Field dependence of the magnetic spectrum in anisotropic and Dzyaloshinskii-Moriya antiferromagnets. I. Theory

We consider theoretically the effects of an applied uniform magnetic field on the magnetic spectrum of anisotropic two-dimensional and Dzyaloshinskii-Moriya layered quantum Heisenberg antiferromagnets. The former case is relevant for systems such as the two-dimensional square lattice antiferromagnet ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$, while the latter is known to be relevant to the physics of the layered orthorhombic antiferromagnet ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$. We first establish the correspondence between the low-energy spectrum obtained within the anisotropic nonlinear sigma model and by means of the spin-wave approximation for a standard easy-axis antiferromagnet. Then, we focus on the field-theory approach to calculate the magnetic-field dependence of the magnon gaps and spectral intensities for magnetic fields applied along the three possible crystallographic directions. We discuss the various possible ground states and their evolution with temperature for the different field orientations, and the occurrence of spin-flop transitions for fields perpendicular to the layers (transverse fields) as well as for fields along the easy axis (longitudinal fields). Measurements of the one-magnon Raman spectrum in ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ and ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ and a comparison between the experimental results and the predictions of the present theory will be reported in paper II of this research work [L. Benfatto et al., Phys. Rev. B 74, 024416 (2006)].

Figure 1

Left: the hatched circles represent the ${\mathrm{O}}^{--}$ ions tilted above the $\mathrm{Cu}{\mathrm{O}}_2$ plane; the empty ones are tilted below it; small black circles are ${\mathrm{Cu}}^{++}$ ions. Right: Schematic arrangement of the staggered magnetization (small arrows) and DM vectors (open arrows). Right bottom: definition of the vector ${\mathbf{d}}_{+}={\mathbf{D}}_{+}∕4S$.

Figure 2

(Color online) Field dependence of the in-plane magnon gap for an in-plane field applied at an arbitrary angle $\varphi$ with the $a$ axis, see Eq. (27). The case $\varphi =0$ corresponds to the transverse-field case analyzed in Sec. 4A, Eq. (9), while the case $\varphi =\pi ∕2$ corresponds to a purely longitudinal field, Eq. (10), which has a spin-flop transition at $H=m_a$. The behavior of the gap above the spin-flop transition is described in Sec. 5A below.

Figure 3

(Color online) Spin configuration at zero applied magnetic field in ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$. The arrows with a solid tip represent the staggered spin components ${(-1)}^m{e}^{i\mathbf{Q}\bullet {\mathbf{r}}_i}{\mathbf{n}}_i^m$, the arrows with a two-line tip represent the uniform spin components ${\mathbf{L}}_i^m$, the arrows with an open tip are the full spins ${\mathbf{S}}_i^m$.

Figure 4

(Color online) Momentum dependence of the spectral weight corresponding to the poles at ${\omega }_a(\mathbf{k},k_{\perp })$ or ${\omega }_c(\mathbf{k},k_{\perp })$ in Eq. (39). Here $k_{\perp }=0$, $m_a=0.02J,m_c=0.05J,c=J$, and we used units such that $H=1\mathrm{T}$ corresponds to ${10}^{-3}J$. Inset: momentum dependence of $Z_a,Z_c$ defined in Eq. (41). Observe that even thought $Z_a\approx Z_c$ at $H=10\mathrm{T}$ as $ka\sim 0.5$ the contribution at $k=0$ is always predominant in the momentum sum (39), due to the fact that the $a$ gap softens as the field strength increases.

Figure 5

(Color online) Magnetic-field dependence of the spins ground state and of the $T=0$ average uniform magnetization in the field direction for a field along the $b$ axis. At finite magnetic field the order parameter $\mathbf{n}$ rotates in the $bc$ plane. The spin configuration is determined as usual by ${\mathbf{S}}_i^m∕S={(-1)}^m{e}^{i\mathbf{Q}\bullet {\mathbf{r}}_i}{\mathbf{n}}_i^m+{\mathbf{L}}_i^m$, where the staggered and uniform components are indicated by arrows with different tips as in Fig. 3: the arrows with a two-line tip represent the uniform components ${\mathbf{L}}_m$ and the arrows with a solid tip the staggered components ${(-1)}^m{e}^{i\mathbf{Q}\bullet {\mathbf{r}}_i}{\mathbf{n}}_i^m$. At $H<H_c^{\left(1\right)}\approx 18\mathrm{T}$ (left in the figure) ${\mathbf{n}}_m$ is defined in Eq. (34) and ${\mathbf{L}}_m$ in Eq. (35), so that the spins have both the staggered and uniform components in the $bc$ plane. The uniform components are due both to the magnetic field and to the DM interaction, see Eq. (35), and the sum of ${\mathbf{L}}_m+{\mathbf{L}}_{m+1}$ is aligned along the field. Above the spin-flop transition (right in the figure) the staggered components ${\mathbf{n}}_m$ lie in the $ac$ plane, see Eq. (46) (so that the part along $a$ is orthogonal to the plane of the figure and indicated by a cross), and the uniform components ${\mathbf{L}}_m$ point along $b$, see Eq. (47). The order-parameter values used to evaluate the $⟨\mathbf{L}⟩$ are the same as reported in Fig. 7. Observe that since in our approximation the critical field $H_c^{\left(1\right)}$ is independent of temperature, as well as ${\sigma }_c$, the uniform magnetization is temperature independent below $T_N$.

