Thermodynamics of Shastry-Sutherland Model under Magnetic Field

Abstract

Motivated by the recent experimental discovery of the $T$-linear specific heat in pressurized and magnetized Shastry-Sutherland Mott insulator SrCu$_2$(BO$_3$)$_2$, we perform the state-of-the-art thermal tensor-network computation on the Shastry-Sutherland model under a magnetic field. Our simulation results suggest the existence of a symmetric intermediate phase with $T$-linear specific heat at low temperature, occupying a large parameter space and separating the plaquette-singlet phase and antiferromagnetic phase at low fields and other symmetry-breaking phases at high fields before the system is fully polarized. Such an unexpected novel state bears an astonishing similarity to the experimental findings in the material. It opens the door to further investigations of the possible liberation of deconfined magnetized Dirac spinons by the competing interactions in this highly frustrated quantum magnet model, and by the combined effects of magnetic field and pressure in the the associated Shastry-Sutherland Mott insulator SrCu$_2$(BO$_3$)$_2$.

Figures (6)

• Figure 1: Phase diagram of the SS model under a magnetic field. The previously known zero-field phase boundaries are shown as dashed lines. The dots indicate parameter points studied via XTRG simulations, with the red dots highlighting regions where a linear temperature dependence of the specific heat is observed, serving as a signature for the Dirac spinon state (DSS). The remaining acronyms denote the spin-polarized (SP), dimer singlet (DS), plaquette singlet (PS), spin liquid (SL), antiferromagnetic (AF), and field-induced antiferromagnetic (AF$'$) phases. The inset shows the lattice of the SS model.
• Figure 2: Specific heat in the SS model at zero magnetic field. Temperature dependence of (a) specific heat $C$ and (b) specific heat divided by temperature $C/T$ for different coupling strengths. $g=0.6$ is inside the DS phase, $g=0.7$ is inside the PS phase, $g=0.77,0.8$ are close to the PS-AF transition and $g=0.95$ is inside the AF phase.
• Figure 3: Specific specific heat divided by temperature, $C/T$, as a function of temperature $T$ for various selected magnetic fields $h$, at different coupling constants $g$ for different phases. Panels (a)-(e) display: (a) DS phase ($g=0.6$), (b) PS phase ($g=0.7$), (c) PS--SL phase boundary ($g=0.77$), (d) SL phase ($g=0.8$), and (e) AF phase ($g=0.95$). The red dashed line represents a linear extrapolation of the last three $C/T$ data points based on $C/T = a_0 +a_1T$, which intercepts the vertical axis at a finite value $a_0$. This finite intercept implies a linear dependence of the $C$ against $T$. Noisy data points at very low temperatures were omitted because of truncation errors and inaccuracies in the numerical derivative used to obtain the specific heat.
• Figure 4: Physical observables (entanglement entropy, magnetic structure factor, and magnetization) computed via XTRG across coupling constants $g$: (top) half-system thermal entanglement entropy $S_E$, (middle) spin structure factor at $\mathbf{q}=(\pi,\pi)$, $S^{\pm(z)}(\pi,\pi) = \frac{1}{N} \sum_{i,j=1}^{N} e^{i\mathbf{q}\cdot(\mathbf{r}_i-\mathbf{r}_j)} \langle {S}^{\pm(z)}_i \cdot {S}^{\pm(z)}_j \rangle$, (bottom) normalized magnetization, $M_z/M_{sat}= \frac{2}{N} \left| \sum_{i=1}^N \langle S_i^z \rangle \right|$. Observables are measured at temperature $T\approx0.01$. All quantities are shown as functions of magnetic field $h$, with color scheme indicating phase regimes consistent with Fig. \ref{['fig:fig1']}. The dashed lines in the bottom panel indicate the 1/3 and 1/2 magnetic plateaus.
• Figure 5: Quasi-1D lattice geometry for XTRG simulations. The cylindrical cluster consists of 3×6×4 sites. Black and brown bonds denote the inter-dimer coupling $J$ and intra-dimer coupling $J'$ from Eq. \ref{['eq:hamiltonian']}, respectively. Periodic boundary conditions are applied along the y-direction. The gray snake-shaped line illustrates the mapping of the cluster onto a one-dimensional chain with long-range interactions.