Signatures of a Deconfined Phase Transition on the Shastry-Sutherland Lattice: Applications to Quantum Critical ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$

We study a possible deconfined quantum phase transition in a realistic model of a two-dimensional Shastry-Sutherland quantum magnet, using both numerical and field theoretic techniques. Using the infinite density matrix renormalization group (IDMRG) method, we verify the existence of an intermediate plaquette valence bond solid (PVBS) order, with twofold degeneracy, between the dimer and Néel ordered phases. We argue that the quantum phase transition between the Néel and PVBS orders may be described by a deconfined quantum critical point (DQCP) with an emergent O(4) symmetry. By analyzing the correlation length spectrum obtained from IDMRG, we provide evidence for the DQCP and emergent O(4) symmetry in the lattice model. Such a phase transition has been reported in the recent pressure-tuned experiments in the Shastry-Sutherland lattice material ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$ [Nat. Phys. 13, 962 (2017)]. The nonsymmorphic lattice structure of the Shastry-Sutherland compound leads to extinction points in the scattering, where we predict sharp signatures of a DQCP in both the phonon and magnon spectra associated with the spinon continuum. The effect of weak interlayer couplings present in the three-dimensional material is also discussed. Our results should help guide the experimental study of DQCP in quantum magnets.

Popular Summary

In certain antiferromagnets (where adjacent magnetic moments align in an alternating pattern), defects in the atomic structure can proliferate, triggering a phase transition into a paramagnet. In quantum magnets—magnets where atomic spins strongly interact—an intricate quantum correlation causes defect proliferation to trigger a different ordered phase, one with broken lattice symmetry. Such a continuous phase transition between two distinct ordered phases is called a deconfined quantum critical point (DQCP), and it has been long sought after because of its theoretical importance. However, its experimental realization has been absent. We demonstrate the possibility of the DQCP in a realistic model of the 2D quantum magnet ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$, a compound recently found to transition, under external pressure, between an antiferromagnet and a valence-bond solid (a state where entangled pairs of spins form a specific pattern).

One interesting consequence of the DQCP is the unification of two seemingly independent order parameter fluctuations for an antiferromagnet and valence-bond solid. As a result, a system at the DQCP exhibits interesting symmetry features enlarged from the original microscopic symmetry. Employing both analytical and numerical methods, we find strong evidence for such unification, where the fluctuations of two distinct order parameters are intertwined. Furthermore, we predict sharp signatures of the DQCP in spectra of phonons and magnons, which can be examined by inelastic x-ray or neutron scattering experiments. Finally, we reveal the importance of a 3D interlayer coupling, which can drastically change the fate of DQCPs in general.

Our theoretical analysis for DQCP candidate materials, as well as a full characterization of spectroscopic signatures of the DQCP, will provide a valuable guide for an experimental search.

Figure 1

(a) The Shastry-Sutherland lattice of copper sites (small circles) in ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$, on which the spins reside. The spins are coupled across nearest-neighbor bonds ($J_1$, in blue) and dimer bonds ($J_2$, in red). Each unit cell contains four sites, as shaded in yellow. The glide-reflection ($G_x$ and $G_y$) and the diagonal-reflection (${\sigma }_{xy}$ and ${\sigma }_{x\overline{y}}$) symmetries are indicated on the lattice. (b) Diffraction peaks from copper sites. The darker dot indicates a higher intensity. The extinction points are marked out by red circles. The first Brillouin zone is shaded in yellow, corresponding to the unit cell in (a). Special momentum points $\mathrm{\Gamma }$, $X$, $Y$, and $M$ are defined as labeled. (c) The phase diagram of the spin model Eq. (1). The Néel antiferromagnetic and DVBS phases are separated by the intermediate PVBS phase upon tuning the $J_1/J_2$ ratio. The critical points are determined in Table 1 based on our IDMRG result. The transition between PVBS and Néel phases is likely to be a DQCP (or weakly first-order proximate to a DQCP).

Figure 2

Visualization of bond strengths $⟨{\mathit{S}}_i·{\mathit{S}}_j⟩$ within the IDMRG unit cell. The $\widehat{x}$ direction is along the cylinder, and the $\widehat{y}$ direction is around the circumference. The width of the bond represents the value of ${\mathit{S}}_1·{\mathit{S}}_2$. The line color is black (red) if ${\mathit{S}}_1·{\mathit{S}}_2$ is negative (positive). One can notice that the singlet is formed at a plaquette.

