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Numerical Evidence for Entanglement Time: Scaling Laws and the Past Hypothesis

Alik Gimranov

Published Jun 19, 2026• 0 citations
Publisher

Abstract

We present numerical evidence for the Hilbert Space Refactorization Principle and the Intensive Time Principle proposed in Ref. [1]. Using spin-chain simulations with Heisenberg XXX interaction, we demonstrate three key results: (1) Past Hypothesis without fine-tuning: Refactorization of Hilbert space from (8+2) to (9+1) spin bipartition reduces the effective entropy of the daughter subsystem by a factor of ~5 (ratio S_daughter/S_before ≈ 0.19). (2) Modular time deviation: For N=8 spins initialized in the Néel state, the entanglement time deviates from proper time by Δτ ≈ 1.82 (18% reduction) over t = 10, correlating with the rate of entanglement growth. (3) Scaling laws: Systematic simulations for N = 6, 8, 10, 12 reveal that entanglement entropy scales extensively (ΔS ~ N^0.80), while entanglement time scales sub-extensively (Δτ ~ N^0.27), yielding the non-linear relation Δτ ~ (ΔS)^0.34. This provides numerical confirmation of the Intensive Time Principle: time is an intensive thermodynamic observable, fundamentally distinct from extensive quantities like entropy. These results establish entanglement time as a robust physical effect that survives the thermodynamic limit, supporting the conceptual framework of Ref. [1]. This is Part II of a trilogy. Part I presents the conceptual framework (DOI: 10.5281/zenodo.20760213). Part III will derive observational predictions for gravitational wave detectors.

