Hopping Term Research Articles

Spectral form factor in a minimal bosonic model of many-body quantum chaos.

We study spectral form factor in periodically kicked bosonic chains. We consider a family of models where a Hamiltonian with the terms diagonal in the Fock space basis, including random chemical potentials and pairwise interactions, is kicked periodically by another Hamiltonian with nearest-neighbor hopping and pairing terms. We show that, for intermediate-range interactions, the random phase approximation can be used to rewrite the spectral form factor in terms of a bistochastic many-body process generated by an effective bosonic Hamiltonian. In the particle-number conserving case, i.e., when pairing terms are absent, the effective Hamiltonian has a non-Abelian SU(1,1) symmetry, resulting in universal quadratic scaling of the Thouless time with the system size, irrespective of the particle number. This is a consequence of degenerate symmetry multiplets of the subleading eigenvalue of the effective Hamiltonian and is broken by the pairing terms. In such a case, we numerically find a nontrivial systematic system-size dependence of the Thouless time, in contrast to a related recent study for kicked fermionic chains.

Dirac fermions with plaquette interactions. I. SU(2) phase diagram with Gross-Neveu and deconfined quantum criticalities

We investigate the ground state phase diagram of an extended Hubbard model with $\pi$-flux hopping term at half-filling on a square lattice, with unbiased large-scale auxiliary-field quantum Monte Carlo simulations. As a function of interaction strength, there emerges an intermediate phase which realizes two interaction-driven quantum critical points, with the first between the Dirac semimetal and an insulating phase of weak valence bond solid (VBS) order, and the second separating the VBS order and an antiferromagnetic insulating phase. These intriguing quantum critical points are respectively bestowed with Gross-Neveu and deconfined quantum criticalities, and the critical exponents $\eta_\text{VBS}=0.6(1)$ and $\eta_\text{AF}=0.58(3)$ at deconfined quantum critical point satisfy the CFT Bootstrap bound. We also investigate the dynamical properties of the spin excitation and find the spin gap open near the first transition and close at the second. The relevance of our findings in realizing deconfined quantum criticality in fermion systems and the implication to lattice models with further extended interactions such as those in quantum Moir\'e systems, are discussed.

Interplay of disorder and point-gap topology: Chiral modes, localization, and non-Hermitian Anderson skin effect in one dimension

Symmetry-protected spectral topology in non-Hermitian systems has interesting manifestations such as dynamically anomalous chiral currents and skin effect. We study the interplay between symmetries and disorder in a paradigmatic model for spectral topology - the non-reciprocal Su-Schrieffer-Heeger model. We numerically study the effect of disorder in on-site and non-reciprocal hopping terms. Using a real-space winding number, we investigate the impact of disorder on the spectral topology and the anomalous chiral modes under periodic boundary conditions. We discover a remarkable robustness of chiral current under disorder. The value of the chiral current retains the clean system value, is independent of disorder strength and is tracked completely by the real-space winding number for class A which has no symmetries, and class AIII, which has a sub-lattice symmetry. In class $D^\dagger$, which has $PT$-symmetric on-site gain and loss terms, we find that the disorder-averaged current is not robust while the winding number is robust. We study the localization physics using the inverse participation ratio and local density of states. As the disorder strength is increased, a mobility-edge phase with a finite winding appears. The abrupt vanishing of the winding number marks a transition from a partially localized to a fully localized phase. Under open boundary conditions, we observe a series of transitions through skin effect-partial skin effect-no skin effect phases. Further, we study the non-Hermitian Anderson skin effect (NHASE) for different symmetry classes, where the system without skin effect develops a disorder-driven skin effect at intermediate disorder values. Remarkably while NHASE is present for different classes, the real-space winding number shows a direct correspondence with it only when all symmetries are broken.

