Topological Hamiltonian Research Articles

Anisotropic Josephson Diode Effect in the Topological Hybrid Junctions with the Hexagonal Warping

Recently the diode effect in superconductivity became an active area of research. In particular, the three-dimensional topological insulators may be one of the most suitable materials to implement the superconducting diodes. It is common to consider only linear and quadratic terms of the topological insulator Hamiltonian in the low energy expansion. Typically the effect of the hexagonal warping is neglected. However, the hexagonal warping can be very significant in consideration of the transport properties of the TI materials, such as Bi2Se3 or Bi2Te3. In this theoretical work we present the study of the Josephson diode effect based on the topological insulator weak link. We address the question of the hexagonal warping influence on the Josephson diode effect. We argue that the warping term leads to the anisotropy of the Josephson diode effect.

Programmable integrated photonics for topological Hamiltonians

A variety of topological Hamiltonians have been demonstrated in photonic platforms, leading to fundamental discoveries and enhanced robustness in applications such as lasing, sensing, and quantum technologies. To date, each topological photonic platform implements a specific type of Hamiltonian with inexistent or limited reconfigurability. Here, we propose and demonstrate different topological models by using the same reprogrammable integrated photonics platform, consisting of a hexagonal mesh of silicon Mach-Zehnder interferometers with phase shifters. We specifically demonstrate a one-dimensional Su-Schrieffer-Heeger Hamiltonian supporting a localized topological edge mode and a higher-order topological insulator based on a two-dimensional breathing Kagome Hamiltonian with three corner states. These results highlight a nearly universal platform for topological models that may fast-track research progress toward applications of topological photonics and other coupled systems.

Ensemble-learning error mitigation for variational quantum shallow-circuit classifiers

Classification is one of the main applications of supervised learning. Recent advancements in developing quantum computers have opened a new possibility for machine learning on such machines. Due to the noisy performance of near-term quantum computers, error mitigation techniques are essential for extracting meaningful data from noisy raw experimental measurements. Here, we propose two ensemble-learning error mitigation methods, namely, bootstrap aggregating and adaptive boosting, which can significantly enhance the performance of variational quantum classifiers for both classical and quantum datasets. The idea is to combine several weak classifiers, each implemented on a shallow noisy quantum circuit, to make a strong one with high accuracy. While both of our protocols substantially outperform error-mitigated primitive classifiers, the adaptive boosting shows better performance than the bootstrap aggregating. The protocols have been exemplified for classical handwriting digits as well as quantum phase discrimination of a symmetry-protected topological Hamiltonian, in which we observe a significant improvement in accuracy. Our ensemble-learning methods provide a systematic way of utilizing shallow circuits to solve complex classification problems. Published by the American Physical Society 2024

Multi-terminal Josephson junctions: A road to topological flux networks

Multi-terminal Josephson junctions were recently proposed as a versatile and tunable platform to emulate topological Bloch-like Hamiltonians in arbitrary dimensions. In this perspective article, we will give a brief overview of the subject and recognize these mesoscopic devices as realizations of topological flux networks as the ones envisioned by Avron and coworkers in their seminal works on the early days of the quantum Hall effect. We summarize the current state-of-the-art theoretical and experimental research regarding these Josephson devices, highlighting recent developments and giving an outlook on current trends.

Large-scale geometry obstructs localization

We explain the coarse geometric origin of the fact that certain spectral subspaces of topological insulator Hamiltonians are delocalized, in the sense that they cannot admit an orthonormal basis of localized wavefunctions, with respect to any uniformly discrete set of localization centers. This is a robust result requiring neither spatial homogeneity nor symmetries and applies to Landau levels of disordered quantum Hall systems on general Riemannian manifolds.

