Bulletin of the American Physical Society
APS March Meeting 2022
Volume 67, Number 3
Monday–Friday, March 14–18, 2022; Chicago
Session Q59: Theoretical Methods for Topological InsulatorsRecordings Available

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Sponsoring Units: DCMP Chair: Frank Schindler, Princeton University Room: Hyatt Regency Hotel DuSable AB 
Wednesday, March 16, 2022 3:00PM  3:12PM 
Q59.00001: Manybody quadrupolar sum rule for higherorder topological insulator Wonjun Lee, Gil Young Cho, Byungmin Kang The modern theory of polarization establishes the bulkboundary correspondence for the bulk polarization. We attempt to extend it to a sum rule of the bulk quadrupole moment by employing a manybody 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 manybody operator in different boundary conditions. We demonstrate the validity of the sum rule through extensive numerical computations on various noninteracting tightbinding models. We also observe that individual terms in the sum rule correspond to the bulk quadrupole moment, the edgelocalized polarizations, and the corner charge in the thermodynamic limit on some models. 
Wednesday, March 16, 2022 3:12PM  3:24PM 
Q59.00002: A bulkboundary correspondence for 3D bosonic symmetry protected topological phases Kyle Kawagoe, Michael Levin 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 Fsymbols for string like domain wall excitations in the boundary theory which are then identified with an important bulk topological invariant. 
Wednesday, March 16, 2022 3:24PM  3:36PM 
Q59.00003: Classification of (2+1)D invertible fermionic topological phases with symmetry Naren Manjunath, Maissam Barkeshli, YuAn Chen, Poshen Hsin 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 ω_{2} in the group H^{2}(G_{b}, Z_{2}). Our approach proceeds by gauging fermion parity and classifying the resulting G_{b} symmetryenriched topological orders while keeping track of certain additional data and constraints. We perform this analysis through two perspectives, using Gcrossed braided tensor categories and Spin(2c_{})_{1} ChernSimons 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 10fold way classification of topological insulators and superconductors fits into our scheme, along with general nonperturbative constraints due to certain choices of c_{} and G_{f}. 
Wednesday, March 16, 2022 3:36PM  3:48PM 
Q59.00004: A DMRG study of the extended HaldaneHubbard model Rafael Miksian Magaldi In this work, we study the extended HaldaneHubbard model at halffilling in a honeycomb lattice. We employ the density matrix renormalization group algorithm in the matrix product state formulation (DMRGMPS) to obtain the ground state of the system and generate its phase diagram. In agreement with previous results, we find that strong onsite interactions can create a spin density wave phase, whereas strong nearestneighbor interactions favor a charge density wave phase. A topologically nontrivial phase was also found in the weaklyinteracting regime, by measuring the Chern number using a chargepumping 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. 
Wednesday, March 16, 2022 3:48PM  4:00PM 
Q59.00005: PancharatnamZak Phase for TwoDimensional Fermionic Systems Sepide Mohamadi, Jahanfar Abouie Topological states of matter are usually classified by quantized topological invariants. In onedimensional crystals, the Zak phase is a useful invariant to detect topological states. However, Zak phase is gaugedependent, 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 gaugeinvariant topological numbers. In this work we define PancharatnamZak phase, a gauge independent topological number for twodimensional fermionic systems, and investigate the phase diagram of two topological systems: i) the SuSchriefferHeeger, and ii) the Kitaev models. In twodimensional systems, the Chern number is usually used to examine the topological phases, however in the presence of timereversal and inversion symmetries, the Chern number is vanishing, and we have to employ other invariants. We show that the PhancharatnamZak phase can be employed for characterizing the topological phases of twodimensional fermionic systems. 
Wednesday, March 16, 2022 4:00PM  4:12PM 
Q59.00006: HigherForm Response of 3D Fractional Topological Insulators Benjamin T Moy, Hart Goldman, Ramanjit Sohal, Eduardo H Fradkin Threedimensional fractional topological insulators (FTIs) are exotic symmetry enriched phases hosting extended defects with anyonic statistics. Here we consider the higherform 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 twoform gauge field that probes their electric Z_{N} oneform symmetry, which we show leads to a universal topological index for the FTI state. 