Experimental realization of boundary-obstructed topological insulators using acoustic two-dimensional Su–Schrieffer–Heeger network

Topological insulators that can host special symmetry-protected boundary states and corner states have attracted increasing intention in acoustic engineering. Recently, the concept of the boundary-obstructed topological (BOT) phases has defined a class of topological phases without bulk energy band closing around zero energy, which greatly broadens the applications of the topological states. In this work, based on the two-dimensional Su–Schrieffer–Heeger network, we show that the band degeneracies around zero energy can be removed to open a complete bandgap by judiciously tuning the hopping terms to break C4v symmetry down to C2v symmetry but with the topological phase invariant, which can be directly characterized by the BOT phase. Furthermore, we experimentally propose a rigorous acoustic sample to visualize the hierarchy of the in-gap higher-order topological states exactly. Crucially, by designedly connecting the lattice with outside environment, we show that these spectrally isolated states still response to the specific frequencies robustly. Our results are expected to be helpful for manipulating wave propagation and sound energy harvesting.

Quantum transport in nonlinear Rudner-Levitov models

Quantum transport in a class of nonlinear extensions of the Rudner-Levitov model is numerically studied in this paper. We show that the quantization of the mean displacement, which embodies the quantum coherence and the topological characteristics of the model, is markedly modified by nonlinearities. Peculiar effects such as a ``trivial-nontrivial'' transition and unidirectional long-range quantum transport are observed. These phenomena can be understood on the basis of the dynamic behavior of the effective hopping terms, which are time and position dependent, containing contributions of both the linear and nonlinear couplings.

Kinetic formation of trimers and multimers in a spinless fermionic chain

We study a chain of spinless fermions with a multimer hopping term which kinematically favors the formation of multimers and competes with single-particle hopping. We argue that this model generically stabilizes two different multimer phases, as well as intermediate phases where the free-fermion and multimer fluids coexist and do not spatially separate. Using density-matrix renormalization group techniques, we establish the phase diagram of the model in the case of trimers. For one of the intermediate phases, hybridization between the fermions and trimers liquids does occur and the onset of their correlations is well captured by a generalized BCS-like ansatz. The case of tetramers is finally addressed.

Unconventional superconductivity due to interband polarization

We analyze in detail the superconductivity that arises in an extended Hubbard model describing a multiband system with repulsive interactions. We show that virtual interband processes induce an effective attractive interaction for small momentum transfers, a situation not found in most models of superconductivity from repulsion. This attraction can be traced back, in real space, to the presence of correlated hopping terms induced by interband polarization. We reveal this physics with both strong-coupling expansion and many-body perturbation theory, supplemented by numerical calculations. Finally, we point out interesting similarities with the problem of interacting electrons in twisted bilayer graphene, suggesting the importance of interband contribution to superconductivity.

Efficient product formulas for commutators and applications to quantum simulation

We construct product formulas for exponentials of commutators and explore their applications. First, we directly construct a third-order product formula with six exponentials by solving polynomial equations obtained using the operator differential method. We then derive higher-order product formulas recursively from the third-order formula. We improve over previous recursive constructions, reducing the number of gates required to achieve the same accuracy. In addition, we demonstrate that the constituent linear terms in the commutator can be included at no extra cost. As an application, we show how to use the product formulas in a digital protocol for counterdiabatic driving, which increases the fidelity for quantum state preparation. We also discuss applications to quantum simulation of one-dimensional fermion chains with nearest- and next-nearest-neighbor hopping terms, and two-dimensional fractional quantum Hall phases.

Observation of symmetry-protected corner states in breathing honeycomb topolectrical circuits

We report the experimental observation of second-order corner states in a two-dimensional breathing honeycomb topolectrical circuit with sixfold rotational symmetry C6 through voltage measurements. The topological corner states originate from the nontrivial bulk topology, which can be characterized by the topological invariant associated with the rotation eigenspectrum. We confirm two types of corner states, both originate from the C6 symmetry, while one of them is specially pinned to zero admittance because of the emerging chiral symmetry protection. We then examine the robustness of zero modes in the presence of next-nearest-neighbor hopping terms that destroy chiral symmetry but still preserve C6 symmetry. Our work provides a paradigm in circuit systems to study the exotic topological physics.

