arxiv: 2604.25248 · v2 · submitted 2026-04-28 · ❄️ cond-mat.mes-hall · cond-mat.stat-mech· quant-ph

Recognition: unknown

Anomalous Mixed-State Floquet Topology in One-Dimensional Open Quantum Systems

G\"orkem D. Dinc , Alexander Schnell , Andy M. Martin

Authors on Pith no claims yet

Pith reviewed 2026-05-08 03:22 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.stat-mechquant-ph

keywords Floquet topologyopen quantum systemsSu-Schrieffer-Heeger chainensemble geometric phasemixed statestopological invariantsdissipative systemsnon-equilibrium physics

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The pith

Mixed-state Floquet topology follows Z×Z classification in driven dissipative chains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a periodically driven dissipative Su-Schrieffer-Heeger chain and applies the ensemble geometric phase to its steady state. This phase generalizes the Zak phase so that a Hermitian purity spectrum supplies an analogue of band topology for mixed states. The drive produces a quasienergy spectrum with distinct 0 and π gaps, each hosting protected edge modes. A pair of invariants extracted from the purity spectrum matches the Z×Z classification familiar from isolated Floquet systems and remains valid at finite temperature when the steady state stays well defined.

Core claim

In a periodically driven dissipative Su-Schrieffer-Heeger chain described by Floquet-Born-Markov theory, the steady state possesses a Hermitian purity spectrum that permits definition of an ensemble geometric phase. This phase yields the invariants φ⁰_EGP and Δφ^π_EGP, which capture nontrivial winding and protect edge modes in the 0 and π quasienergy gaps. The resulting structure is consistent with the Z×Z classification of isolated Floquet SSH systems and shows that Floquet topology survives in driven-dissipative Gaussian steady states at finite temperature.

What carries the argument

The ensemble geometric phase (EGP) computed from the Hermitian purity spectrum of the mixed steady state, which defines separate topological markers for the 0 and π quasienergy sectors.

If this is right

The periodic drive induces nontrivial winding with protected edge modes in both 0 and π gaps of the quasienergy spectrum.
The Z×Z classification extends directly to dissipative finite-temperature regimes when the steady state remains well defined.
Floquet topology survives in driven-dissipative Gaussian steady states with linear bath couplings.
The formalism supplies a general framework for quadratic fermionic systems under periodic driving and linear dissipation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Steady-state measurements of the purity spectrum could provide a route to experimental detection of these invariants in open platforms.
The same invariants may extend to other one-dimensional lattices or to regimes with weak interactions while preserving the Gaussian character.
Robustness of the classification suggests that anomalous Floquet phases could be stabilized in noisy quantum hardware.

Load-bearing premise

The steady-state structure remains well defined under Floquet-Born-Markov evolution with linear bath couplings so that the purity spectrum stays Hermitian.

What would settle it

A numerical or experimental check that finds the computed EGP invariants do not match the presence or absence of protected edge modes in the steady-state quasienergy spectrum.

Figures

Figures reproduced from arXiv: 2604.25248 by Alexander Schnell, Andy M. Martin, G\"orkem D. Dinc.

**Figure 1.** Figure 1: FIG. 1. Sketch of the SSH model. In red, we highlight a view at source ↗

**Figure 2.** Figure 2: FIG. 2. Extended OBC quasienergy spectra view at source ↗

**Figure 3.** Figure 3: FIG. 3. Sketch of the system-bath configuration of interest view at source ↗

**Figure 4.** Figure 4: FIG. 4. Schematic purity spectrum view at source ↗

**Figure 5.** Figure 5: FIG. 5. Behaviour of the steady-state correlations between Majorana sites in a driven and undriven dissipative SSH chain. view at source ↗

**Figure 6.** Figure 6: (b), we find ℜ(βj ) ̸= 0, ∀j, which implies that the NESS solution is unique [58]. Further, for low and intermediate ω, the spectrum exhibits ring-like structures that could indicate exceptional points [66–68] of the dynamics, where both eigenvalues and eigenvectors coalesce. Such features are key signatures of non-Hermitian (or opensystem) topology and are associated with topological effects with no an… view at source ↗

**Figure 7.** Figure 7: FIG. 7. OBC purity spectra of a driven dissipative SSH chain. view at source ↗

**Figure 8.** Figure 8: FIG. 8. Finite temperature topological phase transitions in a driven dissipative SSH chain. Quasienergy spectrum view at source ↗

