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Transition between dissipatively stabilized helical states

Simon Essink, Wolff, Stefan, Gunter Markus Schutz, Corinna Kollath, Vladislav Popkov

Published Aug 17, 2020• 3 citations
OpenAlex

Abstract

We analyze a XXZ spin-1/2 chain which is driven dissipatively at its boundaries. The dissipative driving is modeled by Lindblad jump operators which only act on both boundary spins. In the limit of large dissipation, we find that the boundary spins are pinned to a certain value and at special values of the interaction anisotropy, the steady states are formed by a rank-2 mixture of helical states with opposite winding numbers. Contrarily to previous stabilizations of topological states, these helical states are not protected by a gap in the spectrum of the Lindbladian. By changing the anisotropy, the transition between these steady states takes place via mixed states of higher rank. In particular, crossing the value of zero anisotropy a totally mixed state is found as the steady state. The transition between the different winding numbers via mixed states can be seen in light of the transitions between different topological states in dissipatively driven systems. The results are obtained by developing a perturbation theory in the inverse dissipative coupling strength and using the numerical exact diagonalization and matrix product state methods.Received 9 October 2019Accepted 12 March 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.022007Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCold gases in optical latticesDecoherence in quantum gasesQuantum Zeno dynamicsTopological phases of matterPhysical SystemsQuantum spin modelsTechniquesMaster equationGeneral PhysicsAtomic, Molecular & Optical

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Transition between dissipatively stabilized helical states

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We investigate the non-equilibrium topology of the steady state of a periodically driven, dissipative Su-Schrieffer-Heeger (SSH) chain using the ensemble geometric phase (EGP) $φ_{\mathrm{EGP}}$, a generalisation of the Zak phase to mixed states. The coupling to thermal reservoirs is treated microscopically within Floquet-Born-Markov theory without the rotating-wave approximation. We classify the steady state rather than the generator of the dynamics, using its purity spectrum as the analogue of a band structure. We show that, within the EGP framework, the $\mathbb{Z}\times\mathbb{Z}$ structure of isolated chiral-symmetric Floquet SSH systems generally does not carry over to the steady state. Instead, the steady state carries a single invariant whose type is set by an underlying symmetry. When this symmetry holds, the invariant is $\mathbb{Z}_2$-valued, with quantised $φ_{\mathrm{EGP}}=\pmπ/2$ at all times and finite temperatures. The EGP can switch within a driving period, tracking the presence of edge modes in the open-boundary purity spectrum and thereby establishing a bulk-boundary correspondence. The fraction of each period spent in the nontrivial regime varies continuously with the driving frequency, and the switching disappears if the dependence of the dissipator on the drive is neglected. When the symmetry is broken, $φ_{\mathrm{EGP}}$ is no longer quantised, but its winding over a drive period $T$ defines an invariant $ν_T \in \mathbb Z$, with no purity-spectrum bulk-boundary correspondence. We propose that its changes signal temperature- and frequency-driven topological phase transitions. Our formalism extends to general quadratic fermionic systems with linear bath couplings.

