Statistical complexity and the road to equilibrium in many-body chaotic quantum systems

Manuel H. Muñoz-Arias

Phys. Rev. E 106, 044103 – Published 3 October 2022

Abstract

In this work we revisit the problem of equilibration in isolated many-body interacting quantum systems. We pay particular attention to quantum chaotic Hamiltonians, and rather than focusing on the properties of the asymptotic states and how they adhere to the predictions of the Eigenstate Thermalization Hypothesis, we focus on the equilibration process itself, i.e., the road to equilibrium. Along the road to equilibrium the diagonal ensembles obey an emergent form of the second law of thermodynamics and we provide an information theoretic proof of this fact. With this proof at hand we show that the road to equilibrium is nothing but a hierarchy in time of diagonal ensembles. Furthermore, introducing the notions of statistical complexity and the entropy-complexity plane, we investigate the uniqueness of the road to equilibrium in a generic many-body system by comparing its trajectories in the entropy-complexity plane to those generated by a random Hamiltonian. Finally, by treating the random Hamiltonian as a perturbation we analyzed the stability of entropy-complexity trajectories associated with the road to equilibrium for a chaotic Hamiltonian and different types of initial states.

Received 25 May 2022
Revised 12 August 2022
Accepted 14 September 2022

DOI:https://doi.org/10.1103/PhysRevE.106.044103

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics Quantum chaos Quantum quench Quantum statistical mechanics

Condensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsQuantum Information, Science & Technology

Authors & Affiliations

Manuel H. Muñoz-Arias ^*

Center for Quantum Information and Control, CQuIC, Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA

^*mhmunoz@unm.edu

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Vol. 106, Iss. 4 — October 2022

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Images

Figure 1
(a) Long-time average E-C coordinates, $(\bar{H}, \bar{C})$ , of the diagonal ensemble in the Fock basis, for all Fock states as initial states, quenched under ${\hat{H}}_{BH}$ in Eq. (17). We work with a unit filled lattice at $N = L = 8$ and $U = 1.23$ , $J = 2 U$ . (b, c) Dynamics of the diagonal ensemble in the Fock basis shown as trajectories in the entropy-complexity plane. Time flows from $H = 0$ to $H = 1$ , following the monotonicity of the diagonal entropy. In both cases we show the trajectories of some representative initial Fock states, belonging to the states colored blue and green in panel (a), respectively. In all three subplots the red star shows the coordinates of distributions evolved under random unitaries sampled from the circular orthogonal ensemble.
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Figure 2
(a) Energy density of the quench BH Hamiltonian with 8 atoms in 8 lattice sites, and $J = 2 U$ and $U = 1.23$ . The energy density has a Gaussian shape, characteristic of quantum chaotic systems [75]. The bars represent the values of mean energy for all the states in the Fock basis. (b) Inverse participation ratio of the Fock states for the 8 in 8 BH system under study. The IPR is measured on the eigenbasis of ${\hat{H}}_{BH}$ . The dashed red line shows the IPR value averaged over the COE. (c–e) Width in energy of all the Fock states whose energy is shown in panel (a). In all the subplots we use the same color code as in Fig. 1.
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Figure 3
Trajectories in the E-C plane of the diagonal ensemble in the Fock basis, for a single initial Fock state evolved under the Hamiltonian in Eq. (25), for each of the three sets identified in Fig. 1, blue (a), green (b), and black (c). In all panels, the continuous red line corresponds to the unperturbed E-C trajectory, the dashed line to the E-C trajectory obtained evolving under a random Hamiltonian, and the yellow star the COE coordinate. The continuous, blue, green and black lines show 40 different trajectories resulting from 40 different realization of the perturbed evolution, each with a different random Hamiltonian. For the displayed results we use $ε = 0.5$ for the perturbation strength, other parameters are as in Fig. 1.
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Figure 4
(a) Euclidean distance between the diagonal ensemble of Fock states, in the basis of the quench Hamiltonian, to their diagonal ensemble after the action of a random unitary. The normalization factor is given by the distance between the E-C coordinate of ${ρ |}_{diag}^{(t_{0})} = {| c_{l} (0) |}^{2}$ and the E-C coordinate of the COE. Each point is the result of averaging over ten initial Fock states in each of the three sets (blue, green, and black). The distance to the COE coordinate for each of those states is the result of averaging over 100 different trajectories generated by 100 random Hamiltonians sampled from the GOE. The inset shows the $d (ε)$ for all the BH systems considered; the solid red line is the finite size scaling curve giving the infinite size limit. The inset and the main subplot share the $x$ axis. (b–d) Characteristic exponent $η$ of the sensitivity to perturbations as a function of the inverse dimension $1 / D_{BH}$ . In all figures the solid line represents the points in the regime $ε < 0.1$ and the dashed lines the points in the regime $ε > 0.1$ . The point at $1 / D_{BH} = 0$ is obtained from the finite size scaling.
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Figure 5
Equilibrium coordinates in the entropy-complexity plane for all the states in the Fock basis, after evolving under a Hamiltonian quench. The shown coordinates in panels (a), (b), and (c) correspond to B-H systems with filling factors of $N / L = 7 / 7, 8 / 7, 7 / 8$ , respectively. For all the studied BH systems we observe the separation, in the E-C plane, of the Fock basis into the three sets identified in Fig. 1.
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Figure 6
Inverse participation ratios of Fock basis states, measured in the basis of the quench Hamiltonian, ${\hat{H}}_{BH}$ . Results are for the BH systems with $N / L = 7 / 7, 8 / 7, 7 / 8$ in panels (a), (b), and (c), respectively. We have ordered the Fock basis so that states in each of the three sets, blue, green and black will appear together. The dashed red line shows the IPR value average over the COE.
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