Magnons and continua in a magnetized and dimerized spin-$\frac{1}{2}$ chain

We examine the magnetic field dependent excitations of the dimerized spin-1/2 chain, copper nitrate, with antiferromagnetic intradimer exchange $J_1=0.44\left(1\right)\mathrm{meV}$ and exchange alternation $\alpha =J_2/J_1=0.26\left(2\right)$. Magnetic excitations in three distinct regimes of magnetization are probed through inelastic neutron scattering at low temperatures. At low and high fields there are three and two long-lived magnonlike modes, respectively. The number of modes and the antiphase relationship between the wave-vector dependent energy and intensity of magnon scattering reflect the distinct ground states: A singlet ground state at low fields ${\mu }_{0}H<{\mu }_0{H}_{c1}=2.8\mathrm{T}$ and an $S_z=1/2$ product state at high fields ${\mu }_{0}H>{\mu }_0{H}_{c2}=4.2\mathrm{T}$. In the intermediate-field regime, a continuum of scattering for $\hslash \omega \approx J_1$ is indicative of a strongly correlated gapless quantum state without coherent magnons.

Figure 1

Crystal structure of CN. (a) Perspective view along the $b\text{axis}$. (b) View along the $\left[10\overline{1}\right]$ direction. (c) Expanded perspective view along the $\left[10\overline{1}\right]$ direction with 4 + 1 coordination of ${\mathrm{Cu}}^{2+}$ sites highlighted by square pyramidal polyhedra. Vectors ${\mathbf{d}}_1$, ${\mathbf{u}}_0$, and ${\mathbf{u}}_0^{\prime }$ are the respective intradimer and two possible interdimer vectors described in the text. The interdimer exchange path we propose connects dimer units for which the basal planes of the 4 + 1 coordination are nearly coplanar along the [111] direction in (b)–(d) via the ${\mathbf{u}}_0$ interdimer bond and illustrated by $J_1-J_2$ alternating chains in (d) and (e). (d),(e) Schematic of interdimer interactions in CN in adjacent $\left[10\overline{1}\right]$ planes. Intradimer interactions, $J_1$, are represented by circles linked by solid heavy lines. The two possible interdimer interactions are represented as solid ($J_2$) and dashed ($J_2^{\prime }$) lines. In (b) and (c), proton sites D/H(4) and D/H(5), and oxygen sites O(3), O(4), and O(9) as enumerated in Ref. [54] have been omitted for clarity. These omitted atoms participate in bonding with water sites adjoining the copper atom, secondary exchange paths along the $b$ direction, and bonding between the $\left[10\overline{1}\right]$ planes.

Figure 2

Constant $\mathbf{Q}$ scans for CN at $T=1.25$ K. Red solid lines are from a global fit to resolution convolved single modes with dispersion as listed in Eq. (2). Blue dashed lines are from a global fit to resolution convolved single modes with dispersion as listed in Eq. (6).

Figure 3

Normalized scattering intensity, $\tilde{I}(\tilde{q},\hslash \omega )$, for CN at $T=1.8$ K for (a) $H=0$, (b) ${\mu }_{0}H=1.5\mathrm{T}$, and (c) ${\mu }_{0}H=7\mathrm{T}$ from triple-axis measurements. Zero-field results are shown as half the measured/calculated intensity. Panels (d)–(f) are the results of fits described in the text.

Figure 4

(a) Normalized scattering intensity, $\tilde{I}(\tilde{q},\hslash \omega )$, for CN at $T=1.8$ K and ${\mu }_{0}H=3.55\mathrm{T}$ from triple-axis measurements. For clarity, data above the white dashed line ($\hslash \omega >0.7\mathrm{meV}$) have been multiplied by a factor of 2. (b) Peak positions for $\hslash \omega \approx 0.45\mathrm{meV}$ for constant $\tilde{q}$ scans through $H=0$ and ${\mu }_{0}H=3.55\mathrm{T}$ spectra at $T=1.8$ K.

