pseudo-Landau Levels Research Articles

Twist-angle dependent pseudo-magnetic fields in monolayer CrCl2/graphene heterostructures.

The generation of pseudo-magnetic fields in strained graphene leads to quantized Landau levels in the absence of an external magnetic field, providing the potential to achieve a zero-magnetic-field analogue of the quantum Hall effect. Here, we report the realization of a pseudo-magnetic field in epitaxial graphene by building a monolayer CrCl2/graphene heterointerface. The CrCl2 crystal structure exhibits spontaneous breaking of three-fold rotational symmetry, yielding an anisotropic displacement field at the interface. Using scanning tunneling spectroscopy, we have discovered a sequence of pseudo-Landau levels associated with massless Dirac fermions. A control experiment performed on the CrCl2/NbSe2 interface confirms the origin as the pseudo-magnetic field in the graphene layer that strongly interacts with CrCl2. More interestingly, the strength of the pseudo-magnetic fields can be tuned by the twist angle between the monolayer CrCl2 and graphene, with a variation of up to threefold, depending on the twist angle of 0° to 30°. This work presents a rare 2D heterojunction for exploring PMF-related physics, such as the valley Hall effect, with the advantage of easy and flexible implementation.

Topological origin of flat bands as pseudo-Landau levels in uniaxial strained graphene nanoribbons and induced magnetic ordering due to electron-electron interactions

Flat bands play a central role in the presence of correlated phases in moir\'e and other modulated two-dimensional systems. In this paper, flat bands are shown to exist in uniaxially periodic strained graphene. Such strain should be produced for example by a substrate. The model is thus mapped into a one-dimensional effective Hamiltonian and this allows to find the conditions for having flat bands, i.e., a long-wavelength modulation only on each one of the bipartite graphene sublattices, while having a tagged strain field between neighboring carbon atoms. The origin of such flat bands is thus tracked down to the existence of topological localized wavefunctions at domain walls separating different regions, each with a nonuniform Su-Schriffer-Hegger model (SSH) type of coupling. Thereafter, the system is mapped into a continuum model allowing to explain the numerical results in terms of the Jackiw-Rebbi model and of pseudo-Landau levels. Finally, the interplay between the obtained flat bands and electron-electron interaction is explored through the Hubbard model. The numerical results within the mean-field approximation indicate that the flat bands induce N\'eel antiferromagnetic and ferromagnetic domains even for a very weak Hubbard interaction, while the repulsive Hubbard interaction results in an effective electron-electron attraction. Thus this paper provides a simple, analytically solvable but realistic model to understand the physical origin of flat bands, pseudo-Landau levels and their effects on the effective electron-electron interaction.

Strong second-harmonic generation by sublattice polarization in non-uniformly strained monolayer graphene

Despite the potential of graphene for building a variety of quantum photonic devices, its centrosymmetric nature forbids the observation of second harmonic generation (SHG) for developing second-order nonlinear devices. To activate SHG in graphene, extensive research efforts have been directed towards disrupting graphene’s inversion symmetry using external stimuli like electric fields. However, these methods fail to engineer graphene’s lattice symmetry, which is the root cause of the forbidden SHG. Here, we harness strain engineering to directly manipulate graphene’s lattice arrangement and induce sublattice polarization to activate SHG. Surprisingly, the SHG signal is boosted 50-fold at low temperatures, which can be explained by resonant transitions between strain-induced pseudo-Landau levels. The second-order susceptibility of strained graphene is found to be larger than that of hexagonal boron nitride with intrinsic broken inversion symmetry. Our demonstration of strong SHG in strained graphene offers promising possibilities for developing high-efficiency nonlinear devices for integrated quantum circuits.

Theory of quantized photonic spin Hall effect in strained graphene under a sub-Tesla external magnetic field.

