[Paper Review] Symmetry-Preserving Quadratic Lindbladian and Dissipation Driven Topological Transitions in Gaussian States

Liang Mao, Fan Yang+1 more|arXiv (Cornell University)|Jan 11, 2023

Advanced Thermodynamics and Statistical MechanicsPhysics and Astronomy46 references3 citations

TL;DR

This paper proposes that the topology of a density matrix—characterized by the topological invariant of its modular Hamiltonian—can undergo a dynamical transition during Lindbladian evolution in open quantum systems. For fermionic Gaussian states, symmetry-preserving Lindbladians preserve the modular Hamiltonian's symmetry class, enabling topological transitions when the initial state is nontrivial; this is demonstrated in AIII and DIII symmetry classes with signatures in the density matrix eigenvalue spectrum at transition points.

ABSTRACT

The dynamical evolution of an open quantum system can be governed by the Lindblad equation of the density matrix. In this paper, we propose to characterize the density matrix topology by the topological invariant of its modular Hamiltonian. Since the topological classification of such Hamiltonians depends on their symmetry classes, a primary issue we address is determining the requirement for the Lindbladian operators, under which the modular Hamiltonian can preserve its symmetry class during the dynamical evolution. We solve this problem for the fermionic Gaussian state and for the modular Hamiltonian being a quadratic operator of a set of fermionic operators. When these conditions are satisfied, along with a nontrivial topological classification of the symmetry class of the modular Hamiltonian, a topological transition can occur as time evolves. We present two examples of dissipation-driven topological transitions where the modular Hamiltonian lies in the AIII class with U(1) symmetry and the DIII class without U(1) symmetry. By a finite size scaling, we show that this density matrix topology transition occurs at a finite time. We also present the physical signature of this transition.

Motivation & Objective

• To investigate whether the topology of a density matrix can dynamically transition during Lindbladian evolution in open quantum systems.
• To identify conditions under which Lindbladian operators preserve the symmetry class of the modular Hamiltonian in Gaussian states.
• To demonstrate that topological transitions can occur even when the initial modular Hamiltonian is not related to the system Hamiltonian.
• To provide measurable signatures of such transitions through the eigenvalue spectrum of the density matrix.

Proposed method

• The modular Hamiltonian is defined via $ \hat{\rho} = e^{-\hat{K}} $, with $ \hat{K} $ being a quadratic fermionic operator.
• The topological classification of $ \hat{K} $ is analyzed within the Altland-Zirnbauer symmetry classes, focusing on AIII and DIII.
• Symmetry-preserving Lindbladians are derived by ensuring the Lindblad operators commute with the symmetry operations of $ \hat{K} $, using the condition $ \mathcal{S} \hat{L}_{\mu} \mathcal{S}^{-1} = \hat{L}_{\mu} $.
• The time evolution of $ \hat{K} $ is computed under the Lindblad equation, tracking changes in the topological invariant.
• Numerical simulations are performed for two models: one in the AIII class with U(1) symmetry and one in the DIII class without U(1) symmetry.
• The transition is diagnosed by observing the vanishing of the purity gap in the density matrix eigenvalues at critical times.

Experimental results

Research questions

• RQ1 Can the topology of the density matrix, as encoded in the modular Hamiltonian, undergo a dynamical transition during Lindbladian evolution in open quantum systems?
• RQ2 Under what conditions on the Lindblad operators is the symmetry class of the modular Hamiltonian preserved during time evolution?
• RQ3 What are the measurable signatures of a topological transition in the density matrix eigenvalue spectrum?
• RQ4 How does the topological invariant of the modular Hamiltonian evolve when the initial state is topologically nontrivial but the steady state is trivial?
• RQ5 Can such transitions be observed experimentally, and what physical observables can signal them?

Key findings

• A topological transition occurs in the modular Hamiltonian when the initial state is topologically nontrivial and the steady state is trivial, driven by a symmetry-preserving Lindbladian.
• In the AIII class example, the winding number changes from 1 to 0 at a critical time, with the purity gap vanishing as the modular Hamiltonian becomes gapless.
• In the DIII class example, the Fu-Kane $ Z_2 $ invariant transitions from -1 (nontrivial) to +1 (trivial), confirmed by the vanishing of the purity gap at the transition point.
• The eigenvalues of the density matrix show a clear signature at the transition: the largest eigenvalue $ \lambda_0 $ is no longer separated from the rest by a finite gap.
• The transition is robust under the condition that the Lindblad operators preserve the relevant symmetries (U(1), time-reversal, or chiral symmetry), ensuring the symmetry class of $ \hat{K} $ is maintained.
• The results suggest that the ensemble geometric phase could be a measurable probe of the topological invariant in the AIII class, providing a potential experimental signature.