Figure 5

Normalized scattering intensity, $\tilde{I}(\tilde{q}=2\pi ,\hslash \omega )$, for CN at $T=1.8$ K for magnetic fields $0\le {\mu }_{0}H\le 7\mathrm{T}$ from triple-axis measurements. Results have been offset vertically for presentation. Solid lines are fits to resolution limited Gaussians as described in the text. Dotted red lines for $2.8\le {\mu }_{0}H\le 4.2\mathrm{T}$ are fits to resolution limited Gaussians with an additional FWHM added in quadrature as described in the text. Data below $\hslash \omega =0.15\mathrm{meV}$, which are dominated by incoherent elastic nuclear scattering, have been omitted for clarity.

Figure 6

Normalized scattering intensity, $\tilde{I}(\tilde{q}=\pi ,\hslash \omega )$, for CN at $T=1.8$ K for magnetic fields $0\le {\mu }_{0}H\le 7\mathrm{T}$ from triple-axis measurements. Results have been offset vertically for presentation. Solid lines are fits to resolution limited Gaussians as described in the text. Dotted red lines for $2.8\le {\mu }_{0}H\le 4.2\mathrm{T}$ are fits to resolution limited Gaussians with an additional FWHM added in quadrature as described in the text. Data below $\hslash \omega =0.15\mathrm{meV}$, which are dominated by incoherent elastic nuclear scattering, have been omitted for clarity.

Figure 7

Normalized scattering intensity, $\tilde{I}(\tilde{q},\hslash \omega )$ for CN at $T=0.224$ K for (a) $H=0$, (b) ${\mu }_{0}H=2.82\mathrm{T}$, (c) ${\mu }_{0}H=3.13\mathrm{T}$, (d) ${\mu }_{0}H=3.62\mathrm{T}$, (e) ${\mu }_{0}H=4.03\mathrm{T}$, and (f) ${\mu }_{0}H=4.2\mathrm{T}$ from time-of-flight measurements. The data in (a) are shown at half the intensity scale. The trapezoidal area in (a) was covered by the high-resolution Config. 3 while the surrounding area was measured with Config. 1. Data in (b) were measured with Config. 1. Data in (c), (d), and (e) were measured with Config. 2. Data in (f) were measured with Config. 4. False color images of the scattering intensity were produced by binning the data onto an orthogonal grid in $\hslash \omega$ and $\tilde{q}$. The data were then convolved with a normalized Gaussian with FWHM principal axes of $\hslash \omega =31$ $\mu \mathrm{eV}$ and $\tilde{q}/\pi =0.125$. The corresponding FWHM ellipsoid is shown in (a), and approximates the average resolution ellipsoid.

Figure 8

Wave-vector averaged magnetic scattering intensity for CN at $T=0.224$ K as a function of $\hslash \omega$ and applied magnetic field from time-of-flight measurements. The average covered the range from $\tilde{q}=\frac{1}{2}\pi$ to $\tilde{q}=\frac{7}{2}\pi$. Data for different fields are offset vertically for presentation purposes. Solid lines are the result of Gaussian fits as described in the text. The $H=0$ and ${\mu }_{0}H=2.82\mathrm{T}$ data were combined from measurements made in Configs. 1 and 3. The ${\mu }_{0}H=3.13\mathrm{T}$ and ${\mu }_{0}H=4.03\mathrm{T}$ data were acquired using Config. 2. The ${\mu }_{0}H=3.62\mathrm{T}$ data were combined from measurements made in Configs. 1, 2, and 3. The ${\mu }_{0}H=4.2\mathrm{T}$ data are from measurements made in Config. 4.

Figure 9

Constant $\tilde{q}=2\pi$ scattering intensity ($\delta \tilde{q}=0.13\pi$) as a function of $\hslash \omega$ and applied magnetic field for CN at $T=0.224$ K from time-of-flight measurements. The zero-field data have been reduced in intensity by a factor of 3 for presentation purposes. Solid lines are fits to Gaussians as described in the text. Results have been offset vertically for presentation. The $H=0$ and ${\mu }_{0}H=2.82\mathrm{T}$ data were combined from measurements made in Configs. 1 and 3. The ${\mu }_{0}H=3.13\mathrm{T}$ and ${\mu }_{0}H=4.03\mathrm{T}$ data are from Config. 2. The ${\mu }_{0}H=3.62\mathrm{T}$ data were combined from measurements made in Configs. 1, 2, and 3. The ${\mu }_{0}H=4.2\mathrm{T}$ data are from measurements made in Config. 4.

