Exact Liouvillian Spectrum of a One-Dimensional Dissipative Hubbard Model

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.

show abstract

“…allow to efficiently determine the spectrum of L without encountering these scaling problems are extremely useful [92,93].…”

Section: Liouvillian Spectrummentioning

confidence: 99%

Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems

Minganti

Huybrechts

2022

Quantum

View full text Add to dashboard Cite

show abstract

“…Accordingly, topological phases associated with the GKSL equation need to be considered in order to understand the topological phenomena of dissipative quantum systems. Nonequilibrium phenomena in dissipative many-body quantum systems have attracted attention recently [24][25][26][27][28][29][30]. In particular, the topological properties of steady states [31][32][33][34][35][36] and the spectra of Lindbladians [37][38][39][40][41][42][43][44][45] have been studied.…”

Section: Introductionmentioning

confidence: 99%

Topological phases protected by shifted sublattice symmetry in dissipative quantum systems

Kawasaki¹,

Mochizuki²,

Obuse³

2022

Preprint

View full text Add to dashboard Cite

Dissipative dynamics of quantum systems can be classified topologically based on the correspondence between the Lindbladian in the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation and the non-Hermitian Hamiltonian in the Schrödinger equation. While general non-Hermitian Hamiltonians are classified into 38 symmetry classes, the previous work shows that the Lindbladians are classified into 10 symmetry classes due to physical constraints. In this work, however, we unveil a topological classification of Lindbladians based on sublattice symmetry (SLS), which is not previously considered and can increase the number of symmetry classes for the Lindbladians. We introduce shifted SLS so that the Lindbladian can retain this symmetry and take on the same role of SLS for the topological classification. For verification, we construct a model of the dissipative quantum system retaining shifted SLS and confirm the presence of edge states protected by shifted SLS. Moreover, the relationship between the presence of shifted SLS protected edge states and dynamics of an observable quantity is also discussed.

show abstract

“…On the contrary, it has been revealed that dissipation can be used as an efficient tool for preparing novel states in open quantum systems. To date, a lot of theoretical studies have shown that dissipation drastically alters various aspects of quantum manybody physics [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In particular, the interplay between unitary many-body dynamics and nonunitary state evolution due to dissipation leads to unconventional phase transitions [19][20][21][22][23][24][25], quantum critical phenomena [26][27][28][29][30], and measurementinduced entanglement transitions [31][32][33][34][35][36][37][38][39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Universal properties of dissipative Tomonaga-Luttinger liquids: Case study of a non-Hermitian XXZ spin chain

Yamamoto,

Nakagawa,

Tezuka

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We demonstrate the universal properties of dissipative Tomonaga-Luttinger (TL) liquids by calculating correlation functions and performing finite-size scaling analysis of a non-Hermitian XXZ spin chain as a prototypical model in one-dimensional open quantum many-body systems. Our analytic calculation is based on an effective field theory with bosonization, a finite-size scaling approach in conformal field theory, and the Bethe-ansatz solution. We also perform numerical calculations based on the density-matrix renormalization group analysis generalized to non-Hermitian systems (NH-DMRG). We uncover that the model in the massless regime with weak dissipation belongs to the universality class characterized by the complex-valued TL parameter, which is related to the complex generalization of the c = 1 conformal field theory. As the dissipation strength increases, the values of the TL parameter obtained by the NH-DMRG and the Bethe-ansatz analysis start to deviate from each other, indicating that the model becomes massive for strong dissipation. Our results can be tested with ultracold atoms by introducing two-body loss to the two-component Bose-Hubbard system.

show abstract

Exact Liouvillian Spectrum of a One-Dimensional Dissipative Hubbard Model

Cited by 86 publications

References 69 publications

Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems

Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems

Topological phases protected by shifted sublattice symmetry in dissipative quantum systems

Universal properties of dissipative Tomonaga-Luttinger liquids: Case study of a non-Hermitian XXZ spin chain

Contact Info

Product

Resources

About