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Meissner effect

The Meissner effect is the expulsion of a magnetic field from the interior of a superconductor when the material is cooled below its critical temperature in the presence of an external magnetic field, resulting in perfect diamagnetism.^[1] This phenomenon distinguishes superconductors from ordinary perfect conductors, as the latter would trap and maintain any pre-existing magnetic flux rather than actively excluding it.^[1] Discovered in 1933 by German physicists Walther Meissner and Robert Ochsenfeld during experiments on lead and tin samples, the effect was observed when these materials, cooled to superconducting states, unexpectedly reduced the magnetic flux threading through them to zero.^[2] Their findings, published in Die Naturwissenschaften, revealed that this field expulsion occurs regardless of whether the superconductor was in the field before or after cooling, highlighting a fundamental property of superconductivity beyond mere zero electrical resistance.^[3] The Meissner effect is quantitatively described by the phenomenological London equations, proposed by brothers Fritz and Heinz London in 1935, which model superconductors as having a characteristic penetration depth—typically on the order of nanometers—over which the magnetic field decays exponentially inside the material.^[1] Microscopically, the effect arises from the formation of Cooper pairs in the Bardeen-Cooper-Schrieffer (BCS) theory of 1957, where paired electrons respond to the magnetic field by generating persistent screening currents that cancel the applied field within the superconductor.^[4] This property enables practical applications, including magnetic levitation in high-speed trains and sensitive magnetic field sensors in scientific instruments.^[5]

Historical Background

Discovery

The discovery of superconductivity by Heike Kamerlingh Onnes in 1911, who observed the sudden drop to zero electrical resistance in mercury wire at approximately 4.2 K, marked the beginning of research into this quantum phenomenon but initially left its magnetic properties unclear.^[6] In 1933, German physicists Walther Meissner and Robert Ochsenfeld investigated the behavior of superconductors in magnetic fields to better understand these properties.^[7] They focused on cylindrical samples of lead and tin, materials known to exhibit superconductivity below their respective critical temperatures of 7.2 K for lead and 3.7 K for tin.^[8] The experimental setup involved placing the samples in a uniform external magnetic field of approximately 5 gauss and gradually cooling them using liquid helium to temperatures below the critical point while precisely measuring the magnetic field distribution both inside and outside the samples with sensitive fluxmeters.^[2] Contrary to expectations that the magnetic field would remain trapped within the superconductor due to persistent supercurrents—similar to the behavior anticipated from zero-resistance conduction—Meissner and Ochsenfeld observed the complete expulsion of the field from the interior of the samples upon transitioning to the superconducting state.^[2] This surprising result indicated that superconductors act as perfect diamagnets, with the field lines bending around the material rather than penetrating it.^[7] They reported these findings in a seminal paper published in Die Naturwissenschaften in November 1933, establishing the effect now known as the Meissner effect.^[2] This observation distinguished superconductivity from mere zero resistance, revealing it as a distinct thermodynamic equilibrium state where magnetic flux is rigorously excluded, thereby providing crucial evidence for the development of subsequent theories.^[7]

Early Developments

Following the observation of the Meissner effect in 1933, researchers quickly sought to replicate and extend the finding through additional experiments on various materials. Early confirmations included studies on pure elements such as mercury, tin, and lead, where the complete expulsion of magnetic fields was observed upon cooling below the critical temperature in the presence of an applied field. These experiments, building on the initial work, also examined superconducting alloys, demonstrating consistent field expulsion across both pure metals and alloys, thereby establishing the effect's broad applicability. In 1935, brothers Fritz and Heinz London developed the first phenomenological theory to account for the Meissner effect, proposing that superconductors respond to magnetic fields through persistent supercurrents that screen the interior. Central to their model was the introduction of the London penetration depth, denoted λ_L, which quantifies the characteristic length scale (typically on the order of 10–100 nm in conventional superconductors) over which the magnetic field decays exponentially from the surface into the material. This theory treated the superconductor as a medium where the electromagnetic response leads to perfect diamagnetism, with the field B satisfying ∇²B = B / λ_L² inside the superconductor.^[9] Fritz London extended this framework in 1936, emphasizing the rigidity of the superconducting electron fluid as a key feature. He argued that the electrons in the superconducting state behave as a coherent, incompressible fluid with unalterable velocity distribution, akin to a rigid body rotation, which enforces the exclusion of magnetic flux and prevents any acceleration of the supercurrents in response to electric fields. This concept of rigidity provided a deeper physical intuition for the Meissner effect, portraying superconductivity as a state where the electron assembly maintains a fixed phase coherence against perturbations. These theoretical advances were complemented by experiments distinguishing the Meissner effect from flux trapping in ferromagnets or hypothetical perfect conductors. Unlike ferromagnetic materials, where magnetic domains can trap flux due to hysteresis, superconductors actively expel fields during the transition to the superconducting state, even if the field is applied beforehand—a behavior confirmed in mercury and alloy samples cooled in uniform fields. This active expulsion, rather than passive persistence, underscored the thermodynamic nature of the effect. Consequently, the Meissner effect became recognized as the defining magnetic hallmark of superconductivity, shifting emphasis from zero electrical resistance (a kinetic property) to field exclusion as an equilibrium characteristic essential for identifying true superconductors.^[7]

