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arXiv:2512.00464 · Physical Review Research

Characterizing topology at nonzero temperature: Topological invariants and indicators in the extended Su-Schrieffer-Heeger model

Julia D. Hannukainen, Nigel R. Cooper

Published May 9, 2026• 1 citation
arXivHTMLPDF

Abstract

We compare three complementary diagnostics for mixed Gaussian states at nonzero temperature, focusing on the Su-Schrieffer-Heeger (SSH) chain and its inversion-symmetric extension. While the ensemble geometric phase, a mixed-state generalization of the Zak phase, remains well defined at nonzero temperature, the modulus of the corresponding expectation value vanishes in the thermodynamic limit, limiting its practical use. To develop diagnostics suitable for large systems, we introduce local twist operators acting on neighboring sites, whose expectation values provide local indicators of the underlying topological phase. The topological phase is identified from the relative magnitude of these expectation values, which only requires measuring two local expectation values at nonzero temperature, together with one additional nonlocal expectation value when next-nearest-neighbor hopping is included. In addition, we generalize the local chiral marker to mixed Gaussian states, fully determined by its single-particle correlation matrix, with a nonzero purity gap in their effective single-particle Hamiltonian. The presence of a purity gap ensures that the correlation matrix can be flattened to an effective projector. Evaluating the chiral marker with respect to the band-flattened correlation matrix yields a real-space topological invariant that coincides with the winding number in the zero-temperature limit. The ensemble geometric phase, the local twist operators, and the local chiral marker provide complementary methods to characterize topology in the SSH chain beyond pure states.