Figure 3

At $L=10$, $\chi =4000$. (a) Energy ($E$) per site. The horizontal dotted line represents the exact ground state energy of the dimer VBS state, $E_{\text{site}}=-0.375$. (b) $\partial E/\partial J_1$ per site near the transition between plaquette VBS and Néel order. The continuous first-order derivative is a characteristic of the continuous phase transition. Black dashed lines denote phase boundaries among the dimer VBS, plaquette VBS, and Néel order.

Figure 4

The inverse of the largest correlation length for spin-singlet and -triplet operators as a function of $J_1/J_2$ at $L=10$ and $\chi =4000$. Here, the dashed line represents the transition point extracted from the second-order energy derivative. The small plot at the top right shows the plaquette order parameter ($\mathrm{Im}\mathcal{M}$) across the transition. Deep in the VBS phase, the correlation length of a spin-triplet operator is larger than that of a spin-singlet operator as expected by a mean-field theory. As we approach the critical point, we can observe that the correlation length of the spin-singlet sector becomes larger than that of the spin-triplet sector. This behavior agrees with what is expected from the scenario in Fig. 8.

Figure 5

The ideal Shastry-Sutherland lattice, deformed from Fig. 1. Each site $i$ can be labeled by a Cartesian coordinate $(x,y)$. The $x$ and $y$ coordinates are calibrated on the top and right, respectively. The displacement vectors $\widehat{x}=(1,0)$ and $\widehat{y}=(0,1)$ are defined to connect nearest-neighbor sites. $J_1$ and $J_2$ couplings are assigned to the nearest-neighbor (in blue) and dimer (in red) bonds.

Figure 6

Two types of PVBS phases. Thick purple links encircle the plaquette on which the spin singlet is formed. A diamond plaquette contains the diagonal bond, while a square plaquette does not. Each type of the PVBS order induces a corresponding lattice distortion. The undistorted lattice is shown as the background for contrast. Depending on the sign of ${\lambda }_2$, either diamond plaquette or square plaquette PVBS is favored.

Figure 7

Phase diagram of the appearance of O(4) DQCP on perturbing the SO(5) theory. Arrows indicate the RG flow direction.

Figure 8

(a) RG flow of the quadrupled monopole term ${\lambda }_4{\mathcal{M}}^4$ in the square lattice DQCP scenario, which is dangerously irrelevant. For a small deviation of the tuning parameter $J_1/J_2$ toward the VBS phase, ${\lambda }_4$ initially decreases under RG flow until the RG scale reaches the spin-spin correlation length. Only after that does ${\lambda }_4$ begin to increase to reach the VBS fixed point. This result has noticeable consequences for observables in a finite size numerical simulation. (b) Schematic plot for the inverse correlation length ${\xi }^{-1}$ of spin-singlet and -triplet operators as a function of the tuning parameter for the square lattice DQCP scenario with either O(4) or SO(5) emergent symmetries. Note that ${\xi }^{-1}$ is related to the energy (mass gap) of the associated excitation [32]. Here, the skyrmion corresponds to the Higgsed “photon” excitation in the $C{P}^1$ theory. In the VBS phase, this photon excitation manifests as a spin-singlet VBS order parameter fluctuation, i.e., the VBS domain wall thickness. Similarly, the magnon becomes a “triplon” in the VBS phase. The plot assumes the simulation of the DQCP at the finite circumference $L$ and bond dimension $\chi$, which prevents $\xi$ from diverging at the DQCP.

Figure 9

The correlation length spectrum for (a) the $J_1\text{−}{J}_2$ model in the spin-$1/2$ square lattice ($\chi =2000$) and (b) the Shastry-Sutherland model ($\chi =4000$). The correlation length spectrum coincides with the excitation level-crossing spectrum in Fig. 2 of Ref. [50] (after switching $J_2/J_1$ to $J_1/J_2$). For the value of $J_1/J_2$ smaller than the range shown in the figure, we get the collinear striped AFM order (DVBS) for the square (Shastry-Sutherland) lattice. The level-crossing behaviors for both models are similar in the Néel ordered side.

Figure 10

(a) The $\pi$-flux model of fermionic spinons. The spinon hopping on the thick red bond gets a minus sign, such that each plaquette has a $\pi$ flux. Additional spinon interaction terms are applied to blue shaded diamonds. (b) The arrangement of the site indices $[ijkl]$ around each shaded diamond.

