Session Q59: Theoretical Methods for Topological Insulators

Many-body quadrupolar sum rule for higher-order topological insulator

The modern theory of polarization establishes the bulk-boundary correspondence for the bulk polarization. We attempt to extend it to a sum rule of the bulk quadrupole moment by employing a many-body operator introduced in Ref.s 1 and 2. The sum rule consists of the alternating sum of four observables, which are the phase factors of the many-body operator in different boundary conditions. We demonstrate the validity of the sum rule through extensive numerical computations on various non-interacting tight-binding models. We also observe that individual terms in the sum rule correspond to the bulk quadrupole moment, the edge-localized polarizations, and the corner charge in the thermodynamic limit on some models.

[1] Byungmin Kang, Ken Shiozaki, and Gil Young Cho, “Many-body order parameters for multipoles in solids,” Phys. Rev. B 100, 245134 (2019).

[2] William A. Wheeler, Lucas K. Wagner, and Taylor L. Hughes, “Many-body electric multipole operators in extended systems,” Phys. Rev. B 100, 245135 (2019).   

A bulk-boundary correspondence for 3D bosonic symmetry protected topological phases

A universal property of symmetry protected topological (SPT) phases is that they have low energy boundary modes that are protected under the symmetry. This fact inspires an important problem in the theory of SPT phases: How does one identify a bulk SPT phase given a low energy theory of its boundary? This question is particularly challenging in the case of interacting SPT phases where band theory approaches are inapplicable. In this talk, we present a method for solving this problem in the case of gapless boundaries of (3+1)D interacting bosonic systems. This solution centers around the calculation of generalized F-symbols for string like domain wall excitations in the boundary theory which are then identified with an important bulk topological invariant.   

Classification of (2+1)D invertible fermionic topological phases with symmetry

We provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups G_f and general values of the chiral central charge c_-. Here G_f is a central extension of a bosonic symmetry group G_b by fermion parity, (-1)^F, specified by a second cohomology class ω₂ in the group H²(G_b, Z₂). Our approach proceeds by gauging fermion parity and classifying the resulting G_b symmetry-enriched topological orders while keeping track of certain additional data and constraints. We perform this analysis through two perspectives, using G-crossed braided tensor categories and Spin(2c_-)₁ Chern-Simons theory coupled to a background G gauge field. These results give a way to characterize and classify invertible fermionic topological phases in terms of a concrete set of data and consistency equations, which is more physically transparent and computationally simpler than the more abstract methods using cobordism theory and spectral sequences.  We show how the 10-fold way classification of topological insulators and superconductors fits into our scheme, along with general non-perturbative constraints due to certain choices of c_- and G_f.   

A DMRG study of the extended Haldane-Hubbard model

In this work, we study the extended Haldane-Hubbard model at half-filling in a honeycomb lattice. We employ the density matrix renormalization group algorithm in the matrix product state formulation (DMRG-MPS) to obtain the ground state of the system and generate its phase diagram. In agreement with previous results, we find that strong on-site interactions can create a spin density wave phase, whereas strong nearest-neighbor interactions favor a charge density wave phase. A topologically non-trivial phase was also found in the weakly-interacting regime, by measuring the Chern number using a charge-pumping scheme in a finite cylinder geometry. We compare this phase diagram to previous results in the literature obtained via other methods, such as exact diagonalization in small clusters and infinite DMRG, highlighting the similarities and differences in the regions where the phase transitions happen.   

Pancharatnam-Zak Phase for Two-Dimensional Fermionic Systems

Topological states of matter are usually classified by quantized topological invariants. In one-dimensional crystals, the Zak phase is a useful invariant to detect topological states. However, Zak phase is gauge-dependent, and this dependence can sometimes lead to mistakes in the detection of topological phases in more complex systems. To prevent such a mistake, we should look for gauge-invariant topological numbers. In this work we define Pancharatnam-Zak phase, a gauge independent topological number for two-dimensional fermionic systems, and investigate the phase diagram of two topological systems: i) the Su-Schrieffer-Heeger, and ii) the Kitaev models. In two-dimensional systems, the Chern number is usually used to examine the topological phases, however in the presence of time-reversal and inversion symmetries, the Chern number is vanishing, and we have to employ other invariants. We show that the Phancharatnam-Zak phase can be employed for characterizing the topological phases of two-dimensional fermionic systems.   

