Microscopic mechanism for the $\frac{1}{8}$ magnetization plateau in SrCu${}_2$(BO${}_3$)${}_2$

The frustrated quantum magnet SrCu${}_2$(BO${}_3$)${}_2$ shows a remarkably rich phase diagram in an external magnetic field including a sequence of magnetization plateaux. The by far experimentally most studied and most prominent magnetization plateau is the $\frac{1}{8}$ plateau. Theoretically, one expects that this material is well described by the Shastry-Sutherland model. But, recent microscopic calculations indicate that the $\frac{1}{8}$ plateau is energetically not favored. Here, we report on a very simple microscopic mechanism which naturally leads to a $\frac{1}{8}$ plateau for realistic values of the magnetic exchange constants. We show that the $\frac{1}{8}$ plateau with a square unit cell benefits most compared to other plateau structures from quantum fluctuations which, to a large part, are induced by Dzyaloshinskii-Moriya interactions. Physically, such couplings result in kinetic terms in an effective hard-core-boson description leading to a renormalization of the energy of the different plateau structures which we treat in this work on the mean-field level. The stability of the resulting plateaux are discussed. Furthermore, our results indicate a series of stripe structures above $\frac{1}{8}$ and a stable magnetization plateau at $\frac{1}{6}$. Most qualitative aspects of our microscopic theory agree well with a recently formulated phenomenological theory for the experimental data of SrCu${}_2$(BO${}_3$)${}_2$. Interestingly, our calculations point to a rather large ratio of the magnetic couplings in the Shastry-Sutherland model such that nonperturbative effects become essential for the understanding of the frustrated quantum magnet SrCu${}_2$(BO${}_3$)${}_2$.

Figure 1

(Left) Illustration of the Shastry-Sutherland lattice and of the two-body interactions $V_{\delta }{\widehat{n}}_j{\widehat{n}}_{j+\delta }$. Thick solid lines (dotted-dashed lines) correspond to the magnetic exchange coupling $J$ ($J^{\prime }$). The two-body interaction labeled by $V_{\delta }$ are defined as the density-density interaction between the thick dimer labeled as $j$ and the dimer labeled $V_{\delta }$. (Right) The hopping amplitudes $t_1$ and $t_2$ are illustrated.

Figure 2

Amplitude $V_{\delta }$ of the extrapolated two-body interactions as a function of $J^{\prime }/J$. Inset: Different dlogPadé extrapolants (solid lines) as well as the bare series (dashed lines) for $V_3^{\prime }$ and $V_5$. The displayed curves correspond to the data given in Ref. 18.

Figure 3

Lowest classical energies $E_{\mathrm{cl}}/\left(JN\right)$ per dimer of all plateaux in the density regime $\frac{1}{9}$ $\le n\le$ $\frac{1}{6}$ as a function of $\mu /J$ for (a) $J^{\prime }/J=0.5$ and (b) $J^{\prime }/J=0.68$ using Eq. (10) setting all kinetic terms to zero. Additionally, the structures of the realized low-energy plateaux at densities $\frac{1}{9}$, $\frac{2}{15}$, and $\frac{1}{6}$ as well as the lowest $\frac{1}{8}$ plateau are explicitly given as little pictograms. Note that the energies of the different $\frac{2}{15}$ and $\frac{1}{6}$ structures shown are exactly degenerate. The circle on a dimer denotes the presence of a particle.

Figure 4

Classical plateaux which are relevant at low densities. The unit cells of the different structures are shown in dark gray (blue). Note that not all plotted structures are realized within the CA. This includes all plateaux at density $n=$ $\frac{1}{8}$. Additionally, for the structures $\frac{2}{15}$-b2 ($\frac{2}{15}$-big) is built alternatingly by three $\frac{1}{8}$-square (three $\frac{1}{8}$-rhomboid) and one $\frac{1}{6}$. Each $\frac{1}{8}$-stripe is shaded in light gray.