Wednesday, March 16, 2022 4:12PM  4:24PM 
Q59.00007: Fractional hinge and corner charges in various crystal shapes with cubic symmetry katsuaki naito 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 vertextransitive polyhedra with the cubic symmetry such as a cube, an octahedron and a cuboctahedron [1]. We derive realspace 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 momentumspace 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 highsymmetry momenta are not sufficient to determine the corner charge. In order to resolve this issue, we introduce an additional Wilsonloop invariant. We show that the momentumspace formulas for the corner charges can be obtained by combining the EBR matrix with the new Wilsonloop invariant. 
Wednesday, March 16, 2022 4:24PM  4:36PM 
Q59.00008: Dirac Fractals on the Surface of Topological Insulators Lakshmi Pullasseri Madom Narayana Iyer, Daniel Shaffer, Luiz H Santos 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 symmetryprotected 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 timereversal 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 noninteger Hausdorff dimension on the surface of TIs. This novel set of Dirac fractals opens a fruitful venue to explore the fate of the bulkboundary correspondence in TIs and other topological phases when the surface state dimensionality is nontrivial. 
Wednesday, March 16, 2022 4:36PM  4:48PM 
Q59.00009: Universality of Boundary Charge Fluctuations Clara S Weber, Kiryl Piasotski, Mikhail Pletyukhov, Jelena Klinovaja, Daniel Loss, Herbert Schoeller, Dante M Kennes We establish the quantum fluctuations ΔQ_{B}^{2} 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 nontopological band, Anderson, and Mott insulators alike. In the vicinity of the phase transition we find a universal scaling ΔQ_{B}^{2}(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 Diraclike bulk spectrum we demonstrate a scaling with the inverse gap in one dimension and a logarithmic one in two dimensions. 
Wednesday, March 16, 2022 4:48PM  5:00PM 
Q59.00010: Statistical learning of engineered topological phases in the kagome superlattice of AV_{3}Sb_{5} Paul Wunderlich, Thomas Mertz, Shinibali Bhattacharyya, Francesco Ferrari, Roser Valenti Recently, the kagome metals AV_{3}Sb_{5} (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. 
Wednesday, March 16, 2022 5:00PM  5:12PM 
Q59.00011: Fillingenforced topological crystalline insulators Xu Yang, Yuanming Lu 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 nontrivial 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 fillingenforced nontrivial topological multipole insulators. We first study a specific 1D model with 2 electrons per unitcell 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 nontrivial 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 nontrivial quadrupole insulators. 
Wednesday, March 16, 2022 5:12PM  5:24PM 
Q59.00012: Classification of locality preserving unitaries with symmetry Carolyn Zhang, Michael Levin 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. 
Wednesday, March 16, 2022 5:24PM  5:36PM 
Q59.00013: Cubic 3D Chern photonic insulators with orientable large Chern vectors Chiara Devescovi, Mikel García Díez, Maia G Garcia Vergniory, Iñigo Robredo, María Blanco de Paz, Barry Bradlyn, Juan Luis Mañes, Aitzol GarcíaEtxarri 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 multimodal 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, nonzero Chern vectors can be achieved at reduced magnetization conditions, interestingly for photonic applications in the frequency regime where the magnetic response is weak. 
Wednesday, March 16, 2022 5:36PM  5:48PM 
Q59.00014: Modulation of a 3D Chern photonic crystal by means of Group Theory Mikel García Díez, Chiara Devescovi, Maia G Garcia Vergniory, Iñigo Robredo, María Blanco de Paz, Barry Bradlyn, Juan Luis Mañes, Aitzol GarcíaEtxarri

Wednesday, March 16, 2022 5:48PM  6:00PM 
Q59.00015: Anatomy of twobodypropagator topology Dimitri Pimenov, Moshe Goldstein, Alex Kamenev One convenient way to define topological quantum numbers for interacting topological insulators, is to directly analyze the singleparticle propagator. Here, we extend this idea to twobody propagators, which characterize both bound and scattering states of two topological particles interacting via shortranged 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 commonlyassumed strongcoupling limit. We comment on topological phase transitions and the bulkboundary correspondence in the twobody case. 
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