Stabilizing multiple topological fermions on a quantum computer

In classical and single-particle settings, non-trivial band topology always gives rise to robust boundary modes. For quantum many-body systems, however, multiple topological fermions are not always able to coexist, since Pauli exclusion prevents additional fermions from occupying the limited number of available topological modes. In this work, we show, through IBM quantum computers, how one can robustly stabilize more fermions than the number of topological modes through specially designed 2-fermion interactions. Our demonstration hinges on the realization of BDI- and D-class topological Hamiltonians on transmon-based quantum hardware, and relied on a tensor network-aided circuit recompilation approach. We also achieved the full reconstruction of multiple-fermion topological band structures through iterative quantum phase estimation (IQPE). All in all, our work showcases how advances in quantum algorithm implementation enable noisy intermediate-scale quantum (NISQ) devices to be exploited for topological stabilization beyond the context of single-particle topological invariants.

Subgap states at ferromagnetic and spiral-ordered magnetic chains in two-dimensional superconductors. II. Topological classification

We investigate the topological classification of the subgap bands induced in a two-dimensional superconductor by a densely packed chain of magnetic moments with ferromagnetic or spiral alignments. The wave functions for these bands are composites of Yu-Shiba-Rusinov-type states and magnetic scattering states and have a significant spatial extension away from the magnetic moments. We show that this spatial structure prohibits a straightforward extraction of a Hamiltonian useful for the topological classification. To address the latter correctly we construct a family of spatially varying topological Hamiltonians for the subgap bands adapted for the broken translational symmetry caused by the chain. The spatial dependence in particular captures the transition to the topologically trivial bulk phase when moving away from the chain by showing how this, necessarily discontinuous, transition can be understood from an alignment of zeros with poles of Green's functions. Through the latter the topological Hamiltonians reflect a characteristic found otherwise primarily in strongly interacting systems.

Topological holographic quench dynamics in a synthetic frequency dimension

The notion of topological phases extended to dynamical systems stimulates extensive studies, of which the characterization of nonequilibrium topological invariants is a central issue and usually necessitates the information of quantum dynamics in both the time and momentum dimensions. Here, we propose the topological holographic quench dynamics in synthetic dimension, and also show it provides a highly efficient scheme to characterize photonic topological phases. A pseudospin model is constructed with ring resonators in a synthetic lattice formed by frequencies of light, and the quench dynamics is induced by initializing a trivial state, which evolves under a topological Hamiltonian. Our key prediction is that the complete topological information of the Hamiltonian is encoded in quench dynamics solely in the time dimension, and is further mapped to lower-dimensional space, manifesting the holographic features of the dynamics. In particular, two fundamental time scales emerge in the dynamical evolution, with one mimicking the topological band on the momentum dimension and the other characterizing the residue time evolution of the state after the quench. For this, a universal duality between the quench dynamics and the equilibrium topological phase of the spin model is obtained in the time dimension by extracting information from the field evolution dynamics in modulated ring systems in simulations. This work also shows that the photonic synthetic frequency dimension provides an efficient and powerful way to explore the topological nonequilibrium dynamics.

Elementary band representations for the single-particle Green's function of interacting topological insulators

We discuss the applicability of elementary band representations (EBRs) to diagnose spatial- and time-reversal-symmetry protected topological phases in interacting insulators in terms of their single-particle Green's functions. We do so by considering an auxiliary non-interacting system $H_{\textrm{T}}(\mathbf{k}) = -G^{-1}(0, \mathbf{k})$, known as the topological Hamiltonian, whose bands can be labeled by EBRs. This labeling is stable if neither (i) the gap in the spectral function at zero frequency closes, (ii) the Green's function has a zero at zero frequency or (iii) the Green's function breaks a protecting symmetry. We demonstrate the use of EBRs applied to the Green's function on the one-dimensional Su-Schrieffer-Heeger model with Hubbard interactions, which we solve by exact diagonalization for a finite number of unit cells. Finally, the use of EBRs for the Green's function to diagnose so-called symmetry protected topological phases is discussed, but remains an open question.

