Imaginary-time Quantum Relaxation Critical Dynamics with Semi-ordered Initial States

TL;DR

This paper investigates the complex imaginary-time relaxation dynamics near quantum critical points starting from semi-ordered initial states, revealing rich spatially-dependent scaling behaviors and developing a comprehensive scaling theory.

Contribution

It introduces a new scaling framework for quantum relaxation dynamics with semi-ordered initial states, highlighting the role of an interfacial region and the exponent eta_1.

Findings

Local order parameter decays with different exponents in interfacial and bulk regions.

Scaling functions are analytical for the quantum Ising model.

The exponent eta_1 is not independent in the quantum Ising model.

Abstract

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 $M\propto \tau^{-\beta/\nu z}$, 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 $M\propto \tau^{-\beta_1/\nu z}$, with $\beta_1$ being an additional dynamic critical exponent. Far away from the interfacial region the local order parameter decays as $M\propto \tau^{-\beta/\nu z}$ in the short-time stage, then crosses over to the scaling behavior of $M\propto \tau^{-\beta_1/\nu z}$ 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 $\beta_1$ is not an independent exponent.