Entanglement dynamics from universal low-lying modes

TL;DR

This paper identifies a universal mechanism behind the late-time growth of Renyi entropies in chaotic quantum many-body systems by mapping Lorentzian dynamics to a Euclidean evolution generated by a multi-copy superhamiltonian $P_{2n}$. In Brownian time-dependent models without symmetries, the low-energy spectrum consists of gapped, plane-wave-like domain-wall excitations between ferromagnetic ground states labeled by permutations in ${\mathcal{S}}_n$, whose dispersion relations $E(k)$ determine the entanglement membrane tension ${\mathcal{E}}(v)$ via a Legendre transform. The second Renyi entropy exhibits a robust membrane picture across large-$q$ and finite-$q$ Brownian-GUE models, with tensions obeying physical constraints, while the third Renyi entropy exhibits phase transitions in ${\mathcal{E}}_3(v)$ arising from multi-domain-wall/scattering effects; a coherent inclusion of scattering states yields a consistent membrane description. The framework generalizes to fixed-coupling Brownian models and higher spatial dimensions, suggesting a universal, symmetry-determined set of low-energy modes that govern entanglement dynamics and potentially extend to time-independent Hamiltonians and holographic contexts.

Abstract

Information-theoretic quantities such as Renyi entropies show a remarkable universality in their late-time behaviour across a variety of chaotic many-body systems. Understanding how such common features emerge from very different microscopic dynamics remains an important challenge. In this work, we address this question in a class of Brownian models with random time-dependent Hamiltonians and a variety of different microscopic couplings. In any such model, the Lorentzian time-evolution of the $n$-th Renyi entropy can be mapped to evolution by a Euclidean Hamiltonian on 2$n$ copies of the system. We provide evidence that in systems with no symmetries, the low-energy excitations of the Euclidean Hamiltonian are universally given by a gapped quasiparticle-like band. The eigenstates in this band are plane waves of locally dressed domain walls between ferromagnetic ground states associated with two permutations in the symmetric group $S_n$. These excitations give rise to the membrane picture of entanglement growth, with the membrane tension determined by their dispersion relation. We establish this structure in a variety of cases using analytical perturbative methods and numerical variational techniques, and extract the associated dispersion relations and membrane tensions for the second and third Renyi entropies. For the third Renyi entropy, we argue that phase transitions in the membrane tension as a function of velocity are needed to ensure that physical constraints on the membrane tension are satisfied. Overall, this structure provides an understanding of entanglement dynamics in terms of a universal set of gapped low-lying modes, which may also apply to systems with time-independent Hamiltonians.

Figures (19)

• Figure 1: Left: Example of a candidate line appearing in the minimization of \ref{['membrane_form']}. Right: A cartoon of the membrane tension function. The first two constraints in \ref{['const']} are equivalent to the fact the ${{\mathcal{E}}}(v)$ is tangent to the $y=v$ line at some $v=v_B$.
• Figure 2: $\Tr[\rho_A(t)^n]$ can be represented as a Lorentzian path integral on $2n$ copies of the theory, with $n$ forward and $n$ backward evolutions. The initial conditions of the path integral are determined by $\rho_0$, and the final conditions are determined by the pattern of traces in $\Tr[\rho_A(t)^n]$.
• Figure 3: We consider a family of random time-dependent Hamiltonians, where individual terms have a spatially local structure.
• Figure 4: Left: Example of a domain wall final state relevant for the evolution of the $n$-th Renyi entropy for region $R$ in two spatial dimensions. Right: We show the pattern of entanglement between different copies in the states $\ket{e}$ and $\ket{\eta}$ for $n=3$.
• Figure 5: The red data points show the momentum-resolved low-energy spectra for $P_4$ in \ref{['aleft_def_mt']} or \ref{['eq:PEspin_mt']} for $L = 14$ for (a) $q = 2$, (b) $q = 3$, and (c) $q = 4$. These energies are found using exact diagonalization with the twisted boundary conditions described in Appendix \ref{['app:num']}. The ground states are at $E = 0$ and the low-energy spectrum is almost unchanged between $L=12$ and $L=14$. These are compared to the blue curves, which are the dispersion relations obtained from the minimization of the expectation value of $P_4$ in the states \ref{['phikdef']} for $\Delta=4$, $L=40$ with open boundary conditions.