Session F08: Interacting Topological Systems

Time-dependent Gutzwiller simulation of Floquet topological superconductivity

Periodically driven systems provide a novel route to control the topology of quantum materials. In particular, Floquet theory allows an effective band description of periodically-driven systems through the Floquet Hamiltonian. Along this direction, it was theoretically predicted that d-wave cuprate superconductors irradiated with circularly-polarized light (CPL) exhibit Floquet topological superconductivity purely from the many-body effect by employing the high frequency expansion (HFE) and deriving Floquet t–J model [Kitamura and Aoki, Commun. Phys. (2022)]. Here, we study the time evolution of d-wave superconductors irradiated with CPL [Anan et al., arXiv: 2309.06069]. We observe the development of the id_xy-wave pairing amplitude along with the original d_x²−y²-wave order upon gradual increasing of the field amplitude, owing to the three-site term with broken time-reversal symmetry. We further numerically construct the Floquet Hamiltonian for the steady state, with which we identify the system as the fully-gapped d+id-wave superconducting phase with a nonzero Chern number. We explore the low-frequency regime where the HFE breaks down, and find that the topological gap of an experimentally-accessible size can be achieved at much lower laser intensities.   

Gapped boundaries of 3D bosonic topological orders from mapping class group representations

Topologically ordered (TO) phases display interesting bulk-boundary relationships: a striking example of this is a chiral edge mode of a fractional quantum Hall phase. Non-chiral TOs, by contrast, allow gapped boundary conditions. Gapped boundaries of 2D TOs have been classified in terms of Lagrangian algebras. A similar systematic understanding in 3D has not been developed. In this work, we take steps in that direction. We consider a Dijkgraaf-Witten gauge theory defined on the spacetime T³ x S¹, where spatial coordinates form a 3-torus T³. The ground states of this 3D TO form a representation of the mapping class group of T³, isomorphic to SL(3,Z). In the quasiparticle basis, the eigenvectors stabilized by the S and T matrices – generators of SL(3,Z) – correspond to vector-valued partition functions of gapped boundary states. We demonstrate this for the simplest example, the 3D Z₂ TO, and find that our method reproduces the known rough and smooth boundaries. Further, we consider Z₂×Z₂ TO and find 5 classes of gapped boundaries. We relate our results to gapped phases of 2D symmetric quantum models via the Symmetry-TO correspondence. We propose a way to systematically identify symmetry-enforced gaplessness of 2D bosonic theories.   

Signatures of Supersymmetry in the ν=5/2 Fractional Quantum Hall Effect

The Moore-Read state, one of the leading candidates for describing the fractional quantum Hall effect at filling factor ν=5/2, is a paradigmatic p-wave superconductor with non-Abelian topological order. Among its many exotic properties, the state hosts two collective modes: a bosonic density wave and a neutral fermion mode that arises from an unpaired electron in the condensate. It has recently been proposed that the descriptions of the two modes can be unified by postulating supersymmetry (SUSY) that relates them in the long-wavelength limit. Here we extend the SUSY description to construct wave functions of the two modes on closed surfaces, such as the sphere and torus, and we test the resulting states in large-scale numerical simulations. We demonstrate the equivalence in the long-wavelength limit between SUSY wave functions and previous descriptions of collective modes based on the Girvin-MacDonald-Platzman ansatz, Jack polynomials, and bipartite composite fermions. Leveraging the first-quantized form of the SUSY wave functions, we study their energies using the Monte Carlo method and show that realistic ν=5/2 systems are close to the putative SUSY point, where the two collective modes become degenerate in energy.   

Probing THz magnetic optical response in two-dimensional correlated Mott insulator using time-dependent Wannier functions

The study of light-matter coupling is of great theoretical and experimental interest, between its potential to create new non-equilibrium states and ability to experimentally probe unconventional quantum phases. Unlike conventional THz spectroscopy, which relies on coupling spins to magnetic fields, we demonstrate another mechanism for spin-photon coupling in correlated insulators due to poor electron localization bounded by quantum geometry. Here, photons may excite a magnetic response by perturbing the wavefunction shape of local moments via a THz electric field. We evaluate the optical conductivity in two-dimensional Mott insulators while accounting for the time-dependent deformation of the Wannier functions [1]. We discuss ramifications on realistic materials with non-trivial quantum geometry.

[1] W. T. Tai and M. Claassen, Quantum-Geometric Light-Matter Coupling in Correlated Quantum Materials, arXiv:2303.01597.   

