Topological nature of nonlinear optical effects in solids

Photon-drag photovoltaic effects and quantum geometric nature

The bulk photovoltaic effect (BPVE) generates a direct current photocurrent under uniform irradiation and is a nonlinear optical effect traditionally studied in noncentrosymmetric materials. The two main origins of BPVE are the shift and injection currents, arising from transitions in electron position and electron velocity during optical excitation, respectively. Recently, it was proposed that photon-drag effects could unlock BPVE in centrosymmetric materials. However, experimental progress remains limited. In this work, we provide a comprehensive theoretical analysis of photon-drag effects inducing BPVE (photon-drag BPVE). Notably, we find that photon-drag BPVE can be directly linked to quantum geometric tensors. Additionally, we propose that photon-drag shift currents can be fully isolated from other current contributions in nonmagnetic centrosymmetric materials. We apply our theory explicitly to the 2D topological insulator 1T'-WTe2. Furthermore, we investigate photon-drag BPVE in a centrosymmetric magnetic Weyl semimetal, where we demonstrate that linearly polarized light generates photon-drag shift currents.

Sensitive and Anisotropic Room-Temperature Terahertz Photodetection in Quasi-1D Nodal-Line Semimetal NbNiTe5

Low-dimensional topological materials merge the benefits of reduced dimensionality with the nontrivial topological phases, garnering significant attention as promising candidates for next-generation optoelectronic devices. The quasi-1D nodal-line semimetal NbNiTe5 showcases distinct in-plane anisotropy alongside robust Dirac nodal-line points, rendering it a fascinating platform for exploring the intricate interplay between novel quantum states of matter and low-energy radiation. Here, sensitive and anisotropic terahertz photodetection driven by Dirac fermions and the intrinsic anisotropic properties of NbNiTe5 are presented. Leveraging the enhanced carrier transport characteristics derived from nontrivial band topology, room-temperature responsivity of 1.36 A W-1, noise equivalent power of 5.31 × 10-11 W Hz-1/2 as well as fast photoresponse speed of 4.5 µs are achieved. The exceptionally high anisotropic photoresponse ratio of 84 highlights the potential for improving the performance of polarization-sensitive photodetectors. This work is crucial for advancing the understanding of nontrivial topology and the development of terahertz photodetector technologies.

Optical sorting: past, present and future

Optical sorting combines optical tweezers with diverse techniques, including optical spectrum, artificial intelligence (AI) and immunoassay, to endow unprecedented capabilities in particle sorting. In comparison to other methods such as microfluidics, acoustics and electrophoresis, optical sorting offers appreciable advantages in nanoscale precision, high resolution, non-invasiveness, and is becoming increasingly indispensable in fields of biophysics, chemistry, and materials science. This review aims to offer a comprehensive overview of the history, development, and perspectives of various optical sorting techniques, categorised as passive and active sorting methods. To begin, we elucidate the fundamental physics and attributes of both conventional and exotic optical forces. We then explore sorting capabilities of active optical sorting, which fuses optical tweezers with a diversity of techniques, including Raman spectroscopy and machine learning. Afterwards, we reveal the essential roles played by deterministic light fields, configured with lens systems or metasurfaces, in the passive sorting of particles based on their varying sizes and shapes, sorting resolutions and speeds. We conclude with our vision of the most promising and futuristic directions, including AI-facilitated ultrafast and bio-morphology-selective sorting. It can be envisioned that optical sorting will inevitably become a revolutionary tool in scientific research and practical biomedical applications.

Ultrafast photocurrent hysteresis in photoferroelectric α-In2Se3 diagnosed by terahertz emission spectroscopy

Nonvolatile control over the physical state of polar materials through all-optical methods has been a long-standing objective pursued in optoelectronics. Photoferroelectric semiconductors exhibit immense potential in capturing multimodal nonvolatile states, attributed to their spontaneous and reversible in-plane and out-of-plane polarizations. Herein, we uncover an unprecedented nonvolatile, zero-bias, ultrafast photocurrent hysteresis response with an innovative all-optical approach, discerned by analyzing in-plane and out-of-plane terahertz (THz) waves emitted from photoferroelectric α-In2Se3. The mechanism underlying such ultrafast photocurrent hysteresis arises from anomalous linear and circular photovoltaic effects synchronously fueled by a localized rearrangement of polarization. By harnessing the anisotropic photoferroelectric kinetics-induced relative phase between the in-plane and out-of-plane polarizations, we further demonstrate the flexible selection of chirality, tunable rotational angle, and optimizable ellipticity of THz waves. Our findings present a unique ultrafast and nondestructive strategy for investigating photoferroelectric hysteresis, empowering dynamic polarization manipulation of THz waves for a wide range of THz applications.

Strain-Reduced Inversion Symmetry in Ultrathin SnP2Se6 Crystals for Giant Bulk Piezophotovoltaic Generation

With the potential to surpass the Shockley-Queisser (S-Q) limitation for solar energy conversion, the bulk photovoltaic (BPV) effect, which is induced by the broken inversion symmetry of the lattice, presents prospects for future light-harvesting technologies. However, the development of BPV is largely limited by the low solar spectrum conversion efficiency of existing noncentrosymmetric materials with wide band gaps. This study reports that the strain-induced reduction of inversion symmetry can enhance the second-order nonlinear susceptibility (χ(2)) of SnP2Se6 crystals by an order of magnitude, which contributes to an extremely high value of 1.3 × 10-8 m·V-1 under 1550 nm excitation, and is high among two-dimensional (2D) crystals. More importantly, owing to the orientation-dependent reduction of lattice symmetry, the BPV generation induced by strain, referred to as the bulk piezophotovoltaic effect, is demonstrated in the SnP2Se6 crystal with strong in-plane anisotropy. The strain along the Se zigzag direction greatly facilitates the generation of the giant photocurrent covering an extended spectrum ranging from 400 to 1100 nm, resulting in leading-level values of the BPV coefficient among noncentrosymmetric crystals, while the BPV effect is barely modulated along the Se armchair direction even with a large strain of 0.57%. This study highlights the potential of the bulk piezophotovoltaic effects for energy conversion efficiency and offers a promising strategy for the design of next-generation light-harvesting devices.