DOI
10.5281/zenodo.20761692

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We present numerical evidence for the Hilbert Space Refactorization Principle and the Intensive Time Principle proposed in Ref. [1]. Using spin-chain simulations with Heisenberg XXX interaction, we demonstrate three key results: (1) Past Hypothesis without fine-tuning: Refactorization of Hilbert space from (8+2) to (9+1) spin bipartition reduces the effective entropy of the daughter subsystem by a factor of ~5 (ratio S_daughter/S_before ≈ 0.19). (2) Modular time deviation: For N=8 spins initialized in the Néel state, the entanglement time deviates from proper time by Δτ ≈ 1.82 (18% reduction) over t = 10, correlating with the rate of entanglement growth. (3) Scaling laws: Systematic simulations for N = 6, 8, 10, 12 reveal that entanglement entropy scales extensively (ΔS ~ N^0.80), while entanglement time scales sub-extensively (Δτ ~ N^0.27), yielding the non-linear relation Δτ ~ (ΔS)^0.34. This provides numerical confirmation of the Intensive Time Principle: time is an intensive thermodynamic observable, fundamentally distinct from extensive quantities like entropy. These results establish entanglement time as a robust physical effect that survives the thermodynamic limit, supporting the conceptual framework of Ref. [1]. This is Part II of a trilogy. Part I presents the conceptual framework (DOI: 10.5281/zenodo.20760213). Part III will derive observational predictions for gravitational wave detectors.
We provide rigorous numerical verification of the modular time definition through Tomita–Takesaki theory and establish a scaling relation between entanglement time τ_ent = dS/dt and modular time τ_mod = √Var(K_ρ) in Heisenberg XXX spin chains. Extensive simulations for system sizes N = 6–20 and systematic robustness checks reveal that τ_ent ∝ τ_mod^γ with exponent γ∞ = 2.07 ± 0.04 for N ≥ 14, consistent with diffusion-like growth of entanglement in modular time. We verify that the modular flow satisfies all four axioms of a Tomita–Takesaki automorphism group to numerical precision ~10⁻¹⁵. Furthermore, we demonstrate that emergent time is intensive (non-additive): τ(2N)/(2τ(N)) = 0.76 ± 0.17 ≠ 1. These results provide strong numerical evidence that different notions of time in quantum many-body systems are related by a scale-invariant law, with potential implications for emergent gravity.
We investigate the scaling relation between the physical entanglement time τent and the modular time τmod in the dynamics of the integrable Heisenberg XXX spin chain. Contrary to the hypothesis of a stable universal exponent γ ≈ 2, we find no evidence for a time-independent power-law scaling. Instead, the effective exponent γ(t) exhibits a systematic drift toward smaller values at longer evolution times, following a robust power-law relaxation γ(t) −γ∞ ∼ t−α with α ≈ 0.65, supported by an excellent data collapse with relative spread below 6%. Remarkably, the amplitude of this relaxation is independent of system size N within the accessible window N ∈ {14,16,18,20}. We interpret this behavior as an effective renormalization group flow of the scaling exponent in time, suggesting that γ is not a fixed scaling dimension but a running quantity controlled by a marginally irrelevant operator. Our results demonstrate that previously reported values γ ≈ 2 originate from preasymptotic regimes rather than a true scaling law, and reveal a dynamical mechanism by which simple scaling breaks down in quantum entanglement evolution.
Characterising the universal aspects of non-equilibrium quantum many-body dynamics is one of the key goals of this century's physics research. Progress, however, is hindered by the lack of general theoretical frameworks for studying interacting quantum matter far from equilibrium. A recent breakthrough has been the realization that several key non-equilibrium quantities, such as the rate of growth of entanglement or the fluctuations of conserved charges within finite subsystems, can be related to equilibrium properties through a space-time duality that effectively exchanges the roles of space and time. This observation effectively enables the study of non-equilibrium phenomena using tools and concepts borrowed from equilibrium statistical mechanics and thermodynamics. A first proof of principle of this framework, dubbed space-time duality approach (SDA), was provided by interacting integrable systems, where thermodynamic properties can often be characterized exactly, while dynamical quantities typically remain beyond analytical reach. Subsequent developments, however, revealed that the SDA suffered from an intrinsic ambiguity, restricting its applicability to homogeneous quenches and to charge fluctuations arising from symmetric initial states. Here we resolve this ambiguity from first principles and derive closed-form predictions for entanglement growth and charge fluctuations after general quantum quenches. We benchmark our results against the exact analytical solution of the Rule 54 quantum cellular automaton and extensive TEBD simulations of the XXZ chain. Moreover we show that, when specialised to the entanglement entropy, our framework naturally reproduces the predictions of the quasiparticle picture.
Abstract If an asymmetry in time does not arise from the fundamental dynamical laws of physics, it may be found in special boundary conditions. The argument normally goes that since thermodynamic entropy in the past is lower than in the future according to the Second Law of Thermodynamics, then tracing this back to the time around the Big Bang means the universe must have started off in a state of very low thermodynamic entropy: the Thermodynamic Past Hypothesis. In this paper, we consider another boundary condition that plays a similar role, but for the decoherent arrow of time, i.e. the subsystems of the universe are more mixed in the future than in the past. According to what we call the Entanglement Past Hypothesis, the initial quantum state of the universe had very low entanglement entropy. We clarify the content of the Entanglement Past Hypothesis, compare it with the Thermodynamic Past Hypothesis, and identify some challenges and open questions for future research.
We analyze numerically the dynamical generation of quantum entanglement in a system of two interacting particles, started in a coherent separable state, for decreasing values of ℏ. As ℏ→0 the entanglement entropy, computed at any finite time, converges to a finite nonzero value. The limit law that rules the time dependence of entropy is well reproduced by purely classical computations. Its general features can be explained by simple classical arguments, which expose the different ways entanglement is generated in systems that are classically chaotic or regular.
We present numerical evidences for the logarithmic scaling of the entanglement entropy in critical random spin chains. Very large scale exact diagonalizations performed at the critical XXXXXX point up to L=2000L=2000L=2000 spins 12\frac{1}{2}21​ lead to a perfect agreement with recent real-space renormalization-group predictions of [Refael and Moore Phys. Rev. Lett. 93, 260602 (2004)] for the logarithmic scaling of the entanglement entropy in the random singlet phase with an effective central charge $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}=c\ifmmode\times\else\texttimes\fi{}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2$. Moreover, we provide the first visual proof of the existence of the random singlet phase with the help of quantum entanglement concept.