Type-II Weyl points in a synthetic three-dimensional acoustic lattice

We propose to realize type-II Weyl points in a one-dimensional Aubry–Andre–Harper model with cosine modulation in both hopping and on-site terms, which together form a synthetic three-dimensional parameter space. By constructing a one-dimensional acoustic lattice comprising multiple coupled cavities with two adjustable structural parameters, we implement the acoustical analogue of the type-II Weyl semimetal. Good agreement is observed between the theoretical predictions and numerical simulations, with both displaying the tilted dispersion and Fermi arc. Our study enables the exploration of high-dimensional topological physics by constructing a low-dimensional physical system and may open up possibilities for the design of novel acoustic devices.

Finite temperature mean-field theory with intrinsic non-Hermitian structures for Bose gases in optical lattices

We reveal a divergent issue associated with the mean-field theory for Bose gases in optical lattices constructed by the widely used straightforward mean-field decoupling of the hopping term, where the corresponding mean-field Hamiltonian generally assumes no lower energy bound once the spatial dependence of the mean-field superfluid (SF) order parameter is taken into account. Via a systematic functional integral approach, we solve this issue by establishing a general finite temperature mean-field theory that can treat any possible spatial dependence of the order parameter without causing the divergent issue. Interestingly, we find the theory generally assumes an intrinsic non-Hermitian structure that originates from the indefiniteness of the hopping matrix of the system. Within this theory, we develop an efficient approach for investigating the physics of the system at finite temperature, where properties of the system can be calculated via straightforward investigation on the saddle points of an effective potential function for the order parameter. We illustrate our approach by investigating the finite temperature SF transition of Bose gases in optical lattices. Since the underlying finite temperature mean-field theory is quite general, this approach can be straightforwardly applied to investigate the finite temperature properties of related systems with phases possessing complex spatial structures.

On chain models for contact electrification

An exact analytical model of charge dynamics for a chain of atoms with asymmetric hopping terms is presented. Analytic and numeric results are shown to give rise to similar dynamics in both the absence and presence of electron interactions. The chain model is further extended to the case of two atoms per cell (a perfect alloy system). This extension is further applied to contact electrification between two different atomic chains and the effect of increasing the magnitude of the contact transfer matrix element is studied.

Long-range hopping and indexing assumption in one-dimensional topological insulators

In this paper, we show that the introduction of long-range hoppings in 1D topological insulator models implies that different possibilities of site indexing must be considered when determining the bulk topological invariants in order to avoid the existence of hidden symmetries. The particular case of the extended SSH chain is addressed as an example where such behavior occurs. In this model, the introduction of long-range hopping terms breaks the bipartite property and a band inversion occurs in the band structure as the relative values of the hopping terms change, signaling a crossover between hopping parameter regions of "influence" of different chiral symmetries. Furthermore, edge states become a linear combination of edge-like states with different localization lengths and reflect the gradual transition between these different chiral symmetries.

Non-Hermitian mosaic dimerized lattices

Non-Hermitian systems have attracted much attention during the past few years, both theoretically and experimentally. The existence of non-Hermiticity can induce multiple exotic phenomena that cannot be observed in Hermitian systems. In this work, we introduce a new non-Hermitian system called the non-Hermitian mosaic dimerized lattice. Unlike the regular nonreciprocal lattices where asymmetric hoppings are imposed on every hopping term, here in the mosaic dimerized lattices the staggered asymmetric hoppings are only added to the nearest-neighboring hopping terms with equally spaced sites. By investigating the energy spectra, the non-Hermitian skin effect (NHSE), and the topological phases in such lattice models, we find that the period of the mosaic asymmetric hopping can influence the system’s properties significantly. For a system with real system parameters, we find that as the strength of asymmetric hopping increases, the energy spectra of the system under open boundary conditions will undergo a real-imaginary or real-complex transition. As to the NHSE, we find that when the period is odd, there appears no NHSE in the system and the spectra under open boundary conditions (OBCs) and periodic boundary conditions (PBCs) are the same (except for the topological edge modes under OBCs). If the period of the mosaic asymmetric hopping is even, the NHSE will emerge and the spectra under different boundary conditions exhibit distinctive structures. The PBC spectra form loop structures, indicating the existence of point gaps that are absent in the spectra under OBCs. The point gap in the PBC spectrum is shown to be the topological origin of the NHSE under OBCs, which also explains the NHSE in our mosaic dimerized lattices. To distinguish whether the bulk states of the system under OBCs are shifted to the left or right end of the one-dimensional lattice due to the NHSE, we define a new variable called the directional inverse participation ratio (dIPR). The positive dIPR indicates that the state is localized at the right end while the negative dIPR corresponds to the states localized at the left end of the one-dimensional lattice. We further study the topological zero-energy edge modes and characterize them by calculating the Berry phases based on the generalized Bloch Hamiltonian method. In addition, we also find that the topological edge modes with nonzero but constant energy can exist in the system. Our work provides a new non-Hermitian lattice model and unveils the exotic effect of mosaic asymmetric hopping on the properties of non-Hermitian systems.

