Yang-Baxter integrable Lindblad equations

Ziolkowska, Aleksandra A.; Eßler, Fabian H. L.

doi:10.21468/scipostphys.8.3.044

Cited by 60 publications

(65 citation statements)

References 74 publications

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“…For later use, we define "eigenvalues" and "eigenvectors" of the Liouvillian L which can be thought of as a non-Hermitian matrix in a doubled Hilbert space, Ket ⊗ Bra. With the following isomorphism, the density matrix is mapped to a vector in the doubled Hilbert space [72][73][74][75][76][77][78][79][80][81][82][83][84]:…”

Section: B Vectorized Density Matrices In the Doubled Hilbert Spacementioning

confidence: 99%

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Fate of fractional quantum Hall states in open quantum systems: Characterization of correlated topological states for the full Liouvillian

Yoshida

Kudo

Katsura

et al. 2020

Phys. Rev. Research

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Despite previous extensive analysis of open quantum systems described by the Lindblad equation, it is unclear whether correlated topological states, such as fractional quantum Hall states, are maintained even in the presence of the jump term. In this paper, we introduce the pseudospin Chern number of the Liouvillian which is computed by twisting the boundary conditions only for one of the subspaces of the doubled Hilbert space. The existence of such a topological invariant elucidates that the topological properties remain unchanged even in the presence of the jump term, which does not close the gap of the effective non-Hermitian Hamiltonian (obtained by neglecting the jump term). In other words, the topological properties are encoded into an effective non-Hermitian Hamiltonian rather than the full Liouvillian. This is particularly useful when the jump term can be written as a strictly block-upper (-lower) triangular matrix in the doubled Hilbert space, in which case the presence or absence of the jump term does not affect the spectrum of the Liouvillian. With the pseudospin Chern number, we address the characterization of fractional quantum Hall states with two-body loss but without gain, elucidating that the topology of the non-Hermitian fractional quantum Hall states is preserved even in the presence of the jump term. This numerical result also supports the use of the non-Hermitian Hamiltonian which significantly reduces the numerical cost. Similar topological invariants can be extended to treat correlated topological states for other spatial dimensions and symmetry (e.g., one-dimensional open quantum systems with inversion symmetry), indicating the high versatility of our approach.

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Section: B Vectorized Density Matrices In the Doubled Hilbert Spacementioning

confidence: 99%

“…The number of unit cells is L. We imposed the periodic boundary condition c † Lα = c † 0α . The above open quantum system is mapped to the closed system, which has been discussed for the specific choice of t (t = t) [79,83]. The Liouvillian reads…”

Section: A Mapping the Open Quantum System To A Closed Systemmentioning

confidence: 99%

Fate of fractional quantum Hall states in open quantum systems: Characterization of correlated topological states for the full Liouvillian

Yoshida

Kudo

Katsura

et al. 2020

Phys. Rev. Research

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“…The Lindblad evolution, fully Markovian, is a priori simpler, although of course not equivalent to the fermionic reservoirs, and as such has been widely used coupled with a Liouvillian formalism [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48], to tackle out-of-equilibrium issues. Beyond experimental interest, for which the Lindblad coupling is the proper microscopic description, on the theory side, this approach has allowed to unveil nontrivial properties of highly excited and correlated systems: Integrable structures, traditionally restrained to closed systems in the quantum realm [49][50][51][52][53][54][55][56]; the existence of ballistics spin transport [36,57,58] and anomalous diffusion [59][60][61] in the integrable XXZ model, thus allowing for the discovery of Kardar-Parisi-Zhang correlations [62,63] in the quantum realm [64][65][66][67]. Additionally, it has allowed to characterize the anomalous transport properties of disordered [68,69] and quasiperiodic [70] interacting systems, the persistence of ballistic transport in the presence of level repulsion induced by single impurities [...…”

Section: Introductionmentioning

confidence: 99%

Generic transport formula for a system driven by Markovian reservoirs

2020

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We present a generic, compact formula for the current flowing in interacting and noninteracting systems which are driven out of equilibrium by biased reservoirs described by Lindblad jump operators. We show that, in the limit of high temperature and chemical potential, our formula is equivalent to the well-known Meir-Wingreen formula, which describes the current flowing through a system connected to fermionic baths, therefore bridging the gap between the two formalisms. Our formulation gives a systematic way to address the transport properties of correlated systems strongly driven out of equilibrium. As an illustration, we provide explicit calculations of the current in three cases : (i) a single-site impurity, (ii) a free-fermionic chain, and (iii) a fermionic chain with loss and gain terms along the chain. In this last case, we find that the current across the system has the same behavior for loss or gain terms and depends on the loss/gain rate in a nonmonotonic way.

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“…A second direction concerns the relation with other physical observables : even though we have observed in Section 5.3 that the most natural local observables as well as the von Neumann entropy are insensitive to the presence of a cutoff, namely they do not develop a sharp jump as the trace-norm distance to equilibrium does at the mixing times t mix (L), it remains an intriguing question whether the cutoff phenomenon might transpire into other physical quantities. On a more technical level, it would of course be interesting to move on to interacting systems, for instance using the mapping of some interacting Liouvillians onto Bethe ansatz solvable quantum Hamiltonians [41][42][43][44]. The study of mixing in this framework however seems to be a challenging issue.…”

Section: Discussionmentioning

confidence: 99%

Mixing times and cutoffs in open quadratic fermionic systems

Vernier

2020

SciPost Phys.

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In classical probability theory, the term cutoff describes the property of some Markov chains to jump from (close to) their initial configuration to (close to) completely mixed in a very narrow window of time. We investigate how coherent quantum evolution affects the mixing properties in two fermionic quantum models (the ``gain/loss'' and ``topological'' models), whose time evolution is governed by a Lindblad equation quadratic in fermionic operators, allowing for a straightforward exact solution. We check that the cutoff phenomenon extends to the quantum case and examine how the mixing properties depend on the initial state. In the topological case, we further show how the mixing properties are affected by the presence of a long-lived edge zero mode when taking open boundary conditions.

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Yang-Baxter integrable Lindblad equations

Cited by 60 publications

References 74 publications

Fate of fractional quantum Hall states in open quantum systems: Characterization of correlated topological states for the full Liouvillian

Fate of fractional quantum Hall states in open quantum systems: Characterization of correlated topological states for the full Liouvillian

Generic transport formula for a system driven by Markovian reservoirs

Mixing times and cutoffs in open quadratic fermionic systems

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