DOI
10.1103/pwkc-lqq6

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We compare three complementary diagnostics for mixed Gaussian states at nonzero temperature, focusing on the Su-Schrieffer-Heeger (SSH) chain and its inversion-symmetric extension. While the ensemble geometric phase, a mixed-state generalization of the Zak phase, remains well defined at nonzero temperature, the modulus of the corresponding expectation value vanishes in the thermodynamic limit, limiting its practical use. To develop diagnostics suitable for large systems, we introduce local twist operators acting on neighboring sites, whose expectation values provide local indicators of the underlying topological phase. The topological phase is identified from the relative magnitude of these expectation values, which only requires measuring two local expectation values at nonzero temperature, together with one additional nonlocal expectation value when next-nearest-neighbor hopping is included. In addition, we generalize the local chiral marker to mixed Gaussian states, fully determined by its single-particle correlation matrix, with a nonzero purity gap in their effective single-particle Hamiltonian. The presence of a purity gap ensures that the correlation matrix can be flattened to an effective projector. Evaluating the chiral marker with respect to the band-flattened correlation matrix yields a real-space topological invariant that coincides with the winding number in the zero-temperature limit. The ensemble geometric phase, the local twist operators, and the local chiral marker provide complementary methods to characterize topology in the SSH chain beyond pure states.
The topological properties of the Su-Schrieffer-Heeger (SSH) model in the presence of nearest-neighbor interaction are investigated by means of a topological marker, generalized from a noninteracting one by utilizing the single-particle Green's function of the many-body ground state. We find that despite the marker not being perfectly quantized in the presence of interactions, it always remains finite in the topologically nontrivial phase while converging to zero in the trivial phase when approaching the thermodynamic limit, and hence correctly judges the topological phases in the presence of interactions. The marker also correctly captures the interaction-driven, second-order phase transitions between a topological phase and a Landau-ordered phase, which is a charge-density wave order in our model with a local-order parameter, as confirmed by the calculation of entanglement entropy and the many-body Zak phase. Our paper thus points to the possibility of generalizing topological markers to interacting systems through Green's function, which may be feasible for topological insulators in any dimension and symmetry class.
We study two-particle states in a Su-Schrieffer-Heeger (SSH) chain with periodic boundary conditions and nearest-neighbor (NN) interactions. The system is mapped into a problem of a single particle in a two-dimensional (2D) SSH lattice with potential walls along specific edges. The 2D SSH model has a trivial Chern number but a non-trivial Zak's phase, the one-dimensional (1D) topological invariant, along specific directions of the lattice, which allow for the presence of topological edge states. Using center-of-mass and relative coordinates, we calculate the energy spectrum of these two-body states for strong interactions and find that, aside from the expected appearance of doublon bands, two extra in-gap bands are present. These are identified as bands of topological states localized at the edges of the internal coordinate, the relative distance between the two particles. As such, the topological states reported here are intrinsically many-body in what concerns their real space manifestation, having no counterpart in single-particle states derived from effective models. Finally, we compare the effect of Hubbard interactions with that of NN interactions to show how the presence of the topological bound states is specific to the latter case.
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.
While topology is a property of a quantum state itself, most existing methods for characterizing the topology of interacting phases of matter require direct knowledge of the underlying Hamiltonian. We offer an alternative by utilizing the one-particle density matrix formalism to extend the concept of the Chern, chiral, and Chern-Simons markers to include interactions. The one-particle density matrix of a free-fermion state is a projector onto the occupied bands, defining a Brillouin zone bundle of the given topological class. This is no longer the case in the interacting limit, but as long as the one-particle density matrix is gapped, its spectrum can be adiabatically flattened, connecting it to a topologically equivalent projector. The corresponding topological markers thus characterize the topology of the interacting phase. Importantly, the one-particle density matrix is defined in terms of a given state alone, making the local markers numerically favorable, and providing a valuable tool for characterizing topology of interacting systems when only the state itself is available. To demonstrate the practical use of the markers we use the chiral marker to identify the topology of midspectrum eigenstates of the Ising-Majorana chain across the transition between the ergodic and many-body localized phases. We also apply the chiral marker to random states with a known topology, and compare it with the entanglement spectrum degeneracy. Published by the American Physical Society 2024
Despite extensive studies on the one-dimensional Su-Schrieffer-Heeger-Hubbard (SSHH) model, the variant incorporating second-nearest neighbor hopping remains largely unexplored. Here, the topological classification of the extended SSH model is analyzed within the framework of the BDI symmetry class, using the winding number as the corresponding topological invariant. We investigate the ground-state properties of this extended SSHH model using the constrained-path auxiliary-field quantum Monte Carlo (CP-AFQMC) method. We show that this model exhibits rich topological phases, characterized by robust edge states against interaction. We quantify the properties of these edge states by analyzing spin correlation and second-order Rényi entanglement entropy. The system exhibits long-range spin correlation and near-zero Rényi entropy at half-filling. Besides, there is an anti-ferromagnetic order at quarter-filling. Interestingly, an external magnetic field disrupts this anti-ferromagnetic order, restoring long-range spin correlation and near-zero Rényi entropy. Furthermore, our work provides a paradigm for studying topological properties in large interacting systems via the CP-AFQMC algorithm.
The ground states of noninteracting fermions in one-dimension with chiral symmetry form a class of topological band insulators, described by a topological invariant that can be related to the Zak phase. Recently, a generalization of this quantity to open systems---known as the ensemble geometric phase (EGP)---has emerged as a robust way to describe topology at nonzero temperature. By using this quantity, we explore the nature of topology allowed for dissipation beyond a Lindblad description, to allow for coupling to external baths at finite temperatures. We introduce two main aspects to the theory of open-system topology. First, we discover topological phase transitions as a function of the temperature TTT, manifesting as changes in differences of the EGP accumulated over a closed loop in parameter space. We characterize the nature of these transitions and reveal that the corresponding nonequilibrium steady state can exhibit a nontrivial structure---contrary to previous studies where it was found to be in a fully mixed state. Second, we demonstrate that the EGP itself becomes quantized when key symmetries are present, allowing it to be viewed as a topological marker which can undergo equilibrium topological transitions at nonzero temperatures.