Universality of equilibration dynamics after quantum quenches
Abstract
We investigate the distribution of the eigenvalues of the reduced density matrix (entanglement spectrum) after a global quantum quench. We show that in an appropriate scaling limit the lower part of the entanglement spectrum exhibits ``universality''. In the scaling limit and at asymptotically long times the distribution of the entanglement spectrum depends on two parameters that can be determined from the Rényi entropies. We show that two typical scenarios occur. In the first one, the distribution of the entanglement spectrum levels is similar to the one describing the ground-state entanglement spectrum in Conformal Field Theories. In the second scenario, the lower levels of the entanglement spectrum are highly degenerate and their distribution is given by a series of Dirac deltas. We benchmark our analytical results in free-fermion chains, such as the transverse field Ising chain and the XX chain, in the rule 54 chain, and in Bethe ansatz solvable spin models.
Summary
This paper investigates the universality of the entanglement spectrum after global quantum quenches in one-dimensional systems. The authors analyze the distribution of eigenvalues of the reduced density matrix (entanglement spectrum) in the short-time and long-time regimes, focusing on the lower part of the spectrum. Their main finding is that, in an appropriate scaling limit, the lower part of the entanglement spectrum exhibits universality, meaning its distribution depends on only two parameters derived from Rényi entropies. Two scenarios are identified: one where the distribution resembles the ground-state entanglement spectrum in Conformal Field Theories (CFTs), and another where the lower levels are highly degenerate, following a series of Dirac delta functions. The authors benchmark their analytical results using free-fermion chains (transverse field Ising chain, XX chain), the rule 54 chain, and Bethe ansatz solvable spin models (XXZ chain). The methodology involves deriving an analytical expression for the probability density function P(λ) of the entanglement spectrum eigenvalues based on the large-α expansion of the Rényi entropies. They relate the moments of the reduced density matrix to the distribution P(λ) using a generating function and Cauchy's theorem. The validity of this approach is assessed by comparing the resulting distribution with numerical results for various integrable models. They analyze the large-alpha expansion of the Renyi entropies for these models, and based on the form of the expansion (specifically, the presence or absence of a 1/alpha term), predict either a CFT-like entanglement spectrum or a "staircase" entanglement spectrum. The paper demonstrates that the structure of the entanglement spectrum in the scaling limit encodes information about the underlying physics and the quench protocol.
Key Insights
- •The lower part of the entanglement spectrum exhibits universality after a quantum quench, characterized by a distribution P(λ) that depends on only two parameters derived from the Rényi entropies: the largest eigenvalue λ_m and a parameter r_1 related to the large-α behavior of the Rényi entropies.
- •Two typical scenarios emerge: a CFT-like distribution P(λ) ≈ δ(λ - λ_m) + (b r_1 I_1(2r_1 ξ))/(λ ξ) with ξ = sqrt(b ln(λ_m/λ)), and a "staircase" distribution P(λ) = Σ δ(λ - λ_m exp(-d_1 k)), where b = -ln(λ_m) and d_1 is a parameter related to the exponential decay in the Renyi entropies.
- •The parameter r_1 is model-dependent and encodes non-universal information, while in CFTs, r_1 = 1, making the cumulative distribution function n(λ) super-universal and dependent only on the central charge (in the ground state).
- •For the TFIC, quenches to the critical point (h=1) lead to a CFT-like structure, while off-critical quenches result in a "staircase" structure. The parameter *a1* in the large-alpha expansion dictates which behavior is observed.
- •In the XXZ chain, the authors derive coupled integral equations for the slope of the Rényi entropy growth and analyze their large-α behavior. They identify conditions under which the entanglement spectrum exhibits either a CFT-like or staircase structure based on the initial state (Néel or Majumdar-Ghosh) and the anisotropy parameter Δ.
- •The authors provide an analytical formula (Eq. 16) for the distribution P(λ) and its cumulative distribution function n(λ) (Eq. 17) in the CFT-like scenario and show how to include subleading corrections (Eq. 22).
- •They find a "duality" between space and time, in the sense that the parameter a_1 in the large-alpha expansion is the same in the short-time and long-time regimes, with the substitution of length *l* with twice the time, *2t*.
Practical Implications
- •This research provides a framework for understanding the dynamics of entanglement in quantum many-body systems, which is crucial for the development of quantum technologies such as quantum computers and quantum simulators.
- •The analytical expressions derived for the entanglement spectrum can be used to benchmark numerical simulations of quantum quenches and to validate theoretical models of equilibration and thermalization.
- •Practitioners can use the identified universality classes to classify the behavior of different quantum systems after a quench, potentially simplifying the analysis and prediction of their dynamics.
- •The connection between Rényi entropies and the entanglement spectrum provides a practical way to extract information about the system's state from experimentally measurable quantities.
- •Future research directions include extending this framework to higher-dimensional systems, non-integrable models, and more complex quench protocols, as well as exploring the implications of these findings for quantum information processing and quantum control.