[Paper Review] Canonical Universality

Anatoly Dymarsky, Hong Liu|arXiv (Cornell University)|Feb 24, 2017

Quantum many-body systems16 references17 citations

TL;DR

This paper proposes Canonical Universality (CU), a hypothesis stating that in chaotic quantum many-body systems, all pure states within a sufficiently narrow energy band exhibit thermal behavior for any local or extensive observable. The key result is that deviations from thermal equilibrium scale with a smooth, volume-dependent function ΔE(x), ensuring all states in narrow bands are thermal up to exponentially small corrections, contrasting with ETH's focus on individual eigenstates.

ABSTRACT

Isolated quantum system in a pure state may be perceived as thermal if only substantially small fraction of all degrees of freedom is probed. We propose that in a chaotic quantum many-body system all states with sufficiently small energy fluctuations are approximately thermal. We refer to this hypothesis as Canonical Universality (CU). The CU hypothesis complements the Eigenstate Thermalization Hypothesis (ETH) which proposes that for chaotic systems individual energy eigenstates are thermal. Integrable and MBL systems do not satisfy CU. We provide theoretical and numerical evidence supporting the CU hypothesis.

Motivation & Objective

• To identify conditions under which pure quantum states in narrow energy bands behave thermally, beyond the Eigenstate Thermalization Hypothesis (ETH).
• To establish a universal framework for thermalization in chaotic systems that applies to all states in a narrow energy window, not just individual eigenstates.
• To quantify the minimal energy width ΔE(x) required for a state to deviate from thermal equilibrium by a tolerance x.
• To distinguish chaotic systems (which satisfy CU) from integrable and many-body localized (MBL) systems (which do not), via the behavior of ΔE(x).

Proposed method

• Define A^max(E,ΔE) and A^min(E,ΔE) as the maximal and minimal expectation values of an observable A over all normalized states in a narrow energy band of width ΔE.
• Introduce ΔE(x) as the minimal energy band width such that at least one state in the band deviates from the microcanonical expectation A^micro by tolerance x.
• Propose that in chaotic systems, ΔE(E,x) ≈ γ(E,x) + O(1/Ω), where γ(x) is a smooth, monotonically non-decreasing function for x > 0 and non-increasing for x < 0.
• Use canonical typicality and random matrix theory to argue that typical states in narrow bands are thermal with exponential precision.
• Analyze volume dependence of ΔE(x) using quench protocols and subsystem decomposition, showing ΔE(x) remains finite in the thermodynamic limit for small x.
• Derive analytic bounds on ΔE(x) for averaged observables, showing linear scaling with volume for large deviations and non-trivial scaling for small x.

Experimental results

Research questions

• RQ1 Under what conditions do all pure quantum states in a narrow energy band of a chaotic system behave thermally, even if they are not energy eigenstates?
• RQ2 How does the minimal energy width ΔE(x) required to deviate from thermal equilibrium scale with system size and tolerance x in chaotic systems?
• RQ3 Why do integrable and MBL systems fail to satisfy the Canonical Universality hypothesis despite satisfying ETH?
• RQ4 What is the functional form of ΔE(x) in the thermodynamic limit, and how does it depend on the observable A and system parameters?
• RQ5 Can the volume dependence of ΔE(x) be analytically constrained for local and extensive observables?

Key findings

• For chaotic systems, all states in a narrow energy band ΔE < γ(x) are approximately thermal, with deviations bounded by x, implying thermalization is universal across the band.
• The function ΔE(x) is described by a smooth function γ(x) + O(1/Ω), with δ = 2 for generic operators in the small-x limit, indicating γ(x) ∝ x² for small x.
• For small x, ΔE(x) scales as L^−a for some a ≥ 0, ensuring ΔE(x) > 0 for x ≠ 0 and ΔE(0) = 0, meaning thermal behavior is robust for sufficiently narrow bands.
• In the thermodynamic limit, ΔE(x) remains finite for small x when x is achieved via subsystem quench protocols, showing non-trivial scaling independent of volume.
• For large deviations, ΔE(x) scales linearly with system size, consistent with maximal possible deviations requiring broad energy bands.
• Numerical results for spin chains confirm that matrix elements of local operators are normally distributed in central bands, supporting the validity of the thermal approximation.