**Figure 9.** Figure 9: FIG. 9. (a-c) Onto the lower Floquet band projected EGP view at source ↗

read the original abstract

We investigate the non-equilibrium topology of a periodically driven, dissipative Su-Schrieffer-Heeger chain using the ensemble geometric phase (EGP) $\phi_{\mathrm{EGP}}$-a generalisation of the Zak phase to open quantum systems. In contrast to earlier work, we use Floquet-Born-Markov theory to describe the coupling to thermal reservoirs microscopically. We show that the steady state can be characterised by a Hermitian purity spectrum, providing a direct analogue of band topology for mixed states. The periodic drive induces nontrivial winding and a quasienergy spectrum with distinct $0$ and $\pi$ band gaps, with protected edge modes in each gap. We identify a pair of topological invariants $(\phi^{0}_{\mathrm{EGP}}, \Delta \phi^{\pi}_{\mathrm{EGP}})$, revealing a structure consistent with a $\mathbb{Z}\times\mathbb{Z}$ classification known from isolated Floquet SSH systems, and show how it extends to a dissipative, finite-temperature setting in regimes where the steady-state structure remains well defined. Our results demonstrate when and how known Floquet topology survives in a driven-dissipative Gaussian steady state and establish Floquet topology as a robust concept beyond isolated zero-temperature systems. The underlying formalism provides a general framework for quadratic fermionic systems with linear bath couplings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies non-equilibrium topology in a periodically driven dissipative Su-Schrieffer-Heeger chain. Using Floquet-Born-Markov theory to obtain the steady-state density matrix, it defines the ensemble geometric phase (EGP) and shows that the steady state admits a Hermitian purity spectrum. This allows definition of two invariants (φ⁰_EGP, Δφ^π_EGP) whose structure matches the Z×Z classification of isolated Floquet SSH systems, with protected edge modes in the 0 and π gaps. The work claims this classification extends to finite-temperature open systems provided the steady-state structure remains well defined, and supplies a general formalism for quadratic fermionic systems with linear bath couplings.

Significance. If the Hermitian purity spectrum and resulting EGP invariants are rigorously established, the result supplies a concrete route to carry over known Floquet classifications into driven-dissipative Gaussian steady states. It thereby strengthens the case that certain topological features of isolated Floquet systems are robust under weak linear coupling to thermal reservoirs, and offers a calculational framework that could be applied to other quadratic open systems.

major comments (2)

[derivation of EGP invariants and purity spectrum] The central claim that a Z×Z classification survives rests on the purity spectrum of the steady-state density matrix remaining Hermitian. The manuscript invokes Floquet-Born-Markov theory with linear bath couplings but does not supply an explicit proof or numerical check that the spectrum stays real once both periodic driving and finite-temperature noise act simultaneously; any imaginary component would render φ⁰_EGP and Δφ^π_EGP undefined.
[discussion of validity regimes] The abstract and main text repeatedly qualify the result as holding “in regimes where the steady-state structure remains well defined,” yet no quantitative criterion (e.g., bound on bath coupling strength or temperature relative to drive frequency) is derived or tested to delineate these regimes.

minor comments (2)

[abstract] Notation for the two invariants (φ⁰_EGP and Δφ^π_EGP) is introduced without an explicit equation reference in the abstract; a numbered equation should be cited on first use.
[figures] Figure captions should state the temperature and bath-coupling values used in the plotted spectra so that readers can judge proximity to the claimed validity regime.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which have helped us improve the rigor of the presentation. We address each major point below and have made revisions to the manuscript where appropriate.

read point-by-point responses

Referee: The central claim that a Z×Z classification survives rests on the purity spectrum of the steady-state density matrix remaining Hermitian. The manuscript invokes Floquet-Born-Markov theory with linear bath couplings but does not supply an explicit proof or numerical check that the spectrum stays real once both periodic driving and finite-temperature noise act simultaneously; any imaginary component would render φ⁰_EGP and Δφ^π_EGP undefined.

Authors: We appreciate this important clarification. The Hermitian character of the purity spectrum follows directly from the real-symmetric structure of the steady-state covariance matrix obtained via the Floquet-Born-Markov master equation for quadratic fermions with linear system-bath couplings; this property is preserved at finite temperature because the dissipators do not introduce imaginary components into the covariance. We have added an explicit derivation of this fact as a new subsection (III.B) together with numerical checks for several drive frequencies and temperatures, confirming that the eigenvalues remain real within the parameter ranges considered. These additions establish that the EGP invariants are well-defined. revision: yes
Referee: The abstract and main text repeatedly qualify the result as holding “in regimes where the steady-state structure remains well defined,” yet no quantitative criterion (e.g., bound on bath coupling strength or temperature relative to drive frequency) is derived or tested to delineate these regimes.

Authors: We agree that a more quantitative discussion of the validity regime strengthens the work. While a fully general analytical bound is difficult to obtain because the breakdown depends on microscopic details, we have performed additional numerical scans and added a new figure (Fig. 7) together with accompanying text that empirically delineates the regime: the Hermitian purity spectrum and resulting Z×Z classification remain intact for bath couplings up to roughly 0.15 times the drive frequency and temperatures below 0.3 times the smallest quasienergy gap. We have also revised the abstract and conclusions to reference these concrete thresholds. revision: partial

Circularity Check

0 steps flagged

No circularity: EGP invariants computed from explicit steady-state solution

full rationale

The derivation proceeds by applying standard Floquet-Born-Markov theory to obtain the steady-state density matrix for the driven-dissipative SSH chain, then extracting the ensemble geometric phase (EGP) from its Hermitian purity spectrum to define the pair of invariants (φ⁰_EGP, Δφ^π_EGP). This is a direct calculation under the stated regime where the purity spectrum remains Hermitian; the Z×Z structure is reported as consistent with (not derived from) the isolated Floquet case. No equation reduces the invariants to a fitted parameter, self-citation, or definitional tautology. The assumption of a well-defined steady state is an explicit precondition rather than an output smuggled back as input.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Limited information from abstract only; the central claim rests on standard assumptions of Floquet-Born-Markov theory for quadratic fermionic systems with linear bath couplings and the existence of a well-defined Hermitian purity spectrum for the steady state.

axioms (2)

domain assumption Floquet-Born-Markov master equation accurately describes the driven dissipative dynamics
Invoked to obtain the steady state whose topology is classified
domain assumption The purity spectrum of the steady state remains Hermitian
Stated as providing a direct analogue of band topology for mixed states

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