Figure 5

One-triplon hopping amplitude $t_2/J$ as a function of $J^{\prime }/J$. Solid line represents the bare series of order 17, while the dashed curves correspond to various dlogPadé extrapolations.

Figure 6

Illustration of the additional magnetic couplings beyond the ones included in the Shastry-Sutherland model: (a) interdimer DM interaction $D_{\mathrm{z}}$ in $z$ direction. (b), (c) Heisenberg exchanges to next-nearest-neighbor dimers are denoted by $J_{\mathrm{c}}$ (chainlike coupling) and $J_{\mathrm{l}}$ (ladderlike coupling).

Figure 7

Illustrations of the regions ${\mathcal{R}}_r$ having the dimension dim(${\mathcal{R}}_r)=25$. Left: Figure displays two regions ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ of two triplets (black dimers) having an overlap ${\mathcal{O}}_{12}$ (shaded area). Right: Illustration of the regions ${\mathcal{R}}_r$ for the classical plateau at density $n=$ $\frac{1}{9}$. The regions with an odd respectively even number are periodically equivalent. Note that regions ${\mathcal{R}}_r$ with $r$ odd and $r$ even display exactly the same behavior due to rotational symmetry of the lattice.

Figure 8

Qualitative illustration of the renormalization factors due to the overlap of different regions. (a) We consider the initial situation that a particle on dimer 0 hops to the dimer 1 which is part of the overlap with a second region. All dimers contained in this second region are shaded in dark gray (blue) and the number attached on top of these dimers corresponds to the probability (density) that the particle in this second region is on a specific dimer. (b) If all renormalization factors are absent, the particle which has moved to dimer 1 is unaffected by the $10\%$ probability that another particle of the second region is already present on dimer 1. As a consequence, the particle does not pay the potential energy resulting from this $10\%$ of the density distribution on dimer 1. Because the hopping amplitude is $i{t}_1$, it would be attractive to hop on dimers with a large density of another particle in order to avoid high potential barriers. This is clearly unphysical. (c), (d) These two figures illustrate the same situation but with the two renormalization factors for the kinetic and the interaction parts of the mean-field Hamiltonian. The hopping amplitude $i{t}_1$ for the hopping from dimer 0 to dimer 1 is renormalized to $(1-\langle {\widehat{n}}_1\rangle )i{t}_1$. At the same time, this leads to a change of the density distribution of the particle inside the second region. The density expectation value $\langle {\widehat{n}}_k\rangle$ of the particle in the second region is effectively increased by $\langle {\widehat{n}}_k\rangle /(1-\langle {\widehat{n}}_1\rangle )$.

Figure 9

Schematic illustration of the effective observables ${\widehat{S}}_{1,j}^{\mathrm{eff},z}$ and ${\widehat{S}}_{2,j}^{\mathrm{eff},z}$. Left figure represents a local density $\widehat{n}$ from a (mean-field) state of the low-energy description. The right figure corresponds to the distribution of the local magnetization in the effective model originating from this finite density. The radius of the circles is proportional to the square root of the density (left)/local magnetization (right) on the dimer (left)/spin sites (right). In the right figure, filled (empty) circles denote a local magnetization pointing outside (inside) the plane.

Figure 10

(a) The energy gain per dimer $E_{\mathrm{gain}}/\left(JN\right)$ is plotted as a function of $t_1/J$ and $t_2/J$ for density $\frac{1}{8}$ at $J^{\prime }/J=0.65$. The three considered structures are $\frac{1}{8}$-square (triangle down), $\frac{1}{8}$-rhomboid (circle), and $\frac{1}{8}$-ca (triangle up). The combination of both kinetics terms favors the structure $\frac{1}{8}$-square with a square unit cell. For other $J^{\prime }/J$ values (not shown), the energy gain $E_{\mathrm{gain}}/\left(JN\right)$ has a very similar behavior. (b) The mean-field energy per dimer $E_{\mathrm{mf}}/\left(JN\right)$ is plotted for $\frac{1}{8}$-square (triangle down), $\frac{1}{8}$-rhomboid (circle), and $\frac{1}{8}$-ca (triangle up) as a function of $t_1/J$ and $t_2/J$. Note that in the shown parameter regime, always one of these three structures corresponds to the global minimum of the mean-field energy.