Figure 10

Constant $\tilde{q}=\pi$ scattering intensity ($\delta \tilde{q}=0.13\pi$) as a function of $\hslash \omega$ and applied magnetic field for CN at $T=0.224$ K from time-of-flight measurements. The zero-field data have been reduced in intensity by a factor of 3. Solid lines are fits to Gaussians as described in the text. Results have been offset vertically for clarity. The $H=0$ and ${\mu }_{0}H=2.82\mathrm{T}$ data were combined from measurements made in Configs. 1 and 3. The ${\mu }_{0}H=3.13\mathrm{T}$ and ${\mu }_{0}H=4.03\mathrm{T}$ data are from Config. 2. The ${\mu }_{0}H=3.62\mathrm{T}$ data were combined from measurements made in Configs. 1, 2, and 3. The ${\mu }_{0}H=4.2\mathrm{T}$ data are from measurements made in Config. 4.

Figure 11

Zero-temperature exact diagonalization calculations for $N=6$ (a)–(c) and $N=16$ (d)–(f) $\alpha =0.25$ alternating antiferromagnetic Heisenberg spin chains as a function of applied magnetic field at zero wave vector. Panels (a) and (d) are the calculated field-dependent energy levels. Only the lowest-energy multiplet at each total spin $S$ is shown. States illustrated in the same color in both (a) and (b) have the same $S$. Panel (d) also shows states with $S=4$ (olive), $S=5$ (light blue), and $S=8$ (pink). Multiplets with $S=6$ and $S=7$ are omitted for clarity. Panels (b) and (e) are the lowest energy dipole active transitions. Dotted and dashed lines are extrapolations of the field dependence of excitations in the intermediate-field regime. Panels (c) and (f) are the calculated magnetization based on the total moment of the ground state. The dotted line in these panels is the measured $T=0.1$ K field dependent magnetization of CN [75]. Panels (a) and (b) also identify transitions in the basis of total spin $S$ and the $z$ component of spin $S_z$ using lower case Roman numerals (i)–(vi) as listed in the table in panel (c) and described in Ref. [79]. Reduced units are used for the bottom axes, and units of applied magnetic field for the case of $J_1\approx 0.44$ are used for the top axes.

Figure 12

Fitted peak position in the $\tilde{q}=\pi$ magnetic neutron-scattering spectrum of CN as a function of applied magnetic field for $T=1.8$ K (open symbols, triple-axis measurements) and $T=0.224$ K (closed symbols, time-of-flight measurements). The raw data and fits are shown in Figs. 5 and 9. ${\mu }_0{H}_{c1}=2.8\mathrm{T}$ and ${\mu }_0{H}_{c2}=4.2\mathrm{T}$ are indicated by dashed vertical lines. For excitations near $\hslash \omega \approx 0.45\mathrm{meV}$ at intermediate fields, fits were performed with an additional FWHM parameter $2\Gamma$ added in quadrature to the resolution width. The time-of-flight constant $\tilde{q}$ scans in the intermediate field range shown in Figs. 9 and 10 are fit to Gaussian peaks. The light (dark) shaded area corresponds to $k_{B}T=0.155\left(0.019\right)\mathrm{meV}$. Symbol styles correspond to different excitations as noted in the legend. The data points plotted with an open square with a cross inset for ${\mu }_{0}H=4.6$ and $5\mathrm{T}$ corresponds to the measured lower-energy excitation determined from higher energy thermally populated transitions as described in the text. Solid lines are theoretical mode energies based on finite chain calculations. Dotted and dashed grey lines in the intermediate-field regime are the limiting behavior of mode energies based on finite chain calculations.

Figure 13

Zero field (a) and intermediate field (b)–(f) dispersion relations for the $S_z=0$ mode near $J_1=0.44\mathrm{meV}$ extracted by fitting the time-of-flight measurements shown in Figs. 7–7(f) as described in the text. The horizontal error bar represents the range of integration in $\frac{\tilde{q}}{\pi }$ used for the constant wave-vector scans which were fit to extract the dispersion values. Multiple data sets for a given magnetic field were extracted from the different configurations used during the measurement. The blue triangles correspond to fitting the data from configuration 4. The green inverted triangles were measured in configuration 2. The solid (open) black (red) circles (squares) correspond to fits to data acquired in configuration 3(1). The solid lines in (a) are fits of the data to the zero-field dispersion as described in the text.