Topological semimetals are unique quantum materials with protected energy band crossings. They bridge the gap between topological insulators and conventional metals, exhibiting exotic electronic properties that provide insights into fundamental quantum phenomena.

These materials possess nontrivial band topology, linear energy dispersion, and symmetry-protected features. They display unique surface states and bulk-boundary correspondence, leading to fascinating transport properties and potential applications in next-generation electronics and quantum computing.

Fundamentals of topological semimetals

• Topological semimetals represent a unique class of quantum materials in condensed matter physics characterized by protected crossings of energy bands
• These materials bridge the gap between topological insulators and conventional metals, exhibiting exotic electronic properties
• Understanding topological semimetals provides insights into fundamental quantum phenomena and potential applications in next-generation electronics

Band structure characteristics

• Possess nontrivial band topology with protected band crossings at discrete points or along lines in momentum space
• Exhibit linear energy dispersion near the crossing points, resembling massless Dirac or Weyl fermions
• Feature bulk band degeneracies that cannot be removed by small perturbations without breaking certain symmetries
• Display unique low-energy excitations distinct from conventional metals or semiconductors

Symmetry protection mechanisms

• Rely on crystalline symmetries (inversion, rotation, mirror) to protect topological features
• Time-reversal symmetry plays a crucial role in certain types of topological semimetals (Dirac semimetals)
• Require the presence of specific symmetries to maintain the stability of band crossings
• Breaking of symmetries can lead to transitions between different topological phases or to trivial semimetals

Bulk-boundary correspondence

• Establishes a fundamental relationship between bulk topological properties and surface state characteristics
• Manifests as unique surface states (Fermi arcs) connecting bulk band crossing points
• Guarantees the existence of topologically protected surface states immune to backscattering
• Provides a powerful tool for experimental verification of topological semimetal phases

Types of topological semimetals

• Topological semimetals encompass various subclasses with distinct band structures and symmetry requirements
• Each type exhibits unique physical properties and potential applications in condensed matter physics
• Understanding the differences between these types aids in material design and experimental investigations

Weyl semimetals

• Feature isolated band crossing points called Weyl nodes acting as monopoles of Berry curvature
• Require breaking of either time-reversal or inversion symmetry
• Exhibit Fermi arc surface states connecting projections of bulk Weyl nodes
• Display the chiral anomaly leading to negative magnetoresistance (TaAs, NbAs)

Dirac semimetals

• Possess four-fold degenerate band crossing points protected by crystalline symmetries
• Maintain both time-reversal and inversion symmetry
• Can be viewed as two copies of Weyl semimetals with opposite chirality
• Exhibit linear energy dispersion in all three momentum directions (Cd3As2, Na3Bi)

Nodal line semimetals

• Characterized by band crossings along closed loops or lines in momentum space
• Protected by combinations of symmetries such as mirror reflection and time-reversal
• Display drumhead surface states nested inside the projection of bulk nodal lines
• Exhibit unique quantum oscillations and optical responses (PbTaSe2, ZrSiS)

Topological invariants

• Topological invariants serve as mathematical tools to classify and characterize topological phases in condensed matter systems
• These quantities remain unchanged under continuous deformations of the system, providing robust indicators of topological properties
• Understanding topological invariants aids in predicting and analyzing the behavior of topological semimetals

Berry phase and curvature

• Berry phase quantifies the geometric phase acquired by a quantum state under adiabatic evolution
• Berry curvature acts as an effective magnetic field in momentum space, influencing electron dynamics
• Integral of Berry curvature over a closed surface yields topological invariants (Chern numbers)
• Plays a crucial role in determining the transport properties of topological semimetals

Chern number

• Integer-valued topological invariant characterizing the global topology of band structures
• Calculated by integrating the Berry curvature over a closed two-dimensional manifold in momentum space
• Determines the number and chirality of edge states in quantum Hall systems and Weyl semimetals
• Non-zero Chern number indicates the presence of topologically protected surface states

Z2 invariant

• Characterizes time-reversal invariant topological insulators and some classes of topological semimetals
• Takes values of either 0 (trivial) or 1 (topological) for each independent momentum direction
• Determined by the parity eigenvalues of occupied bands at time-reversal invariant momenta
• Predicts the existence of protected surface states in topological insulators and Dirac semimetals

Fermi arcs and surface states

• Fermi arcs and surface states represent unique manifestations of bulk topology in topological semimetals
• These features provide experimental signatures for identifying and characterizing topological semimetal phases
• Understanding the properties of surface states aids in developing novel devices exploiting topological properties

Formation of Fermi arcs

• Arise as topologically protected surface states connecting projections of bulk Weyl nodes
• Result from the bulk-boundary correspondence principle in topological semimetals
• Form open contours in the surface Brillouin zone, unlike closed Fermi surfaces in conventional metals
• Length and shape of Fermi arcs depend on the separation and position of bulk Weyl nodes

Experimental observations

• Directly visualized using angle-resolved photoemission spectroscopy (ARPES)
• Appear as distinct features in the surface electronic structure of Weyl and Dirac semimetals
• Exhibit spin-momentum locking, with spin orientation determined by the chirality of bulk nodes
• Observed in various materials (TaAs, Cd3As2) confirming theoretical predictions of topological semimetal phases

