[Paper Review] Scaling of the thermal spectral function for quantum critical bosons in one dimension

Thomas Barthel, Ulrich Schollwöck+1 more|arXiv (Cornell University)|Dec 14, 2012

Quantum and electron transport phenomena5 references3 citations

TL;DR

This paper presents a novel time-dependent density matrix renormalization group (tDMRG) scheme that doubles the maximum reachable times for finite-temperature simulations in one-dimensional quantum systems. The method enables precise computation of the thermal spectral function for quantum critical bosons with dynamic critical exponent z=2, confirming its universal scaling form S(k,ω) = (1/T)·Φ_S(k/√T, ω/T) and providing a numerically accurate determination of the scaling function Φ_S.

ABSTRACT

We present an improved scheme for the precise evaluation of finite-temperature response functions of strongly correlated systems in the framework of the time-dependent density matrix renormalization group. The maximum times that we can reach at finite temperatures T are typically increased by a factor of two, when compared against the earlier approaches. This novel scheme, complemented with linear prediction, allows us now to evaluate dynamic correlators for interacting bosons in one dimension. We demonstrate that the considered spectral function in the quantum critical regime with dynamic critical exponent z=2 is captured by the universal scaling form S(k,omega)=(1/T)*Phi(k/sqrt(T),omega/T) and calculate the scaling function precisely.

Motivation & Objective

• To develop a more efficient tDMRG scheme for finite-temperature response functions in strongly correlated 1D quantum systems.
• To enable the precise evaluation of dynamic correlators for interacting bosons in one dimension at finite temperatures.
• To test and confirm the universal scaling form of the thermal spectral function in the z=2 quantum critical regime.
• To determine the scaling function Φ_S(k/√T, ω/T) with high numerical accuracy.
• To demonstrate universality of Φ_S across different values of nearest-neighbor interaction V.

Proposed method

• The authors introduce a new tDMRG scheme (scheme C) that significantly reduces entanglement growth during time evolution, increasing the maximum reachable time by a factor of two compared to previous methods.
• The scheme employs optimized truncation parameters (ε_β = 10⁻¹², ε_t = 10⁻¹⁰) to maintain high precision while managing bond dimensions.
• Linear prediction is applied to extend the time evolution data and extract the spectral function S(k,ω) from the real-time correlators.
• The thermal spectral function is computed via S(k,ω) = ∑_x e^{ikx} ∫ dt e^{iωt} S(x,t), with S(x,t) derived from the thermal density matrix and time-evolved operators.
• The scaling hypothesis S(k,ω) ≈ (1/T)·Φ_S(k/√T, ω/T) is tested by rescaling data in k/√T and ω/T, with collapse confirming the universal form.
• Analytical asymptotic results from the Riemann-Hilbert problem formalism are used to benchmark the numerical results at large times and distances.

Experimental results

Research questions

• RQ1 Does the thermal spectral function for quantum critical bosons in one dimension obey a universal scaling form S(k,ω) = (1/T)·Φ_S(k/√T, ω/T)?
• RQ2 What is the precise functional form of the scaling function Φ_S(k/√T, ω/T) in the z=2 quantum critical regime?
• RQ3 Is the scaling function Φ_S universal across different values of the nearest-neighbor interaction V?
• RQ4 How does the new tDMRG scheme improve the accuracy and reach of finite-temperature dynamic correlation functions compared to previous methods?
• RQ5 To what extent do finite-size effects and linear prediction influence the extraction of Φ_S at low temperatures?

Key findings

• The thermal spectral function S(k,ω) for one-dimensional quantum critical bosons with z=2 exhibits the universal scaling form S(k,ω) = (1/T)·Φ_S(k/√T, ω/T), confirmed by data collapse across different temperatures and system sizes.
• The scaling function Φ_S is universal across different values of the nearest-neighbor interaction V, as evidenced by the collapse of rescaled S(k,ω) curves for V=0, 1/4, and 1/2.
• For k=0, the scaling function is well-fit by Φ_S(0, ω/T) = 0.649·(ω/T) / (1 + 0.826·(ω/T)³), with high precision in the low-frequency regime.
• The numerical results for the real-space correlator g(ξ,τ) show excellent agreement with the analytical asymptotic formula derived from the Riemann-Hilbert problem formalism for V=0.
• The new tDMRG scheme enables simulations with maximum times roughly doubled compared to earlier approaches, drastically reducing errors from linear prediction and finite-size effects at low temperatures.
• The study confirms that the scaling function Φ_S decays as ~ω⁻² at large frequencies, consistent with theoretical expectations for z=2 criticality.