Nonlinearity-induced transition in the nonlinear Su-Schrieffer-Heeger model and a nonlinear higher-order topological system

We study the topological physics in nonlinear Schr\"{o}dinger systems on lattices. We employ the quench dynamics to explore the phase diagram, where a pulse is given to a lattice point and we analyze its time evolution. There are two system parameters $\lambda $ and $\xi $, where $\lambda $ controls the hoppings between the neighboring links and $\xi $ controls the nonlinearity. The dynamics crucially depends on these system parameters. Based on analytical and numerical studies, we derive the phase diagram of the nonlinear Su-Schrieffer-Heeger (SSH) model in the ($\lambda ,\xi $) plane. It consists of four phases. The topological and trivial phases emerge when the nonlinearity $\xi $ is small. The nonlinearity-induced localization phase emerges when $\xi $ is large. We also find a dimer phase as a result of a cooperation between the hopping and nonlinear terms. A similar analysis is made of the nonlinear second-order topological system on the breathing Kagome lattice, where a trimer phase appears instead of the dimer phase.

Quantum dot-like plasmonic modes in twisted bilayer graphene supercells

We derive a material-realistic real-space many-body Hamiltonian for twisted bilayer graphene from first principles, including both single-particle hopping terms for p z electrons and their long-range Coulomb interaction. By disentangling low- and high-energy subspaces of the electronic dispersion, we are able to utilize state-of-the-art constrained random phase approximation calculations to reliably describe the non-local background screening from the high-energy s, p x , and p y electron states which we find to be independent of the bilayer stacking and thus of the twisting angle. The twist-dependent low-energy screening from p z states is subsequently added to obtain a full screening model. We use this modeling scheme to study plasmons in electron-doped twisted bilayer graphene supercells. We find that the finite system size yields discretized plasmonic levels, which are controlled by the system size, doping level, and twisting angle. This tunability together with atomic-like charge distributions of some of the excitations renders these plasmonic excitations remarkably similar to the electronic states in electronic quantum dots. To emphasize this analogy in the following we refer to these supercells as plasmonic quantum dots. Based on a careful comparison to pristine AB-stacked bilayer graphene plasmons, we show that two kinds of plasmonic excitations arise, which differ in their layer polarization. Depending on this layer polarization the resulting plasmonic quantum dot states are either significantly or barely dependent on the twisting angle. Due to their tunability and their coupling to light, these plasmonic quantum dots form a versatile and promising platform for tailored light-matter interactions.

Time evolution of energies and populations in germanium perturbed by a near-infrared pulse on the atto-second scale

Implementing time-resolved spectroscopy is one of the challenging topics in solid-state physics: indeed, it can be a crucial step to unveil the early time dynamics of fundamental processes in systems such as semiconductors, metals, superconductors and topological materials. Then, it is essential to build theoretical tools in order to understand the experimental results and what actually happens in solid-state systems on a new and finer timescale: the atto-second one. Accordingly, we have developed a theoretical–numerical tool in order to study the time evolution of the electronic populations on the atto-second scale in materials perturbed by a laser pulse. After finding the ab-initio band structure of the material, we compute the maximally-localized Wannier functions and obtain the hopping terms. Then, we study the effect of the laser pulse using the Peierls substitution method and obtain the equation of motion for the electronic populations. Finally, we obtain a dynamical DOS of the system and the corresponding dynamical electronic populations. In this work, as a relevant case study, we report our results for germanium. We identify oscillations in the eigenenergies and in the populations which roughly follow either the field or its square, with some phase shifts with respect to the field.

