Exotic magnetization plateaus in a quasi-two-dimensional Shastry-Sutherland model

We find unconventional Mott insulators in a quasi-2D version of the Shastry-Sutherland model in a magnetic field. In our realization on a four-leg tube geometry, these are stabilized by correlated hopping of localized magnetic excitations. Using perturbative continuous unitary transformations (pCUTs, plus classical approximation or exact diagonalization) and the density matrix renormalisation group method (DMRG), we identify prominent magnetization plateaus at magnetizations $M=1/8,3/16,1/4$, and $1/2$. While the plateau at $M=1/4$ can be understood in a semiclassical fashion in terms of diagonal stripes, the plateau at $M=1/8$ displays highly entangled wheels in the transverse direction of the tube. Finally, the $M=3/16$ plateau is most likely to be viewed as a classical 1/8 structure on which additional triplets are fully delocalized around the tube. The classical approximation of the effective model fails to describe all these plateau structures which benefit from correlated hopping. We relate our findings to the full 2D system, which is the underlying model for the frustrated quantum magnet $\mathrm{SrCu}{\left({\mathrm{BO}}_3\right)}_2$.

Figure 1

Illustration of the four-leg Shastry-Sutherland tube. Solid red bonds denote interdimer couplings $J$ and dashed blue lines refers to the intradimer coupling $J^{\prime }$. The tube is periodically coupled in the vertical direction as indicated by the blue dashed lines at the upper end. The unit cell is shown as thin black box covering four dimers.

Figure 2

Magnetization $M/M_{\mathrm{s}}$ as a function of the magnetic field $h$ obtained with pCUT(+ED) and DMRG. (a) Results from pCUT(+CA). Inset shows magnetization profile at $M=7/24$. (b) Results from pCUT(+ED) on a $8\times 4$ with periodic (PBC) and open (OBC) boundary conditions for $J^{\prime }/J=0.3$ as well as $J^{\prime }/J=0.5$ in the inset. (c) DMRG results for systems with $8\times 4$ and $16\times 4$ dimers for $J^{\prime }/J=0.3$. (d) DMRG results for systems with $8\times 4$ dimers for various values of $J^{\prime }/J$.

Figure 3

Local magnetization of the 1/8 plateau using DMRG. Upper (lower left) panel corresponds to an open tube with 64 dimers (32 dimers) and $J^{\prime }/J=0.3 \left(J^{\prime }/J=0.6\right)$. The diameters of circles relate to the local magnetization, which is also given explicitly as numbers. Positive (negative) values denote a local magnetization parallel (antiparallel) to the field direction. The lower right panel gives an illustration of the dominant order-two correlated hopping processes $t^{\prime }$ and $t^{\prime \prime }$.

Figure 4

Illustration of the plateau with $M=1/8$ obtained by pCUT(+CA). Thin red (thick dark) lines correspond to empty (filled) dimers. The most relevant two-particle interactions $V_3^{\prime }$ and $V_6$ are shown as grey/cyan dashed lines with arrows on both sides pointing to the two involved filled dimers.

Figure 5

Local magnetization of the 1/8 plateau on an open tube with 16 (left) 32 (right) dimers for $J^{\prime }/J=0.3$ using DMRG (top), pCUT(+ED) (middle), and ${\mathrm{pCUT}}_{\mathrm{finite}}(+\mathrm{ED})$ (bottom). The diameters of circles relate to the local magnetization, which is also given explicitly as numbers. Positive (negative) values denote a local magnetization parallel (antiparallel) to the field direction. One observes the realization of one (left) and two (right) quantum wheels.

Figure 6

Local magnetization of the 1/4 plateau on an open tube with 64 dimers using DMRG. One observes the superposition of four diagonal stripes. Top (bottom) panel corresponds to $J^{\prime }/J=0.3 \left(J^{\prime }/J=0.6\right)$. The diameters of circles relate to the local magnetization, which is also given explicitly as numbers. Positive (negative) values denote a local magnetization parallel (antiparallel) to the field direction.

Figure 7

Illustration of the plateau with $M=1/4$ obtained by pCUT(+CA). Thin red (thick dark) lines correspond to empty (filled) dimers. The most relevant two-particle interactions $V_2,V_4,V_3^{\prime }$, and $V_7$ are shown as grey/cyan dashed lines with arrows on both sides pointing to the two involved filled dimers.

Figure 8

Illustration of one semiclassical stripe on vertical dimers relevant for $M=1/4$. Thin red (thick dark) lines correspond to empty (filled) dimers. The most relevant two-particle interactions $V_2$ and $V_5$ are shown as grey/cyan dashed lines with arrows on both sides pointing to the two involved filled dimers. Each particle of the stripe can fluctuate by the dominant order-two correlated hopping process $t^{\prime }$ to the closest horizontal dimers above and below the stripe, which is sketched for the particle in the second row.

Figure 9

Local magnetization of the 1/4 plateau on an open tube with 16 (left) 32 (right) dimers for $J^{\prime }/J=0.3$ using DMRG (top), pCUT(+ED) (middle), and ${\mathrm{pCUT}}_{\mathrm{finite}}(+\mathrm{ED})$ (bottom). The diameter of circles relates to the local magnetization which is also given explicitly as numbers. Positive (negative) values denote a local magnetization parallel (antiparallel) to the field direction.

Figure 10

The upper panel illustrates a crystal of three-particle wheels having $M=1/8$. Wheels are highlighted by black boxes. The lower panel shows a plateau with $M=3/16$ consisting of a classical $M=1/8$ diamond structure, which is doped by a third particle in each unit cell which delocalize around the tube by correlated hopping. Thin red (thick dark) lines correspond to empty (filled) dimers. Delocalized particles are sketched as thick dark lines of half length.

Figure 11

Local magnetization of the 3/16 plateau on a tube with 32 dimers for $J^{\prime }/J=0.3$ using pCUT(+ED) (bottom) and DMRG (topl) with periodic boundary conditions. The diameter of circles relates to the local magnetization, which is also given explicitly as numbers. Positive (negative) values denote a local magnetization parallel (antiparallel) to the field direction.

Figure 12

Local magnetization of the 1/8 plateau on a four-leg system with 64 dimers for $J^{\prime }/J=0.3$ using open boundary conditions in both directions obtained by DMRG. The diameters of circles relate to the local magnetization, which is also given explicitly as numbers. Positive (negative) values denote a local magnetization parallel (antiparallel) to the field direction.