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Condensed matter physics

Condensed matter physics is the branch of physics that studies the properties of large collections of atoms composing both natural and synthetic materials, particularly in their condensed phases such as solids, liquids, and other complex states, employing principles from quantum mechanics and statistical mechanics to explain macroscopic behaviors arising from microscopic interactions.^[1] This field encompasses a broad scope, including the investigation of electronic, magnetic, optical, and structural properties of materials ranging from crystalline solids and amorphous glasses to polymers, composites, and artificially structured systems at atomic scales.^[1] Key subfields include solid-state physics, which focuses on ordered structures like semiconductors and metals; soft matter physics, dealing with liquids, colloids, and biological materials; and quantum matter, exploring exotic states such as superconductors and topological insulators.^[2] It overlaps significantly with materials science, nanotechnology, and biophysics, addressing phenomena like phase transitions, emergent behaviors in many-body systems, and nonequilibrium dynamics.^[3] Condensed matter physics is the largest subfield of physics, with the American Physical Society's Division of Condensed Matter Physics boasting nearly 6,700 members as of 2019—almost double the size of the next largest divisions—and representing about one-third of physicists in the United States.^[4] Its importance lies in driving technological innovations, including the invention of the transistor in 1947, which enabled modern electronics; magnetic resonance imaging (MRI) for medical diagnostics; and optical fibers for telecommunications.^[1] Recent advances continue to fuel progress in quantum computing,^[5] high-temperature superconductivity discovered in 1986, and nanomaterials like graphene,^[6] underscoring its role in the information revolution and interdisciplinary applications in energy, biology, and engineering.^[1]^[4] Historically rooted in early 20th-century quantum mechanics discoveries, such as the explanation of superfluid helium in the 1930s, the field evolved rapidly after World War II, leading to the establishment of dedicated societies and a surge in research on solid-state phenomena.^[1] Today, it remains highly diverse and vibrant, with experimental techniques like nanofabrication and computational simulations enabling atomic-scale manipulation and prediction of material behaviors.^[1]

Overview

Definition and scope

Condensed matter physics is the branch of physics that studies the macroscopic and microscopic properties of matter in its condensed phases, particularly solids and liquids, where particles are densely packed and interact strongly to produce collective behaviors such as electrical conductivity, magnetism, and superfluidity.^[1] This field emphasizes how the collective interactions among vast numbers of atoms or molecules give rise to emergent properties that cannot be predicted from the behavior of isolated particles alone.^[3] Unlike atomic or molecular physics, which focuses on the properties of individual atoms or small clusters, condensed matter physics investigates the complex phenomena arising in systems with 10^{23} or more particles, where quantum and statistical effects dominate on large scales.^[1] The scope of condensed matter physics encompasses a wide range of materials and states, including crystalline and amorphous solids, liquids, and soft matter such as polymers, colloids, granular materials, and even biological structures.^[1] It addresses both equilibrium properties, like phase transitions and thermodynamic stability, and non-equilibrium dynamics, such as transport processes and response to external fields.^[7] The field deliberately excludes dilute gases, where particle interactions are weak and negligible, and plasmas, which involve ionized particles at high temperatures; instead, it centers on dense systems where short-range forces lead to ordered or disordered structures.^[1] The term "condensed matter physics" was coined in 1967 by Philip W. Anderson and Volker Heine while renaming their research group at the University of Cambridge's Cavendish Laboratory, aiming to unify studies of solid-state physics with low-temperature phenomena and to broaden the focus beyond just solids to include liquids and other dense phases.^[8] This naming shift highlighted the field's emphasis on fundamental scientific questions over applied technologies, distinguishing it from narrower labels like "solid-state physics" and fostering a more inclusive approach to understanding matter's collective states.^[9]

Relation to other disciplines

Condensed matter physics intersects significantly with chemistry, particularly in the study of molecular solids and surfaces, where principles from both fields elucidate the electronic and structural properties of materials at the atomic scale. For instance, understanding chemical bonding in solids bridges solid-state chemistry and physics, enabling insights into band structures and surface reactions that influence material functionality.^[10] This overlap extends to condensed-matter chemistry, which examines the physics and chemistry of materials in condensed states, including multi-level structures from molecules to living organisms.^[11] The discipline also forms a core foundation for materials science, providing theoretical and experimental frameworks for designing novel materials with tailored properties, such as superconductors and nanomaterials. Condensed matter physics contributes to the discovery and optimization of these materials by analyzing their structural, electronic, and magnetic behaviors, which directly informs engineering applications in energy storage and electronics.^[12] Furthermore, it relies heavily on statistical mechanics to describe thermodynamic properties of condensed systems, including phase transitions and collective behaviors in solids, liquids, and amorphous materials. This integration allows for predictive models of macroscopic phenomena from microscopic interactions.^[13] In quantum computing, condensed matter physics provides essential models for solid-state qubits, notably in superconducting circuits where quantum coherence is maintained at low temperatures to perform computational operations. Research in this area has advanced the realization of scalable quantum processors by leveraging phenomena like Josephson junctions and circuit quantum electrodynamics.^[14] These developments highlight the field's role in bridging fundamental quantum mechanics with practical information technologies. Soft matter principles from condensed matter physics find applications in biophysics, particularly in modeling protein folding and lipid membranes, where entropic and elastic effects govern molecular assembly and cellular processes. For example, the dynamics of lipid-protein interactions in membranes are analyzed using soft matter concepts to understand stability and function in biological environments.^[15] Such connections extend to self-assembly mechanisms in proteins, treating them as soft condensed matter systems to predict folding pathways.^[16] The economic impact of condensed matter physics is profound, underpinning technologies like semiconductors and advanced materials that drive global productivity. In 2016, physics-based industries contributed 12.6% to U.S. GDP.^[17] For instance, the U.S. semiconductor industry had a total economic impact of $246 billion in 2020, with broader effects amplifying this figure.^[18] As of 2025, the global consumer electronics market is projected to reach $1tn annually.^[19] Recent U.S. investments under the CHIPS Act have spurred over $500 billion in semiconductor-related projects.^[20] Worldwide, these advancements illustrate the field's transformative role in economic growth.

History

Classical foundations

The foundations of condensed matter physics in the classical era were laid through applications of Newtonian mechanics to the mechanical properties of solids and fluids, beginning with Isaac Newton's Philosophiæ Naturalis Principia Mathematica published in 1687.^[21] In this work, Newton extended his laws of motion to describe the behavior of elastic bodies, including the propagation of sound waves through dense media like air and solids, where he derived the speed of sound as proportional to the square root of the medium's elasticity divided by its density.^[22] This approach treated solids as compressible elastic continua, providing an early framework for understanding deformation and vibration without invoking microscopic structure. In the early 19th century, Thomas Young advanced these ideas by quantifying elasticity in solids through his 1807 introduction of the modulus of elasticity, now known as Young's modulus, which measures a material's stiffness under tensile or compressive stress.^[23] Young's work built on Newtonian principles to model wave propagation in elastic solids, such as longitudinal sound waves, emphasizing how material properties like density and elasticity govern speed and attenuation.^[24] Complementing this, Augustin-Jean Fresnel in the 1810s and 1820s developed the wave theory of light by analogizing it to transverse vibrations in an elastic solid ether, deriving equations for wave speed and refraction that mirrored elastic wave behavior in anisotropic media.^[25] Fresnel's elasticity surface, a mathematical construct for wave propagation in biaxial elastic solids, predicted phenomena like double refraction, linking macroscopic elasticity to optical properties.^[26] Thermodynamic insights into condensed phases emerged with J. Willard Gibbs's 1876 paper "On the Equilibrium of Heterogeneous Substances," which introduced the phase rule governing the coexistence of solids, liquids, and gases at equilibrium. The rule, expressed as F = C - P + 2 where F is degrees of freedom, C is components, and P is phases, provided a classical criterion for phase stability under varying temperature, pressure, and composition, without reference to atomic details.^[27] This framework enabled systematic mapping of phase boundaries in multi-component systems, foundational for understanding condensed matter equilibria. Crystal physics progressed with Auguste Bravais's 1850 memoir "Mémoire sur les systèmes formés par des points distribués régulièrement sur un plan ou dans l'espace," identifying the 14 unique three-dimensional lattices that describe the periodic arrangements in crystalline solids.^[28] Bravais's classification, based on symmetry and geometric constraints, enumerated lattice types from cubic to triclinic, emphasizing how translational and point symmetries dictate macroscopic crystal forms.^[29] Building on this, the Curie brothers—Pierre and Jacques—discovered piezoelectricity in 1880 while studying crystals like quartz and tourmaline, observing that mechanical stress induces electric polarization, and vice versa, in non-centrosymmetric lattices.^[30] Their experiments quantified charge generation proportional to applied strain, revealing electromechanical coupling in solids as a classical consequence of crystal asymmetry. Early metallurgy applied these classical concepts to alloys, developing empirical phase diagrams to predict microstructures without quantum theory. For instance, 19th-century studies of iron-carbon systems mapped eutectic and peritectic reactions using thermal analysis, guiding heat treatments for steel production based on Gibbs's equilibrium principles.^[31] These diagrams illustrated how cooling paths determine phase distributions, such as austenite decomposition, enabling practical control of material properties like hardness through diffusion and solidification.^[32] However, classical models struggled to explain specific heat anomalies and electronic conductivity in metals, hinting at the need for microscopic theories beyond macroscopic thermodynamics.