3-Loop Braiding and Topological Charge from Membrane Operators in 3+1d Twisted Lattice Gauge Theory

Membrane operators can be used to explicitly construct the loop-like excitations in toy models for topological phases. We discuss the 3+1d twisted lattice gauge theory model, which is a Hamiltonian realization of the Dijkgraaf-Witten topological field theory. By constructing the membrane operators that produce flux loops that are linked to an existing base flux loop, we find the three-loop braiding relations, which describe what happens when two loop-like excitations are exchanged while linked to a third loop. This explicit treatment gives us further insight into how non-Abelian three-loop braiding may arise even for Abelian groups. The membrane operators can also be used to find the conserved topological charges in the model, which are related to the ground state degeneracy.   

Topological Green's function zeros

The interplay of topology and strong correlations manifests itself in a plethora of exotic phenomena. Specifically, topological bands of Green's function zeros have recently attracted substantial interest. Here, we present an analytically tractable model displaying such topological bands of zeros in the fermionic Green's function when the system is tuned to a topologically ordered phase. We further demonstrate the existence of "edge states" of zeros and discuss their experimental implications, in particular when proximitized to edge states of non-interacting topological insulators. If time permits, we will also discuss transport signatures   

Berry curvature and topological characterization for strongly correlated electron systems

Characterizing the topological nature of electronic states in strongly correlated quantum materials,

especially in cases with nodal excitations, is a problem of extensive current interest [1,2]. In non-

interacting cases, Bloch functions specify the Berry curvature and quantized topological charge of

symmetry protected band crossings. However, in strongly correlated systems, in particular where the

quasiparticle picture fails, Bloch functions cannot be used and no suitable alternative approaches are

available. Here we fill this void by developing a Green’s function approach. We illustrate our

approach in the extreme correlation limit with nodal states. Our formalism offers a way to

systematically characterize electronic topology in strongly correlated settings [3].

[1] Hu, H., Chen, L., Setty, C. Garcia-Diez, M., Grefe, S.E., Prokofiev, A., Kirchner, S., Vergniory,

M.G., Paschen, S., Cano, J. and Si, Q., 2021. arXiv preprint arXiv:2110.06182.

[2] Setty, C., Sur, S., Chen, L., Xie, F., Hu, H., Paschen, S., Cano, J. and Si, Q., 2023. arXiv preprint

arXiv:2301.13870.

[3] Setty, C et al (in prep)   

Weyl-Kondo Semimetals in the Quantum Critical Regime

The interplay between interaction and topology is of extensive current interest [1]. In insulating systems, strong correlations are known to produce new topological phases, as exemplified by fractional quantum Hall effect. Whether correlations can drive novel gapless topological phases is an open question. Recently [2, 3], we have established that symmetry constraints also apply to the Green’s function, leading to the possibility of non-Fermi liquid topological semimetal states with correlated-induced emergent excitations. The first such realization happens in a quantum critical phase [2]. Here, we show [3] that Weyl-Kondo semimetal develops near a quantum critical point in a Kondo lattice system. We present the properties and topological signatures of this state. Our work charts a broad path to explore the relationship between gapless electronic topology and strong correlations.

 

 

[1] S. Paschen & Q. Si, Nat. Rev. Phys. 3, 9 (2021).

 

[2] H. Hu et al., arXiv:2110.06182

 

[3] L. Chen, H. Hu, et al., unpublished (2023)   

Density oscillation of Laughlin Quasihole

Quasiholes are emergent elementary particles in the fractional quantum Hall effect. Due to the correlation between electrons, they are not point objects but extended ones which exhibit rich internal structures. In this work, we show that the complex density distribution of Laughlin quasiholes in real space can be modeled with a simple damped oscillation in the occupation-number space, which are fully determined by two characteristic length scales, the decay length and oscillation wave vector. In this way, the Laughlin quasihole, Laughlin edge and the pair correlation function of Laughlin ground state can be studied on an equal footing. Moreover, by tuning the interaction from the parent Hamiltonian to Coulomb interaction, we find the density oscillation of quasihole will be primarily modified by the low-energy quasihole-magnetoroton states, leading to an additional pair of oscillation and decay length for more realistic quasiholes in the experiments. Our results can also be useful for the large scale numerical computation of ground state and quasihole state variational energies.   

Duality on the lattice and application to gapped/gapless topological phases

Topological phases of matter are classified based on the global symmetries of physical systems. When a system has a global symmetry, we can define duality transformations, such as gauging and fermionization. These transformations are also defined for lattice models. The dual model exhibits a dual global symmetry, while the original symmetry becomes local after the duality transformation. In this talk, we introduce the way to construct lattice models for various types of topological phases using dualities. Our models exhibit the same behavior as the usual dual model in the infrared (IR) regime. To achieve this, we enforce the Gauss law constraint as an energy cost. We point out that the model is generally different from the usual dual model in terms of topological phases. This difference arises from the requirement to treat an original global symmetry, which becomes local in the IR, as a global symmetry in our construction. As an application, we discuss a systematic approach for constructing models representing gapless topological phases.   