Skin Effect of Nonlinear Optical Responses in Antiferromagnets

Nonlinear optics plays important roles in the research of fundamental physics and the applications of high-performance optoelectronic devices. The bulk nonlinear optical responses arise from the uniform light absorption in noncentrosymmetric crystals, and hence are usually considered to be the collective phenomena of all atoms. Here we show, in contrast to this common expectation, the nonlinear optical responses in antiferromagnets can be selectively accumulated near the surfaces, representing a skin effect. This is because the inversion symmetry, despite being broken globally by magnetism, is barely violated locally deeply inside these antiferromagnets. Using A-type layered antiferromagnets as the representatives, we predict that the spatial-dependent nonlinear optical responses, such as the bulk photovoltaic effect and second harmonic generation, are notable in the top- and bottommost layers and decay rapidly when moving away from the surfaces. Such a phenomenon is strongly associated with the antiferromagnetism and exists in a broad range of antiferromagnets composed of centrosymmetric sublattices, offering promising device applications using these antiferromagnets. Our work uncovers a previously overlooked property of nonlinear optical responses and opens new opportunities for high-performance antiferromagnetic optospintronics.

Unveiling Intrinsic Bulk Photovoltaic Effect in Atomically Thin ReS2

The bulk photovoltaic effect (BPVE) offers a promising avenue to surpass the efficiency limitations of current solar cell technology. However, disentangling intrinsic and extrinsic contributions to photocurrent remains a significant challenge. Here, we fabricate high-quality, lateral devices based on atomically thin ReS2 with minimal contact resistance, providing an optimal platform for distinguishing intrinsic bulk photovoltaic signals from other extrinsic photocurrent contributions originating from interfacial effects. Our devices exhibit large bulk photovoltaic performance with intrinsic responsivities of ∼1 mA/W in the visible range, without the need for external tuning knobs such as strain engineering. Our experimental findings are supported by theoretical calculations. Furthermore, our approach can be extrapolated to investigate the intrinsic BPVE in other noncentrosymmetric van der Waals materials, paving the way for a new generation of efficient light-harvesting devices.

Strongly enhanced shift current at exciton resonances in a noncentrosymmetric wide-gap semiconductor

Excitons are fundamental quasiparticles that are ubiquitous in photoexcited semiconductors and insulators. Despite causing a sharp and strong photoabsorption near the interband absorption edge, charge-neutral excitons do not yield photocurrent in conventional photovoltaic processes unless dissociated into free charge carriers. Here, we experimentally demonstrate that excitons can directly contribute to photocurrent generation through a nonlinear light-matter interaction in a noncentrosymmetric semiconductor CuI. Epitaxial thin films of CuI exhibit a substantial enhancement of photocurrent at exciton resonance energies even below the bandgap. From the light polarization dependence, this photocurrent is identified to be shift current, a nonlinear photocurrent driven by the change in the geometric Berry phase of electron wave functions upon the optical transition. The shift current at the exciton resonance is much larger than that induced above the band gap by free electron-hole excitation, and their signs are opposite. First-principles calculations elucidate that the sign and magnitude of the exciton shift current are strongly dependent on the strain in the thin film. The present study reveals the crucial role of excitons in enhancing the shift current magnitude and its strain sensitivity, and will open an unprecedented route for efficient manipulation of nonlinear optical effects.

Quantized Integrated Shift Effect in Multigap Topological Phases

We show that certain three-dimensional multigap topological insulators can host quantized integrated shift photoconductivities due to bulk invariants that are defined under reality conditions imposed by additional symmetries. We recast the quantization in terms of the integrated torsion tensor and the non-Abelian Berry connection constituting Chern-Simons forms. Physically, we recognize that the topological quantization emerges purely from virtual transitions contributing to the optical response. Our findings provide another quantized electromagnetic dc response due to the nontrivial band topology, beyond the quantum anomalous Hall effect of Chern insulators and quantized circular photogalvanic effect found in Weyl semimetals.

Optical Conductivity and Photo-Induced Polaronic Formation in Co2MnGa Topological Semimetal

Topological materials occupy an important place in the quantum materials family due to their peculiar low-energy electrodynamics, hosting emergent magneto-electrical, and nonlinear optical responses. This manuscript reports on the optical responses for the magnetic topological nodal semimetal Co2MnGa, studied in a thin film geometry at various thicknesses. The thickness-dependent optical conductivity is investigated, observing a substantial dependence of the electronic band structure on thickness. Additionally, details on the ultrafast response of the low energy excitations in the terahertz frequency are reported by employing optical pump-terahertz probe (OPTP) spectroscopy. In particular, the photocarrier dynamics of Co2MnGa thin films is studied at varying pump fluence, pump wavelength, and film thickness, observing a negative THz photoconductivity which is assigned to a dynamical formation of large polarons in the material.

Polarization photodetectors with configurable polarity transition enabled by programmable ferroelectric-doping patterns

Advances in symmetry-breaking engineering of heterointerfaces for optoelectronic devices have garnered significant attention due to their immense potential in tunable moiré quantum geometry and enabling polarization light detection. Despite several proposed approaches to breaking the symmetry of low-dimensional materials, there remains a lack of universal methods to create materials with prominent polarization detection capabilities. Here, we introduce a reliable strategy for manipulating the symmetry of low-dimensional materials through a programmable ferroelectric-doping patterns technique. This method introduces a spontaneous photocurrent and enables the detection of linearly polarization light in isotropic 2H-MoTe2. The 2H-MoTe2 photodetector exhibits a significant short-circuit photocurrent intensity (Jsc = 29.9 A/cm2) and open-circuit voltage Voc of 0.12 V ( ~ 3 × 105 V/cm). Under a specific bias, the polarization ratio transitions from 1 to ∞/-∞, shifting from a positive state (unipolar regime) to a negative state (bipolar regime). These findings underscore the potential of ferroelectric-doping patterns as a promising approach to creating composite materials with artificial bulk photovoltaic effect and achieving high-performance polarization light detection.

Controllable orbital angular momentum monopoles in chiral topological semimetals

The emerging field of orbitronics aims to generate and control orbital angular momentum for information processing. Chiral crystals are promising orbitronic materials because they have been predicted to host monopole-like orbital textures, where the orbital angular momentum aligns isotropically with the electron's crystal momentum. However, such monopoles have not yet been directly observed in chiral crystals. Here, we use circular dichroism in angle-resolved photoelectron spectroscopy to image orbital angular momentum monopoles in the chiral topological semimetals PtGa and PdGa. The spectra show a robust polar texture that rotates around the monopole as a function of photon energy. This is a direct consequence of the underlying magnetic orbital texture and can be understood from the interference of local atomic contributions. Moreover, we also demonstrate that the polarity of the monopoles can be controlled through the structural handedness of the host crystal by imaging orbital angular moment monopoles and antimonopoles in the two enantiomers of PdGa, respectively. Our results highlight the potential of chiral crystals for orbitronic device applications, and our methodology could enable the discovery of even more complicated nodal orbital angular momentum textures that could be exploited for orbitronics.