Quantum mechanical developments

The transition to quantum mechanics in the 1920s fundamentally transformed the study of solids, resolving paradoxes in electronic properties that classical theories could not explain. The Drude-Lorentz model of 1900, which modeled conduction electrons as a classical gas colliding with ions, accurately predicted the ratio of electrical to thermal conductivity (Wiedemann-Franz law) but failed to account for the observed linear temperature dependence of resistivity at low temperatures and the molar specific heat of metals, which approached zero rather than the classical value of $3R/2 per electron as required by equipartition. These discrepancies arose because classical statistics ignored quantum restrictions on electron states, building briefly on earlier thermodynamic insights into heat capacities. Quantum corrections emerged through the free electron gas model incorporating Fermi-Dirac statistics. In 1926, Paul Dirac formulated a quantum statistical mechanics for systems obeying the exclusion principle, where fermions fill antisymmetric wavefunctions, limiting each state to two electrons of opposite spin. Enrico Fermi independently derived similar statistics for ideal gases, emphasizing quantization in phase space. Arnold Sommerfeld applied these in 1928 to metals, treating electrons as a degenerate Fermi gas at room temperature with a high Fermi energy (around 5–10 eV for typical metals), yielding a specific heat contribution \gamma T (where \gamma \approx 1 mJ/mol·K²) from excitations near the Fermi surface, while the bulk remains frozen out, and correcting conductivity to show weak temperature dependence below the Debye temperature. A pivotal advance was Felix Bloch's 1928 theorem, which addressed electrons in the periodic lattice potential of crystals by solving the time-independent Schrödinger equation. Bloch demonstrated that eigenfunctions take the form \psi_{\mathbf{k}}(\mathbf{r}) = u_{\mathbf{k}}(\mathbf{r}) e^{i \mathbf{k} \cdot \mathbf{r}}, where u_{\mathbf{k}} has the lattice periodicity, allowing electrons to propagate as waves modulated by the crystal structure rather than scattering incoherently. This quasi-classical picture enabled the treatment of band formation from overlapping atomic orbitals. Complementing this, Léon Brillouin in 1930 introduced Brillouin zones in reciprocal k-space, defined as Wigner-Seitz cells around reciprocal lattice points, which demarcate allowed wavevector ranges for plane waves; boundaries correspond to Bragg reflection conditions, leading to energy gaps that distinguish metals from insulators. These developments also illuminated exotic phenomena like superconductivity. Heike Kamerlingh Onnes discovered in 1911 that pure mercury abruptly loses all electrical resistance below 4.2 K, a zero-resistance state persisting in persistent currents, initially interpreted classically as perfect conduction but later recognized as a quantum phase coherence effect. In 1933, Walther Meissner and Robert Ochsenfeld observed that superconducting tin and lead expel applied magnetic fields entirely (perfect diamagnetism), with the flux threading only surface currents up to a critical field H_c, distinguishing superconductivity from mere zero resistance and highlighting its quantum electromagnetic response.^[33]

Modern era and key discoveries

The modern era of condensed matter physics, beginning after World War II, shifted focus toward understanding complex many-body interactions in quantum systems, building on earlier quantum foundations to explain emergent phenomena in solids and liquids. A pivotal advancement was Lev Landau's Fermi liquid theory, which described the behavior of interacting electrons in metals as a quasi-classical fluid of quasiparticles, providing a framework for low-temperature properties like specific heat and susceptibility despite strong correlations.^[34] This theory, published in 1956, enabled quantitative predictions for systems like liquid helium-3 and electron gases in metals.^[34] Shortly thereafter, in 1957, John Bardeen, Leon Cooper, and John Robert Schrieffer developed the microscopic BCS theory of superconductivity, explaining the pairing of electrons via phonon-mediated attraction, which leads to zero-resistance states below a critical temperature.^[35] This breakthrough resolved a long-standing puzzle from the 1911 discovery of superconductivity in mercury and earned the trio the 1972 Nobel Prize in Physics.^[35] The theory predicted key observables, such as the energy gap and isotope effect, and laid the groundwork for applications in magnets and quantum computing. In the 1970s, Kenneth Wilson's renormalization group methods revolutionized the study of critical phenomena near phase transitions, allowing systematic analysis of scaling behaviors in interacting systems by integrating out short-wavelength fluctuations.^[36] Introduced in 1971, this approach clarified universality classes and fixed points, earning Wilson the 1982 Nobel Prize and enabling precise calculations for magnetic and fluid transitions.^[36] The 1980s brought the surprise discovery of high-temperature superconductors by J. Georg Bednorz and K. Alex Müller, who observed superconductivity at 35 K in a barium-lanthanum-copper-oxide ceramic, far exceeding BCS predictions for conventional materials. This 1986 finding, confirmed and extended to over 130 K in yttrium-based cuprates by 1987, sparked intense research into unconventional pairing mechanisms involving strong electron correlations and d-wave symmetry, though a full microscopic theory remains elusive. Their work, awarded the 1987 Nobel Prize, transformed the field by highlighting cuprates as a new class of materials for potential room-temperature applications. The 21st century unveiled novel states of matter, starting with the 2004 isolation of graphene—a single atomic layer of carbon—by Andre Geim and Konstantin Novoselov using mechanical exfoliation from graphite.^[37] This two-dimensional material exhibited massless Dirac fermions, exceptional electron mobility, and ballistic transport, enabling studies of quantum Hall effects and relativistic physics at room temperature; their achievement garnered the 2010 Nobel Prize.^[37] Theoretical predictions in 2005 by Charles Kane and Eugene Mele identified the quantum spin Hall effect in graphene edges, heralding topological insulators: materials insulating in the bulk but conducting on surfaces via protected edge states immune to backscattering. Experimental realizations followed, with bismuth-based compounds like Bi_{1-x}Sb_x confirming robust surface states by 2008, opening avenues for spintronics and fault-tolerant quantum computing. Philip W. Anderson's 1973 proposal of quantum spin liquids—disordered ground states of interacting spins without magnetic order, stabilized by quantum fluctuations and geometric frustration—gained experimental traction in the 2010s. Materials like herbertsmithite (ZnCu_3(OH)_6Cl_2) showed fractionalized excitations and spinon continua via neutron scattering, confirming gapless or gapped spin liquid phases resistant to ordering down to millikelvin temperatures.^[38] Parallel developments in ultracold atomic gases provided tunable analogs for condensed matter systems, with the first Bose-Einstein condensate (BEC) observed in 1995 by Eric Cornell, Carl Wieman, and coworkers in dilute rubidium vapor at 170 nK.^[39] This macroscopic quantum state, where atoms occupy the lowest quantum level, mimics superfluidity and enables simulation of Hubbard models, topological phases, and nonequilibrium dynamics using optical lattices through 2025.^[39] BECs have since facilitated precision tests of many-body theories inaccessible in solids. In 2019, superconductivity was reported in thin films of rare-earth infinite-layer nickelates at around 15 K, establishing a new family of unconventional superconductors analogous to cuprates.^[40] Building on this, 2024 experiments confirmed superfluidity in supersolids by observing quantized vortices in ultracold dipolar gases of dysprosium atoms, providing direct evidence for this long-sought quantum state of matter that combines solidity and superfluidity.^[41]

