In this work with Mathias Scheurer, we study a quantum spin liquid subject to dissipation in a manner which renders the full non-unitary dynamics exactly solvable.

Understanding the effects of environmental couplings on quantum many-body systems is a difficult problem. As our quantum state generically will not remain pure under time evolution, one must instead talk about the dynamics of the reduced density matrix describing our system, obtained by tracing out environmental degrees of freedom. A generic Markovian time evolution of this reduced density matrix that respects the unitarity of the underlying (system plus environment) dynamics is given by the Lindblad equation. This Lindblad equation has been extensively studied in the context of atomic physics, usually focusing on the dynamics of a single atom; however, applications to strongly-interacting many-body systems are rare, as many of the powerful tools we have developed for studying closed systems do not immediately transfer over to the Linbladian framework.

In order to use some of the tricks we have from closed systems, it is helpful to think of the Lindbladian superoperator as an ordinary non-Hermitian operator acting on a doubled Hilbert space, i.e. the Hilbert space of operators. In this language, density matrices are states whose time evolution is generated by the Lindbladian operator, much like a Hamitonian generates the time evolution of a pure state. This "doubling" of the Hilbert space has a useful pictoral representation for lattice models with a notion of locality. If our original Hilbert space is a 1D chain of qubits, our doubled Hilbert space is a 1D ladder. A 2D lattice becomes a bilayer model, and so on.

This framing is helpful in constructing exactly solvable Lindbladians. If we can find an ordinary Hamiltonian on a bilayer lattice that is exactly solvable through some clever tricks, there's a good chance that the same tricks will transfer over to an appropriately-defined Lindbladian on a 2D system! This is the approach we take in our paper. Specifically, we construct a model for a quantum spin liquid on a bilayer square lattice, which becomes exactly solvable through a parton construction via similar techniques as the Kitaev honeycomb model. By making some of the couplings non-Hermitian, we can reformulate this as the Lindbladian of a square lattice quantum spin model. Similar tricks have been used prior in a 1D ladder.

This construction means that we can in principle recover the full Lindbladian spectrum. The imaginary energy gap in the Lindbladian spectrum dictates the rate of equilibration to the steady-state solution, and this can be calculated exactly. However, there is clearly more physics going on than just this. In particular, our Lindbladian spectrum admits a natural quasiparticle interpretation in terms of Majorana fermions and \(\mathbb{Z}_2\) gauge fields, just like the Kitaev honeycomb model. One of the main contributions of our paper is developing a physically intuitive picture that connects the existence of quasiparticle excitations in this "doubled" Hilbert space with the thermalization of different classes of operators. Each quasiparticle has its own energy gap, which can be associated with the thermalization timescale of a distinct class of observables. In particular, a parametric separation of quasiparticle energy gaps likewise implies a separation of thermalization timescales. As exaclty-solvable Lindbladians tend to be a consequence of such a quasiparticle description, we believe that this physical picture will prove to be quite fruitful. In our model, the fractionalized nature of our quasiparticle excitations leads to some of the classes of operators being "string-like" and having anomalously long thermalization timescales relative to simple single-site operators.