Surface state properties

• Display robustness against non-magnetic impurities due to topological protection
• Exhibit high mobility and unique transport characteristics distinct from bulk states
• Contribute to novel quantum oscillations in magnetotransport measurements
• Offer potential applications in spintronics and low-dissipation electronic devices

Transport properties

• Transport properties of topological semimetals reveal unique signatures of their nontrivial band topology
• These characteristics distinguish topological semimetals from conventional metals and semiconductors
• Understanding transport phenomena aids in developing novel electronic and spintronic devices

Chiral anomaly

• Quantum anomaly leading to non-conservation of chiral charge in the presence of parallel electric and magnetic fields
• Manifests as pumping of electrons between Weyl nodes of opposite chirality
• Results in enhanced conductivity along the direction of applied magnetic field
• Provides a distinctive experimental signature for identifying Weyl and Dirac semimetals

Negative magnetoresistance

• Unusual decrease in electrical resistance with increasing magnetic field
• Arises from the chiral anomaly in Weyl and Dirac semimetals
• Exhibits a characteristic angular dependence, maximized when electric and magnetic fields are parallel
• Serves as a key experimental probe for verifying the topological nature of semimetals

Quantum oscillations

• Periodic oscillations in various physical properties (resistivity, magnetization) as a function of inverse magnetic field
• Reveal information about the Fermi surface topology and electron dynamics
• Display unique features in topological semimetals due to the presence of Weyl or Dirac points
• Provide insights into the Berry phase and non-trivial band topology of the system

Experimental techniques

• Experimental techniques play a crucial role in identifying and characterizing topological semimetals
• These methods provide direct evidence for the unique electronic structure and transport properties of topological materials
• Combining multiple experimental approaches allows for comprehensive understanding of topological semimetal physics

ARPES for band structure

• Angle-resolved photoemission spectroscopy directly maps the electronic band structure of materials
• Reveals linear band crossings and Fermi arc surface states in topological semimetals
• Provides information on the spin texture and momentum-dependence of electronic states
• Enables visualization of bulk and surface states, confirming theoretical predictions (Cd3As2, TaAs)

Magnetotransport measurements

• Probe the response of electrical conductivity to applied magnetic fields
• Reveal signatures of the chiral anomaly through negative magnetoresistance
• Allow for the observation of quantum oscillations providing information on Fermi surface topology
• Enable the study of Berry phase effects and non-trivial band topology in topological semimetals

Scanning tunneling spectroscopy

• Provides local information on the electronic density of states with atomic resolution
• Allows for direct visualization of surface states and their spatial distribution
• Reveals signatures of bulk band topology through quasiparticle interference patterns
• Enables the study of impurity effects and local electronic structure in topological semimetals

Materials and realizations

• Topological semimetals can be realized in various material systems and engineered structures
• Understanding different realizations aids in exploring fundamental physics and developing practical applications
• Diverse material platforms offer opportunities for tuning and optimizing topological properties

Inorganic crystalline materials

• Natural crystalline compounds exhibiting topological semimetal phases
• Include transition metal pnictides (TaAs, NbAs) as Weyl semimetals
• Feature Dirac semimetals in compounds like Cd3As2 and Na3Bi
• Display nodal line semimetal behavior in materials such as PbTaSe2 and ZrSiS

Engineered heterostructures

• Artificially designed layered structures to realize topological semimetal phases
• Include topological insulator/normal insulator superlattices for Weyl semimetal states
• Utilize strain engineering and interface effects to induce topological phase transitions
• Allow for precise control and tuning of topological properties through material composition and layer thicknesses

Photonic and acoustic analogs

• Artificial structures mimicking electronic band structures in electromagnetic or acoustic systems
• Realize Weyl points and nodal lines in specially designed photonic crystals
• Demonstrate topological surface states and bulk-boundary correspondence in acoustic metamaterials
• Provide macroscopic platforms for studying topological physics and developing novel wave-guiding devices

Applications and future prospects

• Topological semimetals offer exciting possibilities for next-generation technologies and fundamental research
• Their unique properties enable novel applications in various fields of condensed matter physics and beyond
• Ongoing research aims to harness the potential of topological semimetals for practical device applications

Spintronics and quantum computing

• Exploit spin-momentum locking of surface states for efficient spin current generation
• Utilize topological protection to create robust quantum bits (qubits) for quantum computation
• Explore Majorana fermions in topological superconductor heterostructures for topological quantum computing
• Develop novel spintronic devices with low power consumption and high efficiency

High-performance electronics

• Leverage high carrier mobility and linear dispersion for ultra-fast electronic devices
• Utilize chiral anomaly for novel magnetoresistive sensors and memory devices
• Explore possibilities for low-dissipation electronics exploiting topological surface states
• Develop terahertz detectors and emitters based on the unique optical properties of topological semimetals

Topological quantum chemistry

• Apply concepts from topological semimetals to predict and design new topological materials
• Develop systematic classification schemes for topological phases in real materials
• Utilize symmetry indicators and band representations to automate the discovery of topological materials
• Explore the interplay between topology and strong correlations in quantum materials