We investigate the dissipative dynamics yielded by the Lindblad equation within the coexistence region around a first-order phase transition. In particular, we consider an exactly solvable, fully connected quantum Ising model with $n$-spin exchange ($n>2$)—the prototype of quantum first-order phase transitions—and several variants of the Lindblad equations. We show that physically sound results, including exotic nonequilibrium phenomena such as the Mpemba effect, can be obtained only when the Lindblad equation involves jump operators defined for each of the coexisting phases, whether stable or metastable.

We generalize the classical thermal Mpemba effect (where an initially hot system relaxes faster to the final equilibrium state than a cold one) to open quantum systems coupled to several reservoirs. We show that, in general, two different types of quantum Mpemba effects are possible. They may be distinguished by quantum state tomography. However, the existence of a quantum Mpemba effect (without determining the type) can already be established by measuring simpler observables such as currents or energies. We illustrate our general results for the experimentally feasible case of an interacting two-site Kitaev model coupled to two metallic leads.

We use the Lindblad equation method to investigate the onset of a mobility edge and the topological phase transition in the disordered SSH chain connected to two external baths in the large bias limit. From the scaling properties of the nonequilibrium stationary current flowing across the system, we recover the localization/delocalization in the disordered chain. To probe the topological phase transition in the presence of disorder, we use the even-odd differential occupancy as a means to discriminate topologically trivial from topologically nontrival phases in the out-of-equilibrium system. Eventually, we argue how to generalize our method to other systems undergoing a topological phase transition in the presence of disorder.

We study the thermal transport through a Majorana island connected to multiple external quantum wires. In the presence of a large charging energy, we find that the Wiedemann-Franz law is nontrivially violated at low temperature, contrarily to what happens for the overscreened Kondo effect and for nontopological junctions. For three wires, we find that the Lorenz ratio is rescaled by a universal factor $2/3$ and we show that this behavior is due to the presence of localized Majorana modes on the island.

We discuss the emergence of zero-energy Majorana modes in a disordered finite-length $p$-wave one-dimensional superconducting ring, pierced by a magnetic flux $\mathrm{\Phi }$ tuned at an appropriate value $\mathrm{\Phi }={\mathrm{\Phi }}_{*}$. In the absence of fermion parity conservation, we evidence the emergence of the Majorana modes by looking at the discontinuities in the persistent current $I\left[\mathrm{\Phi }\right]$ at $\mathrm{\Phi }={\mathrm{\Phi }}_{*}$. By monitoring the discontinuities in $I\left[\mathrm{\Phi }\right]$, we map out the region in parameter space characterized by the emergence of Majorana modes in the disordered ring.