Manipulating Photogalvanic Effects in Two-Dimensional Multiferroic Breathing Kagome Materials

Multiferroic materials, known for their multiple tunable orders, present an exceptional opportunity to manipulate nonlinear optical responses that are sensitive to symmetry. In this study, we propose leveraging electric and magnetic fields to selectively control and switch specific types of photogalvanic effects in two-dimensional multiferroic breathing kagome materials. Taking monolayer Nb3I8 as an example, we demonstrate that the shift current, characterized by the real-space shift of electrons and holes, is predominantly unaffected by magnetic order. In contrast, injection current, featured by quantum metric dipole in momentum space, is closely related to valley polarization, which can be controlled by a magnetic field. Furthermore, both photocurrents can be reversed by an out-of-plane electric field via lattice breathing. Our findings reveal the potential of multiferroic breathing kagome structures for multifunctional optoelectronic applications and sensors.

Theory of topological exciton insulators and condensates in flat Chern bands

Excitons are the neutral quasiparticles that form when Coulomb interactions create bound states between electrons and holes. Due to their bosonic nature, excitons are expected to condense and exhibit superfluidity at sufficiently low temperatures. In interacting Chern insulators, excitons may inherit the nontrivial topology and quantum geometry from the underlying electron wavefunctions. We theoretically investigate the excitonic bound states and superfluidity in flat-band insulators pumped with light. We find that the exciton wavefunctions exhibit vortex structures in momentum space, with the total vorticity being equal to the difference of Chern numbers between the conduction and valence bands. Moreover, both the exciton binding energy and the exciton superfluid density are proportional to the Brillouin-zone average of the quantum metric and the Coulomb potential energy per unit cell. Spontaneous emission of circularly polarized light from radiative decay is a detectable signature of the exciton vorticity. We propose that the vorticity can also be experimentally measured via the nonlinear anomalous Hall effect, whereas the exciton superfluidity can be detected by voltage-drop quantization through a combination of quantum geometry and Aharonov-Casher effect. Topological excitons and their superfluid phase could be realized in flat bands of twisted Van der Waals heterostructures.

Enhancing shift current response via virtual multiband transitions

Materials exhibiting a significant shift current response could potentially outperform conventional solar cell materials. The myriad of factors governing shift-current response, however, poses significant challenges in finding such strong shift-current materials. Here we propose a general design principle that exploits inter-orbital mixing to excite virtual multiband transitions in materials with multiple flat bands to achieve an enhanced shift current response. We further relate this design principle to maximizing Wannier function spread as expressed through the formalism of quantum geometry. We demonstrate the viability of our design using a 1D stacked Rice-Mele model. Furthermore, we consider a concrete material realization - alternating angle twisted multilayer graphene (TMG) - a natural platform to experimentally realize such an effect. We identify a set of twist angles at which the shift current response is maximized via virtual transitions for each multilayer graphene and highlight the importance of TMG as a promising material to achieve an enhanced shift current response at terahertz frequencies. Our proposed mechanism also applies to other 2D systems and can serve as a guiding principle for designing multiband systems that exhibit an enhanced shift current response.

Surface photogalvanic effect in Ag2Te

The bulk photovoltaic effect (BPVE) in non-centrosymmetric materials has attracted significant attention in recent years due to its potential to surpass the Shockley-Queisser limit. Although these materials are strictly constrained by symmetry, progress has been made in artificially reducing symmetry to stimulate BPVE in wider systems. However, the complexity of these techniques has hindered their practical implementation. In this study, we demonstrate a large intrinsic photocurrent response in centrosymmetric topological insulator Ag2Te, attributed to the surface photogalvanic effect (SPGE), which is induced by symmetry reduction of the surface. Through diverse spatially-resolved measurements on specially designed devices, we directly observe that SPGE in Ag2Te arises from the difference between two opposite photocurrent flows generated from the top and bottom surfaces. Acting as an efficient SPGE material, Ag2Te demonstrates robust performance across a wide spectral range from visible to mid-infrared, making it promising for applications in solar cells and mid-infrared detectors. More importantly, SPGE generated on low-symmetric surfaces can potentially be found in various systems, thereby inspiring a broader range of choices for photovoltaic materials.

Berry curvature signatures in chiroptical excitonic transitions

The topology of the electronic band structure of solids can be described by its Berry curvature distribution across the Brillouin zone. We theoretically introduce and experimentally demonstrate a general methodology based on the measurement of energy- and momentum-resolved optical transition rates, allowing to reveal signatures of Berry curvature texture in reciprocal space. By performing time- and angle-resolved photoemission spectroscopy of atomically thin WSe2 using polarization-modulated excitations, we demonstrate that excitons become an asset in extracting the quantum geometrical properties of solids. We also investigate the resilience of our measurement protocol against ultrafast scattering processes following direct chiroptical transitions.

Nonlinear optical signatures of topological Dirac fermion

Dirac fermion in topological materials exhibits intriguing nonlinear optical responses. However, their direct correlation with the linearly dispersed band remains elusive experimentally. Here, we take topological semimetal ZrSiS as a paradigm, unveiling three unique nonlinear optical signatures of Dirac fermion. These signatures include strong quadrupolar response, quantum interference effect, and exponential divergent four-wave mixing (FWM), all of which are described by the prominent third-order nonlinear optical susceptibility. Resonantly enhanced by linear bands, quadrupolar second harmonic generation in centrosymmetric bulk overwhelms the electric-dipole contribution at the surface with inherent inversion symmetry breaking. Furthermore, owing to the interference between multiple resonant transition pathways within linear bands, difference-frequency FWM is several orders of magnitude stronger than sum-frequency FWM and third harmonic generation. The difference-frequency FWM further displays an inverse-square divergence toward degenerate excitation, whose scaling law perfectly matches with the long-sought behavior of Dirac fermion. These signatures lay the solid foundation toward the practical applications of topological materials in nonlinear optoelectronics and photonics.

