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.
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.
Entanglement measures are useful tools in characterizing otherwise unknown quantum phases and indicating transitions between them. Here we examine the concurrence and entanglement entropy in quantum spin chains with random long-range couplings, spatially decaying with a power-law exponent $\ensuremath{\alpha}$. Using the strong disorder renormalization group (SDRG) technique, we find by analytical solution of the master equation a strong disorder fixed point, characterized by a fixed point distribution of the couplings with a finite dynamical exponent, which describes the system consistently in the regime $\ensuremath{\alpha}>\frac{1}{2}$. A numerical implementation of the SDRG method yields a power-law spatial decay of the average concurrence, which is also confirmed by exact numerical diagonalization. However, we find that the lowest-order SDRG approach is not sufficient to obtain the typical value of the concurrence. We therefore implement a correction scheme which allows us to obtain the leading-order corrections to the random singlet state. This approach yields a power-law spatial decay of the typical value of the concurrence, which we derive both by a numerical implementation of the corrections and by analytics. Next, using numerical SDRG, the entanglement entropy (EE) is found to be logarithmically enhanced for all $\ensuremath{\alpha}$, corresponding to a critical behavior with an effective central charge
c=ln(2), independent of $\ensuremath{\alpha}$. This is confirmed by an analytical derivation. Using numerical exact diagonalization (ED), we confirm the logarithmic enhancement of the EE and a weak dependence on $\ensuremath{\alpha}$. For a wide range of partition size
l, the EE fits a critical behavior with a central charge close to
c=1, which is the same as for the clean Haldane-Shastry model with a power-law-decaying interaction with $\ensuremath{\alpha}=2$. Only for small $l\ensuremath{\ll}L$, in a range which increases with the number of spins
N, we find deviations which are rather consistent with the strong disorder fixed point central charge
c=ln(2). Furthermore, we find using ED that the concurrence shows power-law decay, albeit with smaller power exponents than obtained by SDRG. We also present results obtained with DMRG and find agreement with ED for sufficiently small $\ensuremath{\alpha}<2$, whereas for larger $\ensuremath{\alpha}$ DMRG tends to underestimate the entanglement entropy and finds a faster decaying concurrence.
The dynamical behavior of a quantum many-particle system is characterized by the lifetime of its excitations. When the system is perturbed, observables of any nonconserved quantity decay exponentially, but those of a conserved quantity relax to equilibrium with a power law. Such processes are associated with a dynamical exponent
z that relates the spread of correlations in space and time. We present numerical results for the Fredkin model, a quantum spin chain with a three-body interaction term, which exhibits an unusually large dynamical exponent. We discuss our efforts to produce a reliable estimate $z\ensuremath{\doteq}3.16(1)$ through direct simulation of the quantum evolution and to explain the slow dynamics in terms of an excited bond that executes a constrained random walk in Monte Carlo time.
We present power-law relaxation behavior of the local magnetizations in the equilibration dynamics of the spin-1/2 Heisenberg spin chain as an isolated integrable quantum system. We perform the exact time evolution of the expectation values of the local spin operators by evaluating them with the determinant formula of the form factors. We construct such an initial quantum state that has a localized profile of the local magnetizations, and perform the exact time evolution over a very long period of time. We show that the local magnetization relaxes as some power of the time variable with no definite time scale, while the fidelity relaxes very fast with its relaxation time being proportional to the inverse of the energy width, i.e. the Boltzmann time.
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.
The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground-state properties of a system is limited by the size $\ensuremath{\chi}$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S\ensuremath{\sim}\frac{1}{6}\text{log}\text{ }\ensuremath{\chi}$ with high precision. This result can be understood as the emergence of an effective finite correlation length ${\ensuremath{\xi}}_{\ensuremath{\chi}}$ ruling all the scaling properties in the system. We produce six extra pieces of evidence for this finite-$\ensuremath{\chi}$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, the scaling of the entanglement entropy for a finite block of spins, the existence of scaling functions, and the agreement with analogous classical results. All our computations are consistent with a scaling relation of the form ${\ensuremath{\xi}}_{\ensuremath{\chi}}\ensuremath{\sim}{\ensuremath{\chi}}^{\ensuremath{\kappa}}$, with $\ensuremath{\kappa}=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $\ensuremath{\kappa}\ensuremath{\sim}1.37$. We also show how finite-$\ensuremath{\chi}$ scaling allows us to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density-matrix renormalization-group results and their classical analog obtained with the corner transfer-matrix renormalization group.