Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2403.09084 · Physical review. B./Physical review. B

Imaginary-time relaxation quantum critical dynamics in the two-dimensional dimerized Heisenberg model

Jiaqi Cai, Yu-Rong Shu, Xue-Qing Rao, Shuai Yin

Published May 6, 2024• 3 citations
arXivHTMLPDF

Abstract

We study the imaginary-time relaxation critical dynamics of the N\'eel paramagnetic quantum phase transition in the two-dimensional dimerized $S=\frac{1}{2}$ 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.

DOI
10.1103/physrevb.109.184303

Related papers

Scaling corrections in driven critical dynamics: Application to the two-dimensional dimerized quantum Heisenberg model

Jing-Wen 静雯 Liu 刘, Shuai Yin, Yu-Rong Shu

Abstract Driven critical dynamics in quantum phase transitions holds significant theoretical importance, and also has practical applications in fast-developing quantum devices. While scaling corrections have been shown to play important roles in fully characterizing equilibrium quantum criticality, their impact on nonequilibrium critical dynamics has not been extensively explored. In this work, we investigate the driven critical dynamics in a two-dimensional quantum Heisenberg model. We find that in this model the scaling corrections arising from both finite system size and finite driving rate must be incorporated into the finite-time scaling form in order to properly describe the nonequilibrium scaling behaviors. In addition, improved scaling relations are obtained from the expansion of the full scaling form. We numerically verify these scaling forms and improved scaling relations for different starting states using the nonequilibrium quantum Monte Carlo algorithm.

arXiv:2502.04927•Chinese Physics B•Mar 28, 2025•1 citation

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

Yu-Rong Shu, Shuai Yin, Dao‐Xin Yao

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.

arXiv:1705.05931•Physical review. B./Physical review. B•Sep 12, 2017•17 citations

Imaginary-Time Quantum Relaxation Critical Dynamics with Semi-Ordered Initial States

Zhixuan Li, Shuai Yin, Yu-Rong Shu

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.

arXiv:2210.03979•Chinese Physics Letters•Feb 1, 2023•1 citation

Probing universal imaginary-time relaxation critical dynamics with infinite projected entangled pair states

He-Yu Lin, Shuai Yin, Z. Y. Xie +1 more

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 $M\propto τ^{-β/(νz)}$ 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 $M\propto τ^θ$ with $θ=0.1958$ 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.

cond-mat.str-elCross-listedarXiv:2511.10934•Physical review. B./Physical review. B•Nov 17, 2025•0 citations

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

Yu-Rong Shu, Shuai Yin

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.

arXiv:2006.11833•Physical review. B./Physical review. B•Sep 21, 2020•5 citations

Dynamical critical behavior of the three-dimensional Heisenberg model

Pawel Peczak, David Paul Landau

We have carried out extensive Monte Carlo simulations of the dynamic critical behavior of L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic classical Heisenberg models with periodic boundary conditions. The equilibrium relaxation time \ensuremath{\tau} is determined from the long-time exponential decay of the time-displaced correlation function of the magnetization. Using finite-size scaling and corrections to scaling we find that at the critical point ${\mathit{T}}_{\mathit{c}}$=(0.6929${)}^{\mathrm{\ensuremath{-}}1}$, \ensuremath{\tau} grows like \ensuremath{\tau}\ensuremath{\simeq}${\mathit{L}}^{\mathit{z}}$ with z=1.96(6). In the paramagnetic region we show that the relaxation time is represented by a universal scaling function \ensuremath{\tau}${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{z}}$ vs (T/${\mathit{T}}_{\mathit{c}}$-1)${\mathit{L}}^{1/\ensuremath{\nu}}$ with z=1.96(6). This estimate is consistent with theoretical predictions and results of experimental studies concerning the dynamical critical behavior of Heisenberg magnets perturbed by order-parameter nonconserving dipolar interactions. We also describe how a simple over-relaxation algorithm of Creutz and of Brown and Woch can be incorporated into the Metropolis sampling method for the classical Heisenberg model, thereby significantly reducing the critical slowing down; the resultant correlation time varies as \ensuremath{\tau}\ensuremath{\sim}(T/${\mathit{T}}_{\mathit{c}}$-1${)}^{\mathrm{\ensuremath{-}}0.77}$, which implies z\ensuremath{\simeq}1.10.

Physical review. B, Condensed matter•Jun 1, 1993•45 citations
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.
Abstract Driven critical dynamics in quantum phase transitions holds significant theoretical importance, and also has practical applications in fast-developing quantum devices. While scaling corrections have been shown to play important roles in fully characterizing equilibrium quantum criticality, their impact on nonequilibrium critical dynamics has not been extensively explored. In this work, we investigate the driven critical dynamics in a two-dimensional quantum Heisenberg model. We find that in this model the scaling corrections arising from both finite system size and finite driving rate must be incorporated into the finite-time scaling form in order to properly describe the nonequilibrium scaling behaviors. In addition, improved scaling relations are obtained from the expansion of the full scaling form. We numerically verify these scaling forms and improved scaling relations for different starting states using the nonequilibrium 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.
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.
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 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 qc=0.170(14){q}_{\mathrm{c}}=0.170(14)qc​=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 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 have carried out extensive Monte Carlo simulations of the dynamic critical behavior of L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic classical Heisenberg models with periodic boundary conditions. The equilibrium relaxation time \ensuremath{\tau} is determined from the long-time exponential decay of the time-displaced correlation function of the magnetization. Using finite-size scaling and corrections to scaling we find that at the critical point Tc{\mathit{T}}_{\mathit{c}}Tc​=(0.6929${)}^{\mathrm{\ensuremath{-}}1}$, \ensuremath{\tau} grows like \ensuremath{\tau}\ensuremath{\simeq}Lz{\mathit{L}}^{\mathit{z}}Lz with z=1.96(6). In the paramagnetic region we show that the relaxation time is represented by a universal scaling function \ensuremath{\tau}${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{z}}$ vs (T/Tc{\mathit{T}}_{\mathit{c}}Tc​-1)${\mathit{L}}^{1/\ensuremath{\nu}}$ with z=1.96(6). This estimate is consistent with theoretical predictions and results of experimental studies concerning the dynamical critical behavior of Heisenberg magnets perturbed by order-parameter nonconserving dipolar interactions. We also describe how a simple over-relaxation algorithm of Creutz and of Brown and Woch can be incorporated into the Metropolis sampling method for the classical Heisenberg model, thereby significantly reducing the critical slowing down; the resultant correlation time varies as \ensuremath{\tau}\ensuremath{\sim}(T/Tc{\mathit{T}}_{\mathit{c}}Tc​-1${)}^{\mathrm{\ensuremath{-}}0.77}$, which implies z\ensuremath{\simeq}1.10.