[Paper Review] Universal term of Entanglement Entropy in the $π$-flux Hubbard model

Yuan Da Liao|arXiv (Cornell University)|Jul 20, 2023

Machine Learning in Materials Science11 citations

TL;DR

The paper introduces a controllable incremental algorithm within determinant quantum Monte Carlo to accurately compute the second Rényi entanglement entropy (EE) for interacting fermions, addressing exponential variance issues and enabling parallelizable, scalable EE calculations without extra configuration space.

ABSTRACT

Researchers in physical science aim to uncover universal features in strongly interacting many-body systems, often hidden in complicated observables like entanglement entropy (EE). The non-local nature of EE makes it challenging to compute numerically, necessitating the development of an unbiased and convenient algorithm. In this paper, we use quantum Monte Carlo to reveal that the coefficient of variation in direct EE calculations increases exponentially with system size, leading to inaccuracies. To address this issue, we develop a power incremental algorithm and a technique for straightforwardly calculating the universal term of EE, successfully evaluating the EE of a 2D Hubbard model. Our numerical results demonstrate the consistency of the universal coefficient of EE from sharp corners at the Gross-Neveu quantum critical point and for free Dirac fermions. Our method can also be applied to other unstable observables, such as partition functions, entanglement spectra, and negativity, thereby fostering computational and theoretical progress.

Motivation & Objective

• Seek universal features of entanglement entropy in strongly correlated systems.
• Develop a statistically stable and practical method to compute the second Rényi EE for interacting fermions in DQMC.
• Eliminate reliance on additional configuration spaces while providing a quantitative handle on the number of incremental steps.
• Extend the approach to other observables related to determinants of Green functions (e.g., entanglement spectrum, negativity).

Proposed method

• Analyze the instability of det g_A^{s1,s2} in Grover’s EE definition and identify exponential variance with system size.
• Introduce a controllable incremental scheme using Z(λ_k) with W(λ_k, det g_A^{s1,s2}) = (det g_A^{s1,s2})^{λ_k} and λ_k = k/N_λ.
• Show that log det g_A^{s1,s2} is normally distributed, fit its mean μ(L) and variance σ^2(L) as power laws, and set N_λ ≈ |μ| to control variance.
• Derive a parallelizable incremental ratio Z(λ_{k+1})/Z(λ_k) that uses (det g_A^{s1,s2})^{λ_k} and (det g_A^{s1,s2})^{1/N_λ} to keep variance manageable.
• Quantitatively determine N_λ ∼ 0.5 L^{1.35} from μ(L) scaling and implement the approach within projector DQMC without adding extra configuration space.
• Demonstrate improved accuracy and efficiency by comparing with prior methods and show EE convergence as N_λ grows.

Experimental results

Research questions

• RQ1 Can the entanglement entropy of interacting fermions be computed accurately in DQMC without introducing additional configuration space?
• RQ2 How does the distribution of det g_A^{s1,s2} behave, and can normal-distribution properties of log det g be exploited to control variance?
• RQ3 What is the scaling of the optimal number of incremental steps N_λ with system size L, and how does this affect computational cost?
• RQ4 Can the method be extended to other observables dependent on determinants of Green functions, such as entanglement spectra and negativity?

Key findings

• The direct evaluation of det g_A^{s1,s2} exhibits rare spikes and unstable statistics as coupling grows and system size increases.
• log det g_A^{s1,s2} follows a normal distribution with μ(L) ~ -0.50 L^{1.35} and σ(L) ~ 0.67 L^{0.77}.
• Coefficient of variation of det g grows exponentially, but the CV of (det g)^{1/N_λ} or of log det g decays with system size when N_λ scales as L^{1.35}.
• An incremental scheme using Z(λ_k) with (det g)^{λ_k} and (det g)^{1/N_λ} enables accurate EE without extra configuration space and can be parallelized across increments.
• The method achieves reliable 2nd Rényi EE estimates with CPU time comparable to standard DQMC and provides a quantitative way to choose N_λ (≈ 0.5 L^{1.35}).
• The approach is extendable to other observables tied to determinants of Green functions, such as entanglement spectra and negativity.