Spin Correlations and Dynamical Structure Factors

The dynamical spin structure factor is the bridge from a magnetic Hamiltonian to scattering and local-probe spectra. It locates elastic order, magnon poles, multiparticle continua, and redistribution of spectral weight. Its normalization and sum rules are essential: a visually convincing dispersion that violates them is not a complete magnetic description.

Required background. Spin Waves and Magnons supplies one-magnon states and coherence factors; Spectral Moments and Sum Rules supplies Lehmann positivity and frequency moments. Helpful background. Collective Scattering Spectroscopy supplies instrumental resolution, form factors, and inversion limits.

Definition and exact relations

For $N$ spins at positions $\mathbf r_i$, choose

$$S^{\alpha\beta}(\mathbf q,\omega)=\frac1{2\pi N} \int_{-\infty}^{\infty}dt\,e^{i\omega t} \sum_{ij}e^{-i\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} \langle S_i^\alpha(t)S_j^\beta(0)\rangle.$$

Its Lehmann representation is nonnegative for diagonal components at positive spectral weights. Frequency integration returns the equal-time correlator,

$$\int d\omega\,S^{\alpha\beta}(\mathbf q,\omega) =\frac1N\sum_{ij}e^{-i\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} \langle S_i^\alpha S_j^\beta\rangle.$$

For fixed-length spins and a complete discrete Brillouin zone,

$$\frac1N\sum_{\mathbf q}\int d\omega \sum_{\alpha=x,y,z}S^{\alpha\alpha}(\mathbf q,\omega)=S(S+1).$$

Covalency, form factors, incomplete energy windows, and itinerant charge fluctuations affect how an experimental integral approaches this local-moment sum rule; they do not justify changing the theoretical normalization silently. Frequency-moment constraints for magnetic chains were derived by Hohenberg and Brinkman 1974, pp. 128–131.

Thermal equilibrium gives detailed balance,

$$S^{\beta\alpha}(-\mathbf q,-\omega) =e^{-\beta\omega}S^{\alpha\beta}(\mathbf q,\omega),$$

and the fluctuation–dissipation relation

$$S^{\alpha\beta}(\mathbf q,\omega) =\frac{\chi_{\alpha\beta}''(\mathbf q,\omega)} {\pi(1-e^{-\beta\omega})}$$

for compatible susceptibility and Fourier conventions.

From magnons to continua

Long-range magnetic order contributes an elastic Bragg term at the ordering wave vector. Linear spin-wave theory gives one-magnon delta functions,

$$S(\mathbf q,\omega)\supset Z_{\mathbf q}\, \delta(\omega-\omega_{\mathbf q}),$$

with a polarization- and Bogoliubov-dependent residue $Z_{\mathbf q}$. Interactions shift and broaden the pole and transfer weight to two- and higher-magnon continua. In a noncollinear magnet, cubic vertices can permit spontaneous decay even at zero temperature.

A continuum can also arise from fractional quasiparticles whose physical spin operator creates pairs or larger sets. Its boundaries, polarization, temperature evolution, momentum-integrated weight, and consistency with thermodynamics are then more informative than breadth alone. The continuum observed in the triangular antiferromagnet Cs$_2$CuCl$_4$ provided an influential fractionalization comparison Coldea et al. 2001, pp. 1335–1338, but subsequent inference in any material must also test magnon decay, disorder, and phonon hybridization.

What magnetic probes actually measure

For unpolarized neutrons, the magnetic cross section contains

$$\lvert F(\mathbf q)\rvert^2 \sum_{\alpha\beta} (\delta_{\alpha\beta}-\widehat q_\alpha\widehat q_\beta) S^{\alpha\beta}(\mathbf q,\omega),$$

times known kinematic constants. The projector removes spin components parallel to momentum transfer, while $F(\mathbf q)$ is the magnetic form factor. Domain populations, $g$-tensor anisotropy, absorption, and resolution convolution must be included before comparing absolute intensity; the full cross-section convention is tabulated in the Lovesey 1986 paperback reprint, chs. 9–11.

NMR and muon relaxation measure momentum-weighted low-frequency fluctuations through probe-specific hyperfine or dipolar form factors. Electron spin resonance emphasizes near-uniform response; resonant x-ray techniques have different polarization and orbital selectivity. Agreement across probes is valuable precisely because their momentum and frequency filters differ.

Inference boundaries

Exact sum rules constrain total weight but do not uniquely decompose it into quasiparticles. Maximum-entropy or other analytic continuation of imaginary-time data is ill conditioned and must report resolution dependence. Background subtraction can create or erase broad continua. A strong interpretation states the measured $(\mathbf q,\omega)$ window, absolute calibration, form factor, resolution kernel, temperature, field, and alternative line-shape models.

Exercises

1. Derive the total-moment sum rule from the definition above.

Solution

Frequency integration sets $t=0$. Then $(1/N)\sum_{\mathbf q}e^{-i\mathbf q\cdot(\mathbf r_i-\mathbf r_j)}=\delta_{ij}$. Summing spin components gives $(1/N)\sum_i\langle\mathbf S_i^2\rangle=S(S+1)$.

2. At low temperature and $\omega>0$, compare energy-loss and energy-gain intensities.

Solution

Detailed balance gives $S(-\mathbf q,-\omega)=e^{-\beta\omega}S(\mathbf q,\omega)$ for a diagonal, inversion-symmetric response. Energy gain is exponentially suppressed relative to energy loss when $\beta\omega\gg1$.

References

• Coldea, R., D. A. Tennant, A. M. Tsvelik, and Z. Tylczynski. “Experimental Realization of a 2D Fractional Quantum Spin Liquid.” Physical Review Letters 86 (2001): 1335–1338. DOI.
• Hohenberg, P. C., and W. F. Brinkman. “Sum Rules for the Frequency Spectrum of Linear Magnetic Chains.” Physical Review B 10 (1974): 128–131. DOI.
• Lovesey, S. W. Theory of Neutron Scattering from Condensed Matter, Volume 2: Polarization Effects and Magnetic Scattering. Oxford: Clarendon Press, 1986 paperback reprint. Publisher.