The quantized photonic spin Hall effect (PSHE) in the strained graphene-substrate system is predicted under a sub-Tesla external magnetic field, which is two orders of magnitude smaller than required to produce the quantized effect in the conventional graphene-substrate system. It is found that in-plane and transverse spin-dependent splittings in the PSHE, exhibit different quantized behaviors and are closely related to the reflection coefficients. Unlike the quantized PSHE in the conventional graphene-substrate system formed by the splitting of real Landau levels, the quantized PSHE in the strained graphene-substrate system is attributed to the splitting of pseudo-Landau levels caused by the pseudo-magnetic field and the lifting of valley degeneracy of the n ≠ 0 pseudo-Landau levels induced by the sub-Tesla external magnetic field. At the same time, the pseudo-Brewster angles of the system are also quantized with the change of Fermi energy. The sub-Tesla external magnetic field and the PSHE appear as quantized peak values near these angles. The giant quantized PSHE is expected to be used for direct optical measurements of the quantized conductivities and pseudo-Landau levels in the monolayer strained graphene.

Heavy-fermion representation for twisted bilayer graphene systems

We construct a heavy fermion representation for twisted bilayer graphene (TBG) systems. Two local orbitals (per spin/valley) are analytically found, which are exactly the maximally localized zero modes of the continuum Hamiltonian near the AA-stacking center. They have similar properties to the Wannier functions found in a recent study, but also have a clear interpretation as the zeroth pseudo Landau levels (ZLL) of Dirac fermions under the uniform strain field created by twisting. The electronic states of TBG can be viewed as the hybridization between these ZLL orbitals and other itinerant states, which can be obtained following the standard procedure of orthogonalized plane wave method. The ``heavy fermion'' model for TBG separates the strongly correlated components from the itinerant components and provides a solid base for the comprehensive understanding of the exotic physics in TBG.

Observing the quantum topology of light.

Topological photonics provides a powerful platform to explore topological physics beyond traditional electronic materials and shows promising applications in light transport and lasers. Classical degrees of freedom are routinely used to construct topological light modes in real or synthetic dimensions. Beyond the classical topology, the inherent quantum nature of light provides a wealth of fundamentally distinct topological states. Here we implement experiments on topological states of quantized light in a superconducting circuit, with which one- and two-dimensional Fock-state lattices are constructed. We realize rich topological physics including topological zero-energy states of the Su-Schrieffer-Heeger model, strain-induced pseudo-Landau levels, valley Hall effect, and Haldane chiral edge currents. Our study extends the topological states of light to the quantum regime, bridging topological phases of condensed-matter physics with circuit quantum electrodynamics, and offers a freedom in controlling the quantum states of multiple resonators.

Structural and electronic property engineering in graphene nanowrinkles via scanning tunneling microscopy

Graphene nanowrinkles (GNWs) are easily formed and manipulated by extrinsic perturbation in a continuous graphene nanosheet, which provides a suitable platform to study strain engineering. The deformation-induced pseudo-Landau levels (PLLs) and the wide bandgap states in GNWs were studied separately in different samples, which makes it hard to distinguish their difference, and the direct switch between them has not been observed yet. Here, using scanning tunneling microscopy (STM) tip manipulation, we realize the transition from PLL states to bandgap states through tuning the structure of the nanowrinkles in the same graphene nanosheet. Taking the close scan method, the GNWs are observed to be flattened with their PLL states turned off. Further applying a STM tip pulse, new nanowrinkles are rebuilt in the same graphene nanosheet. The distance between the ravine region of the newly formed GNWs and the substrate is dramatically shortened by the strong tip force. Therefore, the wide bandgap states are detected in the GNWs because of the resulting strong coupling of the substrate. Our result opens an avenue to engineer the quantum phenomena in strained graphene and provides more possibilities for the application of graphene nanostructures on microelectronic devices in the future.