Fundamental concepts

Atomic and lattice structures

In condensed matter, the atomic structure of solids is fundamentally characterized by the periodic arrangement of atoms in crystal lattices. A crystal lattice consists of a repeating array of points in space, each occupied by an identical basis of atoms. The possible three-dimensional lattice types are described by the 14 Bravais lattices, which classify all unique ways to arrange points with translational symmetry, ranging from simple cubic to more complex forms like face-centered cubic and hexagonal close-packed. These lattices form the backbone for the full symmetry of crystals, incorporating not only translations but also rotations, reflections, and inversions, leading to the 230 distinct space groups that enumerate all possible crystal symmetries.^[42] Associated with each real-space lattice is its reciprocal lattice, a geometric construct in momentum space where lattice vectors are transformed via Fourier duality, enabling the analysis of wave-like phenomena in crystals. The first Brillouin zone is the primitive cell of this reciprocal lattice, defined as the Wigner-Seitz cell around the origin—geometrically, a polyhedron formed by planes perpendicularly bisecting lines to the nearest reciprocal lattice points, enclosing the region closest to the origin and crucial for describing periodic boundary conditions in wave propagation.^[43] This zone's boundaries mark points where wave scattering becomes significant, influencing the overall periodicity of lattice excitations.^[44] Not all solids exhibit perfect long-range translational order. Amorphous solids, such as glasses, lack this periodicity, featuring short-range order similar to crystals but no repeating lattice over large distances; their structure can be modeled as a continuous random network where atoms satisfy local bonding rules without global symmetry.^[45] The glass transition marks the reversible change from a viscous liquid to a rigid solid upon cooling, occurring without crystallization due to kinetic arrest in the disordered state.^[45] Quasicrystals represent an intermediate case, displaying sharp diffraction patterns indicative of long-range order but with aperiodic arrangements forbidden in traditional lattices, such as icosahedral symmetry observed in rapidly solidified Al-Mn alloys.^[46] Real crystals invariably contain defects that disrupt ideal periodicity, profoundly affecting material properties. Point defects, including vacancies (missing atoms) and interstitials (extra atoms squeezed into lattice sites), occur at concentrations around 10^{-6} to 10^{-3} in typical solids and facilitate atomic diffusion by providing pathways for atom hopping, with vacancy-mediated diffusion dominating self-diffusion rates via an Arrhenius process.^[47] Line defects, known as dislocations, are linear distortions where the lattice plane terminates abruptly; edge dislocations involve an extra half-plane of atoms, while screw types feature a shear offset, and both enable plastic deformation under stress by allowing dislocation glide, which controls yield strength and work hardening in metals.^[48] Planar defects, such as grain boundaries between crystalline domains and stacking faults within a single crystal, introduce interfaces that scatter phonons and electrons, influencing mechanical toughness by pinning dislocations and promoting fracture resistance through energy absorption.^[49] Lattice vibrations, quantized as phonons, arise from collective atomic displacements in the crystal. In the harmonic approximation, these are normal modes of the lattice, with phonons behaving as bosonic quasiparticles carrying energy \hbar \omega. For acoustic phonon branches in simple models, the dispersion relation near the zone center is linear, given by

\omega(\mathbf{k}) = v |\mathbf{k}|,

where v is the speed of sound and \mathbf{k} the wave vector, reflecting long-wavelength sound waves where adjacent atoms move in phase.^[50] These vibrations underpin thermal and elastic properties, and the lattice structure subtly modulates electronic band structures by imposing periodicity on electron waves.

Electronic properties of matter

The electronic properties of matter in condensed systems arise primarily from the collective behavior of electrons interacting with the periodic potential of atomic lattices. In metals, electrons form a nearly free gas that enables high electrical conductivity, while in insulators and semiconductors, the arrangement of energy levels leads to restricted electron mobility, resulting in poor or temperature-dependent conduction. These distinctions emerge from the underlying atomic structure, where valence electrons occupy bands whose filling determines transport properties. The classical Drude model describes conduction in metals by treating electrons as a gas of charged particles subject to scattering by lattice ions, yielding a conductivity σ = ne²τ / m, where n is the electron density, e the charge, τ the relaxation time, and m the mass. This model successfully explains the temperature dependence of resistivity but fails to account for specific heats and the Wiedemann-Franz law quantitatively. Refinements by Sommerfeld incorporated quantum statistics, replacing Maxwell-Boltzmann distributions with Fermi-Dirac, which better predicts the linear temperature dependence of resistivity at low temperatures and the ratio of thermal to electrical conductivity.^[51] The density of states g(E), representing the number of electron states per unit energy interval, plays a crucial role in these properties; in the free-electron approximation for metals, g(E) increases with energy as g(E) ∝ √E, allowing Fermi-level electrons to dominate transport.^[51] Insulators feature a large energy gap between valence and conduction bands, preventing thermal excitation of electrons across the gap, whereas semiconductors have a smaller gap (typically 0.1–3 eV), enabling conduction via thermal or optical excitation, as seen in silicon with a gap of 1.12 eV at room temperature. Magnetism in condensed matter stems from the alignment of electron magnetic moments, leading to distinct responses to external fields. Diamagnetism occurs in all materials due to induced orbital currents opposing the field, producing a weak negative susceptibility independent of temperature, as described by Larmor diamagnetism. Paramagnetism arises in materials with unpaired spins, where moments align randomly with the field, following Curie's law χ = C/T above the Curie temperature, with C the Curie constant proportional to the moment density.^[52] Ferromagnetism, observed in materials like iron, involves spontaneous alignment of moments below the Curie temperature due to exchange interactions, resulting in net magnetization; the susceptibility follows the Curie-Weiss law χ = C / (T - T_N), where T_N is the Néel temperature for antiferromagnets or Curie temperature for ferromagnets, indicating mean-field-like cooperative effects. Dielectrics exhibit polarization in response to electric fields, where atoms or molecules develop induced dipoles, enhancing the internal field. The macroscopic dielectric constant ε_r relates to microscopic polarizability α via the Clausius-Mossotti relation: (ε_r - 1) / (ε_r + 2) = (4π/3) N α, with N the number density, bridging local and bulk responses. Local fields, such as the Lorentz field E_loc = E + (4π/3) P (with P the polarization), exceed the applied field E due to contributions from surrounding dipoles, influencing the effective polarizability in dense media. Excitons represent bound electron-hole pairs formed upon optical excitation in insulators and semiconductors, where the Coulomb attraction localizes the pair over a few lattice sites in Frenkel excitons, as opposed to larger Wannier-Mott excitons in wide-gap materials. These quasiparticles mediate energy transfer without net charge transport, with binding energies around 0.1–1 eV in molecular crystals. Plasmons, conversely, are collective oscillations of the electron density in metals and semiconductors, behaving as longitudinal waves with frequency ω_p = √(4π n e² / m) in the long-wavelength limit, screened by the lattice and leading to phenomena like surface plasmon polaritons at interfaces.

Many-body effects

In condensed matter physics, many-body effects arise from the interactions among numerous particles, such as electrons, which lead to collective behaviors that cannot be described by independent single-particle approximations alone. These interactions modify the effective potential experienced by each particle, resulting in phenomena like screening, correlation, and pairing that underpin the properties of solids. Building briefly on the single-particle electronic properties discussed earlier, many-body theory accounts for how Coulomb repulsion and exchange effects alter the electronic structure beyond simple band filling.^[53] A foundational approach to incorporating many-body interactions is the Hartree-Fock approximation, which treats the system using a self-consistent mean field where each electron moves in the average potential created by all others, including exchange terms to enforce antisymmetry. Developed by Douglas Hartree in 1928 and refined by Vladimir Fock in 1930, this method approximates the many-electron wavefunction as a single Slater determinant, minimizing the energy variationally while neglecting higher-order correlations. Despite its limitations in capturing strong correlations, Hartree-Fock provides a starting point for understanding exchange-correlation effects, which are central to more advanced theories like density functional theory (DFT). In DFT, the Hohenberg-Kohn theorems establish that the ground-state electron density uniquely determines the external potential and thus the total energy, allowing interactions to be incorporated via an exchange-correlation functional.^[53] Screening represents another key many-body effect, where the charge of an impurity or perturbation is partially compensated by the redistribution of surrounding electrons, reducing the effective interaction range. The Thomas-Fermi model, introduced independently by Llewellyn Thomas and Enrico Fermi in 1927–1928, provides a semiclassical description of this process by treating electrons as a degenerate Fermi gas and solving the Poisson equation self-consistently with a local density approximation for the chemical potential. This leads to an exponential decay of the screened potential, characterized by the Thomas-Fermi screening length \lambda_{TF} = \left( \frac{\epsilon_0 E_F}{e^2 n} \right)^{1/2}, where E_F is the Fermi energy and n is the electron density. However, quantum corrections introduce Friedel oscillations, oscillatory modulations in the electron density around impurities with a period related to the Fermi wavelength, as predicted by Jacques Friedel in 1952 through perturbation theory in the presence of a sharp Fermi surface. These oscillations, decaying as $1/r^3 in three dimensions, arise from the sharp cutoff in the density of states and influence properties like resistivity in metals. In superconductors, many-body effects manifest dramatically through the formation of Cooper pairs, where electrons with opposite spins and momenta bind via an attractive interaction mediated by lattice phonons, overcoming Coulomb repulsion at low energies. Leon Cooper demonstrated in 1956 that, in the presence of an attractive potential, two electrons near the Fermi surface form a bound state with binding energy exponentially sensitive to the interaction strength, setting the stage for the Bardeen-Cooper-Schrieffer (BCS) theory. In BCS, this pairing extends coherently across the system, opening a superconducting gap and enabling zero-resistance transport below the critical temperature. For strongly correlated systems, the Hubbard model captures essential many-body physics through a minimal Hamiltonian describing electrons on a lattice with hopping t between nearest neighbors and on-site Coulomb repulsion U:

H = -t \sum_{\langle i,j \rangle, \sigma} (c_{i\sigma}^\dagger c_{j\sigma} + \text{h.c.}) + U \sum_i n_{i\uparrow} n_{i\downarrow},

where c_{i\sigma}^\dagger creates an electron at site i with spin \sigma, and n_{i\sigma} = c_{i\sigma}^\dagger c_{i\sigma}. Proposed by John Hubbard in 1963, this model highlights competition between kinetic energy (delocalization) and interaction energy (localization), leading to Mott insulators at half-filling for large U/t, where double occupancy is suppressed despite a half-filled band. Extensions like the t-J model, derived in the strong-coupling limit (U \gg t) by projecting out double occupancies, further describe doped Mott insulators relevant to high-temperature superconductors, with superexchange J = 4t^2/U favoring antiferromagnetic order.^[54]

Theoretical approaches

Band theory and electronic structure

Band theory provides the foundational quantum mechanical description of electron behavior in crystalline solids, where the periodic arrangement of atoms leads to the formation of energy bands that determine electrical, optical, and thermal properties of materials. In this framework, electrons are treated as waves propagating through a periodic potential, resulting in allowed and forbidden energy regions known as bands and band gaps, respectively. This theory, developed in the early 20th century, explains why materials exhibit metallic, insulating, or semiconducting behavior based on the filling of these bands relative to the Fermi level. The Bloch theorem, formulated by Felix Bloch in 1928, states that the wavefunction of an electron in a periodic potential can be expressed as a product of a plane wave and a periodic function: \psi(\mathbf{r}) = u_{\mathbf{k}}(\mathbf{r}) e^{i \mathbf{k} \cdot \mathbf{r}}, where u_{\mathbf{k}}(\mathbf{r}) has the same periodicity as the lattice and \mathbf{k} is the wavevector in the Brillouin zone. This theorem implies that electron states are labeled by \mathbf{k}, and the energy eigenvalues E(\mathbf{k}) form continuous bands within the first Brillouin zone, enabling the use of periodic boundary conditions to solve the Schrödinger equation efficiently. The theorem's validity relies on the infinite extent of the crystal, approximating real finite systems well for bulk properties. To compute band structures, approximate models are employed, such as the tight-binding approximation, which assumes electrons are localized on atomic orbitals and hop between nearest neighbors. In this model, the dispersion relation for a simple cubic lattice is given by E(\mathbf{k}) = -t \sum_{\delta} \cos(\mathbf{k} \cdot \delta), where t is the hopping integral and \delta are nearest-neighbor vectors; this yields narrow bands for strongly bound electrons, as in semiconductors. Conversely, the nearly free electron model treats electrons as nearly plane waves perturbed by weak periodic potentials, leading to band gaps at Brillouin zone boundaries via Bragg scattering, with gap sizes proportional to the Fourier components of the potential. These models complement each other: tight-binding for insulators and d-band metals, nearly free electron for simple metals like alkali metals. The density of states (DOS), g(E), quantifies the number of electron states per energy interval, derived from the band structure as g(E) = \frac{1}{V} \sum_{\mathbf{k},n} \delta(E - E_n(\mathbf{k})), where V is the volume. In three dimensions, the DOS varies with energy, exhibiting van Hove singularities—logarithmic or power-law divergences—at critical points where the band extremum occurs, influencing properties like specific heat and superconductivity. The Fermi surface, defined by E(\mathbf{k}) = E_F at zero temperature, separates occupied and unoccupied states; its shape dictates metallic conductivity via the Boltzmann transport equation. A notable example is the Peierls distortion in one-dimensional chains, where lattice softening at half-filling opens a band gap, stabilizing a charge-density wave, as observed in materials like K_{0.3}MoO_3. Density functional theory (DFT) has become the workhorse for calculating band structures and electronic properties, building on the Hohenberg-Kohn theorems that express the ground-state energy as a functional of the electron density. The Kohn-Sham equations, \left[ -\frac{\hbar^2}{2m} \nabla^2 + V_{\text{eff}}(\mathbf{r}) \right] \psi_i(\mathbf{r}) = \epsilon_i \psi_i(\mathbf{r}), map the interacting system to a non-interacting one with an effective potential including Hartree, exchange-correlation, and external terms, solved self-consistently for band energies. DFT excels in predicting ground-state properties like lattice constants and band gaps for thousands of materials, as in the Materials Project database, but underestimates gaps due to approximate exchange-correlation functionals (e.g., LDA) and struggles with excited states or strong correlations, requiring extensions like GW or hybrid functionals.

Symmetry and order parameters

In condensed matter physics, ordered phases often arise from spontaneous symmetry breaking, where the ground state of a system selects a configuration that is not invariant under the full symmetry group of the Hamiltonian, leading to the emergence of long-range order. The order parameter serves as a quantitative measure of this broken symmetry, distinguishing the ordered phase from the disordered one; it is typically zero in the high-temperature disordered phase and acquires a nonzero value below the transition temperature. This framework, central to understanding phenomena like magnetism and superconductivity, allows for a phenomenological description of phase transitions without microscopic details.^[55] Landau theory provides a systematic approach to model these transitions by expanding the free energy in powers of the order parameter near the critical point. The free energy is expressed as F = F_0 + a(T - T_c) \phi^2 + b \phi^4 + \cdots, where \phi is the order parameter, T_c is the critical temperature, a > 0, and b > 0 to ensure stability; minimization yields \phi = 0 for T > T_c and \phi \neq 0 for T < T_c, capturing the second-order transition. This expansion assumes the order parameter transforms under an irreducible representation of the symmetry group and is valid when fluctuations are small.^[55] A classic example is ferromagnetism, where the order parameter is the magnetization vector \mathbf{M}, which breaks rotational symmetry by aligning spins in a preferred direction below the Curie temperature. In nematic liquid crystals, the order parameter is the director \mathbf{n}, a headless unit vector describing the average molecular orientation, breaking rotational but not reflection symmetry. These scalar or vector order parameters quantify the extent of alignment in the respective systems. When a continuous symmetry is spontaneously broken, Goldstone's theorem predicts the existence of gapless excitations, known as Goldstone modes, corresponding to each broken generator; these modes restore the symmetry locally through low-energy fluctuations, such as magnons in ferromagnets. The theorem arises from the degeneracy of the ground state and the conservation of the symmetry current, leading to massless bosons in the effective theory. Beyond conventional symmetry breaking, topological order describes quantum phases where long-range entanglement and ground-state degeneracy on topologically nontrivial manifolds define the order, without local order parameters. A seminal example is the fractional quantum Hall state at filling factor \nu = 1/3, described by Laughlin's wavefunction \psi = \prod_{i<j} (z_i - z_j)^3 \exp\left(-\sum_i |z_i|^2 / 4\ell^2\right), which exhibits quasiparticles with fractional charge e/3 and anyonic statistics. This paradigm, realized in two-dimensional electron gases under strong magnetic fields, highlights orders robust against perturbations.^[56]