Octupolar order and Ising quantum criticality tuned by strain and dimensionality: Application to d -orbital Mott insulators

Recent experiments have discovered multipolar orders in a variety of $d$-orbital Mott insulators. Motivated by uncovering the exchange interactions which underlie octupolar order proposed in the osmate double perovskites, we study a two-site model using exact diagonalization on a five-orbital Hamiltonian, incorporating spin-orbit coupling (SOC) and interactions, and including both intra-orbital and inter-orbital hopping. Using an exact Schrieffer-Wolff transformation, we then extract an effective pseudospin Hamiltonian for the non-Kramers doublets, uncovering dominant ferrooctupolar coupling driven by the interplay of two distinct intra-orbital hopping terms. Using classical Monte Carlo simulations on the face-centered cubic lattice, we obtain a ferrooctupolar transition temperature which is in good agreement with experiments on the osmate double perovskites. We also explore the impact of uniaxial strain and dimensional tuning via ultrathin films, which are shown to induce a transverse field on the Ising octupolar order. This suppresses $T_c$ and potentially allows one to access octupolar Ising quantum critical points. We discuss possible implications of our results for a broader class of materials which may host such non-Kramers doublet ions.

Exact solutions of non-Hermitian chains with asymmetric long-range hopping under specific boundary conditions

We study the one-dimensional general non-Hermitian models with asymmetric long-range hopping and explore how to analytically solve the systems under some specific boundary conditions. Although the introduction of long-range hopping terms prevents us from finding analytical solutions for arbitrary boundary parameters, we identify the existence of exact solutions when the boundary parameters fulfill some constraint relations, which give the specific boundary conditions. Our analytical results show that the wave functions take simple forms and are independent of hopping range, while the eigenvalue spectra display rich model-dependent structures. Particularly, we find the existence of a special point coined as pseudo-periodic boundary condition, for which the eigenvalues are the same as those of the periodical system when the hopping parameters fulfill certain conditions, whereas the eigenstates display the non-Hermitian skin effect.

Gutzwiller projected states for the J1−J2 Heisenberg model on the Kagome lattice: Achievements and pitfalls

We assess the ground-state phase diagram of the $J_1$-$J_2$ Heisenberg model on the kagome lattice by employing Gutzwiller-projected fermionic wave functions. Within this framework, different states can be represented, defined by distinct unprojected fermionic Hamiltonians that include hopping and pairing terms, as well as a coupling to local Zeeman fields to generate magnetic order. For $J_2=0$, the so-called U(1) Dirac state, in which only hopping is present (such as to generate a $\pi$-flux in the hexagons), has been shown to accurately describe the exact ground state [Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Phys. Rev. B 87, 060405 (2013); Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Phys. Rev. X 7, 031020 (2017)]. Here, we show that its accuracy improves in presence of a small antiferromagnetic super-exchange $J_2$, leading to a finite region where the gapless spin liquid is stable; then, for $J_2/J_1=0.11(1)$, a first-order transition to a magnetic phase with pitch vector ${\bf q}=(0,0)$ is detected, by allowing magnetic order within the fermionic Hamiltonian. Instead, for small ferromagnetic values of $|J_2|/J_1$, the situation is more contradictory. While the U(1) Dirac state remains stable against several perturbations in the fermionic part (i.e., dimerization patterns or chiral terms), its accuracy clearly deteriorates on small systems, most notably on $36$ sites where exact diagonalization is possible. Then, upon increasing the ratio $|J_2|/J_1$, a magnetically ordered state with $\sqrt{3} \times \sqrt{3}$ periodicity eventually overcomes the U(1) Dirac spin liquid. Within the ferromagnetic $J_{2}$ regime, evidence is shown in favor of a first-order transition at $J_2/J_1=-0.065(5)$.