Strain-induced pseudo magnetic field in the α−T3 lattice

We investigate the effects of a nonuniform uniaxial strain and a triaxial strain on the $\alpha-{\cal T}_3$ lattice. The analytical expressions of the pseudo-Landau levels (pLLs) are derived based on low-energy Hamiltonians, and are verified by tight-binding calculations. We find the pseudo-magnetic field leads to the oscillating density of states, and the first pLL is sublattice polarized, which is distributed on only two of the three sets of sublattices. For the nonuniform uniaxial strain, we show that pLLs become dispersive due to the renormalization of the Fermi velocity, and a valley polarized current emerges. Our results generalize the study of pseudo-magnetic field to the $\alpha-{\cal T}_3$ lattice, which will not only deepen the understanding of the intriguing effects of mechanical strains, but also provide theoretical foundation for possible experimental studies of the effects of strain on the $\alpha-{\cal T}_3$ lattice.

Strain-induced collapse of Landau levels in real Weyl semimetals

The collapse of Landau levels under an electric field perpendicular to the magnetic field is one of the distinctive features of Dirac materials. So is the coupling of lattice deformations to the electronic degrees of freedom in the form of gauge fields which allows the formation of pseudo-Landau levels from strain. We analyze the collapse of Landau levels induced by strain on realistic Weyl semimetals hosting anisotropic, tilted Weyl cones in momentum space. We perform first-principles calculations, to establish the conditions on the external strain for the collapse of Landau levels in TaAs which can be experimentally accessed.

Tunable Sample-Wide Electronic Kagome Lattice in Low-Angle Twisted Bilayer Graphene.

Overlaying two graphene layers with a small twist angle θ can create a moiré superlattice to realize exotic phenomena that are entirely absent in a graphene monolayer. A representative example is the predicted formation of localized pseudo-Landau levels (PLLs) with kagome lattice in tiny-angle twisted bilayer graphene (TBG) with θ<0.3° when the graphene layers are subjected to different electrostatic potentials. However, this was shown only for the model of rigidly rotated TBG, which is not realized in reality due to an interfacial structural reconstruction. It is believed that the interfacial structural reconstruction strongly inhibits the formation of the PLLs. Here, we systematically study electronic properties of the TBG with 0.075°≤θ<1.2° and demonstrate, unexpectedly, that the PLLs are quite robust for all the studied TBG. The structural reconstruction suppresses the formation of the emergent kagome lattice in the tiny-angle TBG. However, for the TBG around the magic angle, the sample-wide electronic kagome lattices with tunable lattice constants are directly imaged by using a scanning tunneling microscope. Our observations open a new direction to explore exotic correlated phases in moiré systems.

Realizing One-Dimensional Electronic States in Graphene via Coupled Zeroth Pseudo-Landau Levels.

Strain-induced pseudomagnetic fields can mimic real magnetic fields to generate a zero-magnetic-field analog of the Landau levels (LLs), i.e., the pseudo-Landau levels (PLLs), in graphene. The distinct nature of the PLLs enables one to realize novel electronic states beyond what is feasible with real LLs. Here, we show that it is possible to realize exotic electronic states through the coupling of zeroth PLLs in strained graphene. In our experiment, nanoscale strained structures embedded with PLLs are generated along a one-dimensional (1D) channel of suspended graphene monolayer. Our results demonstrate that the zeroth PLLs of the strained structures are coupled together, exhibiting a serpentine pattern that snakes back and forth along the 1D suspended graphene monolayer. These results are verified theoretically by large-scale tight-binding calculations of the strained samples. Our result provides a new approach to realizing novel quantum states and to engineering the electronic properties of graphene by using localized PLLs as building blocks.