Higher-Form Response of 3D Fractional Topological Insulators

Three-dimensional fractional topological insulators (FTIs) are exotic symmetry enriched phases hosting extended defects with anyonic statistics. Here we consider the higher-form global symmetries and anomalies of a special family of bulk FTI models based on Z_N lattice gauge theory with a fractional theta term, which were originally explored by Cardy and Rabinovici. These models host dyon excitations that condense to form the FTI phase. We examine the 't Hooft anomalies and response of these theories to an external two-form gauge field that probes their electric Z_N one-form symmetry, which we show leads to a universal topological index for the FTI state.   

Fractional hinge and corner charges in various crystal shapes with cubic symmetry

Recent studies suggested that even topologically trivial insulators may exhibit fractionally quantized charges localized at hinges or corners. We study hinge and corner charges for five crystal shapes of vertex-transitive polyhedra with the cubic symmetry such as a cube, an octahedron and a cuboctahedron [1]. We derive real-space formulas for the hinge and corner charges in terms of the electric charges associated with bulk Wyckoff positions. We find that both the hinge and corner charges can be predicted from the bulk perspective only modulo certain fractions depending on the crystal shape, because the relaxation near boundaries of the crystal may affect the fractional parts. We also derive momentum-space formulas for the hinge and corner charges by using the method of elementary band representation (EBR) matrix. It turns out that the irreducible representations of filled bands at high-symmetry momenta are not sufficient to determine the corner charge. In order to resolve this issue, we introduce an additional Wilson-loop invariant. We show that the momentum-space formulas for the corner charges can be obtained by combining the EBR matrix with the new Wilson-loop invariant.

[1] K. Naito, R. Takahashi, H. Watanabe, and S. Murakami, arXiv:2109.14914 (2021).   

Dirac Fractals on the Surface of Topological Insulators

Dimensionality plays a fundamental role in the classification of phases of matter. A canonical example is the surface of 3D topological insulators (TIs) that hosts symmetry-protected 2D Dirac fermions. In this talk, we will discuss a different class of topological surface states that results from the interplay of surface Dirac fermions and fractal geometry. Specifically, we study the consequences of coupling Dirac fermions to a time-reversal symmetric Sierpinski fractal deposited on the surface of a 3D TI with the goal of identifying topological surfaces hosting quantum states with fractal dimension. Employing a numerical analysis of the local density of states and an effective theory, we present evidence for the emergence of the Dirac fractals of non-integer Hausdorff dimension on the surface of TIs. This novel set of Dirac fractals opens a fruitful venue to explore the fate of the bulk-boundary correspondence in TIs and other topological phases when the surface state dimensionality is non-trivial.   

Universality of Boundary Charge Fluctuations

We establish the quantum fluctuations ΔQ_B² of the charge Q_B accumulated at the boundary of an insulator as an integral tool to characterize phase transitions where a direct gap closes (and reopens), typically occurring for insulators with topological properties. The power of this characterization lies in its capability to treat different kinds of insulators on equal footing; being applicable to transitions between topological and non-topological band, Anderson, and Mott insulators alike. In the vicinity of the phase transition we find a universal scaling ΔQ_B²(E_g) as function of the gap size E_g and determine its generic form in various dimensions. For prototypical phase transitions with a massive Dirac-like bulk spectrum we demonstrate a scaling with the inverse gap in one dimension and a logarithmic one in two dimensions.   

Statistical learning of engineered topological phases in the kagome superlattice of AV3Sb5

Recently, the kagome metals AV₃Sb₅ (A=K,Rb,Cs) have gained intense research interest, as they display a wide spectrum of exotic topological properties, in addition to superconductivity, charge, orbital momentum and spin density waves. 

Motivated by a plethora of experimental evidence for unconventional charge orders in the enlarged (2×2) unit-cell of the vanadium based kagome metals, we investigate the type of topological phases that can manifest within the electronic parameter space of such kagome superlattices. 

Unlike conventional theoretical approaches, we employ a recently introduced statistical method capable of constructing topological models for any generic lattice, in an unbiased way without prior knowledge of its phase diagram. 

By extracting physically meaningful information from large datasets of randomized hopping parameters for the kagome superlattice, we find possible real-space manifestations of charge and bond modulations and associated flux patterns for different topological classes. 

