Session G19: Chern Insulators, Symmetry Protected Topology, and Entanglement

Edge modes, Hall conductivity and topological features of a band deformed Dice lattice

The evolution of the topological nature of a Chern insulator induced by the deformation of the bandstructure has been studied on a dice lattice. Such band deformation can be achieved by tuning the hopping amplitude (say, t₁) along one of the three neighbors (between A - B sublattices) compared to the other two (which we may denote as t). Further, the time-reversal symmetry of the system is broken by a complex second neighbour hopping. For the isotropic case, that is, for t₁ = t, the system shows gap at the two Dirac points, namely, the K and K^′ points along with the appearance of a zero energy flat band. As one increases t₁, the flat bands get distorted and the band extrema, either at K or at K^′ point are shifted. At a particular value of t₁, namely, t₁ ≈ 1.7t, the shifted band extrema points touch the distorted flat band, and hence the spectral gap vanishes. Beyond this value (that is, for t₁ >1.7t), the gap reopens. A closer inspection of the topological properties of such a manipulated bandstructure reveals that a phase transition occurs at the critical point (t₁ = 1.7t), beyond which the system continues to behave as a band insulator. The topological robustness is demonstrated via computing the edge modes, anomalous Hall conductivity and the Chern number. The latter aids in arriving at the phase diagram in the relevant parameter space, which conclusively shows the vanishing of the Chern insulating phase with Chern number, |C| = 2 to a trivial insulator with |C| = 0 at the critical point through a gap closing transition. Further, to decipher the properties of the edge modes, we study a semi-infinite nanoribbon that yields a pair of chiral edge modes at each edge for t₁ < 1.7t. The results receive robust support from the phase diagram obtained by us. Finally, the vanishing of the plateaus at 2e²/h in the Hall conductivity provides support to the topological phase transition occurring at t₁ = 1.7t.   

Localization properties of an amorphous photonics-inspired system

Anderson showed that disorder from impurities localizes electronic eigenstates in 2D mesoscopic quantum systems [2]. In topological quantum systems, such as integer quantum Hall systems, this disorder localizes almost all states except for the chiral edge state [5] and is an important stabilizing mechanism for topological phases. Recent work [1, 3, 6] has suggested that topological phases also arise in amorphous systems: those where impurities are absent and the disorder is purely geometric. However, the possibility of localization purely from geometric disorder has been largely unexplored. We present evidence for a metal-insulator transition in an amorphous version of a Chern insulator, inspired by simulations of an experimental topological photonics system [4]. We present results from finite size scaling and level-spacing distributions to corroborate this, and compare with localization from impurity disorder.

[1] A. Agarwala and V. Shenoy. Topological insulators in random lattices. Phys. Rev. Lett., 118:1–6, 2017.

[2] P. W. Anderson. Absence of diffusion in certain random lattices. Phys. Rev., 109:1492, 1958.

[3] P. Corbae, S. Ciocys, D. Varjas, E. Kennedy, S. Zeltmann, M. Molina-Ruiz, S. Griffin, C. Jozwiak, Z. Chen, L. Wang, M. Scott, A. Grushin, A. Lanzara, and F. Hellman. 

Evidence for topological surface states in amorphous Bi2Se3.  arXiv:1910.13412

[4] Z. Jia, M. Secli, A. Avdoshkin, W. Redjem, E. J. Dresselhaus, J. E. Moore, and B. Kante. Disordered topological graphs enhancing nonlinear phenomena. Under review.

[5] A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein. Integer quantum Hall transition: An alternative approach and exact results. Phys. Rev. B, 50:7526, 1994.

[6] Q. Marsal, D. Varjas, and A. Grushin. Topological Weaire–Thorpe models of amorphous matter. Proc. National Academy of Science, 117, 2020.   

Phase Transitions in Topological Bilayer Chern Insulator Model

We investigate a bilayer system where the individual layers obey a Chern Insulator model Hamiltonian coupled together with an interlayer contact interaction between the layers. Due to the interplay between the topology and the interlayer interaction, it is natural to expect the emergence of a coherent excitonic insulating phase. Using Hartree-Fock self-consistent mean field simulations, we report on the phase transitions that occur as a result of varying the interaction strength, the potential difference between the top and bottom layers, and the topology of the layers (each of which can be varied independently). We find a rich phase diagram that supports both metallic and insulating behavior as well as the expected excitonic behavior. We also observe reentrant behavior in the metallic phase.   