Phenomenological Description

Basic Mechanism

The Meissner effect refers to the complete expulsion of a magnetic field from the interior of a superconductor when the material transitions to its superconducting state below the critical temperature T_c in the presence of an applied magnetic field H less than the critical field H_c.^[10] This phenomenon results in the magnetic induction \mathbf{B} = 0 throughout the bulk of the superconductor. The magnetic field is expelled from the interior, penetrating only to a small depth \lambda (the London penetration depth, on the order of nanometers) near the surface, where it decays exponentially.^[1] The effect distinguishes superconductivity from mere perfect conductivity, as it actively excludes the field rather than merely preserving trapped flux.^[11] The underlying process involves the generation of persistent supercurrents on the surface of the superconductor that precisely oppose the applied magnetic field. These supercurrents, flowing without resistance in the superconducting state, produce an internal magnetic field that cancels the external one, preventing significant penetration into the material's interior. This expulsion occurs reversibly: upon heating above T_c, the magnetic field re-enters the material as it returns to the normal state.^[12] In contrast to normal metals, where magnetic fields penetrate fully and uniformly due to the absence of such collective current responses, superconductors exhibit this field exclusion only below T_c.^[11] Normal conductors may trap magnetic flux if cooled in a field due to induced eddy currents that decay over time, but they do not expel the field actively.^[13] Visually, the Meissner effect causes external magnetic field lines to bend and compress around the superconductor, avoiding its interior entirely, much like the behavior around an ideal diamagnetic material. This distortion can be demonstrated by levitating a magnet above a cooled superconductor, where the expelled field creates a repulsive force.^[13]

Relation to Diamagnetism

The Meissner effect is characterized by the superconductor behaving as an ideal diamagnet, exhibiting perfect diamagnetism with a magnetic susceptibility of \chi = -1. This leads to a magnetization \mathbf{M} = -\mathbf{H} that precisely opposes the applied magnetic field \mathbf{H}, resulting in zero magnetic induction \mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M}) = 0 throughout the interior of the material in thermodynamic equilibrium. In their seminal observation, Meissner and Ochsenfeld noted this complete expulsion of magnetic flux, which was later interpreted as evidence of perfect diamagnetism.^[14] This perfect diamagnetism sets superconductors apart from paramagnetic and ferromagnetic materials, which display positive susceptibilities and can retain residual magnetism after field removal due to domain alignment or spin ordering. In contrast, the Meissner effect ensures no residual internal field, as the expulsion arises from a fundamental thermodynamic equilibrium state rather than temporary induced currents or hysteresis effects.^[14] The phenomenon underscores superconductivity as a reversible, equilibrium phase where magnetic response is inherently tied to the superconducting order parameter, enabling thermodynamic treatments without energy dissipation.^[14] The underlying mechanism involves persistent surface currents that screen the applied field, consistent with Lenz's law in that they oppose flux changes per Faraday's induction principle, yet these currents endure indefinitely in the absence of resistance, maintaining the expelled state even in static fields. Unlike transient eddy currents in normal conductors, these supercurrents represent a stable equilibrium configuration. At the superconductor's boundary, the tangential component of the magnetic field is zero inside, leading to field lines that are parallel to the surface just outside. Inside the superconductor, any penetrating field decays exponentially over the penetration depth due to the screening by these surface currents.^[1]

