Local topological markers appear to be a powerful tool in the studies of topological matter allowing to treat inhomogeneous systems. To the best of our knowledge no local marker for interacting matter itself, without a reference non-interacting system have been proposed to date. In this letter we suggest a possible many-body local marker based on the single-particle Green's functions. In the limit of translation-invariant system our marker is equivalent to the Ishikawa Matsuyama invariant. In the limit of vanishing interactions it passes to the local Chern marker. Importantly, our proposal can be readily used in the non-equilibrium in contrast with the topological Hamiltonian based approaches to the problem.
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
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.
Local topological markers, topological invariants evaluated by local expectation values, are valuable for characterizing topological phases in materials lacking translation invariance. The Chern marker-the Chern number expressed in terms of the Fourier transformed Chern character-is an easily applicable local marker in even dimensions, but there are no analogous expressions for odd dimensions. We provide general analytic expressions for local markers for free-fermion topological states in odd dimensions protected by local symmetries: a Chiral marker, a local Z marker which in case of translation invariance is equivalent to the chiral winding number, and a Chern-Simons marker, a local Z_{2} marker characterizing all nonchiral phases in odd dimensions. We achieve this by introducing a one-parameter family P_{ϑ} of single-particle density matrices interpolating between a trivial state and the state of interest. By interpreting the parameter ϑ as an additional dimension, we calculate the Chern marker for the family P_{ϑ}. We demonstrate the practical use of these markers by characterizing the topological phases of two amorphous Hamiltonians in three dimensions: a topological superconductor (Z classification) and a topological insulator (Z_{2} classification).
Topological phase transitions in free fermion systems can be characterized by the closing of single-particle gap and the change in topological invariants. However, in the presence of electronic interactions, topological phase transitions can be more complicated. In paper I of this series [Phys. Rev. B 93, 195163 (2016)], we have proposed an efficient scheme to evaluate the topological invariants based on the single-particle Green's function formalism. Here, in paper II, we demonstrate several interaction-driven topological phase transitions (TPTs) in two-dimensional (2D) interacting topological insulators (TIs) via large-scale quantum Monte Carlo (QMC) simulations, based on the scheme of evaluating topological invariants presented in paper I. Across these transitions, the defining symmetries of the TIs have been neither explicitly nor spontaneously broken. In the first two models, the topological invariants calculated from the Green's function formalism succeed in characterizing the topologically distinct phases and identifying interaction-driven TPTs. However, in the other two models, we find that the single-particle gap does not close and the topological invariants constructed from the single-particle Green's function acquire no change across the TPTs. Unexpected breakdown of the Green's function formalism in constructing the topological invariants is thus discovered. We thence classify the topological phase transitions in interacting TIs into two categories in practical computation: Those that have noninteracting correspondence can be characterized successfully by the topological invariants constructed from the Green's functions, while for the others that do not have noninteracting correspondence, the Green's function formalism experiences a breakdown, but more interesting and exciting phenomena, such as emergent collective critical modes at the transition, arise. Discussion on the success and breakdown of topological invariants constructed from the Green's function formalism in the context of symmetry protected topological (SPT) states is presented.
Local topological markers provide a real-space characterization of topological phases and are suited to noncrystalline models. We derive a local topological marker, the Chern-Simons descendant marker, for characterizing the topology of two-dimensional time-reversal-invariant near-Gaussian states from the one-particle density matrix
ρ. We construct an auxiliary three-dimensional state characterized by the local Chern-Simons marker, and use dimensional reduction to obtain the Chern-Simons descendant marker for the original two-dimensional state. This derivation introduces a local Hermitian involution
S that is odd under time reversal. When
[S,ρ]=0, the local Chern-Simons descendant marker reduces to half the difference of the local Chern markers in the
S=±1 sectors. Constructing an
S that commutes exactly with
ρ generally requires fine-tuning, but this is not needed---the local Chern-Simon marker remains well defined whenever
Γ={ρ,S}−S has a spectral gap around zero, even when
[S,ρ]=0. This leaves the practical problem of identifying a suitable
S for a given state, which, importantly, is straightforward even when spin is not a candidate. We show this by considering both strong Rashba and Dresselhaus spin-orbit coupling, for which knowledge of the physical origin of the state reduces the general problem of choosing
S to a one-parameter family
S(θ). We evaluate the optimal angle
θ maximizing the spectral gap of
Γ analytically assuming translation invariance, and find that the gap of
Γ remains open for a broad range of angles around the optimum. The same value of
θ remains valid in disordered systems, providing an easy way to obtain choice of
S for evaluating the local Chern-Simons descendant marker of the state.