Phase transitions and critical phenomena

Phase transitions in condensed matter systems occur when a material undergoes a change from one phase to another, such as from solid to liquid or from ferromagnet to paramagnet, driven by variations in temperature, pressure, or other control parameters. These transitions are classified using the Ehrenfest scheme, which distinguishes between first-order and higher-order transitions based on the continuity of thermodynamic derivatives. In first-order transitions, the first derivative of the free energy with respect to the control parameter, such as entropy or volume, exhibits a discontinuity, accompanied by latent heat and phase coexistence.^[57] Higher-order transitions, particularly second-order ones, involve discontinuities in higher derivatives, with no latent heat and a continuous change in the order parameter that characterizes the broken symmetry of the low-temperature phase.^[57] Second-order phase transitions, also known as continuous transitions, are marked by critical points where fluctuations diverge, leading to phenomena like critical opalescence and power-law correlations. A seminal model for understanding these is the Ising model, which describes spins on a lattice interacting ferromagnetically, capturing the essence of magnetic ordering. In 1944, Lars Onsager provided an exact solution for the two-dimensional Ising model without an external field, revealing a second-order transition at a finite temperature T_c with spontaneous magnetization below T_c.^[58] Near T_c, the correlation length \xi, which measures the spatial extent of spin correlations, diverges as \xi \sim |T - T_c|^{-\nu}, where \nu is a critical exponent; for the 2D Ising model, \nu = 1.^[58] Critical phenomena exhibit universality, meaning that systems with the same dimensionality, symmetry, and range of interactions belong to the same universality class, sharing identical critical exponents independent of microscopic details. This hypothesis, formalized by Leo Kadanoff in 1966 through block-spin rescaling, implies that large-scale behavior near criticality is governed by a few relevant parameters. The renormalization group (RG) framework, developed by Kenneth Wilson, explains this by showing that irrelevant operators become negligible under coarse-graining, leading to fixed points that dictate universal scaling. For the Ising model and \phi^4 theory, the upper critical dimension is d=4, above which mean-field theory—ignoring fluctuations—provides exact exponents, as the Gaussian fixed point is stable. Below d=4, fluctuations dominate, requiring corrections; Wilson's \epsilon-expansion, where \epsilon = 4 - d, perturbs around the mean-field solution to compute exponents systematically, such as \nu = 1/2 + \epsilon/12 + O(\epsilon^2) for the Ising class. In addition to thermal phase transitions, quantum phase transitions occur at absolute zero temperature (T=0), driven by non-thermal parameters like doping or pressure, where quantum fluctuations play the role of thermal ones. These transitions separate phases with different ground-state orders, often exhibiting quantum critical points with enhanced scaling regions at finite but low temperatures.^[59] A prominent example is the metal-insulator transition in doped semiconductors or correlated materials like cuprates, where tuning the carrier density via doping crosses a quantum critical point, leading to divergent susceptibilities and non-Fermi liquid behavior in the vicinity.^[60] In underdoped high-T_c cuprates, such as YBa_2Cu_3O_{6+x}, quantum oscillations reveal a doping-dependent metal-insulator quantum critical point near optimal doping, influencing superconductivity and pseudogap phenomena.^[60]

Experimental methods

Scattering and diffraction techniques

Scattering and diffraction techniques are essential probes in condensed matter physics for elucidating atomic-scale structures and dynamics of materials, leveraging the wave-like properties of particles such as X-rays, neutrons, and electrons to reveal periodic arrangements and correlations in solids. These methods exploit interference patterns arising from elastic and inelastic scattering events, providing momentum- and energy-resolved information about lattice parameters, defects, and excitations without direct imaging. In crystalline systems, diffraction peaks correspond to reciprocal lattice vectors, enabling the reconstruction of real-space atomic positions and orientations. X-ray diffraction stands as a cornerstone for determining crystal structures in condensed matter, pioneered by the Braggs who established the fundamental relation governing constructive interference. Bragg's law, expressed as n\lambda = 2d \sin\theta, where n is an integer, \lambda the X-ray wavelength, d the interplanar spacing, and \theta the incidence angle, quantifies the conditions for diffraction from lattice planes.^[61] This principle underpins techniques like powder diffraction and single-crystal analysis, widely used to map atomic arrangements in metals, semiconductors, and complex oxides, with resolutions down to angstrom scales. Neutron scattering complements X-rays by offering sensitivity to light elements and magnetic moments, making it invaluable for probing both structural and magnetic properties in condensed matter. Elastic neutron diffraction exploits the spin-dependent interaction of neutrons with atomic magnetic moments to resolve antiferromagnetic or ferrimagnetic order, revealing spin configurations that X-rays cannot detect due to their charge neutrality. Inelastic neutron scattering, developed by Brockhouse, measures energy transfers to study lattice vibrations (phonons) and magnetic excitations (magnons), providing dispersion relations that inform electron-phonon coupling and spin-wave stiffness in materials like superconductors and magnets.^[62] Electron diffraction techniques, leveraging the short de Broglie wavelengths of accelerated electrons, are particularly suited for surface and nanoscale investigations in condensed matter. Low-energy electron diffraction (LEED) probes the topmost atomic layers of single crystals by directing electrons (20-200 eV) at grazing angles, producing diffraction spots that indicate surface reconstruction and adsorbate ordering, as first demonstrated in early experiments on metal surfaces. Transmission electron microscopy (TEM) combined with electron diffraction visualizes defects such as dislocations and grain boundaries in bulk materials, using selected-area diffraction patterns to identify strain fields and phase boundaries with sub-nanometer precision. The dynamic structure factor, S(\mathbf{q}, \omega), encapsulates the spatiotemporal correlations probed by these scattering methods, defined as the Fourier transform of the density-density correlation function and governing the intensity of scattering at momentum transfer \mathbf{q} and energy \omega. In practice, it distinguishes quasielastic scattering from sharp excitations like phonons, enabling quantitative analysis of diffusion, relaxations, and collective modes in disordered or ordered phases of condensed matter.

Spectroscopic and resonance methods

Spectroscopic and resonance methods in condensed matter physics probe the energy levels, electronic transitions, and interactions within materials by measuring responses to electromagnetic radiation or magnetic fields. These techniques provide insights into the microscopic properties of solids, such as band structures, vibrational modes, and spin dynamics, without requiring macroscopic transport measurements. Absorption and emission spectroscopies, for instance, reveal electronic excitations by detecting energy absorbed or emitted as light interacts with matter.^[63] Optical spectroscopy encompasses absorption, where photons excite electrons from valence to conduction bands, and emission, where excited states relax and release photons, allowing characterization of band gaps and excitonic effects in semiconductors and insulators. In Raman spectroscopy, inelastic light scattering probes phonon modes; the Stokes shift corresponds to the energy loss of the scattered photon equal to the phonon creation energy, enabling mapping of lattice vibrations and their symmetries in crystals. This technique has been pivotal in studying phonon dispersions and anharmonic interactions in materials like graphene and transition metal dichalcogenides.^[64]^[65] Nuclear magnetic resonance (NMR) and electron spin resonance (ESR) rely on resonance phenomena in magnetic fields, where nuclear or electron spins precess at the Larmor frequency \omega = \gamma B, with \gamma as the gyromagnetic ratio and B the applied field. Chemical shifts in NMR spectra arise from local electronic environments altering the effective field at the nucleus, providing information on atomic coordination and bonding in solids like polymers and inorganic frameworks. ESR extends this to unpaired electrons, revealing spin interactions and defects in magnetic materials.^[66]^[67] Angle-resolved photoemission spectroscopy (ARPES) directly maps the electronic band structure by measuring the kinetic energy and momentum of photoemitted electrons, yielding dispersion relations E(\mathbf{k}) and imaging the Fermi surface in momentum space. This technique has been essential for visualizing topological bands and quasiparticle lifetimes in quantum materials such as high-temperature superconductors and Weyl semimetals.^[68] Ultrafast spectroscopy, developed prominently since the 1990s with the advent of femtosecond laser pulses, captures non-equilibrium dynamics following photoexcitation, such as carrier relaxation and phase transitions on picosecond timescales. These methods track transient states in materials like perovskites, revealing many-body interactions briefly.^[69]