Magnetic parity violation and parity-time-reversal-symmetric magnets

Parity-time-reversal symmetry (PTsymmetry), a symmetry for the combined operations of space inversion (P) and time reversal (T), is a fundamental concept of physics and characterizes the functionality of materials as well asPandTsymmetries. In particular, thePT-symmetric systems can be found in the centrosymmetric crystals undergoing the parity-violating magnetic order which we call the odd-parity magnetic multipole order. While this spontaneous order leavesPTsymmetry intact, the simultaneous violation ofPandTsymmetries gives rise to various emergent responses that are qualitatively different from those allowed by the nonmagneticP-symmetry breaking or by the ferromagnetic order. In this review, we introduce candidates hosting the intriguing spontaneous order and overview the characteristic physical responses. Various off-diagonal and/or nonreciprocal responses are identified, which are closely related to the unusual electronic structures such as hidden spin-momentum locking and asymmetric band dispersion.

Terahertz photon to dc current conversion via magnetic excitations of multiferroics

Direct conversion from terahertz photon to charge current is a key phenomenon for terahertz photonics. Quantum geometrical description of optical processes in crystalline solids predicts existence of field-unbiased dc photocurrent arising from terahertz-light generation of magnetic excitations in multiferroics, potentially leading to fast and energy-efficient terahertz devices. Here, we demonstrate the dc charge current generation from terahertz magnetic excitations in multiferroic perovskite manganites with spin-driven ferroelectricity, while keeping an insulating state with no free carrier. It is also revealed that electromagnon, which ranges sub-terahertz to 2 THz, as well as antiferromagnetic resonance shows the giant conversion efficiency. Polar asymmetry induced by the cycloidal spin order gives rise to this terahertz-photon-induced dc photocurrent, and no external magnetic and electric bias field are required for this conversion process. The observed phenomena are beyond the conventional photovoltaics in semi-classical regime and demonstrate the essential role of quantum geometrical aspect in low-energy optical processes. Our finding establishes a paradigm of terahertz photovoltaic phenomena, paving a way for terahertz photonic devices and energy harvesting.

Light-induced electronic polarization in antiferromagnetic Cr2O3

In a solid, the electronic subsystem can exhibit incipient order with lower point group symmetry than the crystal lattice. Ultrafast external fields that couple exclusively to electronic order parameters have rarely been investigated, however, despite their potential importance in inducing exotic effects. Here we show that when inversion symmetry is broken by the antiferromagnetic order in Cr2O3, transmitting a linearly polarized light pulse through the crystal gives rise to an in-plane rotational symmetry-breaking (from C3 to C1) via optical rectification. Using interferometric time-resolved second harmonic generation, we show that the ultrafast timescale of the symmetry reduction is indicative of a purely electronic response; the underlying spin and crystal structures remain unaffected. The symmetry-broken state exhibits a dipole moment, and its polar axis can be controlled with the incident light. Our results establish a coherent nonlinear optical protocol by which to break electronic symmetries and produce unconventional electronic effects in solids.

Helicity-Sensitive Terahertz Detection in Monolayer 1T'-WTe2

Valleytronics, identified as electronic properties of the energy band extrema in momentum space, has been intensively revived following the emergence of two-dimensional transition metal dichalcogenides (TMDCs) as their valley information can be controlled and probed through the spin angular momentum of light. Previous optical investigations of valleytronics have been limited to the visible/near-infrared spectral regime through which the carriers of most TMDCs can be excited. Monolayer 1T'-WTe2 with broken time-reversal symmetry provides a fertile platform to study the long-wavelength photonic properties in different valleys. Here, we employed a circularly polarized terahertz (THz) laser to selectively excite the valley of monolayer 1T'-WTe2 and demonstrate that the helicity-dependent photoresponse is generated via the photogalvanic effect (PGE). We also observed that the photocurrent is controlled by circular polarization and the external electric field. Because of the tunable Berry curvature dipole derived from the nontrivial wave functions near the inverted gap edge in monolayer WTe2, the bandgap can be tuned efficiently. Our results provide a versatile venue for controlling, detecting, and processing valleytronics and applications in on-chip THz imaging and quantum information processing.

Nonlinear optical diode effect in a magnetic Weyl semimetal

Diode effects are of great interest for both fundamental physics and modern technologies. Electrical diode effects (nonreciprocal transport) have been observed in Weyl systems. Optical diode effects arising from the Weyl fermions have been theoretically considered but not probed experimentally. Here, we report the observation of a nonlinear optical diode effect (NODE) in the magnetic Weyl semimetal CeAlSi, where the magnetization introduces a pronounced directionality in the nonlinear optical second-harmonic generation (SHG). We demonstrate a six-fold change of the measured SHG intensity between opposite propagation directions over a bandwidth exceeding 250 meV. Supported by density-functional theory, we establish the linearly dispersive bands emerging from Weyl nodes as the origin of this broadband effect. We further demonstrate current-induced magnetization switching and thus electrical control of the NODE. Our results advance ongoing research to identify novel nonlinear optical/transport phenomena in magnetic topological materials and further opens new pathways for the unidirectional manipulation of light.

Floquet Fermi Liquid

We demonstrate the existence of a nonequilibrium "Floquet Fermi liquid" state arising in partially filled Floquet Bloch bands weakly coupled to ideal fermionic baths, which possess a collection of "Floquet Fermi surfaces" enclosed inside each other, resembling matryoshka dolls. We elucidate several properties of these states, including their quantum oscillations under magnetic fields which feature slow beating patterns of their amplitude reflecting the different areas of the Floquet Fermi surfaces, consistent with those observed in microwave induced resistance oscillation experiments. We also investigate their specific heat and thermodynamic density of states and demonstrate how by controlling properties of the drive, such as its frequency, one can tune some of the Floquet Fermi surfaces toward nonequilibrium Van Hove singularities without changing the electron density.

Defect-induced helicity dependent terahertz emission in Dirac semimetal PtTe2 thin films

Nonlinear transport enabled by symmetry breaking in quantum materials has aroused considerable interest in condensed matter physics and interdisciplinary electronics. However, achieving a nonlinear optical response in centrosymmetric Dirac semimetals via defect engineering has remained a challenge. Here, we observe the helicity dependent terahertz emission in Dirac semimetal PtTe2 thin films via the circular photogalvanic effect under normal incidence. This is activated by a controllable out-of-plane Te-vacancy defect gradient, which we unambiguously evidence with electron ptychography. The defect gradient lowers the symmetry, which not only induces the band spin splitting but also generates the giant Berry curvature dipole responsible for the circular photogalvanic effect. We demonstrate that the THz emission can be manipulated by the Te-vacancy defect concentration. Furthermore, the temperature evolution of the THz emission features a minimum in the THz amplitude due to carrier compensation. Our work provides a universal strategy for symmetry breaking in centrosymmetric Dirac materials for efficient nonlinear transport.