Our results agree with contemporary theoretical propositions and experimental observations for the AV₃Sb₅ family.

Simultaneously, we predict new higher-order topological phases that may be realized by appropriately manipulating the currently known systems.    

Filling-enforced topological crystalline insulators

Topological crystalline insulators are topological gapped phases protected by crystal symmetries, which exhibits a variety of quantized electromagnetic phenomena. A large class of topological crystalline insulators are characterized by their non-trivial electric multipole moments, which include 1D systems with fractional polarization, and quadrupole insulators that host fractional quadrupole moments, etc.. In this work we provide theorems for the filling-enforced non-trivial topological multipole insulators. We first study a specific 1D model with 2 electrons per unit-cell protected by a spatial inversion and a glide reflection and show that it must have a fractional polarization. In the case of quadrupole insulators, we have systematically studied all the 17 wallpaper groups and established theorems that identify non-trivial quadrupole insulators enforced by certain space groups and filling conditions. Our theorems have the advantage of being robust to interactions, which serves as a powerful guide in experimental searches of non-trivial quadrupole insulators.   

Classification of locality preserving unitaries with symmetry

Locality preserving unitaries (LPUs) are unitary operators that map local operators to nearby local operators. In a given dimension $d$, some LPUs can be written as finite time evolution of a local $d$-dimensional Hamiltonian. Such LPUs are called finite depth local unitaries (FDLUs). Other LPUs are nontrivial in the sense that they cannot be written in this way, and can only be realized at the boundary of a $(d+1)$ dimensional FDLU. These LPUs have been studied in the context of quantum cellular automata and Floquet topological phases. The study of LPUs without symmetry is already a rich subject, and the classification of nontrivial LPUs in higher dimensions is not well understood. In this talk, I will discuss an easier problem: classifying LPUs that are nontrivial only in the presence of a global symmetry. Parts of the classification are related to symmetry protected topological phases, and other parts of the classification are entirely new.   

Cubic 3D Chern photonic insulators with orientable large Chern vectors

A 3D Chern insulator is a Time Reversal Symmetry (TRS) broken topological phase characterized by a vector of three first Chern invariants, associated with the planes supporting topologically protected surface states. In this work, we devise a general strategy to design 3D Chern Insulating (3D CI) cubic Photonic Crystals (PhCs) with orientable and arbitrarily large Chern vectors, in a reduced TRS broken environment. The strategy proceeds in two steps: formation of photonic Weyl points in a magnetic PhC, and their annihilation via geometric modulation on multifold supercells. The resulting crystals present the following novel characteristics: First, large Chern vectors can be obtained by design, making the PhC ideal for multi-modal operation. Second, full orientability of Chern vectors is achieved in the 3D space, opening up larger 3D CI/3D CI interfacing possibilities as compared to 2D Chern PhCs. Finally, non-zero Chern vectors can be achieved at reduced magnetization conditions, interestingly for photonic applications in the frequency regime where the magnetic response is weak.    

Modulation of a 3D Chern photonic crystal by means of Group Theory

This work supports the computational investigation of a 3D Chern photonic crystal insulator engineered to host orientable, large Chern triplets, by Devescovi et al (see reference), with an analytical model based on Group Theory.

The application of an external magnetic field creates a Weyl dipole that is gapped by a real-space cell modulation that couples the points. We develop a general expansion of the energy bands around a given wavevector respecting the crystal symmetries. First, we model the splitting of the degeneracy by the magnetic field giving to the Weyl points. Then, we describe their gapping into a non-trivial insulating phase. The results facilitate the design of the structure by predicting properties of the system before any time-consuming numerical computation and allows the identification of the residual symmetries as the phase transitions occur. In particular, it correctly identifies what kinds of modulations will gap the dipole based on their symmetry properties and reproduces some topological features of the energy bands such as the chirality of the nodal points.   

Anatomy of two-body-propagator topology

One convenient way to define topological quantum numbers for interacting topological insulators, is to directly analyze the single-particle propagator. Here, we extend this idea to two-body propagators, which characterize both bound and scattering states of two topological particles interacting via short-ranged interactions. Based on explicit examples, we show how to define topological quantum numbers even if the bound state bands have entered the scattering continuum. This allows to go beyond the commonly-assumed strong-coupling limit. We comment on topological phase transitions and the bulk-boundary correspondence in the two-body case.