Theoretical Foundations

London Equations

The London equations constitute a phenomenological framework that quantitatively captures the electromagnetic response of superconductors, particularly the expulsion of magnetic fields known as the Meissner effect. Proposed by brothers Fritz and Heinz London in 1935, these equations treat the superconductor as a medium where a fraction of the electrons, denoted by density n_s, behave collectively without dissipation.^[9] The first London equation describes the dynamics of the superconducting current density \mathbf{J}_s in response to an electric field \mathbf{E}:

\frac{d\mathbf{J}_s}{dt} = \frac{n_s e^2}{m} \mathbf{E},

where e is the electron charge and m is the electron mass. This relation arises from the acceleration of the superconducting electrons under the Lorentz force, analogous to the motion of free particles but with zero resistivity, as integrating over time yields \mathbf{J}_s \propto \mathbf{E} in the steady state.^[9] The second London equation links the current density to the magnetic field \mathbf{B}:

\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{B}.

Equivalently, incorporating the magnetic permeability \mu_0,

\nabla \times \mathbf{J}_s = -\frac{\mathbf{B}}{\mu_0 \lambda^2},

where the London penetration depth is given by

\lambda = \sqrt{\frac{m}{\mu_0 n_s e^2}}.

This equation accounts for the Meissner effect by implying that persistent currents are induced to oppose and cancel internal magnetic fields.^[9] The equations derive from the assumption that the superconducting electrons constitute a rigid, inertia-bearing lattice that accelerates as a whole without internal friction or scattering. The Londons modeled this by setting the total canonical momentum of the electron fluid to zero in equilibrium, leading to a constitutive relation where the current responds rigidly to electromagnetic perturbations. Taking the time derivative of the first equation and substituting into Maxwell's equations yields the second, with \lambda emerging as the characteristic length scale over which fields decay.^[9] To illustrate field behavior, consider a superconducting slab of thickness much larger than \lambda, with an external magnetic field \mathbf{H}_0 applied parallel to the surface along the x-direction, and the slab extending from z = 0 (surface) to z > 0 (interior). Combining the second London equation with Ampère's law \nabla \times \mathbf{B} = \mu_0 \mathbf{J}_s (neglecting displacement current for quasistatic fields) results in

\nabla^2 \mathbf{B} = \frac{\mathbf{B}}{\lambda^2}.

The one-dimensional solution inside the superconductor is \mathbf{B}(z) = \mu_0 \mathbf{H}_0 \exp(-z / \lambda), showing exponential decay from the surface. For samples thicker than several \lambda, the field is effectively expelled from the bulk interior, achieving complete diamagnetism with susceptibility \chi = -1.^[9]

Microscopic Theories

The microscopic theory of superconductivity, culminating in the Bardeen-Cooper-Schrieffer (BCS) framework, provides a quantum mechanical explanation for the Meissner effect through the formation of Cooper pairs. In BCS theory, electrons in a metal pair up via attractive interactions mediated by lattice vibrations (phonons), forming bound states known as Cooper pairs with total momentum zero in the ground state.^[15] These pairs condense into a macroscopic quantum state described by a rigid order parameter \psi, representing the superconducting wavefunction, which exhibits phase coherence across the material.^[16] This rigidity ensures that any applied magnetic field induces persistent supercurrents that precisely screen the field from the superconductor's interior, resulting in perfect diamagnetism and the expulsion of magnetic flux, as observed in the Meissner effect.^[17] The Ginzburg-Landau (GL) theory, formulated in 1950 near the critical temperature T_c, offers a complementary phenomenological approach with microscopic underpinnings, expressing superconductivity via a free energy functional minimized to yield equilibrium properties. The functional is given by