Large nonlinear optical magnetoelectric response in a noncentrosymmetric magnetic Weyl semimetal

Weyl semimetals resulting from either inversion (P) or time-reversal (T) symmetry breaking have been revealed to show the record-breaking large optical response due to intense Berry curvature of Weyl-node pairs. Different classes of Weyl semimetals with both P and T symmetry breaking potentially exhibit optical magnetoelectric (ME) responses, which are essentially distinct from the previously observed optical responses in conventional Weyl semimetals, leading to the versatile functions such as directional dependence for light propagation and gyrotropic effects. However, such optical ME phenomena of (semi)metallic systems have remained elusive so far. Here, we show the large nonlinear optical ME response in noncentrosymmetric magnetic Weyl semimetal PrAlGe, in which the polar structural asymmetry and ferromagnetic ordering break P and T symmetry. We observe the giant second harmonic generation (SHG) arising from the P symmetry breaking in the paramagnetic phase, being comparable to the largest SHG response reported in Weyl semimetal TaAs. In the ferromagnetically ordered phase, it is found that interference between this nonmagnetic SHG and the magnetically induced SHG emerging due to both P and T symmetry breaking results in the magnetic field switching of SHG intensity. Furthermore, such an interference effect critically depends on the light-propagating direction. The corresponding magnetically induced nonlinear susceptibility is significantly larger than the prototypical ME material, manifesting the existence of the strong nonlinear dynamical ME coupling. The present findings establish the unique optical functionality of P- and T-symmetry broken ME topological semimetals.

Quantifying the photocurrent fluctuation in quantum materials by shot noise

The DC photocurrent can detect the topology and geometry of quantum materials without inversion symmetry. Herein, we propose that the DC shot noise (DSN), as the fluctuation of photocurrent operator, can also be a diagnostic of quantum materials. Particularly, we develop the quantum theory for DSNs in gapped systems and identify the shift and injection DSNs by dividing the second-order photocurrent operator into off-diagonal and diagonal contributions, respectively. Remarkably, we find that the DSNs can not be forbidden by inversion symmetry, while the constraint from time-reversal symmetry depends on the polarization of light. Furthermore, we show that the DSNs also encode the geometrical information of Bloch electrons, such as the Berry curvature and the quantum metric. Finally, guided by symmetry, we apply our theory to evaluate the DSNs in monolayer GeS and bilayer MoS2 with and without inversion symmetry and find that the DSNs can be larger in centrosymmetric phase.

Giant intrinsic photovoltaic effect in one-dimensional van der Waals grain boundaries

The photovoltaic effect lies at the heart of eco-friendly energy harvesting. However, the conversion efficiency of traditional photovoltaic effect utilizing the built-in electric effect in p-n junctions is restricted by the Shockley-Queisser limit. Alternatively, intrinsic/bulk photovoltaic effect (IPVE/BPVE), a second-order nonlinear optoelectronic effect arising from the broken inversion symmetry of crystalline structure, can overcome this theoretical limit. Here, we uncover giant and robust IPVE in one-dimensional (1D) van der Waals (vdW) grain boundaries (GBs) in a layered semiconductor, ReS2. The IPVE-induced photocurrent densities in vdW GBs are among the highest reported values compared with all kinds of material platforms. Furthermore, the IPVE-induced photocurrent is gate-tunable with a polarization-independent component along the GBs, which is preferred for energy harvesting. The observed IPVE in vdW GBs demonstrates a promising mechanism for emerging optoelectronics applications.

Light-induced shift current vortex crystals in moiré heterobilayers

Transition metal dichalcogenide (TMD) moiré superlattices provide an emerging platform to explore various light-induced phenomena. Recently, the discoveries of novel moiré excitons have attracted great interest. The nonlinear optical responses of these systems are however still underexplored. Here, we report investigation of light-induced shift currents (a second-order response generating DC current from optical illumination) in the WSe2/WS2 moiré superlattice. We identify a striking phenomenon of the formation of shift current vortex crystals-i.e., two-dimensional periodic arrays of moiré-scale current vortices and associated magnetic fields with remarkable intensity under laboratory laser setup. Furthermore, we demonstrate high optical tunability of these current vortices-their location, shape, chirality, and magnitude can be tuned by the frequency, polarization, and intensity of the incident light. Electron-hole interactions (excitonic effects) are found to play a crucial role in the generation and nature of the shift current intensity and distribution. Our findings provide a promising all-optical control route to manipulate nanoscale shift current density distributions and magnetic field patterns, as well as shed light on nonlinear optical responses in moiré quantum matter and their possible applications.

Disorder-induced bulk photovoltaic effect in a centrosymmetric van der Waals material

Sunlight is widely seen as one of the most abundant forms of renewable energy, with photovoltaic cells based on pn junctions being the most commonly used platform attempting to harness it. Unlike in conventional photovoltaic cells, the bulk photovoltaic effect (BPVE) allows for the generation of photocurrent and photovoltage in a single material without the need to engineer a pn junction and create a built-in electric field, thus offering a solution that can potentially exceed the Shockley-Queisser efficiency limit. However, it requires a material with no inversion symmetry and is therefore absent in centrosymmetric materials. Here, we demonstrate that breaking the inversion symmetry by structural disorder can induce BPVE in ultrathin PtSe2, a centrosymmetric semiconducting van der Waals material. Homogenous illumination of defective PtSe2 by linearly and circularly polarized light results in a photoresponse termed as linear photogalvanic effect (LPGE) and circular photogalvanic effect (CPGE), which is mostly absent in the pristine crystal. First-principles calculations reveal that LPGE originates from Se vacancies that act as asymmetric scattering centers for the photo-generated electron-hole pairs. Our work emphasizes the importance of defects to induce photovoltaic functionality in centrosymmetric materials and shows how the range of materials suitable for light sensing and energy-harvesting applications can be extended.

Berry Curvature Spectroscopy from Bloch Oscillations

Artificial crystals such as moiré superlattices can have a real-space periodicity much larger than the underlying atomic scale. This facilitates the presence of Bloch oscillations in the presence of a static electric field. We demonstrate that the optical response of such a system, when dressed with a static field, becomes resonant at the frequencies of Bloch oscillations, which are in the terahertz regime when the lattice constant is of the order of 10 nm. In particular, we show within a semiclassical band-projected theory that resonances in the dressed Hall conductivity are proportional to the lattice Fourier components of the Berry curvature. We illustrate our results with a low-energy model on an effective honeycomb lattice.