Figure 11

(a) Spin excitation spectrum (dynamical structural factor) $S_{\text{magnon}}(\omega ,\mathit{q})$ at the DQCP. A darker color indicates a higher intensity. The dashed line traces out the lower edge of the continuum which is described by Eq. (13). (b) The frequency dependence of the spectral intensity at the extinction point $X$. At the low-frequency limit, the spectral intensity grows with frequency linearly, which manifests the conserved current associated to the emergent O(4) symmetry. (c) Schematic illustration of the phonon spectrum $S_{\text{phonon}}(\omega ,\mathit{q})$. The bare phonon dispersion is inferred from Ref. [67]. The VBS-phonon coupling leads to a continuum in the phonon spectrum. (d) The frequency dependence of the phonon spectral intensity at the extinction point $X$. The intensity falls off in a power law with the frequency, whose exponent should be the same as that of the spin fluctuation at $M$ point.

Figure 12

(a) Three-dimensional interlayer AFM coupling $J_3$ among spins located at the cross between upper and lower diagonal bonds, mediated by the interpenetrating oxygen atom. The other interlayer couplings are negligible. Preferred stacking of (b) the Néel ordered phase with ${\mathit{n}}^{(l)}={\mathit{n}}^{(l+1)}$ (red and blue dots for opposite spins), and (c) the diamond PVBS phase with $\mathrm{Re}{\mathcal{M}}^{(l)}=\mathrm{Re}{\mathcal{M}}^{(l+1)}$ and (d) square PVBS phase with $\mathrm{Im}{\mathcal{M}}^{(l)}=-\mathrm{Im}{\mathcal{M}}^{(l+1)}$ (thick purple bonds mark singlet plaquettes). For the (c) and (d) cases, energetically favorable stacking can be deduced by the interlayer dimer resonance.

Figure 13

(a) The monopole operator inserted in the Euclidean spacetime. White arrows represent spin directions. The imaginary time trajectory of each spin is represented by a colored line. White arrows in the black trajectory show how the direction of the spin changes along the imaginary time when the monopole is inserted. (b) The solid angle $\omega ({\mathit{n}}_r)$ sweeps through the imaginary time with respect to the reference vector ${\mathit{n}}_0$. (c) The dual lattice where a monopole operator resides. Here, plaquette centers correspond to the spin sites. The lattice spin acts as an alternating flux pattern $(-1{)}^{r_x+r_y}4\pi S$ for monopoles. The hopping amplitude along each black arrow gives a phase factor of $e^{i\pi S}=i$ in our $S=1/2$ case.

Figure 14

(a) Square lattice $J_1\text{−}{J}_1^{\prime }$ model with a dimer coupling $J_1^{\prime }$. $J_1^{\prime }$ explicitly breaks the square lattice symmetry. (b) Square lattice $J_1\text{−}{J}_2$ model. (c),(d) Correlation length spectra for the model in (a),(b).

Figure 15

(a) $A$, $B$, $C$, and $D$ label four copper sites in each unit cell (marked out in dashed lines). $k_1$ and $k_2$ are the stiffnesses of the two types of bonds (nearest neighbor and dimer). $X$ and $Y$ label the two types of plaquettes. (b) Schematic illustration of the phonon spectrum. A continuum emerges at the $X$ point due to the VBS-phonon coupling.

Figure 16

The IDMRG simulation results with $\mathrm{\Delta }{J}_1=0.002$ at $L=10$, shown for a range of the IDMRG bond dimension $\chi$. Bond dimension scalings of the (a) energy per site $E$, (b) energy derivative per site $\partial E/\partial J_1$ (data for $\chi =500$, 1000 are not shown here, as these data points behave irregularly and cover the other data points, which can happen at a low bond dimension, for which the IDMRG simulation does not converge properly), (c) truncation error $p$, (d) correlation length of spin-singlet operator ${\xi }_{S=0}$, and (e) PVBS order parameters. Here, (c)–(e) share the same labels for different bond dimensions. Note that the correlation length is plotted, not the inverse as in Fig. 4. In (f), we plot PVBS order parameters as functions of truncation errors for a range of the tuning parameter $J_1$. The blue dotted line is a linear fitting for the three data points at $\chi =2000$, 3000, 4000.

Figure 17

The IDMRG simulation results with $\mathrm{\Delta }{J}_1=0.002$ (0.001 for $\chi =4000$) at $L=8$, shown for a range of MPS bond dimension $\chi$. Different observables are labeled as in Fig. 16. Here, again (c)–(e) share the same labels for different bond dimensions. Note that the PVBS order parameter vanishes at $J_1=0.726$, which is smaller than the value for $L=10$.

Figure 18

Comparison between the $L=8$ and $L=10$ results at $\chi =4000$ for the spin-singlet and -triplet correlation lengths. For $L=8$, the peak of the ${\xi }_{S=0}$ is located at $J_1=0.727$, while for $L=10$ the peak is located at $J_1=0.762$. Unlike the result at $L=10$, the spin-triplet correlation length at $L=8$ does not grow much after the peak.