$ Only saved if saveV is true.
• Sabp Dynamical structure factor in the rotating frame, dimensions are $$[3\times 3\times n_{mode}\times n_{Q}]$$, but the number of modes are equal to twice the number of magnetic atoms.
• formfact Cell containing the labels of the magnetic ions if form factor in included in the spin-spin correlation function.
• cmplxBase The local coordinate system on each magnetic moment is defined by the complex magnetic moments: \begin{align} e_1 &= \Im(\hat{M})\\ e_3 &= Re(\hat{M})\\ e_2 &= e_3\times e_1 \end{align}

• hkl Contains the input $$Q$$ values, dimensions are $$[3\times n_{Q}]$$.
• hklA Same $$Q$$ values, but in $$Å^{-1}$$ unit, in the lab coordinate system, dimensins are $$[3\times n_{Q}]$$.
• incomm Logical value, tells whether the calculated spectra is incommensurate or not.
• obj The copy (clone) of the input obj, see spinw.copy.

The number of magnetic modes (labeled by nMode) for commensurate structures is double the number of magnetic atoms in the magnetic cell. For incommensurate structures this number is tripled due to the appearance of the $$(Q\pm k_m)$$ Fourier components in the correlation functions. For every $$Q$$ points in the following order: $$(Q-k_m,Q,Q+k_m)$$.

If several twins exist in the sample, omega and Sab are packaged into a cell, that contains $$n_{twin}$$ number of matrices.