Multiplicative topological phases

Symmetry-protected topological phases of matter have challenged our understanding of condensed matter systems and harbour exotic phenomena promising to address major technological challenges. Considerable understanding of these phases of matter has been gained recently by considering additional protecting symmetries, different types of quasiparticles, and systems out of equilibrium. Here, we show that symmetries could be enforced not just on full Hamiltonians, but also on their components. We construct a large class of previously unidentified multiplicative topological phases of matter characterized by tensor product Hilbert spaces similar to the Fock space of multiple particles. To demonstrate our methods, we introduce multiplicative topological phases of matter based on the foundational Hopf and Chern insulator phases, the multiplicative Hopf and Chern insulators (MHI and MCI), respectively. The MHI shows the distinctive properties of the parent phases as well as non-trivial topology of a child phase. We also comment on a similar structure in topological superconductors as these multiplicative phases are protected in part by particle-hole symmetry. The MCI phase realizes topologically-protected gapless states that do not extend from the valence bands to the conduction bands for open boundary conditions, which respect the symmetries protecting topological phase, which serves as a blueprint for exotic band connectivity.   

Intrinsically average symmetry-protected topological phases

Symmetry-protected topological (SPT) phases are topologically nontrivial gapped phases of quantum many-body systems protected by certain global symmetry. The notion of SPT order has been recently generalized to systems where the symmetry is broken by quenched disorder, but preserved on average by the disorder ensemble, In this work we present a systematic classification of such SPT phases using a decorated domain wall approach. In particular, we show that there are average SPT phases which do not exist in clean systems, thus are intrinsically disordered. We will present examples of intrinsically average SPT phases and discuss their boundary states.   

Topological invariants for SPT entanglers

We develop a framework for classifying locality preserving unitaries (LPUs) with internal, unitary symmetries in $d$ dimensions, based on $(d-1)$-dimensional "flux insertion operators" which are easily computed from the unitary. Using this framework, we obtain formulas for topological invariants of LPUs that prepare, or entangle, symmetry protected topological phases (SPTs). These formulas serve as edge invariants for Floquet topological phases in $(d+1)$ dimensions that "pump" $d$-dimensional SPTs. For 1D SPT entanglers and certain higher dimensional SPT entanglers, our formulas are completely closed-form.   

Symmetry Protected Topological phases under Decoherence

We study symmetry protected topological (SPT) phases under various types of decoherence, which drives a pure SPT state into a mixed state. We demonstrate that the system can still retain the nontrivial topological information from the SPT ground state even under decoherence, which can arise from noise or weak measurement. The main quantity that we investigate is the ``strange correlator" proposed previously as a diagnosis for the SPT ground states, and in this work we generalize the notion of strange correlator to the mixed state density matrices. Using both exact calculations of the stabilizer Hamiltonians, as well as field theory evaluation, we demonstrate that under decoherence the characteristic behaviors of the SPT states can persist in the ``type-II" strange correlators. Furthermore, in some cases, the ``type-I'' strange correlators exhibit transition behaviors depending on the strength of decoherence, which alerts the presence of information-theoretic transition. We show that in the regime where type-I strange correlators are non-trivial, topological information of the decohered SPT state can be efficiently identified from experiments.   

Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase

We study the (3+1)d boundary theory of Z_N beyond cohomology symmetry protected topological (SPT) phase in (4+1)d, which exhibits a new kind of quantum anomaly in (3+1)d systems with Z_N symmetry. We show that when N ∉ {2,4,8,16}, there cannot be any symmetry-preserving gapped boundary states. When N ∈ {2,4,8,16}, we study a related system with C_N symmetry via the crystalline correspondence principle, and explicitly construct gapped boundary topological order for the beyond-cohomology SPT states. The boundary of the C_N SPT state can be disentangled everywhere except on the rotation axis, where there are chiral edge states described by the (E₈)₁ theory. To gap them out, we introduce N copies of (2+1)d gapped layers all terminating at the rotation axis. We study the problem of symmetry-preserving gapped interface at the rotation axis and in particular show that for N ∈ {2,4,8,16}, if each layer has the topological order of Spin(16/N)₁, the (3+1)d boundary can be completely gapped out while preserving the C_N symmetry.   

