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We study the influence of short-range quantum correlations and classical spatial correlations on the phase diagram of the dissipative XYZ model by using a Gutzwiller Monte Carlo method and a cluster Gutzwiller ansatz for the wave function. Considering lattices of finite size we confirm the emergence of a region with ferromagnetic correlations and two paramagnetic regions and show the emergence of a region with a buildup of ferromagnetic correlations completely missed by mean-field theory. The inclusion of short-range quantum correlations causes drastic alterations of the mean-field correlation functions but our results show that the inclusion of long-range quantum correlations or the use of more sophisticated methods is needed to quantitatively match the exact results. A study of the susceptibility tensor shows that reciprocity is broken, a feature not observed in closed quantum systems. Furthermore, increasing the magnetic field suppresses the magnetization; this is also in contrast with closed quantum systems.

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The standard approach of the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity is to introduce a self-consistent mean-field approximation, and a variational ansatz for the many-body ground state. The resulting mean-field Hamiltonian no longer commutes with the total number operator, and the variational search takes place in Fock space rather than in a Hilbert space of states with fixed number of particles. This is a disadvantage when studying small systems where the canonical ensemble predictions differ from the corresponding grand-canonical results. To remedy this, alternative approaches such as Richardson's method have been put forward. Here, we derive the exact many-body ground state of a model Hamiltonian corresponding to the deep-BCS or flat-band regime, without having to resort to Richardson's set of coupled nonlinear equations. This allows to write the exact many-body ground state in a way that makes the difference with the BCS variational wave function particularly clear. We show that the exact wave function consists of a superposition of many-pair states in such a way that the mean-field averaging corresponds to a summation over these many-pair states. This explains why many expectation values calculated with the BCS variational wave function give the same result as when calculated with the exact wave function, even though these wave functions are different. In the canonical (fixed-number) approach, pairing is investigated using the second-order reduced density matrix and calculating its largest eigenvalue. When interpreted as the order parameter of the superconducting state, this can be compared directly to the behavior of the mean-field gap. Finally, we show that a clear difference between the canonical approach and the BCS grand canonical estimates appears when evaluating pair condensate fluctuations as well as the pair entanglement entropy.

The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbladian evolution (spin, fermions, bosons, …), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other diagonalization techniques and retrieves the Liouvillian low-lying spectrum even for system sizes for which it would be impossible to perform exact diagonalization.