F = \alpha |\psi|^2 + \frac{\beta}{2} |\psi|^4 + \frac{1}{2m^*} \left| \left( -i \hbar \nabla - 2e \mathbf{A} \right) \psi \right|^2 + \frac{B^2}{2 \mu_0},

where \psi is the order parameter, \alpha and \beta are temperature-dependent coefficients (\alpha \propto (T - T_c)), m^* is the effective mass of Cooper pairs, \mathbf{A} is the vector potential, and B = \nabla \times \mathbf{A}. Minimizing F with respect to \psi and \mathbf{A} leads to the condition that magnetic fields B are expelled from the bulk, with penetration confined to a surface layer of thickness \lambda (the London penetration depth).^[18] At the heart of the Meissner effect in these theories is gauge invariance, which mandates that the superconducting state responds to electromagnetic fields through supercurrents that cancel the vector potential \mathbf{A} inside the material, ensuring B = \nabla \times \mathbf{A} = 0 in equilibrium.^[19] This response arises naturally from the phase rigidity of [\psi](/page/Psi): any spatial variation in the phase of [\psi](/page/Psi) induced by \mathbf{A} generates a supercurrent \mathbf{j}_s \propto |\psi|^2 (\nabla \phi - 2e \mathbf{A}/\hbar), where \phi is the phase, adjusting to neutralize \mathbf{A}.^[19] The microscopic theories thus elevate the phenomenological London equations to a quantum foundation, where the effect emerges from the coherent, gap-opened excitation spectrum.^[16] The Meissner effect's occurrence in thermodynamic equilibrium underscores that superconductivity represents the true ground state of the system, rather than a metastable configuration, as the field expulsion minimizes the total free energy without hysteresis in the reversible transition. This equilibrium property distinguishes the superconducting phase and validates the stability of the BCS condensate against thermal fluctuations above T_c.^[20]

Experimental and Observational Features

Verification Methods

The Meissner effect is classically verified using superconducting quantum interference device (SQUID) magnetometers, which precisely measure changes in the magnetic field or magnetization of a sample during cooling in an applied external magnetic field H. In a typical setup, a superconductor is cooled below its critical temperature T_c while exposed to a weak applied field, and the SQUID detects the expulsion of the field from the interior, manifesting as a sharp increase in diamagnetic magnetization corresponding to perfect diamagnetism. This method has been employed since the development of commercial SQUIDs in the 1970s, allowing sensitive detection of field expulsion even in small samples, such as single indium particles where the Meissner screening was quantified by observing the diamagnetic response proportional to the particle volume.^[21] AC susceptibility measurements provide another key verification technique, utilizing mutual inductance coils to apply a small alternating magnetic field and detect the resulting susceptibility \chi. Below T_c, the real part of the AC susceptibility approaches \chi = -1 in SI units (or \chi = -1/(4\pi) in cgs units), indicating complete field expulsion consistent with the Meissner effect, while the imaginary part reveals losses due to flux dynamics. This method is particularly useful for identifying bulk superconductivity, as deviations from \chi = -1 can signal incomplete screening or granularity, and has been applied to high-T_c cuprates to confirm the onset of Meissner screening at the superconducting transition.^[22] Flux quantization experiments further distinguish the Meissner effect by demonstrating that magnetic flux through a superconducting loop is expelled or trapped in discrete quanta of \Phi_0 = h / 2e \approx 2.07 \times 10^{-15} Wb, rather than varying continuously, confirming the perfect diamagnetism and phase coherence of the superconducting state. In pioneering work, thin-walled superconducting cylinders of tin were cooled in a magnetic field, and the persistent currents induced to maintain quantized flux were measured via SQUID detection of the trapped flux after field removal, revealing steps at integer multiples of \Phi_0 and verifying the expulsion mechanism.^[23] Modern techniques such as neutron scattering and muon spin rotation (\muSR) enable probing of internal magnetic fields in complex geometries like thin films, providing depth-resolved verification of field expulsion. Small-angle neutron scattering (SANS) visualizes the Meissner state by detecting the absence of magnetic scattering signals inside the superconductor, as demonstrated in ferromagnetic superconductors where bulk domain screening was confirmed through rocking curve analysis showing suppressed neutron depolarization below T_c. Similarly, low-energy \muSR implants spin-polarized muons to measure local field distributions, revealing near-zero internal fields in the Meissner phase of underdoped cuprates, with relaxation rates dropping sharply upon entering the superconducting state to quantify screening penetration depths on the nanometer scale.^[24]^[25]