Transport and magnetic measurements

Transport and magnetic measurements in condensed matter physics probe the response of materials to applied electric, thermal, and magnetic fields, providing insights into charge carrier dynamics, thermal properties, and magnetic ordering. These techniques measure macroscopic flows of charge, heat, and spin, distinguishing them from local spectroscopic probes by revealing bulk transport behaviors influenced by electronic structure. For instance, conductivity in metals arises from the partially filled conduction band, enabling delocalized electron motion under electric fields.^[70] Electrical transport measurements extend Ohm's law, J = \sigma E, to characterize resistivity \rho = 1/\sigma in solids, where deviations from simple metallic behavior occur due to scattering mechanisms like phonons or impurities in semiconductors and insulators. In the Hall effect, discovered in 1879, a transverse voltage E_y develops perpendicular to an applied current j_x and magnetic field B_z, quantified by the Hall coefficient R_H = E_y / (j_x B_z). For single-carrier systems, R_H = -1/(n e) for electrons or +1/(p e) for holes, allowing direct determination of carrier density n or p, typically in the range of $10^{16} to $10^{20} cm^{-3} for doped semiconductors.^[71]^[71] Thermal transport assesses heat conduction, primarily electronic in metals, via thermal conductivity \kappa, often linked to electrical conductivity by the Wiedemann-Franz law, L = \kappa / (\sigma T) = (\pi^2 / 3) (k_B / e)^2, where L is the Lorenz number, valid for Fermi liquids at temperatures much below the Fermi energy. Violations occur in strongly correlated systems or at low temperatures due to inelastic scattering. Thermoelectric measurements, such as the Seebeck coefficient S = -\Delta V / \Delta T, evaluate materials for energy conversion, with high ZT = S^2 \sigma T / \kappa figures of merit in bismuth telluride-based compounds enabling efficient cooling devices.^[72]^[72] Magnetotransport techniques apply magnetic fields to reveal quantum oscillations in resistivity. Shubnikov-de Haas oscillations in longitudinal resistivity \rho_{xx} arise from Landau level quantization as the Fermi level crosses discrete levels with increasing field, enabling mapping of the Fermi surface topology and effective masses through the oscillation frequency F = (\hbar / 2\pi e) A(k_F), where A(k_F) is the extremal cross-sectional area.^[73] The quantum Hall effect manifests as plateaus in Hall resistivity \rho_{xy} = h / (\nu e^2), with vanishing longitudinal resistivity. The integer quantum Hall effect, observed in 1980 in GaAs heterostructures at millikelvin temperatures and fields above 10 T, features filling factors \nu = 1, 2, \dots, arising from filled Landau levels and edge state conduction. The fractional quantum Hall effect, discovered in 1982 under similar conditions, shows plateaus at \nu = 1/3, 2/5, \dots, attributed to electron correlations forming quasiparticles with fractional charge e/3.^[74]^[75] Magnetic measurements employ superconducting quantum interference devices (SQUIDs) to detect minute magnetic moments down to $10^{-15} Wb in flux, exploiting Josephson junction interference in a superconducting loop. In superconductivity studies, SQUIDs quantify flux quantization in \Phi = n (h / 2e) units, confirming the Meissner effect and critical fields in type-I and type-II superconductors, as demonstrated in early niobium-based devices.^[76]

Subfields and emerging areas

Hard condensed matter

Hard condensed matter physics encompasses the study of crystalline solids, emphasizing their electronic, magnetic, and structural properties under equilibrium conditions. This subfield, rooted in solid-state physics, investigates how atomic arrangements in periodic lattices give rise to collective behaviors such as electrical conductivity and magnetism. Key phenomena include the formation of energy bands that determine material properties, with applications spanning from basic device physics to advanced materials design. Unlike softer or nanoscale systems, hard condensed matter prioritizes rigid, ordered structures where long-range coherence dominates. Semiconductors form a cornerstone of hard condensed matter, where controlled impurity addition—known as doping—modifies carrier concentrations to achieve desired electrical characteristics. In n-type doping, donor atoms like phosphorus introduce extra electrons to the conduction band, while p-type doping with acceptors like boron creates holes in the valence band, enabling tunable conductivity. The p-n junction, formed at the interface of these doped regions, creates a depletion zone with a built-in electric field that rectifies current flow, underpinning diodes and transistors. Band engineering further refines these properties by alloying or straining the lattice to adjust the bandgap, as seen in III-V compounds like GaAs, where bandgap tuning from 1.42 eV in GaAs to 0.36 eV in InAs supports optoelectronic devices.^[77] Superconductors in hard condensed matter are classified into Type I and Type II based on their response to magnetic fields. Type I materials, such as pure lead, expel fields completely up to a critical value H_c via the Meissner effect, transitioning abruptly to the normal state. Type II superconductors, including niobium alloys, allow partial field penetration above a lower critical field H_{c1}, forming quantized flux vortices in a mixed state until an upper critical field H_{c2}, enabling higher current densities for practical magnets. Magnetic properties extend to spintronics, where the 1988 discovery of giant magnetoresistance (GMR) in Fe/Cr multilayers demonstrated resistance changes up to 50% at low temperatures under applied fields, due to spin-dependent scattering. This effect, observed independently by groups led by Albert Fert and Peter Grünberg, revolutionized data storage in hard drives by enabling read heads sensitive to magnetic orientations.^[78] Correlated electron systems highlight strong interactions among electrons in crystalline lattices, leading to emergent behaviors beyond simple band theory. The Kondo effect, identified in 1964, explains resistance minima in dilute magnetic alloys like Cu-Mn, where localized spins scatter conduction electrons, forming a many-body singlet that screens the impurity spin at low temperatures and enhances resistivity logarithmically.^[79] Heavy fermion systems, discovered in 1979 with the superconductor CeCu₂Si₂, feature f-electrons hybridizing with conduction bands to yield quasiparticles with effective masses up to 1000 times the bare electron mass, manifesting in large specific heats and weak magnetism near quantum critical points. Recent advances in the 2020s have illuminated correlated insulators in two-dimensional crystalline systems, notably magic-angle twisted bilayer graphene. At twist angles near 1.1°, flat bands emerge, fostering strong electron correlations that yield insulating states at half-filling with charge gaps of approximately 0.3 meV, attributed to valley-polarized order rather than simple band insulation. These states, observed in 2018, exhibit tunable magnetism and proximity to superconductivity, offering a platform for exploring high-temperature correlated phases in atomically thin solids.^[80]

Soft and biological matter

Soft matter physics encompasses a broad class of materials that exhibit complex behaviors driven by weak interactions, thermal fluctuations, and entropy, including liquids, polymers, colloids, and biological systems. These materials are characterized by their deformability and responsiveness to external stimuli, often displaying intermediate structures between atomic scales and macroscopic dimensions. Unlike rigid crystalline solids, soft matter relies on many-body statistical mechanics to describe collective phenomena such as self-assembly and phase transitions. Liquid crystals represent a quintessential example of soft matter, where rod-like molecules align to form ordered phases while retaining fluidity. In nematic phases, molecules orient along a preferred direction without positional order, enabling anisotropic optical properties exploited in liquid crystal displays (LCDs). Smectic phases introduce layered positional ordering, enhancing stability for applications in sensors and electro-optics. The discovery of these phases by Georges Friedel in 1922 laid foundational work, with modern applications in flat-panel displays stemming from the twisted nematic configuration developed by Martin Schadt and Wolfgang Helfrich in 1971. Polymers and colloidal systems further illustrate soft matter's diversity, where long-chain molecules or suspended particles interact via entropic forces leading to viscoelastic behavior. Glassy dynamics in these systems arise from arrested diffusion near the glass transition, where cooperative rearrangements slow dramatically, as modeled by mode-coupling theory. Viscoelasticity combines elastic recovery with viscous flow, crucial for applications in adhesives and gels. The Flory-Huggins theory, introduced by Paul Flory and Maurice Huggins in the 1940s, provides a mean-field framework for polymer phase separation, predicting miscibility gaps based on interaction parameters χ, where χ > 0.5 drives demixing into polymer-rich and solvent-rich phases. Colloidal analogs, such as polystyrene spheres in suspension, mimic atomic systems but at micron scales, allowing direct visualization of crystallization and gelation. In biological contexts, soft matter principles govern structures like lipid bilayers, which self-assemble into cell membranes via amphiphilic interactions, forming bilayers with hydrophobic cores and hydrophilic surfaces. DNA packing in viruses and chromatin involves hierarchical folding driven by electrostatics and entropic penalties, achieving compaction ratios up to 10,000-fold through toroidal or spool-like configurations. Active matter extends these ideas to living systems, where self-propelled particles—such as motor proteins or bacteria—generate non-equilibrium dynamics, leading to phenomena like flocking and motility-induced phase separation in cellular environments. These processes underpin intracellular transport and tissue mechanics. Emerging research in 2025 has increasingly applied soft matter frameworks to environmental challenges, particularly microplastics, which behave as colloidal pollutants in aquatic systems. Studies reveal how microplastic particles aggregate via polymer-like bridging flocculation, influencing sediment transport and bioavailability in ecosystems. These investigations, often using soft matter simulations, highlight viscoelastic responses under shear flows, informing mitigation strategies for plastic pollution.^[81]