Figure 11

Density distribution of the mean-field solution for density $\frac{1}{8}$ at $J^{\prime }/J=0$.65, $t_1/J=0.01$, and $t_2/J$ $=-0.005$. Note that the densities of all dimers are rounded to the third decimal digit. The suppression of certain fluctuation channels is clearly visible. The radius of the plotted circles is proportional to the square root of the density on each dimer. The red circles denote densities which are larger than 0.5.

Figure 12

Minima of the mean-field energy are plotted as a function of $t_1/J$ and of $t_2/J$. The dashed blue line separates the parameter regime where the classical structure $\frac{1}{8}$-ca is favored from the regime where the $\frac{1}{8}$-square structure is realized. This line is obtained by fitting the lowest energies of $\frac{1}{8}$-square and $\frac{1}{8}$-ca along the displayed grid. Symbols $\alpha$ ($\beta$) represent mean-field solutions of $\alpha$ type ($\beta$ type). Mean-field solutions which differ by less than ${10}^{-4}$ J are considered to be degenerate. In this case, all such structures are displayed.

Figure 13

The figure displays for $J^{\prime }/J=0.68$ the range in the chemical potential $\Delta \mu /J$ where a $\frac{1}{8}$ plateau is realized in the full low-density phase diagram compared to all other plateau structures at different densities. The stabilized $\frac{1}{8}$-plateau changes its structure from the classical $\frac{1}{8}$-ca structure (dashed lines, present for small values of $t_1/J$ and $-t_2/J$) to the $\frac{1}{8}$-square structure (shown as solid black line) when both hoppings $t_1/J$ and $-t_2/J$ are of the order $0.01$.

Figure 14

The energy gain per dimer $E_{\mathrm{gain}}/\left(JN\right)$ is plotted as a function of $t_1/J$ and $t_2/J$ for fixed density $n=$ $\frac{1}{6}$ at $J^{\prime }/J=0.65$. In contrast to the structure $\frac{1}{6}$-new (squares), the classical structure $\frac{1}{6}$-ca (triangles down) has a lower energy gain for all considered parameter values. The structure $\frac{1}{6}$-square ($\frac{1}{6}$-stripe) is denoted by triangles up (circles). If the kinetic terms become large $t_1/J\approx 0.02$ and $t_2/J\approx -0.02$, an unphysical plateau of $\beta$ type has the lowest energy.

Figure 15

The structures at fixed density $n=$ $\frac{1}{6}$ having the lowest mean-field energy $E_{\mathrm{mf}}$ are plotted as a function of $t_1/J$ and $t_2/J$ for (a) $J^{\prime }/J=0.65$ and for (b) $J^{\prime }/J=0.68$. The most important difference between $J^{\prime }/J=0.65$ and $J^{\prime }/J=0.68$ is the change from the $\frac{1}{6}$-ca structure to the $\frac{1}{6}$-new structure. Symbols $\beta$ represent mean-field solutions of $\beta$ type. Mean-field solutions which differ by less than ${10}^{-4}$ J are considered to be degenerate. In this case, all such structures are displayed.

Figure 16

The structures at fixed density $n=2/15$ having the lowest mean-field energy $E_{\mathrm{mf}}$ are plotted as a function of $t_1/J$ and $t_2/J$ for (a) $J^{\prime }/J=0.65$ and for (b) $J^{\prime }/J=0.68$. The dashed blue line separates the region favoring the structure $\frac{2}{15}$-rect present for small values of the kinetic terms from the regime where the structure $\frac{2}{15}$-b${}_2$ is realized. This line is obtained by fitting the lowest energies of $\frac{2}{15}$-rect and $\frac{2}{15}$-b${}_2$ along the displayed grid. Symbols $\alpha$ ($\beta$) represent mean-field solutions of $\alpha$ type ($\beta$ type). Mean-field solutions which differ by less than ${10}^{-4}$ J are considered to be degenerate. In this case, all such structures are displayed.