Berry Curvature Dipole Induced Giant Mid-Infrared Second-Harmonic Generation in 2D Weyl Semiconductor

Due to its inversion-broken triple helix structure and the nature of Weyl semiconductor, 2D Tellurene (2D Te) is promising to possess a strong nonlinear optical response in the infrared region, which is rarely reported in 2D materials. Here, a giant nonlinear infrared response induced by large Berry curvature dipole (BCD) is demonstrated in the Weyl semiconductor 2D Te. Ultrahigh second-harmonic generation response is acquired from 2D Te with a large second-order nonlinear optical susceptibility (χ(2) ), which is up to 23.3 times higher than that of monolayer MoS2 in the range of 700-1500 nm. Notably, distinct from other 2D nonlinear semiconductors, χ(2) of 2D Te increases extraordinarily with increasing wavelength and reaches up to 5.58 nm V-1 at ≈2300 nm, which is the best infrared performance among the reported 2D nonlinear materials. Large χ(2) of 2D Te also enables the high-intensity sum-frequency generation with an ultralow continuous-wave (CW) pump power. Theoretical calculations reveal that the exceptional performance is attributed to the presence of large BCD located at the Weyl points of 2D Te. These results unravel a new linkage between Weyl semiconductor and strong optical nonlinear responses, rendering 2D Te a competitive candidate for highly efficient nonlinear 2D semiconductors in the infrared region.

Quantum Optics Measurement Scheme for Quantum Geometry and Topological Invariants

We show how a quantum optical measurement scheme based on heterodyne detection can be used to explore geometrical and topological properties of condensed matter systems. Considering a 2D material placed in a cavity with a coupling to the environment, we compute correlation functions of the photons exiting the cavity and relate them to the hybrid light-matter state within the cavity. Different polarizations of the intracavity field give access to all components of the quantum geometric tensor on contours in the Brillouin zone defined by the transition energy. Combining recent results based on the metric-curvature correspondence with the measured quantum metric allows us to characterize the topological phase of the material. Moreover, in systems where S_{z} is a good quantum number, the procedure also allows us to extract the spin Chern number. As an interesting application, we consider a minimal model for twisted bilayer graphene at the magic angle, and discuss the feasibility of extracting the Euler number.

Extrinsic Nonlinear Kerr Rotation in Topological Materials under a Magnetic Field

Topological properties in quantum materials are often governed by symmetry and tuned by crystal structure and external fields, and hence, symmetry-sensitive nonlinear optical measurements in a magnetic field are a valuable probe. Here, we report nonlinear magneto-optical second harmonic generation (SHG) studies of nonmagnetic topological materials including bilayer WTe2, monolayer WSe2, and bulk TaAs. The polarization-resolved patterns of optical SHG under a magnetic field show nonlinear Kerr rotation in these time-reversal symmetric materials. For materials with 3-fold rotational symmetric lattice structure, the SHG polarization pattern rotates just slightly in a magnetic field, whereas in those with mirror or 2-fold rotational symmetry, the SHG polarization pattern rotates greatly and distorts. These different magneto-SHG characters can be understood by considering the superposition of the magnetic field-induced time-noninvariant nonlinear optical tensor and the crystal-structure-based time-invariant counterpart. The situation is further clarified by scrutinizing the Faraday rotation, whose subtle interplay with crystal symmetry accounts for the diverse behavior of the extrinsic nonlinear Kerr rotation in different materials. Our work illustrates the application of magneto-SHG techniques to directly probe nontrivial topological properties, and underlines the importance of minimizing extrinsic nonlinear Kerr rotation in polarization-resolved magneto-optical studies.

Photocurrent Induced by a Bicircular Light Drive in Centrosymmetric Systems

A bicircular light (BCL) consists of left and right circularly polarized lights with different frequencies, and draws a roselike pattern with a rotational symmetry determined by the ratio of the two frequencies. Here we show that an application of a BCL to centrosymmetric systems allows a photocurrent generation through introduction of an effective polarity to the system. We derive formulas for the BCL-induced photocurrent from a standard perturbation theory, which is then applied to a simple 1D model and 3D Dirac and Weyl semimetals. A nonperturbative effect with strong light intensity is also discussed with the Floquet technique.

Giant room-temperature nonlinearities in a monolayer Janus topological semiconductor

Nonlinear optical materials possess wide applications, ranging from terahertz and mid-infrared detection to energy harvesting. Recently, the correlations between nonlinear optical responses and certain topological properties, such as the Berry curvature and the quantum metric tensor, have attracted considerable interest. Here, we report giant room-temperature nonlinearities in non-centrosymmetric two-dimensional topological materials-the Janus transition metal dichalcogenides in the 1 T' phase, synthesized by an advanced atomic-layer substitution method. High harmonic generation, terahertz emission spectroscopy, and second harmonic generation measurements consistently show orders-of-the-magnitude enhancement in terahertz-frequency nonlinearities in 1 T' MoSSe (e.g., > 50 times higher than 2H MoS2 for 18th order harmonic generation; > 20 times higher than 2H MoS2 for terahertz emission). We link this giant nonlinear optical response to topological band mixing and strong inversion symmetry breaking due to the Janus structure. Our work defines general protocols for designing materials with large nonlinearities and heralds the applications of topological materials in optoelectronics down to the monolayer limit.

Berry curvature dipole generation and helicity-to-spin conversion at symmetry-mismatched heterointerfaces

The Berry curvature dipole (BCD) is a key parameter that describes the geometric nature of energy bands in solids. It defines the dipole-like distribution of Berry curvature in the band structure and plays a key role in emergent nonlinear phenomena. The theoretical rationale is that the BCD can be generated at certain symmetry-mismatched van der Waals heterointerfaces even though each material has no BCD in its band structure. However, experimental confirmation of such a BCD induced via breaking of the interfacial symmetry remains elusive. Here we demonstrate a universal strategy for BCD generation and observe BCD-induced gate-tunable spin-polarized photocurrent at WSe2/SiP interfaces. Although the rotational symmetry of each material prohibits the generation of spin photocurrent under normal incidence of light, we surprisingly observe a direction-selective spin photocurrent at the WSe2/SiP heterointerface with a twist angle of 0°, whose amplitude is electrically tunable with the BCD magnitude. Our results highlight a BCD-spin-valley correlation and provide a universal approach for engineering the geometric features of twisted heterointerfaces.

