Mixed-State Topology in One-Dimensional Open Floquet Systems

summary

The gist

The steady state of a periodically driven, dissipative Su-Schrieffer-Heeger chain can be characterized by a Hermitian purity spectrum, providing an analogue to band topology for mixed states.

In short

This work investigates topological properties of a periodically driven, dissipative Su-Schrieffer-Heeger chain in its steady state. By analyzing the Hermitian purity spectrum, researchers found that Floquet topology persists in this mixed state system. They established two invariants, $\phi_0$ EGP and $\Delta\phi_{\pi}$ EGP, which characterize a $Z \times Z$ topology, allowing for the detection of edge states in both the 0 and $\pi$ gaps.

Key concepts

Hermitian Purity Spectrum: This spectrum is used to describe the topological properties of a mixed quantum state. It consists of eigenvalues ($\lambda_k$) that range from -1 to +1, where values of +1 or -1 indicate pure states, and 0 indicates fully mixed Gaussian states. A gap in this spectrum is key for defining topological invariants.
Ensemble Geometric Phase (EGP): The EGP ($\phi_{EGP}$) is a phase calculated from the trace of the time-evolved reduced density matrix. It serves as a tool to characterize the topology of mixed states in driven systems. It is crucial for defining topological invariants that reveal underlying structure in the system's dynamics.
Topological Invariants ($\phi_0$ EGP, $\Delta\phi_{\pi}$ EGP): These are specific quantities derived from the EGP associated with the 0 and $\pi$ gaps of the quasienergy spectrum. They act as topological fingerprints that reveal a $Z \times Z$ classification, analogous to those found in isolated systems. These invariants help detect topological phase transitions.
Winding Number ($\nu_{\pi}$): The winding number is a second invariant derived from the change in the EGP over one driving period ($T$). It is defined as $\nu_{\pi} \equiv 1/2\pi \Delta\phi_{\pi}$ and counts how many times the phase encircles the origin. This quantity quantifies topological features related to the $\pi$ gap.

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This episode discusses

The paper

Mixed-State Topology in One-Dimensional Open Floquet Systems · Read on arXiv

School of Physics, University of Melbourne · Technische Universiät Berlin, Institut für Physik und Astronomie

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 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 Z times 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 Z 2-valued, with quantised ϕ EGP= plus or minusπ/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 EGP is no longer quantised, but its winding over a drive period T defines an invariant ν T in 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.

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Mixed-State Topology in One-Dimensional Open Floquet Systems".

Mira: Here's the gist of the paper. The steady state of a periodically driven, dissipative Su-Schrieffer-Heeger chain can be characterized by a Hermitian purity spectrum, providing an analogue to band topology for mixed states.

Kai: This work demonstrates that Floquet topology survives in this driven-dissipative Gaussian steady state and establishes Floquet topology as a robust concept beyond isolated zero-temperature systems Model and Framework The investigation focuses on a periodically driven, dissipative Su-Schrieffer-Heeger (SSH) chain using the ensemble geometric phase (EGP) ϕEGP The system is modeled as an open quantum system decomposed according to H(t) = HS(t) ⊗ 1R + 1S ⊗ HR + λXl Xl ⊗ Yl The steady state is obtained using the Floquet-Born-Markov theory, which allows for a microscopic description of the system in presence of thermal reservoirs For quadratic fermionic Hamiltonians HS(t) and linear bath coupling operators Xl, a mapping onto Majorana operators is performed to obtain the steady-state solution as a general Gaussian quadratic state The time evolution of the reduced density operator ϱS(t) is governed by the Born-Markov master equation Mixed State Topology Characterization The topology of mixed states is addressed by characterizing the steady state through its purity spectrum, which is defined as the eigenvalues λk of HC The purity eigenvalues lie in the interval λk ∈

−1, +1: , where states with λk = ±1 correspond to pure and states with λk = 0 correspond to fully mixed Gaussian states A finite purity gap ∆Gap in the purity spectrum is crucial for the definition of invariants like the EGP The Ensemble Geometric Phase (EGP) ϕEGP is defined as ϕEGP = I log Tr ϱ Tt Topological Invariants The paper identifies a pair of EGP-based topological invariants, (ϕ0 EGP, ∆ϕπ EGP), associated with the 0 and π gaps of the quasienergy spectrum These invariants reveal a structure consistent with a Z × Z classification known from isolated Floquet SSH systems The invariant ϕ0 EGP is defined as ϕ0 EGP = I logZ(t0) for ω > ωc, where t0 is an arbitrary time within a period The second invariant, ∆ϕπ EGP, is defined via the winding number νπ ≡ 1/2π ∆ϕπ EGP = 1/2π Z T0 ∂t ϕEGP(t) dt This winding number counts how many times Z π(t) encircles the origin in the complex plane as t traverses one driving period T Results and Interpretation The analysis of correlations between Majorana quasi-particles shows that for low- and intermediate driving frequencies ω, long-range correlations can be engineered through periodic modulation For high frequencies, the full correlation matrix shows a distinct (sign-)alternating pattern in proximity to the main diagonal, implying that short-range correlations are dominantly present The purity spectra show clear 0 and π band gaps and a very similar curving of the bands as seen in the imaginary part of the rapidity spectrum In Fig.

