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Emergent symmetry is one of the characteristic phenomena in deconfined quantum critical point (DQCP). As its nonequilibrium generalization, the dual dynamic scaling was recently discovered in the nonequilibrium imaginary-time relaxation dynamics in the DQCP of the $J$-$Q_3$ model. In this work, we study the nonequilibrium imaginary-time relaxation dynamics in the $J$-$Q_2$ model, which also hosts a DQCP belonging to the same equilibrium universality class. We not only verify the universality of the dual dynamic scaling at the critical point, but also investigate the breakdown and the vestige of the dual dynamic scaling when the tuning parameter is away from the critical point. We also discuss its possible experimental realizations in devices of quantum computers., 18 pages, 11 figures

Deconfined quantum critical point (DQCP) characterizes a kind of exotic phase transition beyond the usual Landau-Ginzburg-Wilson paradigm. Here we study the nonequilibrium imaginary-time dynamics of the DQCP in the two-dimensional J-Q_{3} model. We explicitly show the deconfinement dynamic process and identify that it is the spinon confinement length, rather than the usual correlation length, that increases proportionally to the time. Moreover, we find that, in the relaxation process, the order parameters of the Néel and the valence-bond-solid orders can be controlled by different length scales, although they satisfy the same equilibrium scaling forms. A dual dynamic scaling theory is then proposed. Our findings not only constitute a new realm of nonequilibrium criticality in DQCP, but also offer a controllable knob by which to investigate the dynamics in strongly correlated systems. Possible realizations in foreseeable quantum computers are also discussed.

We study the short-imaginary-time quantum critical dynamics (SITQCD) in the J-Q$_3$ spin chain, which hosts a quasi-long-range-order phase to a valence bond solid transition. By using the scaling form of the SITQCD with a saturated ordered phase, we are able to locate the critical point at $q_{\rm c}=0.170(14)$. We also obtain the critical initial slip exponent $\theta=-0.507(3)$ and the static exponent $\beta/\nu=0.498(2)$. More strikingly, we find that the scaling dimension of the initial order parameter $x_{0}$ is close to zero, which suggests that the initial order parameter is a marginal operator. As a result, there is no initial increase behavior of the order parameter in the short-imaginary-time relaxation process for this model, which is very different from the relaxation dynamics in the Ising-type phase transitions. Our numerical results are realized by the projector quantum Monte Carlo algorithm., Comment: 8 pages, 8 figures

We use numerical techniques to study dynamical properties at finite temperature ($T$) of the Heisenberg spin chain with random exchange couplings, which realizes the random singlet (RS) fixed point in the low-energy limit. Specifically, we study the dynamic spin structure factor $S(q,\omega)$, which can be probed directly by inelastic neutron scattering experiments and, in the limit of small $\omega$, in nuclear magnetic resonance (NMR) experiments through the spin-lattice relaxation rate $1/T_1$. Our work combines three complementary methods: exact diagonalization, matrix-product-state algorithms, and stochastic analytic continuation of quantum Monte Carlo results in imaginary time. Unlike the uniform system, whose low-energy excitations at low $T$ are restricted to $q$ close to $0$ and $\pi$, our study reveals a continuous narrow band of low-energy excitations in $S(q,\omega)$, extending throughout the Brillouin zone. Close to $q=\pi$, the scaling properties of these excitations are well captured by the RS theory, but we also see disagreements with some aspects of the predicted $q$-dependence further away from $q=\pi$. Furthermore we find spin diffusion effects close to $q=0$ that are not contained within the RS theory but give non-negligible contributions to the mean $1/T_1$. To compare with NMR experiments, we consider the distribution of the local $1/T_1$ values, which is broad, approximately described by a stretched exponential. The mean value first decreases with $T$, but starts to increase and diverge below a crossover temperature. Although a similar divergent behavior has been found for the static uniform susceptibility, this divergent behavior of $1/T_1$ has never been seen in experiments. Our results show that the divergence of the mean $1/T_1$ is due to rare events in the disordered chains and is concealed in experiments, where the typical $1/T_1$ value is accessed., Comment: 19 pages, 14 figures

We investigate the short time quantum critical dynamics in the imaginary time relaxation processes of finite size systems. Universal scaling behaviors exist in the imaginary time evolution and in particular, the system undergoes a critical initial slip stage characterized by an exponent $\theta$, in which an initial power-law increase emerges in the imaginary time correlation function when the initial state has zero order parameter and vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo simulations. In the one-dimensional case, we locate the quantum critical point at $(h/J)_{c}=1.00003(8)$ \thirdrevise{in the thermodynamic limit}, and estimate the critical initial slip exponent $\theta=0.3734(2)$, static exponent $\beta/\nu=0.1251(2)$ \thirdrevise{by analyzing data on chains of length $L=32\sim 256$ and $L=48\sim 256$, respectively}. For the two-dimensional square-lattice system, the critical coupling ratio is given by $3.04451(7)$ \thirdrevise{in the thermodynamic limit} while the critical exponents are \thirdrevise{$\theta=0.209(4)$ and $\beta/\nu=0.518(1)$ estimated by data on systems of size $L=24\sim 64$ and $L=32\sim 64$, correspondingly.} Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts, owing to the essential differences between classical and quantum dynamics. The short time critical dynamics and the imaginary time relaxation QMC approach can be readily adapted to various models., Comment: 12 pages, 8 figures

We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study S=1/2 spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multi-spin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that can not be generated by the SDRG scheme. In the model with multi-spin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties., 15 pages; v2 new material on log corrections added