Fluctuation-induced ferrimagnetism in sublattice-imbalanced antiferromagnets with application to ${\mathrm{SrCu}}_2$(${\mathrm{BO}}_3{)}_2$ under pressure

We show that a collinear Heisenberg antiferromagnet, whose sublattice symmetry is broken at the Hamiltonian level, becomes a fluctuation-induced ferrimagnet at any finite temperature $T$ below the Néel temperature $T_{\mathrm{N}}$. We demonstrate this using a layered variant of a square-lattice $J_1\text{−}{J}_2$ model. Linear spin-wave theory is used to determine the low-temperature behavior of the uniform magnetization, and nonlinear corrections are argued to yield a temperature-induced qualitative change of the magnon spectrum. We then consider a layered Shastry-Sutherland model, describing a frustrated arrangement of orthogonal dimers. This model displays an antiferromagnetic phase for large intradimer couplings. A lattice distortion which breaks the glide symmetry between the two types of dimers corresponds to broken sublattice symmetry and hence gives rise to ferrimagnetism. Given indications that such a distortion is present in the material $\mathrm{Sr}{\mathrm{Cu}}_2{\left(\mathrm{B}{\mathrm{O}}_3\right)}_2$ under hydrostatic pressure, we suggest the existence of a fluctuation-induced ferrimagnetic phase in pressurized $\mathrm{Sr}{\mathrm{Cu}}_2{\left(\mathrm{B}{\mathrm{O}}_3\right)}_2$. We predict a nonmonotonic behavior of the uniform magnetization as function of temperature.

Figure 1

Qualitative temperature dependence of the fluctuation-induced uniform magnetization in sublattice-imbalanced Heisenberg antiferromagnets, with the critical exponent $\beta =0.37$ [23, 24] in $d=3$ dimensions.

Figure 2

Layered square-lattice Heisenberg antiferromagnet with two inequivalent second-neighbor couplings $J_a^{\prime }$ (green) and $J_b^{\prime }$ (red). Thermal fluctuations induce a finite uniform magnetization for $0<T<T_{\mathrm{N}}$ once $J_a^{\prime }\ne J_b^{\prime }$.

Figure 3

Spin-wave dispersion for the toy model (1) along a path in the Brillouin zone and parameters (a) $J=100, J_a^{\prime }=10, J_{\perp }=2$ and (b) $J=0.1, J_a^{\prime }=10, J_{\perp }=1$ in units of $J_b^{\prime }$, both with $\eta =1$. Blue (red) curves correspond to ${\omega }_{\vec{k}+}$ (${\omega }_{\vec{k}-}$), respectively. The inset in (b) shows a zoom into the low-energy part of the dispersion near $\vec{k}=(0,0,0)$.

Figure 4

Fluctuation-induced uniform magnetization, $m_{\mathrm{tot}}$, as function of temperature, together with the finite-temperature corrections to the sublattice magnetizations, $\mathrm{\Delta }{m}_{A,B}=m_{A,B}\left(T\right)-m_{A,B}\left(0\right)$. The coupling constants were set to (a) $J=100, J_a^{\prime }=10, J_{\perp }=2$ and (b) $J=0.1, J_a^{\prime }=10, J_{\perp }=1$ in units of $J_b^{\prime }$. In both plots, the horizontal grid line marks the value $1/2$, whereas the vertical grid line in (b) indicates the temperature $T=JS$.

Figure 5

Fluctuation-induced magnetization as in Fig. 4, but now as function of $J_a^{\prime }/J_b^{\prime }$ at fixed finite temperature. Parameters were set to (a) $T/S=50, J=100, J_{\perp }=2$ and (b) $T/S=1, J=0.1, J_{\perp }=1$, all in units of $J_b^{\prime }$, and are such that the results with the ratio $J_a^{\prime }/J_b^{\prime }=10$ correspond to data in Fig. 4 at the specified temperatures.

Figure 6

(a) Shastry-Sutherland model with intradimer couplings $J_{a,b}$ and interdimer coupling $J^{\prime }$. The dashed lines indicate the unit cell. (b) Ground-state phase diagram of the $S=1/2$ Shastry-Sutherland model with $J_a=J_b$, as reported in Ref. [10]. An intermediate spin-liquid (SL) phase (shaded) [30] has been recently proposed in Ref. [11]. In the present work, the focus is on the antiferromagnetic phase at large $J^{\prime }/J$ shown in red.

Figure 7

Layered Shastry-Sutherland model (9) used to describe ${\mathrm{SrCu}}_2$(${\mathrm{BO}}_3{)}_2$. An orthorhombic distortion is assumed to generate different intradimer couplings $J_a, J_b$.

Figure 8

Spin-wave dispersion of the distorted Shastry-Sutherland model (9) along a path in the Brillouin zone for parameters $J_a=0.97, J_b=0.9$, and $J_{\perp }=0.2$ in units of $J^{\prime }$.

Figure 9

Uniform magnetization, $m_{\mathrm{tot}}$, and thermal corrections, $\mathrm{\Delta }{m}_{A,B}$, to the sublattice magnetization of the distorted Shastry-Sutherland model, with parameters (a) $J_a=0.9, J_b=0.8, J_{\perp }=0.1$ and (b) $J_a=0.97, J_b=0.9, J_{\perp }=0.01$ in units of $J^{\prime }$.

Figure 10

Same quantities as in Fig. 9, but now as a function of $J_a/J_b$ at fixed temperature $T=J^{\prime }S$ and with $J_a=0.9$ (and varying $J_b$) and $J_{\perp }=0.1$ in units of $J^{\prime }$. The ratio $J_a/J_b=1.125$ reproduces the data in Fig. 9 at the specified temperature.