Figure 17

The structures at fixed density $n=$ $\frac{1}{9}$ having the lowest mean-field energy $E_{\mathrm{mf}}$ are plotted as a function of $t_1/J$ and $t_2/J$ for (a) $J^{\prime }/J=0.65$ and for (b) $J^{\prime }/J=0.68$. The dashed blue line separates the parameter regime where the $\frac{1}{9}$ plateau has a positive mean-field energy at $\mu ={\mu }_0$ from the parameter region where the mean-field energy becomes negative at $\mu ={\mu }_0$, which is clearly unphysical. This unphysical behavior is denoted by $\alpha$.

Figure 18

The density $n$ (the magnetization $M$) is plotted as a function of the chemical potential $\mu /J$ (the magnetic field $B/J$) for a vanishing nearest-neighbor hopping $t_1=0$ for different values of $J^{\prime }/J$ and $t_2/J$. The value of the diagonal hopping $t_2$ is consistent with the extrapolated order-17 series of the pure Shastry-Sutherland model. Additionally, the most important structures are shown.

Figure 19

The density $n$ (the magnetization $M$) is plotted as a function of the chemical potential $\mu /J$ (the magnetic field $B/J$) for a nearest-neighbor hopping $t_1=0.01$ and different values of $J^{\prime }/J$ and $t_2/J$. The value of the diagonal hopping $t_2$ is consistent with the extrapolated order-17 series of the pure Shastry-Sutherland model for $J^{\prime }/J\ge 0.65$. Additionally, the relevant structures are shown.

Figure 20

Structures with the lowest mean-field energy are plotted in comparison to all other densities as a function of $t_1/J$ and $t_2/J$ for the density (a) $\frac{1}{8}$ and for the density (b) $\frac{2}{15}$. The black symbol $\times$ marks a parameter regime where both plateaux are realized in the phase diagram. In the relevant regime $t_1/J\approx 0.015$ and $t_2/J\approx -0.01$, the $\frac{2}{15}$-b${}_2$ plateau is only realized in a small region in contrast to the $\frac{1}{8}$-square. Symbols $\alpha$ ($\beta$) represent mean-field solutions of $\alpha$ type ($\beta$ type). Mean-field solutions which differ by less than ${10}^{-4}$ $J$ are considered to be degenerate. In this case, all such structures are displayed.

Figure 21

(a) The density $n$ (the magnetization $M$) is plotted as a function of the chemical potential $\mu /J$ (the magnetic field $B/J$) for $J^{\prime }/J=0.68$, $t_1/J=0.016$, and $t_2/J=-0.01$. Plateaux shown in red correspond to valid Wigner crystals. Mean-field solutions in the regimes $\alpha$ and $\beta$ are unphysical: In $\alpha$, the mean-field energies at $\mu ={\mu }_0$ are negative. In $\beta$, particles are completely delocalized among various dimers inside the crystalline solution. The structure and the magnetization of the plateau at $\frac{1}{9}$ are given in (b) and (c). The corresponding figures for $\frac{1}{8}$ and $\frac{1}{6}$ are shown in (d), (e) and (f), (g).

Figure 22

Figure illustrates the local density of two unphysical mean-field states classified as $\beta$ at densities $\frac{1}{7}$ (a) and $\frac{3}{20}$ (b) shown in Fig. 21a.

Figure 23

Figure shows all stripe structures present in the low-density phase diagram for $J^{\prime }/J=0.68$, $t1/J=0.016$, and $t2/J=-0.01$ as displayed in Fig. 21a. All Wigner crystals contain stripes built by the $\frac{1}{8}$-square structure (shaded in gray).

Figure 24

Illustrations of the relevant correlated hopping processes for the structures at density $\frac{1}{8}$ (top) and density $\frac{1}{6}$ (down).