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arXiv:1705.05931 · Physical review. B./Physical review. B

Universal short-time quantum critical dynamics of finite-size systems

Yu-Rong Shu, Shuai Yin, Dao‐Xin Yao

Published Sep 12, 2017• 17 citations
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Abstract

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 ${(h/J)}_{c}=1.000\phantom{\rule{0.16em}{0ex}}03(8)$ 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 $L=32--256$ and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by $3.044\phantom{\rule{0.16em}{0ex}}51(7)$ 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 $L=24$--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.

DOI
10.1103/physrevb.96.094304

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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 \ensuremathθ\ensuremath{\theta}\ensuremathθ, 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 \ensuremathθ=0.3734(2)\ensuremath{\theta}=0.3734(2)\ensuremathθ=0.3734(2) and the static exponent \ensuremathβ/\ensuremathν=0.1251(2)\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)\ensuremathβ/\ensuremathν=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 \ensuremathθ=0.209(4)\ensuremath{\theta}=0.209(4)\ensuremathθ=0.209(4) and \ensuremathβ/\ensuremathν=0.518(1)\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)\ensuremathβ/\ensuremathν=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.
We propose a scaling theory for the universal imaginary-time quantum critical dynamics for both short and long times. We discover that there exists a universal critical initial slip related to a small initial order parameter M0{M}_{0}M0​. In this stage, the order parameter MMM increases with the imaginary time $\ensuremath{\tau}$ as $M\ensuremath{\propto}{M}_{0}{\ensuremath{\tau}}^{\ensuremath{\theta}}$ with a universal initial-slip exponent $\ensuremath{\theta}$. For the one-dimensional transverse-field Ising model, we estimate $\ensuremath{\theta}$ to be 0.3730.3730.373, which is markedly distinct from its classical counterpart. Apart from the local order parameter, we also show that the entanglement entropy exhibits universal behavior in the short-time region. As the critical exponents in the early stage and in equilibrium are identical, we apply the short-time dynamics method to determine quantum critical properties. The method is generally applicable in both the Landau-Ginzburg-Wilson paradigm and topological phase transitions.
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 investigate the imaginary-time relaxation critical dynamics of the two-dimensional transverse-field Ising model using infinite projected entangled pair states (iPEPS) with the full-update strategy. Simulating directly in the thermodynamic limit, we explore the relaxation process near the critical point with two types of initial states: a fully polarized state and a product state with a small magnetization. For the fully polarized state, the magnetization shows a power law scaling M∝τ−β/(νz)M\propto τ^{-β/(νz)}M∝τ−β/(νz) in the imaginary-time evolution, from which both the critical point and critical exponent can be determined with high accuracy. For the nearly paramagnetic state, the relaxation process exhibits a behavior of M∝τθM\propto τ^θM∝τθ with θ=0.1958θ=0.1958θ=0.1958 being the critical initial-slip exponent, which is in good agreement with that obtained from the dynamic scaling of the self-correlation in quantum Monte Carlo method. These universal features emerge well before the system converges to the ground state, demonstrating the efficiency of imaginary-time evolution for probing quantum criticality. Our results demonstrate that iPEPS can serve as a robust and scalable method for studying dynamical critical phenomena in two-dimensional quantum many-body systems.
We study the imaginary-time relaxation critical dynamics of a quantum system with a vanishing initial correlation length and an arbitrary initial order parameter M0. We find that in quantum critical dynamics, the behavior of M0 under scale transformations deviates from a simple power law, which was proposed for very small M0 previously. A universal characteristic function is then suggested to describe the rescaled initial magnetization, similar to classical critical dynamics. This characteristic function is shown to be able to describe the quantum critical dynamics in both short- and long-time stages of the evolution. The one-dimensional transverse-field Ising model is employed to numerically determine the specific form of the characteristic function. We demonstrate that it is applicable as long as the system is in the vicinity of the quantum critical point. The universality of the characteristic function is confirmed by numerical simulations of models belonging to the same universality class.
We explore the imaginary-time relaxation dynamics near quantum critical points with semi-ordered initial states. Different from the case with homogeneous ordered initial states, in which the order parameter M decays homogeneously as M ∝ τ −β/νz , here M depends on the location x, showing rich scaling behaviors. Similar to the classical relaxation dynamics with an initial domain wall in model A, which describes the purely dissipative dynamics, here as the imaginary time evolves, the domain wall expands into an interfacial region with growing size. In the interfacial region, the local order parameter decays as M ∝ τ −β/νz , with β 1 being an additional dynamic critical exponent. Far away from the interfacial region the local order parameter decays as M ∝ τ −β/νz in the short-time stage, then crosses over to the scaling behavior of M ∝ τ −β/νz when the location x is absorbed in the interfacial region. A full scaling form characterizing these scaling properties is developed. The quantum Ising model in both one and two dimensions are taken as examples to verify the scaling theory. In addition, we find that for the quantum Ising model the scaling function is an analytical function and β 1 is not an independent exponent.
Quantum criticality within Dirac fermions harbors a plethora of exotic phenomena, attracting sustained attention in the past decades. Here, we explore the imaginary-time relaxation dynamics in a typical Dirac quantum criticality belonging to chiral Heisenberg universality class. Performing large-scale quantum Monte Carlo simulation, we unveil rich nonequilibrium critical phenomena from different initial states. In particular, we identify a nonstationary initial slip evolution characterized by an unconventional negative critical exponent θ=-0.84(4), corroborating the significant impact of fermionic critical fluctuations. Furthermore, we generalize the nonequilibrium scaling theory to incorporate both fermionic and bosonic critical modes, capturing their distinct relaxation behaviors. Armed with the scaling theory, we establish a new framework to investigate fermionic quantum criticality based on short-time dynamics, paving a promising avenue to fathoming quantum criticality in diverse fermionic systems with high efficiency.