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arXiv:2006.11833 · Physical review. B./Physical review. B

Short-imaginary-time quantum critical dynamics in the J−Q3 spin chain

Yu-Rong Shu, Shuai Yin

Published Sep 21, 2020• 5 citations
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Abstract

We study the short-imaginary-time quantum critical dynamics (SITQCD) in the $J\ensuremath{-}{Q}_{3}$ 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 ${q}_{\mathrm{c}}=0.170(14)$. We also obtain the critical initial slip exponent $\ensuremath{\theta}=\ensuremath{-}0.507(3)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.498(2)$. 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.

DOI
10.1103/physrevb.102.104425

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We study the short-imaginary-time quantum critical dynamics (SITQCD) in the J\ensuremath−Q3J\ensuremath{-}{Q}_{3}J\ensuremath−Q3​ 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 qc=0.170(14){q}_{\mathrm{c}}=0.170(14)qc​=0.170(14). We also obtain the critical initial slip exponent \ensuremathθ=\ensuremath−0.507(3)\ensuremath{\theta}=\ensuremath{-}0.507(3)\ensuremathθ=\ensuremath−0.507(3) and the static exponent \ensuremathβ/\ensuremathν=0.498(2)\ensuremath{\beta}/\ensuremath{\nu}=0.498(2)\ensuremathβ/\ensuremathν=0.498(2). More strikingly, we find that the scaling dimension of the initial order parameter x0{x}_{0}x0​ 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.
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. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a 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 (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at (h/J)c=1.00003(8){(h/J)}_{c}=1.000\phantom{\rule{0.16em}{0ex}}03(8)(h/J)c​=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length L=32−−256L=32--256L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by 3.04451(7)3.044\phantom{\rule{0.16em}{0ex}}51(7)3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size L=24L=24L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due 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.
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While imaginary-time evolution has long served as a standard paradigm for ground-state preparation in numerical simulations and quantum devices, its intrinsic dynamical properties has been largely overlooked. Here, we investigate the short-imaginary-time critical dynamics in quantum systems with boundaries. A universal scaling theory is developed and verified in the two-dimensional quantum Ising model, uncovering rich dynamic critical behaviors dictated by boundary universality classes. For ordered initial states, the boundary order parameter MsM_sMs​ decays with imaginary time τττ as Ms∝τ−β1/νzM_s \propto τ^{-β_1/νz}Ms​∝τ−β1​/νz, where β1β_1β1​ denotes the boundary order parameter exponent, and ννν and zzz correspond to the correlation length exponent and the dynamic exponent, respectively. For disordered initial states, the autocorrelation of the boundary order parameter is governed by a novel critical exponent θ1θ_1θ1​, which is closely related to the critical initial slip behavior of MsM_sMs​ characterized by the corresponding exponent θ1′θ_1'θ1′​. In contrast to its positive bulk counterpart, the boundary initial-slip exponent θ1′θ_1'θ1′​ is negative for the ordinary transition while remaining positive for the special transition. Although the static universality classes of ddd-dimensional quantum phase transitions generally coincide with those of (d+1)(d+1)(d+1)-dimensional classical phase transitions, we show that θ1θ_1θ1​ does not follow this conventional quantum-classical mapping. We further discuss the implications of our results for more exotic forms of boundary criticality. Our findings provide new physical insights into boundary critical dynamics and offer a novel route for probing exotic boundary critical behaviors in quantum many-body systems.
We study the imaginary-time relaxation critical dynamics of the N\'eel paramagnetic quantum phase transition in the two-dimensional dimerized S=12S=\frac{1}{2}S=21​ Heisenberg model. We focus on the scaling correction in the short-time region. A unified scaling form including both short-time and finite-size corrections is proposed. According to this full scaling form, improved short-imaginary-time scaling relations are obtained. We numerically verify the scaling form and the improved short-time scaling relations for different initial states using projector quantum Monte Carlo algorithm.
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