Topological input-output theory for directional amplification

We present a topological approach to the input-output relations of photonic driven-dissipative lattices acting as directional amplifiers. Our theory relies on a mapping from the optical non-Hermitian coupling matrix to an effective topological insulator Hamiltonian. This mapping is based on the singular value decomposition of non-Hermitian coupling matrices, whose inverse matrix determines the linear input-output response of the system. In topologically non-trivial regimes, the input-output response of the lattice is dominated by singular vectors with zero singular values that are the equivalent of zero-energy states in topological insulators, leading to directional amplification of a coherent input signal. In such topological amplification regime, our theoretical framework allows us to fully characterize the amplification properties of the quantum device such as gain, bandwidth, added noise, and noise-to-signal ratio. We exemplify our ideas in a one-dimensional non-reciprocal photonic lattice, for which we derive fully analytical predictions. We show that the directional amplification is near quantum-limited with a gain growing exponentially with system size, $N$, while the noise-to-signal ratio is suppressed as $1/\sqrt{N}$. This points out to interesting applications of our theory for quantum signal amplification and single-photon detection.

Measuring an interaction-induced topological phase transition via the single-particle density matrix

We discuss the topological phase transition of the spin-$\frac{1}{2}$ fermionic Haldane model with repulsive on-site interaction. We show that the Berry curvature of the topological Hamiltonian, the first Chern number, and the topological phase transition point can be extracted from the single-particle density matrix for this interacting system. Furthermore, we design a tomography scheme for the single-particle density matrix of interacting fermionic two-band models in experimental realizations with cold atoms in optical lattices.

Statistical analysis of the Chern number in the interacting Haldane-Hubbard model

In the context of many-body interacting systems described by a topological Hamiltonian, we investigate the robustness of the Chern number with respect to different sources of error in the self-energy. In particular, we analyze the importance of non-local (momentum dependent) vs. local contributions to the self-energy and show that the local self-energy provides a qualitative description of the topological phase diagrams of many-body interacting systems, whereas the explicit momentum-dependence constitutes a correction to the exact location of the phase transition. For the latter, we propose a statistical analysis, on the basis of which we develop a stochastic upper bound for the uncertainty of the Chern number as a function of the amount of momentum-dependence of the self-energy. We apply this analysis to the Haldane-Hubbard model and discuss the implications of our results for a general class of many-body interacting systems.

Quantum cluster approach to the topological invariants in correlated Chern insulators

We detect the topological properties of Chern insulators with strong Coulomb interactions by use of cluster perturbation theory and variational cluster approach. The common scheme in previous studies only involves the calculation of the interacting bulk Chern number within the natural-unit cell by means of the so-called topological Hamiltonian. With close investigations on a prototype model, the half-filled Haldane Hubbard model, which is subject to both periodic and open boundary conditions, we uncover the unexpected failure of this scheme due to the explicit breaking of the translation symmetry. Instead, we assert that the faithful interacting bulk Chern number in the framework of quantum cluster approaches can be computed in the enlarged unit cell, which is free of the fault caused by the explicit translation symmetry breaking and consistent with the interacting bulk-edge correspondence.

Defining a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition

We address the breakdown of the bulk-boundary correspondence observed in non-Hermitian systems, where open and periodic systems can have distinct phase diagrams. The correspondence can be completely restored by considering the Hamiltonian's singular value decomposition instead of its eigendecomposition. This leads to a natural topological description in terms of a flattened singular decomposition. This description is equivalent to the usual approach for Hermitian systems and coincides with a recent proposal for the classification of non-Hermitian systems. We generalize the notion of the entanglement spectrum to non-Hermitian systems, and show that the edge physics is indeed completely captured by the periodic bulk Hamiltonian. We exemplify our approach by considering the chiral non-Hermitian Su-Schrieffer-Heger and Chern insulator models. Our work advocates a different perspective on topological non-Hermitian Hamiltonians, paving the way to a better understanding of their entanglement structure.