Nanoscale and topological systems

Nanoscale systems in condensed matter physics encompass structures where quantum confinement effects dominate due to spatial restrictions on the scale of the de Broglie wavelength of charge carriers. In quantum wells, electrons are confined in one dimension within layered heterostructures, such as GaAs/AlGaAs superlattices, leading to quantized energy levels that modify the density of states and enable subband formation. This confinement arises from the particle-in-a-box model, where the energy shift is given by

E \sim \frac{\hbar^2}{m L^2},

with \hbar as the reduced Planck's constant, m the effective mass, and L the well width. Seminal work on these structures predicted negative differential conductivity due to miniband transport in periodic potentials. Quantum wires further restrict motion to one dimension, enhancing conductance quantization in ballistic transport regimes, as observed in lithographically patterned InAs nanowires. Quantum dots, zero-dimensional confinements, exhibit discrete atomic-like spectra, with size-tunable optical properties stemming from the inverse dependence of exciton energy on radius, as derived for II-VI semiconductor nanocrystals. Topological insulators represent a class of materials where the bulk is insulating but surfaces or edges host robust, spin-momentum-locked conducting states protected by time-reversal symmetry. The bulk-boundary correspondence principle dictates that the number of protected edge modes equals a topological invariant of the bulk band structure, ensuring dissipationless transport immune to backscattering from non-magnetic impurities. In two dimensions, the quantum spin Hall effect, first proposed in graphene with spin-orbit coupling, features helical edge states characterized by a \mathbb{Z}_2 invariant, distinguishing trivial from nontrivial phases. Three-dimensional topological insulators, such as Bi_2Se_3, extend this to surface Dirac cones. Topological semimetals, including Weyl semimetals discovered in TaAs in 2015, feature bulk Weyl nodes acting as monopoles of Berry curvature, with Fermi arcs on surfaces linking projected nodes and enabling chiral anomaly effects. Two-dimensional materials have revolutionized nanoscale physics through van der Waals heterostructures, where stacking introduces moiré patterns from lattice mismatch or twist angles, creating superlattice potentials that flatten bands and enhance electron correlations. In twisted bilayer graphene, moiré superlattices at "magic" angles near 1.1° yield flat bands near the Fermi level, fostering unconventional superconductivity and correlated insulating states. Hybrid nanowires, such as InSb coated with NbTiN superconductors, host proximity-induced topological superconductivity, where Majorana zero modes—self-conjugate fermions—emerge at wire ends under magnetic fields, signaling potential for non-Abelian braiding in quantum computing. Extensions of the quantum Hall effect to zero-field regimes include the spin Hall effect, where spin currents generate transverse spin accumulation without net charge flow, and the anomalous Hall effect in ferromagnets, both amplified in topological materials via Berry curvature. The quantum anomalous Hall effect, observed in magnetically doped (Bi,Sb)_2Te_3 thin films, realizes quantized Hall conductance without external fields due to nonzero Chern numbers in the band structure. In the 2020s, room-temperature manifestations have emerged, such as unconventional anomalous Hall signals in Fe_3Sn_2 kagome magnets, with Hall resistivities of 3.2 μΩ·cm persisting above 300 K, driven by Weyl fermion contributions and promising spintronic applications.^[82]

Applications and impacts

Electronics and photonics

Condensed matter physics underpins modern electronics through the manipulation of charge carriers in solids, particularly semiconductors, where band structures dictate electrical conductivity. In devices like transistors, the principles of carrier injection, drift, and diffusion govern operation, enabling amplification and switching at nanoscale dimensions. Integrated circuits (ICs) integrate billions of these transistors, leveraging quantum mechanical effects such as tunneling and quantization to achieve high performance. The metal-oxide-semiconductor field-effect transistor (MOSFET) exemplifies these principles, functioning as the cornerstone of ICs. In a MOSFET, a gate electrode separated by an insulating oxide layer modulates the conductivity of a semiconductor channel between source and drain terminals via an applied electric field, controlling the formation of an inversion layer that allows current flow. This field-effect operation relies on the depletion and accumulation of charge carriers in the semiconductor, typically silicon, where doping creates n-type or p-type regions to facilitate electron or hole transport. Complementary metal-oxide-semiconductor (CMOS) technology, combining n- and p-channel MOSFETs, minimizes power dissipation and has driven the proliferation of digital electronics. Gordon Moore's 1965 observation, known as Moore's Law, predicted that the number of transistors on an IC would double approximately every year (later revised to every two years), leading to exponential increases in computing power while costs decreased. This scaling has persisted through decades, fueled by advances in lithography and materials, but by 2025, physical limits such as atomic-scale channel lengths (~1-2 nm) and quantum effects like source-drain leakage pose challenges, prompting shifts toward three-dimensional architectures and novel materials like high-mobility semiconductors. Despite these hurdles, Moore's Law has shaped the electronics industry, enabling devices from smartphones to supercomputers.^[83] Light-emitting diodes (LEDs) and lasers harness electron-hole recombination in semiconductors to convert electrical energy into photons, a process central to photonics. In p-n junction devices, forward biasing injects minority carriers across the junction, where radiative recombination in direct-bandgap materials like gallium arsenide releases photons whose energy matches the bandgap, producing light. LEDs emit incoherent light through spontaneous emission, while lasers achieve coherent output via stimulated emission in a resonant cavity, amplifying light through population inversion. These devices have revolutionized displays, lighting, and communications, with efficiencies exceeding 50% in advanced III-V semiconductors. Quantum dots enhance LED performance by enabling precise color tuning through quantum confinement effects. These nanoscale semiconductor particles, with sizes typically 2-10 nm, exhibit size-dependent emission wavelengths due to discrete energy levels, allowing bandgap engineering for red, green, or blue output without changing material composition. Pioneering work by Bawendi and colleagues demonstrated colloidal synthesis of uniform quantum dots, facilitating their integration into LEDs for wide-color-gamut displays with external quantum efficiencies over 20%. This tunability stems from the inverse relationship between dot radius and emission energy, providing a versatile platform for optoelectronic applications. Photovoltaics apply condensed matter principles to convert sunlight into electricity via the photovoltaic effect in semiconductor junctions. The Shockley-Queisser limit establishes the theoretical maximum efficiency for single-junction solar cells at approximately 33% under standard illumination, arising from thermodynamic constraints on absorption, recombination, and thermalization losses in materials with bandgaps around 1.1-1.3 eV. This limit, derived from detailed balance considerations, assumes blackbody radiation and highlights the trade-off between voltage and current optimized for silicon's indirect bandgap.^[84] Perovskite solar cells, emerging since 2009, have achieved efficiencies up to 27% for single-junction cells as tracked by NREL (as of November 2025) through hybrid organic-inorganic structures that offer tunable direct bandgaps, high absorption coefficients, and low-cost solution processing. These devices exploit ambipolar charge transport and defect tolerance in materials like methylammonium lead iodide. Tandem configurations with silicon have reached up to 34.9%, addressing recombination losses via improved interfaces and passivation techniques.^[85] Optoelectronics benefits from condensed matter structures that control light-matter interactions at subwavelength scales. Photonic crystals, periodic dielectric arrays, create photonic bandgaps analogous to electronic bandgaps in solids, forbidding certain wavelengths and enabling light confinement or guiding without diffraction losses. Seminal demonstrations by Joannopoulos and others showed how these structures manipulate light flow, forming the basis for low-threshold lasers and waveguides in integrated photonics. Metamaterials extend this control through engineered subwavelength resonators, achieving negative refraction, cloaking, and wavefront shaping for applications like superlenses and beam steering. Recent advances integrate active elements for dynamic tuning, enhancing optoelectronic devices such as modulators and sensors.^[86]^[87]

