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

TL;DR

This work addresses nonequilibrium quantum critical dynamics in the two-dimensional transverse-field Ising model by simulating imaginary-time relaxation with full-update iPEPS in the thermodynamic limit. It demonstrates universal short-time scaling: from a saturated initial state, $M(\tau) \propto \tau^{-\beta/(\nu z)}$, and from a small initial magnetization, $M(\tau) \propto \tau^{\theta}$, with the latter exponent $\theta$ converging toward quantum Monte Carlo values as the bond dimension grows. The study yields a precise estimate of the critical field $h_c$ (extrapolated to $h_c \approx 3.0445$) and confirms the universal exponents $-\beta/(\nu z) \approx -0.518$ and $\theta \approx 0.1958$, validating iPEPS as a reliable tool for two-dimensional dynamical critical phenomena. The results highlight the efficiency and scalability of imaginary-time iPEPS for probing quantum criticality and pave the way for exploring more complex models, including frustrated magnets and interacting fermions, without the sign problem.

Abstract

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.

Figures (5)

• Figure 1: Imaginary-time relaxation dynamics for a fully polarized initial state, plotted in log-log scale. Magnetization $M(\tau)$ obtained from iPEPS with different bond dimensions ($D$) and environment bond dimensions ($\chi$) is summarized together. In each calculation, the transverse field leading to the best linear curve is denoted as $h^*$, and for each $h^*$, a linear fitting is performed and is denoted as dashed yellow. (a) For $D=3$ and $\chi=30$. (b) For $D=4$ and $\chi=60$. (c) For $D=5$ and $\chi=60$.
• Figure 2: Extrapolation of the estimated critical field $h^*$ as a function of $1/D$. The data points correspond to the result obtained from $D=3$, $4$, and $5$, which are determined from the best power-law fitting. A simple power fit yields an extrapolated value of ${h_c}\approx 3.0445$, in close agreement with the previous estimation $h_c = 3.04451(7)$Shu2017 (denoted as green dashed line).
• Figure 3: Imaginary-time relaxation dynamics for a product initial state with a tiny magnetization $M_0\approx0.005$, plotted in $\log$-$\log$ scale. Magnetization $M(\tau)/M_0$ obtained from iPEPS with different bond dimensions ($D$) and environment bond dimensions ($\chi$) is summarized together. In each calculation, the transverse field leading to the best linear curve is denoted as $h^*$, and for each $h^*$ a linear fitting is performed and is denoted as dashed yellow. (a) For $D=3$ and $\chi=30$. (b) For $D=4$ and $\chi=60$. (c) For $D=5$ and $\chi=60$.
• Figure 4: The fitting power at $h=h^*$ as a function of $1/D$. The data points correspond to $D = 3$, $4$, and $5$, with critical fields determined from the best power-law fits in Fig. \ref{['fig:Mz2']}. A simple power-law fit yields $\theta \approx 0.19584$ in the large-$D$ limit, consistent with the QMC estimation $0.209(4)$Shu2017 denoted by the green dashed line with the shaded area indicating the statistical error.
• Figure 5: Estimated critical initial-slip exponent $k^*$ as a function of initial magnetization $M_0$, obtained with $D=5$ and $\chi=60$. Each data point is obtained at $h^*=3.044$ as shown in Fig. \ref{['fig:Mz2']} and discussed in the main text. An exponential fit is performed and is denoted by the orange dashed line. Inset: the initial-slip behavior of magnetization as a function of $\tau$ when $M_0$ is small.