Renormalization group flows for Wilson-Hubbard matter and the topological Hamiltonian

Understanding the robustness of topological phases of matter in the presence of interactions poses a difficult challenge in modern condensed matter, showing interesting connections to high energy physics. In this work, we leverage these connections to present a complete analysis of the continuum long-wavelength description of a generic class of correlated topological insulators: Wilson-Hubbard topological matter. We show that a Wilsonian renormalization group (RG) approach, combined with the so-called topological Hamiltonian, provide a quantitative route to understand interaction-induced topological phase transitions that occur in Wilson-Hubbard matter. We benchmark two-loop RG predictions for a quasi-1D Wilson-Hubbard model by means of exhaustive numerical simulations based on matrix product states (MPS). The agreement of the RG predictions with MPS simulations motivates the extension of the RG calculations to higher-dimensional topological insulators.

Anatomy of topological surface states: Exact solutions from destructive interference on frustrated lattices

The hallmark of topological phases is their robust boundary signature whose intriguing properties---such as the one-way transport on the chiral edge of a Chern insulator and the sudden disappearance of surface states forming open Fermi arcs on the surfaces of Weyl semimetals---are impossible to realize on the surface alone. Yet, despite the glaring simplicity of non-interacting topological bulk Hamiltonians and their concomitant energy spectrum, the detailed study of the corresponding surface states has essentially been restricted to numerical simulation. In this work, however, we show that exact analytical solutions of both topological and trivial surface states can be obtained for generic tight-binding models on a large class of geometrically frustrated lattices in any dimension without the need for fine-tuning of hopping amplitudes. Our solutions derive from local constraints tantamount to destructive interference between neighboring layer lattices perpendicular to the surface and provide microscopic insights into the structure of the surface states that enable analytical calculation of many desired properties. We illustrate our general findings on a large number of examples in two and three spatial dimensions. Notably, we derive exact chiral Chern insulator edge states on the spin orbit-coupled kagome lattice, and Fermi arcs relevant for various recently synthesized pyrochlore iridate slabs. Remarkably, each of the pyrochlore slabs exhibit Fermi arcs although only the ones with a magnetic one-in-three-out configuration feature bulk Weyl nodes in realistic parameter regimes. Our approach furthermore signal the absence of topological surface states, which we illustrate for a class of models akin to the trivial surface of Hourglass materials KHg$X$.

Observation of Topological Links Associated with Hopf Insulators in a Solid-State Quantum Simulator**This work was supported by the grants from the Ministry of Science and Technology of China and the Ministry of Education. L.M.D. and S.T.W. acknowledge in addition support from the ARL and the AFOSR MURI programs. D.L.D. is supported by JQI-NSF-PFC and LPS-MPO-CMTC.

Hopf insulators are intriguing three-dimensional topological insulators characterized by an integer topological invariant. They originate from the mathematical theory of Hopf fibration and epitomize the deep connection between knot theory and topological phases of matter, which distinguishes them from other classes of topological insulators. Here, we implement a model Hamiltonian for Hopf insulators in a solid-state quantum simulator and report the first experimental observation of their topological properties, including nontrivial topological links associated with the Hopf fibration and the integer-valued topological invariant obtained from a direct tomographic measurement. Our observation of topological links and Hopf fibration in a quantum simulator opens the door to probe rich topological properties of Hopf insulators in experiments. The quantum simulation and probing methods are also applicable to the study of other intricate three-dimensional topological model Hamiltonians.

