# Universal short time quantum critical dynamics of finite size systems

## TL;DR

This paper studies the universal short-time quantum critical dynamics in finite systems using imaginary time relaxation and quantum Monte Carlo, revealing distinct critical exponents and locating quantum critical points in 1D and 2D transverse field Ising models.

## Contribution

The paper introduces a method to determine quantum critical points and exponents from short-time imaginary time dynamics in finite quantum systems, validated by simulations on 1D and 2D Ising models.

## Key findings

- Quantum critical points accurately located in 1D and 2D models.
- Critical initial slip exponents differ from classical counterparts.
- Universal scaling behaviors observed in short-time quantum dynamics.

## 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 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.