Figure 6

(Color online) Field dependence of the magnon gaps at $T=0$ for a field applied along the $b$ axis. We used $m_a=0.02J,m_c=0.05J$, and units such that $H=1\mathrm{T}$ corresponds to ${10}^{-3}J$, so that the first critical field $H_c^{\left(1\right)}=D_{+}=m_a=20\mathrm{T}$. Below $H_c^{\left(1\right)}$ the in-plane and out-of-plane mode correspond to $a$ and $c$ component fluctuations, whose gaps are given by ${\omega }_a$ and ${\omega }_c$ in Eq. (10), respectively. Above the critical field the in-plane mode corresponds to fluctuations of the $b$ component, and the magnon gaps are defined by Eq. (51).

Figure 7

(Color online) Phase diagram of the system for $\mathbf{H}\Vert b$. The parameter values are chosen to reproduce the data of ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$, i.e., $D_{+}=m_a=0.0116J$, $m_c=0.034J$, $J=130\mathrm{meV}$, ${\rho }_s=0.07J$, and $c=1.3J$ (Ref. 12). For the interlayer coupling we used $\eta =8\times {10}^{-5}{J}^2$, which correctly reproduce the spin-flop transition measured in Ref. 15 for $\mathbf{H}\Vert c$, see end of Sec. 5B. The magnetic field is converted in energy through $H\to g_s{\mu }_{B}H$, where ${\mu }_B$ is the Bohr magneton and $g_s=2.1$ is the gyromagnetic ratio in the $b$ direction. Inset: field evolution of the staggered order parameter components at $T=0$ in the various phases.

Figure 8

(Color online) Magnetic-field dependence of the spin configuration and of the average uniform magnetization at two different temperatures for a field along the $c$ axis. The spin configuration is determined as usual by ${\mathbf{S}}_i∕S={(-1)}^m{e}^{i\mathbf{Q}\bullet {\mathbf{r}}_i}{\mathbf{n}}^m+{\mathbf{L}}^m$, where the staggered and uniform components are indicated by arrows with different tips as in Figs. 3, 5. At $H<H_c$, where $H_c=6\mathrm{T}$ at $T=10\mathrm{K}$ and $H_c=4.6$ at $T=200\mathrm{K}$, see also Fig. 10 below, the system has 3D antiferromagnetic order as described by Eq. (57). The difference between the ${\mathbf{L}}_m$ components (arrows with a two-line tip) in neighboring planes is due to the DM-induced term ${\mathbf{L}}_m^{DM}$ which changes sign from one layer to the next one, while ${\mathbf{L}}_m^H$ is always parallel to $\mathbf{H}$, see Eq. (58). Above the spin flop the spins are ordered ferromagnetically in neighboring layers, see Eq. (60), allowing for the uniform ${\mathbf{L}}_m$ components to align along the field in all the layers. The jump at the transition is proportional to the order parameter ${\sigma }_0$, and decreases as the temperature increases, see the inset of Fig. 10.

Figure 9

(Color online) Low panel: out-of-plane momentum $k_z\equiv k_{\perp }$ dependence of the two solutions ${\omega }_{\pm }$ of Eq. (66) at $\mathbf{k}=0$, compared to the momentum dependence of the function $\omega (\mathbf{k}=0,k_{\perp })=\sqrt{m_a^2+2\eta (1-\cos k_{\perp }d)}$. We used $D_{+}=m_a=0.02J$ and $H=1$ $T={10}^{-3}\mathrm{J}$. Observe the small difference of ${\omega }_{-}$ with respect to $\omega$ at $k_{\perp }=0$. Top panel: momentum dependence of the spectral weight $Z_{\pm }\left(k_{\perp }\right)$ defined in Eq. (67). As one can see, at small momentum $Z_{-}⪢Z_{+}$, while at $k_{\perp }d>\pi ∕2$ the situation is reversed. As a consequence, the momentum sum in Eq. (65) selects always between ${\omega }_{+}$ and ${\omega }_{-}$ the solution which follows most closely the momentum dependence of the solution $\omega \left(k_{\perp }\right)$ of the case $H=0$.

Figure 10

(Color online) $H_c-T$ phase diagram for $\mathbf{H}\parallel c$. We used the same parameter values as Fig. 7, except that in this case $g_s=2.4$, as appropriate for a field in the $c$ direction (Ref. 18). The order parameter ${\sigma }_0\left(T\right)$ has been determined self-consistently through Eq. (62), computing the transverse fluctuations with Eqs. (68, 73) below and above the spin-flop transition, respectively, where the critical field $H_c$ is defined through Eq. (74). Inset: temperature dependence of the staggered order parameter at $H=1\mathrm{T}$.

Figure 11

(Color online) Field dependence of the magnon gaps at $T=0$ for $H\parallel c$. Here $H_c\approx 6.5\mathrm{T}$, so below it the gap decrease according to Eq. (70), while above it they increase according to Eq. (72). The parameter values are the same as in Fig. 10. Inset: field dependence of the zero-temperature order parameter.