Analytic solution to pseudo-Landau levels in strongly bent graphene nanoribbons

Nonuniform elastic strain is known to induce pseudo-Landau levels in Dirac materials. But these pseudo-Landau levels are hardly resolvable in an analytic fashion when the strain is strong because of the emerging complicated space dependence in both the strain-modulated Fermi velocity and the strain-induced pseudomagnetic field. We here analytically characterize the solution to the pseudo-Landau levels in strongly bent graphene nanoribbons by treating the effects of the nonuniform Fermi velocity and pseudomagnetic field on equal footing. The analytic solution is detectable through angle-resolved photoemission spectroscopy and allows quantitative comparison between theories and various experimental signatures of transport, such as the Shubnikov-de Haas oscillation in the complete absence of magnetic fields and the negative strain-resistivity resulting from the valley anomaly. The analytic solution can be generalized to various Dirac materials and will shed light on the related experimental explorations and straintronics applications.

Reverse strain-induced snake states in graphene nanoribbons

Strain can tailor the band structures and properties of graphene nanoribbons (GNRs) with the well-known emergent pseudo-magnetic fields and the corresponding pseudo-Landau levels (pLLs). We design one type of the zigzag GNR (ZGNR) with reverse strains, producing pseudo-magnetic fields with opposite signs in the lower and upper half planes. Therefore, electrons propagate along the interface as "snake states", experiencing opposite Lorentz forces as they cross the zero field border line. By using the Landauer-Buttiker formalism combined with the nonequilibrium Green's function method, the existence and robustness of the reverse strain-induced snake states are further studied. Furthermore, the realization of long-thought pure valley currents in monolayer graphene systems is also proposed in our device.

Electron localization in periodically strained graphene

Pseudo-magnetic field (PMF) in deformed graphene has been proposed as a promising and flexible method to quantum-confine electronic states and create gaps in the local density of states. Motivated by this perspective, we numerically analyze various different configurations leading to electronic localization and band flattening in periodically strained graphene. In particular, we highlight the existence of a fine structure in the pseudo-Landau levels confined in large-PMF regions, the emergence of states confined to PMF nodes as well as of snake-like orbits. In our paper, we further analyze the importance of the relative rotation and asymmetry of the strain lattice with respect to the atomic lattice and show how it can be used to modulate the PMF periodicity and to create localized orbits far from the strain points. Possible implementations and applications of the simulated structures are discussed.

Magnon Landau levels in the strained antiferromagnetic honeycomb nanoribbons

The pseudo-magnetic field created by a non-uniform unaxial strain is introduced into the antiferromagnetic honeycomb nanoribbons. The formation of magnon pseudo-Landau levels, which appear from the upper end of the spectrum and whose level spacings are proportional to the square root of the level index, is revealed by the linear spin-wave theory. The antiferromagnetic order is gradually weakened along the $y$-direction by the strain. At large enough strength, the system is decoupled into isolated zigzag chains near the upper boundary, and demonstrates one-dimensional magnetic property there. While the quantum Monte Carlo simulations also predict such a transition, this exact method gives a critical point deeper in the bulk. We also investigate the $XY$ antiferromagnetic honeycomb nanoribbons, and find similar pseudo-Landau levels and antiferromagnetic evolution. Our results unveil the effect of a non-uniform unaxial strain on the spin excitaions, and may be realized experimentally based on two-dimensional quantum magnetic materials.

Network model and four-terminal transport in minimally twisted bilayer graphene

We construct a two-channel scattering model for the triangular network of valley Hall states in interlayer-biased minimally twisted bilayer graphene from symmetry arguments and investigate electronic transport in a four-terminal setup. In the absence of forward scattering, a single phenomenological parameter tunes the network between a triplet of chiral zigzag modes and pseudo-Landau levels. Moreover, the chiral zigzag modes give rise to robust Aharonov-Bohm resonances in the longitudinal conductance in the presence of a perpendicular magnetic field or an in-plane electric field. Interestingly, we find that when both a magnetic field and an in-plane electric field are applied, the resonances of different zigzag branches split depending on their propagation direction relative to the in-plane electric field. We further demonstrate that while the Hall response vanishes in the chiral zigzag regime, a finite Hall response is obtained without destroying the Aharonov-Bohm resonances in the longitudinal response, by weakly coupling different zigzag branches, which also gives rise to Hofstadter physics at accessible magnetic fields.