cond-mat.mes-hallCross-listedarXiv:2604.25248•Apr 29, 2026•0 citations
We analyze a XXZ spin-1/2 chain which is driven dissipatively at its boundaries. The dissipative driving is modeled by Lindblad jump operators which only act on both boundary spins. In the limit of large dissipation, we find that the boundary spins are pinned to a certain value and at special values of the interaction anisotropy, the steady states are formed by a rank-2 mixture of helical states with opposite winding numbers. Contrarily to previous stabilizations of topological states, these helical states are not protected by a gap in the spectrum of the Lindbladian. By changing the anisotropy, the transition between these steady states takes place via mixed states of higher rank. In particular, crossing the value of zero anisotropy a totally mixed state is found as the steady state. The transition between the different winding numbers via mixed states can be seen in light of the transitions between different topological states in dissipatively driven systems. The results are obtained by developing a perturbation theory in the inverse dissipative coupling strength and using the numerical exact diagonalization and matrix product state methods.Received 9 October 2019Accepted 12 March 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.022007Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCold gases in optical latticesDecoherence in quantum gasesQuantum Zeno dynamicsTopological phases of matterPhysical SystemsQuantum spin modelsTechniquesMaster equationGeneral PhysicsAtomic, Molecular & Optical
We analyze a X X Z spin-1/2 chain which is driven dissipatively at its boundaries.The dissipative driving is modeled by Lindblad jump operators which only act on both boundary spins.In the limit of large dissipation, we find that the boundary spins are pinned to a certain value and at special values of the interaction anisotropy, the steady states are formed by a rank-2 mixture of helical states with opposite winding numbers.Contrarily to previous stabilizations of topological states, these helical states are not protected by a gap in the spectrum of the Lindbladian.By changing the anisotropy, the transition between these steady states takes place via mixed states of higher rank.In particular, crossing the value of zero anisotropy a totally mixed state is found as the steady state.The transition between the different winding numbers via mixed states can be seen in light of the transitions between different topological states in dissipatively driven systems.The results are obtained by developing a perturbation theory in the inverse dissipative coupling strength and using the numerical exact diagonalization and matrix product state methods.
Abstract We investigate the non-equilibrium steady state (NESS) in an open quantum XXZ chain attached at the ends to polarization baths with unequal polarizations. Using the general theory developed in Popkov (2017 Phys. Rev . A 95 052131), we show that in the critical XXZ | Δ | < 1 easy plane case, the steady current in large systems under strong driving shows resonance-like behaviour, by an infinitesimal change of the spin chain anisotropy or other parameters. Alternatively, by fine tuning the system parameters and varying the boundary dissipation strength, we observe a change of the NESS current from diffusive (of order 1 / N , for small dissipation strength) to ballistic regime (of order 1, for large dissipation strength). This drastic change results from an accompanying structural change of the NESS, which becomes a pure spin-helix state characterized by a winding number which is proportional to the system size. We calculate the critical dissipation strength needed to observe this surprising effect.
We show that several models of interacting XXZ spin chains subject to boundary driving and dissipation possess a subtle kind of time-reversal symmetry, making their steady states exactly solvable. We focus on a model with a coherent boundary drive, showing that it exhibits a unique continuous dissipative phase transition as a function of the boundary drive amplitude. This transition has no analog in the bulk closed system or in incoherently driven models. We also show the steady-state magnetization exhibits a surprising fractal dependence on interaction strength, something previously associated with less easily measured infinite-temperature transport quantities (the Drude weight). Our exact solution also directly yields driven-dissipative double-chain models that have pure, entangled steady states that are current carrying.
We investigate dissipation-driven topological phase transitions in one-dimensional quantum open systems governed by the Lindblad equation with linear dissipation operators, which ensure the density matrix retains its Gaussian form throughout the dynamics. By employing the modular Hamiltonian framework, we rigorously demonstrate that the Z2\rm{Z}_2Z2​ topological invariant characterizing steady states in one-dimensional class D systems is exclusively dependent on the dissipation operators, rather than the system Hamiltonian. Through a sudden quench protocol where the system evolves from the steady state of one Lindbladian to another, we reveal that topological transitions can occur at analytically predictable critical times, even when the initial and final steady states share identical topological indices. These transitions are shown, both analytically and numerically, to depend solely on dissipation parameters. Entanglement spectrum analysis demonstrates bulk-edge correspondence in non-equilibrium density matrices via coexisting single-particle gap closures (periodic boundaries) and topologically protected zero modes (open boundaries), directly underpinning the detection of dissipation-induced topology in quantum simulators.
The generation of entanglement in mixed states is relevant to quantum systems coupled to an environment. The dissipative and mixing properties of the environment are unavoidable in physical platforms for quantum simulation and information processing, where entanglement can be a vital resource. In this work, we explore entanglement and heterogeneity in random Lindbladian dynamics describing open quantum systems. We propose a model of a one-dimensional chain of noninteracting, spinless fermions coupled to a local ensemble of baths. The jump operator mediating the interaction with the bath linked to each site has a power-law tail with an exponent p . We show that the system undergoes volume-to-area law entanglement phase transition in the mixed steady state by tuning p which remains stable in the presence of coherent hopping. Unlike the entanglement transition in the pure-state quantum trajectories of open systems, this transition is exhibited by the averaged steady-state density matrix of the Lindbladian. The steady state in the area-law phase is characterized by a spatial heterogeneity in local population imbalance, while the jump operators exhibit a constant participation ratio of the sites they affect. Our work provides a theoretical description of an entanglement transition realized in long-ranged open quantum systems and provides an avenue to stabilize quantum correlations in mixed states. Published by the American Physical Society 2024
We investigate the non-equilibrium topology of the steady state of a periodically driven, dissipative Su-Schrieffer-Heeger (SSH) chain using the ensemble geometric phase (EGP) φEGPφ_{\mathrm{EGP}}φEGP​, a generalisation of the Zak phase to mixed states. The coupling to thermal reservoirs is treated microscopically within Floquet-Born-Markov theory without the rotating-wave approximation. We classify the steady state rather than the generator of the dynamics, using its purity spectrum as the analogue of a band structure. We show that, within the EGP framework, the Z×Z\mathbb{Z}\times\mathbb{Z}Z×Z structure of isolated chiral-symmetric Floquet SSH systems generally does not carry over to the steady state. Instead, the steady state carries a single invariant whose type is set by an underlying symmetry. When this symmetry holds, the invariant is Z2\mathbb{Z}_2Z2​-valued, with quantised φEGP=±π/2φ_{\mathrm{EGP}}=\pmπ/2φEGP​=±π/2 at all times and finite temperatures. The EGP can switch within a driving period, tracking the presence of edge modes in the open-boundary purity spectrum and thereby establishing a bulk-boundary correspondence. The fraction of each period spent in the nontrivial regime varies continuously with the driving frequency, and the switching disappears if the dependence of the dissipator on the drive is neglected. When the symmetry is broken, φEGPφ_{\mathrm{EGP}}φEGP​ is no longer quantised, but its winding over a drive period TTT defines an invariant νT∈Zν_T \in \mathbb ZνT​∈Z, with no purity-spectrum bulk-boundary correspondence. We propose that its changes signal temperature- and frequency-driven topological phase transitions. Our formalism extends to general quadratic fermionic systems with linear bath couplings.