Duality between open systems and closed bilayer systems: Thermofield double states as quantum many-body scars

Abstract

We establish a duality between open many-body systems governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation and satisfying the detailed balance condition on the one side, and closed bilayer systems with a self-adjoint Hamiltonian on the other side. Under this duality, the identity operator on the open system side maps to a quantum many-body scar of the dual Hamiltonian $\mathcal H$. This scar eigenstate has a form of a thermofield double state for a single-body conserved quantity entering the detailed balance conditions. A remarkable feature of this thermofield scar is a tunable single-layer entanglement entropy controlled by the reservoir temperature on the open system side. Further, we identify broad classes of many-body open systems with nontrivial explicit eigen operators $Q$ of the Lindbladian superoperator. The expectation values of the corresponding observables exhibit a simple exponential decay, $\langle Q\rangle_t=e^{-Γt} \langle Q \rangle_0$, irrespectively of the initial state. Under the above duality, these eigen operators give rise to additional (towers of) scars. Finally, we point out that more general superoperators (not necessarily of the GKSL form) can be mapped to self-adjoint Hamiltonians of bilayer systems harbouring scars, and provide an example thereof.

Figures (3)

• Figure 1: Illustration of the duality between an open system and a closed bilayer system. On the left an open system with a Hamiltonian $H$ and Lindblad operators $L_j$, $L_j^\dagger$ is schematically shown. The open system satisfies the detailed balance condition with the inverse temperature $\beta$, see eqs. \ref{['finiteTemperature']},\ref{['annihilationLike']}. On the right the dual closed system with doubled degrees of freedom is depicted. The Hamiltonian of the bilayer system is constructed from the Hamiltonian of the open system and the Lindblad operators, see eq. \ref{['finiteTemperatureHamiltonian']}.
• Figure 2: Entanglement of the thermofield scar \ref{['thermofield double state']} with respect to different bipartitions of a bilayer system. (a) For the bipartition that singles out one layer and traces out another one, the entanglement entropy equals the thermal entanglement entropy for the Hamiltonian $H_0$ at temperature $\beta$. (b) The entanglement entropy vanishes for a transverse bipartition that does not cut the interlayer bonds (here the underlying basis vectors are assumed to be of product form, see the text for details).
• Figure 3: Scars and scarred dynamics for the bilayer system \ref{['dual sigma pm example']} with $N=6$ spins in each layer. The Hamiltonian parameters are $g=1$, $\beta=0.7$, $\omega_{j\,j+1}=1$, $\omega_{j\,j+2}=0.5$ and $\omega_{ij}=0$ otherwise. (a) Rényi entropy $(-\log \mathop{\mathrm{\hbox{tr}}}\nolimits \rho_E^2)$ of eigenstates with respect to the transverse bipartition (see Fig. \ref{['fig 2']}(b)) in two equal parts ($\rho_E$ is the corresponding reduced density matrix of an eigenstate). Highlighted are the thermofield scar $|\mathbb{1} \rangle\rangle_\beta$ and the scars $|Q_n \rangle\rangle_\beta$, $n=1,2,\dots,6$. (b) Overlaps $|\langle E|\Psi_0\rangle|^2$ between the eigenstates and the product initial state $\Psi_0=|\downarrow\downarrow\dots\downarrow\rangle$ (all spins down in both layers). One can see that this initial state is expanded over the thermofield scar and the scars $|Q_n \rangle\rangle_\beta$. (c) Persistent oscillations of the squared total spin polarizations $\left(S^{x,y,z}\right)^2$ after initialization in the state $\Psi_0$.