Materials and energy technologies

Condensed matter physics has significantly advanced the development of superalloys, which are nickel-based alloys exhibiting exceptional phase stability at elevated temperatures, enabling their use in turbine blades and other structural components under extreme conditions. The γ' phase (Ni₃Al), an ordered L1₂-structured precipitate coherent with the γ matrix, provides primary strengthening through mechanisms like antiphase boundary creation during dislocation shearing and Orowan bypassing for larger precipitates, maintaining mechanical integrity up to 1100°C. Alloying with refractory elements such as tungsten and rhenium enhances this stability by reducing topologically close-packed (TCP) phase formation and improving creep resistance; for instance, substituting tungsten for molybdenum in Ni-based superalloys increases creep life by over 130% at 760°C under 259 MPa stress. High-entropy superalloys further leverage configurational entropy to stabilize single-phase solid solutions or dual-phase microstructures, with sluggish diffusion preserving stability during prolonged high-temperature exposure. Phase diagrams, constructed via CALPHAD methods, guide alloy design by predicting phase fractions and transformation temperatures, though challenges persist in multi-component systems. Composites, such as metal-matrix variants reinforced with ceramic particles or fibers, draw from these principles to achieve similar high-temperature phase stability, combining metallic ductility with ceramic rigidity for aerospace applications. In energy storage, solid electrolytes derived from condensed matter studies enable safer, higher-density batteries by facilitating lithium-ion intercalation without flammable liquids. Intercalation involves reversible insertion of Li⁺ ions into layered host structures like graphite or transition metal oxides, governed by diffusion kinetics and lattice strain, which solid electrolytes mitigate through high ionic conductivity materials such as sulfides (e.g., Li₁₀GeP₂S₁₂ with conductivities up to 12 mS/cm) or garnets (e.g., LLZO). These electrolytes suppress dendrite growth in lithium-metal anodes, potentially achieving energy densities exceeding 400 Wh/kg, far surpassing conventional lithium-ion batteries limited to ~300 Wh/kg. Solid-state batteries incorporating these advancements are commercializing in the 2020s, with prototypes from companies like QuantumScape demonstrating over 1000 cycles at 80% capacity retention and plans for EV integration by 2027, though challenges like interfacial resistance and scalability remain. Fuel cells benefit similarly, with solid oxide electrolytes (e.g., yttria-stabilized zirconia) enabling efficient ion transport at 600–800°C, enhancing overall energy conversion efficiency through stable phase behavior under operational stresses. Thermoelectric materials, rooted in condensed matter phenomena like the Seebeck effect, convert waste heat directly to electricity, addressing energy efficiency in industrial and automotive sectors. The Seebeck coefficient (α), quantifying voltage per temperature difference (typically 100–300 μV/K in promising materials), drives this conversion, optimized alongside low thermal conductivity (κ < 1 W/m·K) and high electrical conductivity (σ > 10⁴ S/m) to maximize the figure of merit ZT = α²σT/κ. Skutterudites, exemplified by CoSb₃ with filled voids (e.g., by rare-earth atoms like Ce or Yb), exhibit "rattler" phonons that scatter heat-carrying modes, reducing κ by up to 90% while preserving high α (~200 μV/K) and σ, achieving ZT > 1.4 at 700–800 K ideal for recovering industrial waste heat (e.g., in automotive exhausts, potentially boosting fuel efficiency by 5–10%). These materials enable scalable generators, such as 1 kW modules for heavy-duty vehicles, demonstrating practical viability through nanostructuring and doping to further enhance stability and performance. Recent condensed matter research has propelled metal-organic frameworks (MOFs) as pivotal materials for carbon capture, leveraging their tunable porous structures for selective CO₂ adsorption. MOFs, crystalline networks of metal nodes linked by organic ligands, offer high surface areas (>7000 m²/g) and framework flexibility, enabling capacities up to 30 wt% CO₂ at ambient conditions via physisorption or chemisorption at amine-functionalized sites. Advancements in 2025 include flexible Zn-based MOF films (e.g., Zn₂(BDC)₂DABCO variants) achieving reversible low-pressure uptake (0.4 mmol/g at 25°C and 0.15 bar), suitable for direct air capture, with photo-responsive modifications allowing light-triggered release. Synthesis innovations like mechanochemical and microwave-assisted methods improve scalability and reduce environmental impact, while post-synthetic modifications enhance selectivity over N₂ by factors >100. These developments position MOFs for commercial deployment in post-combustion capture, with pilot-scale modules demonstrating >90% efficiency and regeneration at low energy penalties (~2 GJ/ton CO₂).

Quantum technologies

Quantum technologies leverage the quantum mechanical properties of condensed matter systems to develop advanced devices for computation, simulation, and sensing, where coherence and entanglement play central roles. These technologies exploit phenomena such as superconductivity, topological order, and spin interactions to achieve functionalities unattainable with classical systems. In particular, superconducting circuits, topological encodings, spin-based sensors, and atomic/ionic simulators represent key platforms emerging from condensed matter physics research.^[88] Superconducting qubits, the leading architecture for quantum processors, rely on Josephson junctions—nonlinear superconducting elements that enable tunable anharmonic oscillators for qubit operations. These junctions, formed by thin insulating barriers between superconductors, allow control of quantum states via microwave pulses, with transmon designs minimizing charge noise for improved performance. By 2025, advancements in material purification and fabrication have pushed coherence times beyond 100 μs, with energy relaxation times reaching up to 200 μs in optimized transmon qubits through reduced dielectric losses and improved flux management. Fluxonium qubits, incorporating a large inductor shunting the Josephson junction, have achieved even longer coherence, with a record of 1.43 ms demonstrated in 2023 and further refinements in 2025 enhancing gate fidelities above 99.9%. These milestones, driven by companies like IBM and Google, underscore the scalability potential for error-corrected quantum computing.^[89]^[90]^[91] Topological quantum computing encodes information in non-local quasiparticles, such as anyons, whose braiding operations perform fault-tolerant gates protected from local errors. In condensed matter realizations, Majorana zero modes—exotic excitations at the ends of topological superconductors—serve as anyonic building blocks, with braiding enabling universal computation without overhead from error correction in conventional qubits. Microsoft's efforts since the 2010s culminated in the 2025 Majorana-1 processor, a hybrid platform integrating semiconductor nanowires with superconductors to host and manipulate these modes, achieving initial demonstrations of anyon fusion rules. Concurrently, IBM's collaboration with Cornell validated Fibonacci anyon braiding in a 2025 experiment using fractional quantum Hall states, confirming error-resistant gate implementations with fidelity over 95% for small circuits. These developments highlight the path toward scalable topological processors, though challenges in material quality persist.^[92] In spintronics, nitrogen-vacancy (NV) centers in diamond emerge as premier quantum sensors, exploiting the spin-1 ground state of the defect for high-sensitivity magnetometry. These color centers, created by nitrogen adjacent to a lattice vacancy, exhibit long spin coherence times exceeding 1 ms at room temperature due to the diamond's wide bandgap and low phononic coupling, enabling optically detected magnetic resonance (ODMR) for nanoscale field detection down to 1 nT/√Hz. Recent 2025 advances include multiplexed arrays of over 100 NV ensembles, achieving simultaneous vector magnetometry and temperature mapping with sub-micron resolution, as demonstrated in hybrid diamond devices for biomedical imaging. Such sensors outperform classical magnetometers in sensitivity and spatial precision, finding applications in detecting biomagnetic signals and material defects.^[93]^[94] Quantum simulators using ultracold atoms and trapped ions replicate complex many-body Hamiltonians intractable on classical computers, with the Fermi-Hubbard model—a cornerstone for understanding strongly correlated electrons—serving as a primary target. Ultracold fermions loaded into optical lattices mimic lattice sites and hopping amplitudes, allowing simulation of Mott insulators and superconductivity analogs, as in 2025 experiments observing dynamical phase transitions with filling factors up to six atoms per site. Trapped ions, encoded as effective spins via internal states, implement the Hubbard model through laser-induced couplings, including demonstrations on five-ion chains extracting Rényi entropies to probe entanglement in 1D systems. These platforms, pioneered in seminal works and refined for larger scales, provide insights into high-temperature superconductivity and quantum magnetism without approximations.^[95] Topological phases in these systems offer inherent protection against decoherence, enhancing simulation fidelity.^[96]

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