[Paper Review] Finite-time scaling of dynamic quantum criticality

Shuai Yin, Xizhou Qin+2 more|arXiv (Cornell University)|Jul 6, 2012

Advanced Thermodynamics and Statistical Mechanics2 references6 citations

TL;DR

This paper introduces finite-time scaling (FTS) as a unified framework to study dynamic quantum criticality by balancing three time scales: intrinsic reaction time ($\tau_s$), thermal time ($\tau_T$), and external driving time ($\tau_d$). It demonstrates that FTS enables accurate determination of critical points and exponents at zero and nonzero temperatures, with successful application to the 1D transverse-field Ising model via TEBD and Lindblad master equations, showing robust scaling collapse and agreement with exact results.

ABSTRACT

We develop a theory of finite-time scaling for dynamic quantum criticality by considering the competition among an external time scale, an intrinsic reaction time scale and an imaginary time scale arising respectively from an external driving field, the fluctuations of the competing orders and thermal fluctuations. Through a successful application in determining the critical properties at zero temperature and the solution of real-time Lindblad master equation near a quantum critical point at nonzero temperatures, we show that finite-time scaling offers not only an amenable and systematic approach to detect the dynamic critical properties, but also a unified framework to understand and explore nonequilibrium dynamics of quantum criticality, which shows specificities for open systems.

Motivation & Objective

• To develop a systematic framework for understanding nonequilibrium quantum critical dynamics under time-dependent driving.
• To identify and characterize a new finite-time scaling (FTS) regime dominated by the external driving time scale $\tau_d$.
• To extend the Kibble-Zurek mechanism by providing a scaling description of the impulse regime where adiabaticity breaks down.
• To enable accurate numerical determination of quantum critical points and critical exponents at both zero and finite temperatures.
• To establish the Lindblad master equation as a viable framework for open quantum systems near quantum critical points.

Proposed method

• Introduce three competing time scales: $\tau_s \sim |g|^{-\nu z}$ (reaction), $\tau_T = 1/T$ (thermal), and $\tau_d \sim R^{-z/r}$ (driving), to characterize dynamic quantum criticality.
• Propose a finite-time scaling (FTS) ansatz that unifies equilibrium and nonequilibrium regimes by scaling observables with $\tau_d$ as the dominant time scale in the impulse regime.
• Apply the time-evolving block-decimation (TEBD) algorithm to solve the real-time Schrödinger equation for the 1D transverse-field Ising model at $T=0$.
• Use the Lindblad master equation to model open quantum systems at nonzero temperatures, incorporating dissipation as an independent scaling variable.
• Perform scaling collapses of magnetization data using the FTS form $M_h \sim f(L^{-1}R_z^{-1/r_z}, cR_z^{-z/r_z})$ to extract critical exponents.
• Validate results by comparing numerically extracted critical exponents with exact values from the 1D transverse-field Ising model.

Experimental results

Research questions

• RQ1 How does the competition among the reaction time $\tau_s$, thermal time $\tau_T$, and driving time $\tau_d$ shape the dynamic critical behavior near a quantum phase transition?
• RQ2 Can finite-time scaling (FTS) provide a systematic and unified description of nonequilibrium quantum critical dynamics, including the impulse regime of the Kibble-Zurek mechanism?
• RQ3 How can critical exponents and the critical point be reliably extracted from real-time dynamics using FTS at both zero and nonzero temperatures?
• RQ4 What role does dissipation play in nonequilibrium quantum criticality, and how can it be incorporated as a scaling variable in open systems?
• RQ5 Does the FTS framework remain valid and predictive in the presence of thermal fluctuations and finite-size effects?

Key findings

• The finite-time scaling (FTS) framework successfully identifies a new nonequilibrium regime dominated by the external driving time scale $\tau_d$, which governs nonadiabatic excitations in the impulse regime.
• At $T=0$, the FTS approach yields a critical point estimate of $h_{xc}^{N} = 0.999(2)$, in excellent agreement with the exact value $h_{xc} = 1$.
• Critical exponents extracted via FTS—$\beta = 0.125(11)$, $\delta = 14.9(6)$, $\nu = 0.98(4)$, $z = 1.01(3)$—show strong agreement with exact results.
• At nonzero temperatures, FTS collapses of magnetization data are achieved only when the dissipation rate $c$ is included with scaling dimension $z$, confirming its role as an independent scaling variable.
• The FTS form $M_h \sim f(L^{-1}R_z^{-1/r_z}, cR_z^{-z/r_z})$ enables robust data collapse for $LR_z^{1/r_z} = 1.166$ and $cR_z^{-z/r_z} = 3.603$, validating the scaling ansatz.
• The framework enables direct probing of quantum critical behavior at nonzero temperatures by making thermal effects subordinate, effectively achieving 'dynamic cooling'.