Strong bulk photovoltaic effect in engineered edge-embedded van der Waals structures

Bulk photovoltaic effect (BPVE), a second-order nonlinear optical effect governed by the quantum geometric properties of materials, offers a promising approach to overcome the Shockley-Quiesser limit of traditional photovoltaic effect and further improve the efficiency of energy harvesting. Here, we propose an effective platform, the nano edges embedded in assembled van der Waals (vdW) homo- or hetero-structures with strong symmetry breaking, low dimensionality and abundant species, for BPVE investigations. The BPVE-induced photocurrents strongly depend on the orientation of edge-embedded structures and polarization of incident light. Reversed photocurrent polarity can be observed at left and right edge-embedded structures. Our work not only visualizes the unique optoelectronic effect in vdW nano edges, but also provides an effective strategy for achieving BPVE in engineered vdW structures.

Ultrafast terahertz emission from emerging symmetry-broken materials

Nonlinear optical spectroscopies are powerful tools for investigating both static material properties and light-induced dynamics. Terahertz (THz) emission spectroscopy has emerged in the past several decades as a versatile method for directly tracking the ultrafast evolution of physical properties, quasiparticle distributions, and order parameters within bulk materials and nanoscale interfaces. Ultrafast optically-induced THz radiation is often analyzed mechanistically in terms of relative contributions from nonlinear polarization, magnetization, and various transient free charge currents. While this offers material-specific insights, more fundamental symmetry considerations enable the generalization of measured nonlinear tensors to much broader classes of systems. We thus frame the present discussion in terms of underlying broken symmetries, which enable THz emission by defining a system directionality in space and/or time, as well as more detailed point group symmetries that determine the nonlinear response tensors. Within this framework, we survey a selection of recent studies that utilize THz emission spectroscopy to uncover basic properties and complex behaviors of emerging materials, including strongly correlated, magnetic, multiferroic, and topological systems. We then turn to low-dimensional systems to explore the role of designer nanoscale structuring and corresponding symmetries that enable or enhance THz emission. This serves as a promising route for probing nanoscale physics and ultrafast light-matter interactions, as well as facilitating advances in integrated THz systems. Furthermore, the interplay between intrinsic and extrinsic material symmetries, in addition to hybrid structuring, may stimulate the discovery of exotic properties and phenomena beyond existing material paradigms.

General nonlinear Hall current in magnetic insulators beyond the quantum anomalous Hall effect

Can a generic magnetic insulator exhibit a Hall current? The quantum anomalous Hall effect (QAHE) is one example of an insulating bulk carrying a quantized Hall conductivity while insulators with zero Chern number present zero Hall conductance in the linear response regime. Here, we find that a general magnetic insulator possesses a nonlinear Hall conductivity quadratic to the electric field if the system breaks inversion symmetry, which can be identified as a new type of multiferroic coupling. This conductivity originates from an induced orbital magnetization due to virtual interband transitions. We identify three contributions to the wavepacket motion, a velocity shift, a positional shift, and a Berry curvature renormalization. In contrast to the crystalline solid, we find that this nonlinear Hall conductivity vanishes for Landau levels of a 2D electron gas, indicating a fundamental difference between the QAHE and the integer quantum Hall effect.

Giant Magnetochiral Anisotropy in Weyl Semimetal WTe_{2} Induced by Diverging Berry Curvature

The concept of Berry curvature is essential for various transport phenomena. However, an effect of the Berry curvature on magnetochiral anisotropy, i.e., nonreciprocal magnetotransport, is still elusive. Here, we report that the Berry curvature induces the large magnetochiral anisotropy. In Weyl semimetal WTe_{2}, we observe the strong enhancement of the magnetochiral anisotropy when the Fermi level is located near the Weyl points. Notably, the maximal figure of merit γ[over ¯] reaches 1.2×10^{-6}  m^{2} T^{-1} A^{-1}, which is the largest ever reported in bulk materials. Our semiclassical calculation shows that the diverging Berry curvature at the Weyl points strongly enhances the magnetochiral anisotropy.

Observation of the photovoltaic effect in a van der Waals heterostructure

van der Waals (vdW) heterostructures, which can be assembled with various two-dimensional materials, provide a versatile platform for exploring emergent phenomena. Here, we report an observation of the photovoltaic effect in a WS₂/MoS₂ vdW heterostructure. Light excitation of WS₂/MoS₂ at a wavelength of 633 nm yields a photocurrent without applying bias voltages, and the excitation power dependence of the photocurrent shows characteristic crossover from a linear to square root dependence. Photocurrent mapping has clearly shown that the observed photovoltaic effect arises from the WS₂/MoS₂ region, not from Schottky junctions at electrode contacts. Kelvin probe microscopy observations show no slope in the electrostatic potential, excluding the possibility that the photocurrent originates from an unintentionally formed built-in potential.

Geometric deep optical sensing

Geometry, an ancient yet vibrant branch of mathematics, has important and far-reaching impacts on various disciplines such as art, science, and engineering. Here, we introduce an emerging concept dubbed "geometric deep optical sensing" that is based on a number of recent demonstrations in advanced optical sensing and imaging, in which a reconfigurable sensor (or an array thereof) can directly decipher the rich information of an unknown incident light beam, including its intensity, spectrum, polarization, spatial features, and possibly angular momentum. We present the physical, mathematical, and engineering foundations of this concept, with particular emphases on the roles of classical and quantum geometry and deep neural networks. Furthermore, we discuss the new opportunities that this emerging scheme can enable and the challenges associated with future developments.

Terahertz Nonlinear Hall Rectifiers Based on Spin-Polarized Topological Electronic States in 1T-CoTe2

The zero-magnetic-field nonlinear Hall effect (NLHE) refers to the second-order transverse current induced by an applied alternating electric field; it indicates the topological properties of inversion-symmetry-breaking crystals. Despite several studies on the NLHE induced by the Berry-curvature dipole in Weyl semimetals, the direct current conversion by rectification is limited to very low driving frequencies and cryogenic temperatures. The nonlinear photoresponse generated by the NLHE at room temperature can be useful for numerous applications in communication, sensing, and photodetection across a high bandwidth. In this study, observations of the second-order NLHE in type-II Dirac semimetal CoTe₂ under time-reversal symmetry are reported. This is determined by the disorder-induced extrinsic contribution on the broken-inversion-symmetry surface and room-temperature terahertz rectification without the need for semiconductor junctions or bias voltage. It is shown that remarkable photoresponsivity over 0.1 A W^-1 , a response time of approximately 710 ns, and a mean noise equivalent power of 1 pW Hz^-1/2 can be achieved at room temperature. The results open a new pathway for low-energy photon harvesting via nonlinear rectification induced by the NLHE in strongly spin-orbit-coupled and inversion-symmetry-breaking systems, promising a considerable impact in the field of infrared/terahertz photonics.