Entanglement properties of one-dimensional topological insulators

There is still much to learn about the entanglement properties of symmetry protected topological insulators. Symmetries can be of the spatial form (Inversion, Cn rotation) or of the non-spatial form (chiral, time-reversal, particle-hole). For the non-spatial chiral symmetry, we establish here, for the first time, a rigorous relation between a Z-invariant (winding number) and protected eigenvalues in the entanglement spectrum. For the spatial C2 symmetry, we prove minimum entanglement bounds on two symmetry resolved components: the configurational and number entropy.

    

Parity effects and universal terms of O(1) in the entanglement near a boundary

In the presence of boundaries, the entanglement entropy in lattice models is known to exhibit oscillations with the (parity of the) length of the subsystem, which however decay to zero with increasing distance from the edge. We point out in this article that, when the subsystem starts at the boundary and ends at an impurity, oscillations of the entanglement (as well as of charge fluctuations) appear which do not decay with distance, and which exhibit universal features. We study these oscillations in detail for the case of the XX chain with one modified link (a conformal defect) or two successive modified links (a relevant defect), both numerically and analytically. We then generalize our analysis to the case of extended (conformal) impurities, which we interpret as SSH models coupled to metallic leads. In this context, the parity effects can be interpreted in terms of the existence of non-trivial topological phases.   

Topological Phase Transitions of Generalized Brillouin Zone in non-Hermitian systems

In recent years, non-Hermitian systems have shown unconventional properties and phenomena such as exceptional points, non-Hermitian skin effect, and many more in different research fields without Hermitian analog. In these systems the notion of Brillouin zone (BZ) has been extended to generalized Brillouin zone (GBZ) defined by generalized momenta which can take complex values . A generic feature of topological phases of matter is the bulk-boundary correspondence (BBC) which connects the concept of bulk topology to the emergence of robust boundary states. Recent studies prove that non-Hermitian systems show two types of modified BBC: (i) complex energy topology of the bulk leads to non-Hermitian skin effect, where all bulk states localize at one boundary of the system, and (ii) wave function topology leads to the conventional topological boundary modes.

In this seminar, I will discuss our recent finding of a new type of BBC, which originates from the intrinsic topology of the GBZ. Topologically non-trivial GBZ appear due to general boundary conditions that break the translation symmetry of the system. In our case, the topological phase transition is characterized by the generalized momentum touching of GBZ, which accompanies the emergence of exceptional points.   

Basis-Independent Topological Crystalline Markers for Two-Dimensional C_n-symmetric Obstructed Atomic Insulators and Topological Crystalline Superconductors

Topological crystalline insulators and superconductors are topological phases which are protected by spatial symmetries such as reflections and rotations. These phases have been classified using quantities such as momentum-space rotation invariants and symmetry indicators. Previous works have shown that correspondences exist between two quantities: (i) strong topological invariants such as the Chern number, electromagnetic responses such as the bulk polarization, boundary signatures such as the corner charge and (ii) momentum-space rotation invariants at high-symmetry momenta in the Brillouin zone. In this work, we re-express these previously established relationships for 2D C_n-symmetric (n=2,3,4,6) obstructed atomic insulators and topological crystalline superconductors in terms of topological crystalline markers constructed from projected symmetry operators, which do not require a specific choice of basis such as a momentum-space basis. These topological crystalline markers are similar in form to quantities such as the local Chern marker, which is expressed in terms of the commutator of projected position operators. Our results show that quantities such as the bulk polarization, corner charge, and Chern number, depend on multiple topological crystalline markers based on projected symmetry operators over different symmetry centers. Our results provide a new interpretation of existing topological crystalline invariants that has applications to systems where translational symmetry is broken, but the crystalline symmetry is preserved.