Spin-flip excitation and negative energy dispersion in rotating Bose atoms.

The fractional quantum Hall effect (FQHE) can potentially be studied in rotating Bose-Einstein condensate (BEC) [1]. A two-dimensional (2D) system of rapidly rotating BEC contained in XY-plane creates a fictitious magnetic field along Z-axis, that is perpendicular to the 2D-plane (similar to the magnetic field in a 2D electron system), which forms Landau levels (LL). Then there is a possibility of the FQHE in rapidly rotating system of dilute Bose atoms. I have studied collective spin-flip excitations (SFE) for the first three filling fractions of second and third series (2/(2p+1),3/(3p+1); p is an integer) of Jain’s composite fermion (CF) sequences [2]. I have considered short-ranged Poisson-Teller (PT) interactions between the Bose atoms as well as long range Coulomb interactions to compare the nature of the spectra with FQHE of electrons. Although very short-ranged interaction gives zero energy, as the average separation between particles will be large compared to the width of interaction, the PT interaction potential gives the freedom to control the interaction range. The nature of spectra does not depend on the range of interaction. The interesting fact is that I found anomalous negative dispersion in the excited spectra whereas there is a positive dispersion curve observed only for the Jain’s first series, i.e. for the fractions having form 1/(p+1) [3]. For the higher Jain series SFE shows negative curvature and roton minima at lower momenta; whereas for higher momenta it supports the spin-wave excitation similar as conventional ferromagnets. This phenomenon is produced by mixing of two types of spin-reversed modes: spin-wave modes with no change in LL index and spin-flip modes with decreasing LL index. This combination of two modes produces a spin-flip excitonic state of CF particle-hole pairs.

References:

[1] C. C. Chang, N. Regnault, T. Jolicoeur, J. K. Jain, Phys. Rev. A 72 (2005) 013611.

[2] N. R. Cooper, N.K. Wilkin, Phys. Rev. B 60 (1999) R16279.

[3] M. Indra, D. Majumder, Solid State Communications, 306, art. no. 113796 (2020).   

The Formalism of Conformal Hilbert Spaces and Scale-free Interaction in Fractional Quantum Hall Effect

The fractional quantum Hall (FQH) effect is a family of strongly correlated topological systems in two-dimension, with exotic low lying charge excitations that are anyonic and even non-Abelian. Here we propose a unified framework in understanding the integer and fractional quantum Hall systems via Hilbert space truncation. This framework is closely related to the well-known microscopic pseudopotential and Jack polynomial formalism. The resulting Hilbert spaces with emergent conformal symmetry (i.e. the conformal Hilbert spaces (CHS)) have well-defined topological properties with anyons as "elementary particles". The hierarchical structure of the CHS allows us to reveal internal structures of anyons of the FQH phases (e.g. the Laughlin and Moore-Read phases), and to derive experimentally relevant conditions for such anyons or quasiholes to undergo fractionalisation with the same topological phase. More interestingly, the CHS formalism can also be generalised to the sub-Hilbert spaces of multiple Landau levels (LLs) with scale-free interactions. With such interactions filling factor dependent LL mixing can occur even in the limit of large cyclotron gap. We also propose a novel experimental platform for approximately realising scale-free interactions, that can potentially lead to very robust (non-Abelian) FQH phases from two-body Coulomb-based interaction.

[1] Bo Yang, arXiv: 2307.06361.

[2] Ha Quang Trung and Bo Yang, Phys. Rev. Lett. 127, 046402 (2021).   

Exactly solvable anyonic interferometer on a single edge of a topological liquid

Experimental signatures of anyonic statistics are a key focus in physics. A major milestone in this pursuit has been the direct confirmation of anyonic statistics through interferometry. Within this realm, two configurations of interferometers, namely Fabry-P{'e}rot and Mach-Zehnder setups, have been successfully implemented. However, since the theoretical considerations resort to perturbative treatment, simple theoretical expressions for the electric currents and noises are unavailable at higher visibility of Aharonov-Bohm osculations.

In this talk, we introduce an alternative version of the Mach-Zehnder interferometer [1] in which the quasiparticles tunnel between co-propagating chiral channels on the edges of quantum Hall liquid at the bulk filling factors $ u= n/(2n + 1)$. Unlike the conventional Mach-Zehnder interferometers, this configuration does not require the placement of an Ohmic contact inside the device and directly parallels the optical Mach-Zehnder geometry. Remarkably, this setup admits an easy exact solution for any visibility, including the experimentally optimal regime. The simple expressions found for electric current and noise contain information about fractional charge and fractional statistics.

[1] N. Batra, Z. Wei, S. Vishveshwara, and D. E. Feldman, ArXiv:2308.05236