Field-induced quasi-bound state within the two-magnon continuum of a square-lattice Heisenberg antiferromagnet

Abstract

Quantum magnets in two dimensions display strong quantum interaction effects even when magnetically ordered. Using the metal-organic framework material CuF$_2$(D$_2$O)$_2$(pyz), we investigate the field-dependent spin dynamics of the $S = 1/2$ square-lattice Heisenberg antiferromagnet by high-resolution inelastic neutron scattering to applied fields beyond one third of saturation. We discover an anomalously sharp, dispersive ``shadow mode'' residing within the two-magnon continuum, which shadows the dispersion of the transverse one-magnon branches across the Brillouin zone at an offset equal to the Larmor energy. We perform cylinder matrix-product-state (MPS) calculations that reproduce the field-induced spectrum quantitatively and apply a spectrally consistent $1/S$ spin-wave theory to deduce that the ``Larmor-shadow mode'' is a composite two-magnon resonance: a dispersing magnon at wavevector ${\bf Q}$ couples to the uniform Larmor precession at $Γ$, its small intrinsic linewidth indicating a non-perturbative effect of attractive magnon-magnon interactions. Another quantum-fluctuation phenomenon, the zero-field $(π,0)$ anomaly, is lost at increasing fields, which tighten the spectral weight into the one-magnon and Larmor-shadow modes. To our knowledge, these results constitute the first observation of a sharp quasi-bound state embedded in the continuum of a gapless two-dimensional antiferromagnet.

Figures (22)

• Figure 1: CuF$_2$(D$_2$O)$_2$(pyz) and its field-induced excitation spectrum.a Monoclinic crystal structure of CuF$_2$(D$_2$O)$_2$(pyz), highlighting its quasi-2D nature with square-lattice planes ($bc$) separated along $a$ by organic molecules. b Canted magnetic structure induced by the magnetic field, which is applied with $\hat{B} \parallel \hat{a}$. The magnetic moments are canted by angle $\theta$ towards the field direction, remaining in the $ac^\ast$ plane. The inset represents the experimental geometry, where the momentum transfer Q lies within the $b^\ast c^\ast$ plane, and defines the Cartesian laboratory frame $(x_0,y_0,z_0)$ used in our calculations. c Overview of the reciprocal-lattice plane ${\bf Q} = (0,K,L)$ illustrated by a constant-energy scan at $E = 1.8$ meV and $B = 0$ T. Coloured points mark equivalent positions and their equivalence to the reciprocal-lattice points commonly referred to as $(0,0)$, $(\pi,0)$, $(\pi,\pi)$ and $(\pi/2,\pi/2)$. The blue region is the Q space accessible in our time-of-flight (TOF) experiment with an incident neutron energy of $E_i = 5.5$ meV. Black arrows mark the high-symmetry Q paths shown in Fig. \ref{['fig:Comparison']}. d Isothermal magnetization of CuF$_2$(H$_2$O)$_2$(pyz) at $T = 0.5$ K (data adapted from Ref. manson_chem). Blue arrows indicate the fields at which INS data were collected in our TOF (LET) experiment and red arrows the fields of the triple-axis (TASP) experiment. e, f Overview of spectra measured by neutron spectroscopy (LET) and computed by cylinder matrix-product states (MPS) for a field of $B = 9$ T, showing the two clear low-energy magnon branches accompanied by an emerging higher-lying mode that "shadows" the one-magnon branches at a Q-independent energy offset.
• Figure 2: Comparison of zero-field and field-induced INS and MPS spectra.a- d, i- l Neutron scattering intensity, $I(\mathbf{Q},E)$, measured on LET ($E_i = 5.5$ meV) for the two high-symmetry reciprocal-space directions at the four experimental magnetic fields. $(0,K,0)$ is the ${\rm \Gamma M}$ direction and includes the $(\pi/2,\pi/2)$ point. $(0,-0.5+K,0.5+K)$ is the ${\rm XMX}$ direction and includes the $(\pi,0)$ point. Data were binned using $\delta E = 0.015$ meV and $\delta Q = 0.015$ r.l.u., and were integrated in the perpendicular in-plane direction over $\Delta Q = \pm 0.1$ r.l.u. and in the out-of-plane direction over $\Delta H = \pm 0.4$ r.l.u. We observe the progressive field-induced splitting of the one-magnon excitations into two transverse modes (corresponding to the uniform and staggered parts of the magnetization) and the development of a sharp resonance mode in the multi-magnon continuum at higher energies. The field-independent feature around 1.2 meV (green arrows) is a spurion arising from an additional reflection within the LET magnet sample environment. e- h, m- p Spectral functions calculated by cylinder MPS for both directions at the same four fields, convolved with the polarization factor for direct comparison with the INS data.
• Figure 3: Dispersion and linewidth of single magnons and the Larmor-shadow mode.a, b Dispersion relations for the gapped one-magnon branch (green) and the shadowing high-energy mode (orange) extracted from the INS intensity $I(\mathbf{Q},E)$ at $B = 6$ T (a) and $B = 9$ T (b). Symbols mark the centres of Gaussians fitted at constant ${\bf Q}$ and error bars mark the corresponding Gaussian widths. c, d Energy difference obtained by subtracting the position of the one-magnon branch ($E_{\rm M}$) from that of the shadow mode ($E_{\rm LSM}$). The dashed purple line marks the Larmor energy, $E_{\rm L} = g \mu_{\rm B} \mu_0 B$, for the corresponding field. e, f Full width at half-maximum height (FWHM) of excitations determined from the Gaussian fits. g, h Dispersion and linewidth of the one-magnon and high-energy modes extracted from MPS calculations.
• Figure 4: Field-dependence of zone-boundary excitations. Comparison of the excitation spectra at the two high-symmetry zone-boundary points $(\pi,0)$ and $(\pi/2,\pi/2)$ for all measured magnetic fields. a, b Experimental data acquired using TOF neutron spectroscopy (LET) at $B =$ 0, 3, 6 and 9 T at ${\bf Q} = (0,-0.5,0.5) \equiv (\pi,0)$ and $(0,-0.5,1) \equiv (\pi/2,\pi/2)$. c, d MPS calculations performed for the corresponding fields. e, f Experimental data acquired using triple-axis neutron spectroscopy (TASP) at $B =$ 0, 2, 4, 6, 8, 10 and 12 T at ${\bf Q} = (0,0.5,-0.5) \equiv (\pi,0)$ and $(0,1,-0.5) \equiv (\pi/2,\pi/2)$. g, h$\overline{1/S}$-SWT calculations performed for the corresponding fields.
• Figure 5: Full experimental and calculated spectra at the high-symmetry points for all fields.a- f Experimental data from ToF (LET, blue) and triple-axis (TASP, red) neutron spectroscopy compared with spectra obtained from MPS calculations (black lines with grey shadowing) and $\overline{1/S}$-SWT calculations (green lines) for the $(\pi,0)$ point. The feature at $E \approx 1.2$ meV in the LET data is an artifact of the magnet. g- l Corresponding spectra for the $(\pi/2,\pi/2)$ point. Because the $(\pi/2,\pi/2)$ points $(0, -0.5, 1)$ and $(0, 1, 0.5)$ have inequivalent polarization factors, the full and dashed green lines in panels g and k represent respectively the two different $\overline{1/S}$-SWT predictions for the LET and TASP experiments.