Dynamical signature of fractionalization at a deconfined quantum critical point

Deconfined quantum critical points govern continuous quantum phase transitions at which fractionalized (deconfined) degrees of freedom emerge. Here we study dynamical signatures of the fractionalized excitations in a quantum magnet (the easy-plane J-Q model) that realize a deconfined quantum critical point with emergent O(4) symmetry. By means of large-scale quantum Monte Carlo simulations and stochastic analytic continuation of imaginary-time correlation functions, we obtain the dynamic spin-structure factors in the $S^x$ and $S^z$ channels. In both channels, we observe broad continua that originate from the deconfined excitations. We further identify several distinct spectral features of the deconfined quantum critical point, including the lower edge of the continuum and its form factor on moving through the Brillouin zone. We provide field-theoretical and lattice model calculations that explain the overall shapes of the computed spectra, which highlight the importance of interactions and gauge fluctuations to explain the spectral-weight distribution. We make further comparisons with the conventional Landau O(2) transition in a different quantum magnet, at which no signatures of fractionalization are observed. The distinctive spectral signatures of the deconfined quantum critical point suggest the feasibility of its experimental detection in neutron scattering and nuclear magnetic resonance experiments.

Figure 1

The two lattice models considered in this paper and their schematic phase diagrams. (a) The EPJQ model with two-spin $\left(J\right)$ and six-spin $\left(Q\right)$ couplings preserve all symmetries of the square lattice. We define the tuning parameter chosen as $q=Q/(J+Q)$. The antiferromagnetic XY (AFXY) phase is separated by the DQCP at $q=q_c$ from the columnar VBS phase, which spontaneously breaks lattice symmetries but which has significant fluctuations of the fourfold degenerate dimer pattern close to $q_c$, as indicated. (b) The ${\mathrm{EPJ}}_1{\mathrm{J}}_2$ model, with the tuning parameter $g=J_2/J_1$. The $J_2$ term explicitly pins a columnar dimer pattern and drives the AFXY phase to the spin-disordered trivial (nondegenerate) columnar singlet phase (without spontaneous lattice symmetry breaking) through the 3DXY transition at $g=g_c$.

Figure 2

Expected low-energy features of the dynamic spin-structure factor $S(\mathit{q},\omega )\equiv \mathrm{Im}\chi (\mathit{q},\omega +\mathrm{i}{0}_{+})$ based on the theoretical dynamic spin susceptibility $\chi (\mathit{q},\omega )$ listed in Table 1.

Figure 3

Dynamic spin structure factors $S^x(\mathit{q},\omega )$ (a)–(c) and $S^z(\mathit{q},\omega )$ (d)–(f) obtained from QMC-SAC calculations for the EPJQ model with $L=32$ and $\beta =64$. Here (a) and (d) are inside the AFXY phase, $q=0.2$, (b) and (e) are close to the DQCP, $q=0.6$, and (c) and (f) are inside the VBS phase, $q=0.9$.

Figure 4

Dynamic spin-structure factors $S^x(\mathit{q},\omega )$ (a)–(c) and $S^z(\mathit{q},\omega )$ (d)–(f) obtained from QMC-SAC calculations for the ${\mathrm{EPJ}}_1{\mathrm{J}}_2$ model with $L=32$ and $\beta =64$. Panels (a) and (d) are inside the AFXY phase, $g=2$, (b) and (e) are close to the 3DXY transition point, $g=2.735$, and (c) and (f) are inside the quantum disordered phase, $g=3.6$.

Figure 5

Comparison of the DQCP dynamic spin structure factors between numerics [(a) $S^x$ channel and (c) $S^z$ channel] and theory [(b) $S^x$ channel and (d) $S^z$ channel]. The color map is the same as that in Fig. 3. The dashed curves trace out the upper and low edges of the two-parton continuum, assuming free fermionic partons with the $\pi$-flux state dispersion ${\epsilon }_{\mathit{k}}$ in Eq. (14). The lower edge simply follows ${\epsilon }_{\mathit{k}}$ and the upper edge is given by the maximal two-parton excitation energy $E_{\mathit{q}}={max}_{\mathit{k}\in \text{BZ}}|{\epsilon }_{\mathit{k}}+{\epsilon }_{\mathit{q}-\mathit{k}}|$. The suppressed spectral weight near (0,0) can be captured by matrix element effects.

Figure 6

The (bare) dynamic spin structure factor $S_0(\mathit{q},\omega )$ of the free fermion $\pi$-flux state.

Figure 7

Finite-size scaling of the spin stiffness ${\rho }_s^x$ of the ${\mathrm{EPJ}}_1{\mathrm{J}}_2$ model. (a) Raw data for system sizes $L=8,16,32$, and 64 as a function of $g=J_2/J_1$. (b) The same results rescaled by $L$, so that the crossing points of the curves for different $L$ should approach the critical point, determined here as $g_c=2.735\pm 0.002$. (c) The horizontal axis has been further rescaled as $(g-g_c)L^{1/\nu }$ with the 3D O(2) exponent $\nu$, leading to good data collapse in support of the assumed universality class.

Figure 8

Spin structure factors $S^x(\mathit{q},\omega )$ and $S^z(\mathit{q},\omega )$ obtained from QMC-SAC for the ${\mathrm{EPJ}}_1{\mathrm{J}}_2$ model with $L=32$ and $\beta =2L$. Panels (a) and (c) show data inside the AFXY phase with $g=2$, while (b) and (d) are close to the 3DXY transition point with $g=2.735$. Results are shown along the path $(0,0)-(\pi ,0)-(\pi ,\pi )-(0,0)-(0,\pi )-(\pi ,0)-(0,0)$ through the BZ.