Quantum Monte Carlo study of honeycomb antiferromagnets under a triaxial strain

The honeycomb antiferromagnet under a triaxial strain is studied using the quantum Monte Carlo simulation. The strain dimerizes the exchange couplings near the corners, thus destructs the antiferromagnetic order therein. The antiferromagnetic region is continuously reduced by the strain. For the same strain strength, the exact numerical results give a much smaller antiferromagnetic region than the linear spin-wave theory. We then study the strained $XY$ antiferromagnet, where the magnon pseudo-magnetic field behaves quite differently. The $0$th Landau level appears in the middle of the spectrum, and the quantized energies above (below) it are proportional to $n^{\frac{1}{3}} (n^{\frac{2}{3}})$, which is in great contrast to the equally-spaced ones in the Heisenberg case. Besides, we find the antiferromagnetic order of the $XY$ model is much more robust to the dimerization than the Heisenberg one. The local susceptibility of the Heisenberg case is extracted by the numerical analytical continuation, and no sign of the pseudo-Landau levels is resolved. It is still not sure whether the result is due to the intrinsic problem of the numerical analytical continuation. Thus the existence of the magnon pseudo-Landau levels in the spin-$\frac{1}{2}$ strained Heisenberg Hamiltonian remains an open question. Our results are closely related to the two-dimensional van der Waals quantum antiferromagnets and may be realized experimentally.

Pseudo Landau levels, negative strain resistivity, and enhanced thermopower in twisted graphene nanoribbons

As a canonical response to the applied magnetic field, the electronic states of a metal are fundamentally reorganized into Landau levels. In Dirac metals, Landau levels can be expected without magnetic fields, provided that an inhomogeneous strain is applied to spatially modulate electron hoppings in a way similar to the Aharonov-Bohm phase. We here predict that a twisted zigzag nanoribbon of graphene exhibits strain-induced pseudo Landau levels of unexplored but analytically solvable dispersions at low energies. The presence of such dispersive pseudo Landau levels results in a negative strain resistivity characterizing the ($1+1$)-dimensional chiral anomaly if partially filled and can greatly enhance the thermopower when fully filled.

Coherent states for dispersive pseudo-Landau-levels in strained honeycomb lattices

Dirac fermions in graphene may experiment dispersive pseudo-Landau levels due to a homogeneous pseudomagnetic field and a position-dependent Fermi velocity induced by strain. In this paper, we study the (semi-classical) dynamics of these particles under such a physical context from an approach of coherent states. For this purpose we use a Landau-like gauge to built Perelomov coherent states by the action of a non-unitary displacement operator $D(\alpha)$ on the fundamental state of the system. We analyze the time evolution of the probability density and the generalized uncertainty principle as well as the Wigner function for the coherent states. Our results show how $x$-momentum dependency affects the motion periodicity and the Wigner function shape in phase space.

Probing non-Hermitian phase transitions in curved space via quench dynamics

Non-Hermitian Hamiltonians are relevant to describe the features of a broad class of physical phenomena, ranging from photonics and atomic and molecular systems to nuclear physics and mesoscopic electronic systems. An important question relies on the understanding of the influence of curved background on the static and dynamical properties of non-Hermitian systems. In this work, we study the interplay of geometry and non-Hermitian dynamics by unveiling the existence of curvature-dependent non-Hermitian phase transitions. We investigate a prototypical model of Dirac fermions on a sphere with an imaginary mass term. This exactly-solvable model admits an infinite set of curvature-dependent pseudo-Landau levels. We characterize these phases by computing an order parameter given by the pseudo-magnetization and, independently, the non-Hermitian fidelity susceptibility. Finally, we probe the non-Hermitian phase transitions by computing the (generalized) Loschmidt echo and the dynamical fidelity after a quantum quench of the imaginary mass and find singularities in correspondence of exceptional radii of the sphere.