Nonlinear Hall effect in monolayer phosphorene with broken inversion symmetry

Nonlinear Hall effect (NLHE), a new member of the family of Hall effects, in monolayer phosphorene is investigated. We find that phosphorene exhibits pronounced NLHE, arising from the dipole moment of the Berry curvature induced by the proximity effect that breaks the inversion symmetry of the system. Remarkably, the nonlinear Hall response exhibits central minimum with a width on the order of the band gap, followed by two resonance-like peaks. Interestingly, each resonance peak of the Hall response shifts in the negative region of the chemical potential which is consistent with the shift of valence and conduction bands in the energy spectrum of monolayer phosphorene. It is observed that the two peaks are asymmetric, originated from anisotropy in the band structure of phosphorene. It is shown that the NLHE is very sensitive to the band gap and temperature of the system. Moreover, we find that a phase transition occurs in the nonlinear Hall response and nonlinear spin Hall conductivity of the system under the influence of spin-orbit interaction, tuned by the strength of interaction and band gap induced in the energy spectrum of monolayer phosphorene with broken inversion symmetry.

Visualizing Higher-Fold Topology in Chiral Crystals

Novel topological phases of matter are fruitful platforms for the discovery of unconventional electromagnetic phenomena. Higher-fold topology is one example, where the low-energy description goes beyond standard model analogs. Despite intensive experimental studies, conclusive evidence remains elusive for the multigap topological nature of higher-fold chiral fermions. In this Letter, we leverage a combination of fine-tuned chemical engineering and photoemission spectroscopy with photon energy contrast to discover the higher-fold topology of a chiral crystal. We identify all bulk branches of a higher-fold chiral fermion for the first time, critically important for allowing us to explore unique Fermi arc surface states in multiple interband gaps, which exhibit an emergent ladder structure. Through designer chemical gating of the samples in combination with our measurements, we uncover an unprecedented multigap bulk boundary correspondence. Our demonstration of multigap electronic topology will propel future research on unconventional topological responses.

Giant photon momentum locked THz emission in a centrosymmetric Dirac semimetal

Strong second-order optical nonlinearities often require broken material centrosymmetry, thereby limiting the type and quality of materials used for nonlinear optical devices. Here, we report a giant and highly tunable terahertz (THz) emission from thin polycrystalline films of the centrosymmetric Dirac semimetal PtSe₂. Our PtSe₂ THz emission is turned on at oblique incidence and locked to the photon momentum of the incident pump beam. Notably, we find an emitted THz efficiency that is giant: It is two orders of magnitude larger than the standard THz-generating nonlinear crystal ZnTe and has values approaching that of the noncentrosymmetric topological material TaAs. Further, PtSe₂ THz emission displays THz sign and amplitude that is controlled by the incident pump polarization and helicity state even as optical absorption is only weakly polarization dependent and helicity independent. Our work demonstrates how photon drag can activate pronounced optical nonlinearities that are available even in centrosymmetric Dirac materials.

Giant bulk piezophotovoltaic effect in 3R-MoS2

Given its innate coupling with wavefunction geometry in solids and its potential to boost the solar energy conversion efficiency, the bulk photovoltaic effect (BPVE) has been of considerable interest in the past decade^1-14. Initially discovered and developed in ferroelectric oxide materials², the BPVE has now been explored in a wide range of emerging materials, such as Weyl semimetals^9,10, van der Waals nanomaterials^11,12,14, oxide superlattices¹⁵, halide perovskites¹⁶, organics¹⁷, bulk Rashba semiconductors¹⁸ and others. However, a feasible experimental approach to optimize the photovoltaic performance is lacking. Here we show that strain-induced polarization can significantly enhance the BPVE in non-centrosymmetric rhombohedral-type MoS₂ multilayer flakes (that is, 3R-MoS₂). This polarization-enhanced BPVE, termed the piezophotovoltaic effect, exhibits distinctive crystallographic orientation dependence, in that the enhancement mainly manifests in the armchair direction of the 3R-MoS₂ lattice while remaining largely intact in the zigzag direction. Moreover, the photocurrent increases by over two orders of magnitude when an in-plane tensile strain of ~0.2% is applied, rivalling that of state-of-the-art materials. This work unravels the potential of strain engineering in boosting the photovoltaic performance, which could potentially promote the exploration of novel photoelectric processes in strained two-dimensional layered materials and their van der Waals heterostructures.

Tailoring Bulk Photovoltaic Effects in Magnetic Sliding Ferroelectric Materials

The bulk photovoltaic effect that is intimately associated with crystalline symmetry has been extensively studied in various nonmagnetic materials, especially ferroelectrics with a switchable electric polarization. In order to further engineer the symmetry, one could resort to spin-polarized systems possessing an extra magnetic degree of freedom. Here, we investigate the bulk photovoltaic effect in two-dimensional magnetic sliding ferroelectric (MSFE) systems, illustrated in VSe₂, FeCl₂, and CrI₃ bilayers. The transition metal elements in these systems exhibit intrinsic spin polarization, and the stacking mismatch between the two layers produces a finite out-of-plane electric dipole. Through symmetry analyses and first-principles calculations, we show that photoinduced in-plane bulk photovoltaic current can be effectively tuned by their magnetic order and the out-of-plane dipole moment. The underlying mechanism is elucidated from the quantum metric dipole distribution in the reciprocal space. The ease of the fabrication and manipulation of MSFEs guarantee practical optoelectronic applications.

Resonant Second-Harmonic Generation as a Probe of Quantum Geometry

Nonlinear responses are actively studied as probes of topology and band geometric properties of solids. Here, we show that second harmonic generation serves as a probe of the Berry curvature, quantum metric, and quantum geometric connection. We generalize the theory of second harmonic generation to include Fermi surface effects in metallic systems, and finite scattering timescale. In doped materials the Fermi surface and Fermi sea cause all second harmonic terms to exhibit resonances, and we identify two novel contributions to the second harmonic signal: a double resonance due to the Fermi surface and a higher-order pole due to the Fermi sea. We discuss experimental observation in the monolayer of time reversal symmetric Weyl semimetal WTe_{2} and the parity-time reversal symmetric topological antiferromagnet CuMnAs.