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.
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
We study topological phases of interacting systems in two spatial dimensions in the absence of topological order (i.e., with a unique ground state on closed manifolds and no fractional excitations). These are the closest interacting analogs of integer quantum Hall states, topological insulators, and superconductors. We adapt the well-known Chern-Simons
K-matrix description of quantum Hall states to classify such ``integer'' topological phases. Our main result is a general formalism that incorporates symmetries into the
K-matrix description. Remarkably, this simple analysis yields the same list of topological phases as a recent group cohomology classification, and in addition provides field theories and explicit edge theories for all these phases. The bosonic topological phases, which only appear in the presence of interactions and which remain well defined in the presence of disorder, include (i) bosonic insulators with a Hall conductance quantized to even integers, (ii) a bosonic analog of quantum spin Hall insulators, and (iii) a bosonic analog of a chiral topological superconductor, whose
K matrix is the Cartan matrix of Lie group
E8. We also discuss interacting fermion systems where symmetries are realized in a projective fashion, where we find the present formalism can handle a wider range of symmetries than a recent group super-cohomology classification. Lastly, we construct microscopic models of these phases from coupled one-dimensional systems.
Topological markers provide an efficient and powerful characterization of topological features of many systems, especially when the translation symmetry is broken. Recently, a universal topological marker applicable in different symmetry classes of topological systems is proposed. However, it suffers from irregular behaviors at the boundary and its connection to other topological indexes remains elusive. In this work, we construct regularized universal topological markers that apply to Dirac systems by utilizing position operators that are compatible with periodic boundary conditions. The regularized markers eliminate the obstructive boundary irregularities successfully and give rise to the desired global topological invariants, such as the Chern number, consistently when integrated over all the lattice sites. Furthermore, the regularized form allows us to establish an explicit connection between the markers and some other known topological indices in two dimensions. For instance, it turns out to be equivalent to the Bott index in classes A, D, and C and equivalent to the spin Chern number in classes DIII and AII. We further examine the utility and stability of this marker in disordered scenarios. We find that its variance shows peaks at the phase boundaries, which promotes it as a useful indicator for detecting disorder-induced topological phase transitions.
A universal topological marker has been proposed recently to map the topological invariants of Dirac models in any dimension and symmetry class to lattice sites. Using this topological marker, we examine the conditions under which the global topological order, represented by the spatially averaged topological marker, remains unchanged in the presence of disorder for 1D and 2D systems. We find that if an impurity corresponds to varying a nonzero-matrix element of the lattice Hamiltonian, regardless the element represents hopping, chemical potential, pairing, etc., then the average topological marker is conserved. However, if there are many strong impurities and the average distance between them is shorter than a correlation length, then the average marker is no longer conserved. In addition, strong and dense impurities can be used to continuously interpolate between one topological phase and another. A number of prototype lattice models including Su-Schrieffer-Heeger model, Kitaev chain, Chern insulators, Bernevig-Hughes-Zhang model, and chiral
p-wave superconductors are used to elaborate the ubiquity of these statements.
Topological states of matter were first introduced for noninteracting fermions on an infinite uniform lattice. Since then, substantial effort has been made to generalize these concepts to more complex settings. Recently, local markers have been developed that can describe the topological state of systems without translational symmetry and a well-defined gap. However, no local marker for interacting matter has been proposed yet that is capable of directly addressing an interacting system. Here, we suggest such a many-body local marker based on the single-particle Green's function. Using this marker we identify topological transitions in finite lattices of a Chern insulator with Anderson disorder and Hubbard interactions. Importantly, our proposal can be generalized to nonequilibrium systems.