Destructive quantum interference in electron transport: A reconciliation of the molecular orbital and the atomic orbital perspective

Destructive quantum interference (DQI) in single molecule electronics is a purely quantum mechanical effect and is entirely defined by the inherent properties of the molecule in the junction such as its structure and symmetry. This definition of DQI by molecular properties alone suggests its relation to other more general concepts in chemistry as well as the possibility of deriving simple models for its understanding and molecular device design. Recently, two such models have gained a wide spread attention, where one was a graphical scheme based on visually inspecting the connectivity of the carbon sites in conjugated π systems in an atomic orbital (AO) basis and the other one puts the emphasis on the amplitudes and signs of the frontier molecular orbitals (MOs). There have been discussions on the range of applicability for these schemes, but ultimately conclusions from topological molecular Hamiltonians should not depend on whether they are drawn from an AO or a MO representation, as long as all the orbitals are taken into account. In this article, we clarify the relation between both models in terms of the zeroth order Green’s function and compare their predictions for a variety of systems. From this comparison, we conclude that for a correct description of DQI from a MO perspective, it is necessary to include the contributions from all MOs rather than just those from the frontier orbitals. The cases where DQI effects can be successfully predicted within a frontier orbital approximation we show them to be limited to alternant even-membered hydrocarbons, as a direct consequence of the Coulson-Rushbrooke pairing theorem in quantum chemistry.

From four- to two-channel Kondo effect in junctions of XY spin chains

We consider the Kondo effect in Y-junctions of anisotropic XY models in an applied magnetic field along the critical lines characterized by a gapless excitation spectrum. We find that, while the boundary interaction Hamiltonian describing the junction can be recasted in the form of a four-channel, spin-1/2 antiferromagnetic Kondo Hamiltonian, the number of channels effectively participating in the Kondo effect depends on the chain parameters, as well as on the boundary couplings at the junction. The system evolves from an effective four-channel topological Kondo effect for a junction of XX-chains with symmetric boundary couplings into a two-channel one at a junction of three quantum critical Ising chains. The effective number of Kondo channels depends on the properties of the boundary and of the bulk. The XX-line is a “critical” line, where a four-channel topological Kondo effect can be recovered by fine-tuning the boundary parameter, while along the line in parameter space connecting the XX-line and the critical Ising point the junction is effectively equivalent to a two-channel topological Kondo Hamiltonian. Using a renormalization group approach, we determine the flow of the boundary couplings, which allows us to define and estimate the critical couplings and Kondo temperatures of the different Kondo (pair) channels. Finally, we study the local transverse magnetization in the center of the Y-junction, eventually arguing that it provides an effective tool to monitor the onset of the two-channel Kondo effect.

Experimental realization of the topological Haldane model with ultracold fermions.

The Haldane model on a honeycomb lattice is a paradigmatic example of a Hamiltonian featuring topologically distinct phases of matter. It describes a mechanism through which a quantum Hall effect can appear as an intrinsic property of a band structure, rather than being caused by an external magnetic field. Although physical implementation has been considered unlikely, the Haldane model has provided the conceptual basis for theoretical and experimental research exploring topological insulators and superconductors. Here we report the experimental realization of the Haldane model and the characterization of its topological band structure, using ultracold fermionic atoms in a periodically modulated optical honeycomb lattice. The Haldane model is based on breaking both time-reversal symmetry and inversion symmetry. To break time-reversal symmetry, we introduce complex next-nearest-neighbour tunnelling terms, which we induce through circular modulation of the lattice position. To break inversion symmetry, we create an energy offset between neighbouring sites. Breaking either of these symmetries opens a gap in the band structure, which we probe using momentum-resolved interband transitions. We explore the resulting Berry curvatures, which characterize the topology of the lowest band, by applying a constant force to the atoms and find orthogonal drifts analogous to a Hall current. The competition between the two broken symmetries gives rise to a transition between topologically distinct regimes. By identifying the vanishing gap at a single Dirac point, we map out this transition line experimentally and quantitatively compare it to calculations using Floquet theory without free parameters. We verify that our approach, which allows us to tune the topological properties dynamically, is suitable even for interacting fermionic systems. Furthermore, we propose a direct extension to realize spin-dependent topological Hamiltonians.