Mira: 8, for ω > ωc, ϕ0 EGP correctly predicts the existence of 0-energy edge states seen in the OBC spectra The second invariant ∆ϕπ EGP jumps for low frequencies but correctly predicts edge states in the π gap for intermediate to high frequencies The pair (ϕ0 EGP, νπ), which becomes well-defined in the presence of a finite purity gap, reveals a structure consistent with a Z × Z Floquet topology This provides an extension of Floquet topology into dissipative systems and can be used to detect topological phase transitions in the driven-dissipative SSH chain The findings indicate new ways to engineer and stabilise protected modes using dissipation, and, as a result of the periodic drive, enable richer dynamical phases via independently tunable 0- and π-invariants The gist The steady state of a periodically driven, dissipative Su-Schrieffer-Heeger chain can be characterized by a Hermitian purity spectrum, providing an analogue to band topology for mixed states.

Title and authors: Mira: So we're diving into the specifics of this paper now, "Mixed-State Topology in One-Dimensional Open Floquet Systems," starting with who wrote it and what the title actually implies.

Kai: The authors are Dinc, Schnell, and Martin from the University of Melbourne and Technische Universität Berlin. It’s a team pulling from different backgrounds in quantum physics.

Lev: I’m curious how their specific combination of expertise—from open system theory to condensed matter—leads them to this particular model setup.

Mira: They use the SSH chain as a starting point, which is a one-dimensional lattice model of non-interacting fermions with two bands, where the unit cell has alternating hopping amplitudes t and t prime.

Kai: And they introduce time periodic modulation to those tunneling amplitudes, described by this Hamiltonian HPDSSH(t) with driving strength V and frequency omega.

Lev: That driving term is what keeps things non-equilibrium, forcing us into these Floquet physics regimes where the quasienergy spectrum becomes important.

Mira: They are using the Floquet-Born-Markov theory to describe how this system couples to thermal reservoirs microscopically, which is a key difference from earlier work.

Kai: That’s where they get their steady state solution as a general Gaussian quadratic state, which lets them use these tools for mixed states.

Lev: For us on the hardware side, that means we need to think about how those thermal reservoirs affect the dynamics when we’re trying to maintain coherence in a driven system.

The paper's summary: Kai: So, what's the main message they are trying to convey here? Essentially, they are using this framework to show that even in these open, driven systems at finite temperature, we can still characterize topology.

Mira: They state that the steady state can be characterized by a Hermitian purity spectrum which provides a direct analogue of band topology for mixed states.

Lev: That’s the big conceptual leap—mapping the topological concepts from pure states onto these density matrices.

Kai: And they show that this system, when driven, induces nontrivial winding and a quasienergy spectrum with distinct zero and pi band gaps, along with protected edge modes in each gap <ref:2604.25248#pg1,induces nontrivial winding and a quasienergy spectrum with distinct 0 and>.

Mira: They then identify those two topological invariants, phi0 EGP and delta phi pi EGP, which are associated specifically with the zero and pi gaps of the quasienergy spectrum <ref:2604.25248#pg1>.

Lev: So they’re not just finding a phase; they’re building a measurable quantity that tells you if the system is in a topologically non-trivial state.

Kai: They demonstrate that this pair of invariants reveals a structure consistent with a Z times Z classification known from isolated Floquet SSH systems, but crucially, it extends it to this dissipative, finite-temperature setting.

The paper's improvements: Lev: What about the improvements they suggest over what came before? They seem to be focusing on making the topology detection more practical in these open settings.

Mira: The authors improve by building on the framework of using Floquet-Born-Markov theory to obtain a microscopic description of the system in presence of thermal reservoirs, instead of simpler treatments.

Kai: They also introduce the concept of using a finite purity gap as a prerequisite for defining these topological invariants, which is what makes those invariants well-defined.

Lev: From an engineering standpoint, that finite purity gap is key because if it’s not there, the topological features you're trying to detect just become unreadable noise.

Mira: And they introduce the second invariant delta phi pi EGP defined via the winding number nu pi, which quantifies how many times Z pi encircles the origin in the complex plane as time traverses one driving period T.

Kai: This winding number is a powerful tool because it lets us quantify topological features by counting wraps around that origin over time.

Conclusion: Kai: So to wrap up, they show that this pair of invariants, phi0 EGP and nu pi, which are well-defined due to the finite purity gap, reveal a Z times Z topology in these driven-dissipative systems.

Lev: It means we can detect topological phase transitions in the SSH chain by measuring these invariants.

Mira: This provides an extension of Floquet topology into dissipative systems and shows it’s robust beyond isolated zero-temperature systems.

Kai: They also indicate new ways to engineer and stabilize protected modes using dissipation, and because of the periodic drive, we get richer dynamical phases via those independently tunable zero- and pi invariants <ref:2604.25248#pg1>.

Lev: For real hardware, that means dissipation isn't just noise; it can be used to help protect the topological states from decoherence.

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