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arXiv:2007.07754 · Physical Review A

Lie-algebraic approach to one-dimensional translationally invariant free-fermionic dissipative systems

Lieuwe Bakker, V. I. Yashin, Denis V. Kurlov, Aleksey K. Fedorov, Vladimir Gritsev

Published Nov 19, 2020• 9 citations
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Abstract

We study dissipative translationally invariant free-fermionic theories with quadratic Liouvillians. Using a Lie-algebraic approach, we solve the Lindblad equation and find the density matrix at all times for arbitrary time dependence of the Liouvillian. We then investigate the Liouvillian spectral properties and derive a generic criterion for the closure of the dissipative gap, which is believed to be linked with nonequilibrium dissipative phase transitions. We illustrate our findings with a few exotic examples. Particularly, we show the presence of gapless modes with a linear spectrum for fermions with long-range hopping, which might be related to nonunitary conformal field theories. The predicted effects can be probed in experiments with ultracold atomic and quantum-optical systems using currently available experimental facilities.

DOI
10.1103/physreva.102.052220

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We study dissipative translationally invariant free-fermionic theories with quadratic Liouvillians. Using a Lie-algebraic approach, we solve the Lindblad equation and find the density matrix at all times for arbitrary time dependence of the Liouvillian. We then investigate the Liouvillian spectral properties and derive a generic criterion for the closure of the dissipative gap, which is believed to be linked with nonequilibrium dissipative phase transitions. We illustrate our findings with a few exotic examples. Particularly, we show the presence of gapless modes with a linear spectrum for fermions with long-range hopping, which might be related to nonunitary conformal field theories. The predicted effects can be probed in experiments with ultracold atomic and quantum-optical systems using currently available experimental facilities.
We study dissipative translationally-invariant free fermionic theories with quadratic Liouvillians. Using a Lie-algebraic approach we solve the Lindblad equation and find explicitly the density matrix at all times for arbitrary time dependence of the Liouvillian. We then investigate the Liouvillian spectral properties and derive a generic criterion for the closure of the dissipative gap, which is believed to be linked with nonequilibrium dissipative phase transitions. We illustrate our findings with a few exotic examples. Particularly, we show the presence of gapless modes with a linear spectrum for fermions with a long-range hopping, which might be related to non-unitary conformal field theories. The predicted effects can be probed in experiments with ultracold atomic and quantum-optical systems using currently available experimental facilities.
We study spectral and steady-state properties of generic Markovian dissipative systems described by quadratic fermionic Liouvillian operators of the Lindblad form. The Hamiltonian dynamics is modeled by a generic random quadratic operator, i.e., as a featureless superconductor of class D, whereas the Markovian dissipation is described by M M random linear jump operators. By varying the dissipation strength and the ratio of dissipative channels per fermion, m=M/(2N_F) m = M / ( 2 N F ) , we find two distinct phases where the support of the single-particle spectrum has one or two connected components. In the strongly dissipative regime, this transition occurs for m=1/2 m = 1 / 2 and is concomitant with a qualitative change in both the steady-state and the spectral gap that rules the large-time dynamics. Above this threshold, the spectral gap and the steady-state purity qualitatively agree with the fully generic (i.e., non-quadratic) case studied recently. Below m=1/2 m = 1 / 2 , the spectral gap closes in the thermodynamic limit and the steady-state decouples into an ergodic and a nonergodic sector yielding a non-monotonic steady-state purity as a function of the dissipation strength. Our results show that some of the universal features previously observed for fully random Liouvillians are generic for a sufficiently large number of jump operators. On the other hand, if the number of dissipation channels is decreased the system can exhibit nonergodic features, rendering it possible to suppress dissipation in protected subspaces even in the presence of strong system-environment coupling.
Abstract Markovian open quantum systems are described by the Lindblad master equation ∂ t ρ = L ( ρ ) , where ρ denotes the system’s density operator and L the Liouville super-operator, which is also known as the Liouvillian. For systems with a finite-dimensional Hilbert space, it is a fundamental property of the Liouvillian that the real parts of all its eigenvalues are non-positive. Analogously, for infinite-dimensional Hilbert spaces, the Liouvillian as a map on trace-class operators only has spectral values with non-positive real parts. The usual arguments for these properties are indirect, using that L generates a quantum channel and that quantum channels are contractive. We provide a direct algebraic proof based on the Lindblad form of Liouvillians. Subtleties for infinite-dimensional systems are highlighted. For example, unbounded operators can then be eigenvectors of the adjoint Liouvillian with positive eigenvalues, corresponding to a physical instability of the system. As an illustrative application of the algebraic approach, we prove that a strict minimum can be imposed on the dissipative gap of any unital Markovian spin- s or fermionic many-particle system by adding simple local dissipators. The added dissipators comprise Pauli or generalized Pauli Lindblad operators for spin systems and Majorana Lindblad operators for fermionic systems.
Markovian open quantum systems are described by the Lindblad master equation ∂tρ=L(ρ)\partial_tρ=\mathcal{L}(ρ)∂t​ρ=L(ρ), where ρρρ denotes the system's density operator and L\mathcal{L}L the Liouville super-operator, which is also known as the Liouvillian. For systems with a finite-dimensional Hilbert space, it is a fundamental property of the Liouvillian that the real parts of all its eigenvalues are non-positive. Analogously, for infinite-dimensional Hilbert spaces, the Liouvillian as a map on trace-class operators only has spectral values with non-positive real parts. The usual arguments for these properties are indirect, using that L\mathcal{L}L generates a quantum channel and that quantum channels are contractive. We provide a direct algebraic proof based on the Lindblad form of Liouvillians. Subtleties for infinite-dimensional systems are highlighted. For example, unbounded operators can then be eigenvectors of the adjoint Liouvillian with positive eigenvalues, corresponding to a physical instability of the system. As an illustrative application of the algebraic approach, we prove that a strict minimum can be imposed on the dissipative gap of any Markovian spin-sss or fermionic many-particle system with normal Lindblad operators by adding simple local dissipators. The added dissipators comprise Pauli or generalized Pauli Lindblad operators for spin systems and Majorana Lindblad operators for fermionic systems.
A state of an open quantum system is described by a density matrix, whose dynamics is governed by a Liouvillian superoperator. Within a general framework, we explore fundamental properties of both first-order dissipative phase transitions and second-order dissipative phase transitions associated with a symmetry breaking. In the critical region, we determine the general form of the steady-state density matrix and of the Liouvillian eigenmatrix whose eigenvalue defines the Liouvillian spectral gap. We illustrate our exact results by studying some paradigmatic quantum optical models exhibiting critical behavior.
We study the steady states of translation-invariant open quantum many-body systems governed by Lindblad master equations, where the Hamiltonian is quadratic in the ladder operators, and the Lindblad operators are either linear or quadratic and Hermitian. These systems are called quasifree and quadratic, respectively. We find that steady states of one-dimensional systems with finite-range interactions necessarily have exponentially decaying Green's functions. For the quasifree case without quadratic Lindblad operators, we show that fermionic systems with finite-range interactions are noncritical for any number of spatial dimensions and provide bounds on the correlation lengths. Quasifree bosonic systems can be critical in D>1 dimensions. Last, we address the question of phase transitions in quadratic systems and find that, without symmetry constraints beyond invariance under single-particle basis and particle-hole transformations, all gapped Liouvillians belong to the same phase.