Variations in Superconductors

In Type I superconductors, the Meissner effect results in complete expulsion of applied magnetic fields from the material's interior, maintaining perfect diamagnetism up to the critical magnetic field H_c, at which point superconductivity is abruptly destroyed and the material transitions to the normal state without intermediate phases.^[1] This behavior is characteristic of elemental superconductors with relatively low critical temperatures, such as pure lead (Pb, T_c \approx 7.2 K) and tin (Sn, T_c \approx 3.7 K), where the coherence length \xi exceeds the penetration depth \lambda, leading to a single critical field that governs the full Meissner state. Type II superconductors display a more complex variation of the Meissner effect, with complete field expulsion only below the lower critical field H_{c1}; above H_{c1}, magnetic flux begins to penetrate via quantized Abrikosov vortices arranged in a lattice within the mixed state, allowing partial penetration while preserving superconductivity until the upper critical field H_{c2} is reached, beyond which the normal state ensues.^[26] This intermediate mixed state arises because \lambda > \xi, enabling stable vortex structures as theorized by Abrikosov, and is observed in materials like niobium-titanium alloys (Nb-Ti, T_c \approx 9.5 K) and high-T_c cuprates such as YBa_2Cu_3O_7 (YBCO, T_c \approx 93 K).^[27] The penetration depth \lambda, defining the spatial scale of field screening in the Meissner state, varies significantly with temperature, increasing toward T_c according to \lambda(T) \approx \lambda(0) [1 - (T/T_c)^4]^{-1/2} in the clean limit near the transition, which reduces the effectiveness of flux expulsion as thermal fluctuations weaken the superconducting order parameter.^[28] In high-T_c cuprates, pronounced anisotropy due to their quasi-two-dimensional layered crystal structure causes direction-dependent \lambda, with in-plane penetration depth \lambda_{ab} much smaller than out-of-plane \lambda_c (often by factors of 5–7), leading to weaker field expulsion along the c-axis and influencing overall diamagnetic response.^[29] For thin films and nanostructures, the Meissner effect becomes incomplete when the sample thickness d is comparable to or less than \lambda, as magnetic fields can penetrate across the entire film via non-local screening currents, resulting in reduced diamagnetism and altered susceptibility compared to bulk samples.^[30] In such configurations, proximity effects further modify the behavior, where superconductivity can be induced in adjacent normal metal layers through Cooper pair tunneling, enhancing overall flux expulsion but introducing spatial variations in the superconducting properties.^[31]

Implications and Applications

Physical Consequences

The Meissner effect serves as definitive proof that superconductivity represents a distinct thermodynamic equilibrium state, rather than merely an idealized form of perfect conductivity. In a perfect conductor cooled in a magnetic field, flux lines would remain trapped inside due to the absence of resistance preventing changes in flux linkage, a phenomenon known as flux freezing. However, the observed expulsion of magnetic fields in superconductors below the critical temperature T_c indicates an active reconfiguration to a lower-energy equilibrium phase, where the system minimizes its free energy by excluding the field. This distinction underscores that superconductivity involves a fundamental change in the material's electronic structure, enabling zero resistivity alongside perfect diamagnetism.^[4] A key physical consequence is the existence of a critical magnetic field H_c(T) that delineates the stability of the superconducting state, given phenomenologically by

H_c(T) = H_c(0) \left[1 - \left(\frac{T}{T_c}\right)^2\right],

where H_c(0) is the zero-temperature critical field. This parabolic temperature dependence reflects the competition between the superconducting condensation energy, which favors the ordered state, and the magnetic energy density \frac{1}{2} \mu_0 H_c^2 required to suppress it. The equality of these energies at H = H_c quantifies the energy gain from pairing in the superconducting phase, providing a direct measure of the strength of the superconducting interaction.^[32]^[33] The electromagnetic response in the Meissner state involves supercurrents that flow without dissipation and transport no entropy, ensuring that phase transitions between superconducting and normal states are thermodynamically reversible. Unlike normal currents, which generate heat and entropy through scattering, these coherent supercurrents maintain equilibrium without irreversible losses, allowing the superconductor to respond elastically to applied fields below H_c. This entropy-free nature facilitates precise control in thermodynamic cycles and highlights the reversible character of the superconducting order parameter.^[34] Finally, the Meissner effect fundamentally influences the phase diagrams of superconducting materials, establishing the H-T boundary that separates the zero-field-cooled Meissner phase from the normal state. Below this curve, the system expels fields to achieve equilibrium; crossing it induces a transition driven by the applied field overcoming the condensation energy. In type I superconductors, expulsion is complete up to H_c, whereas type II materials exhibit partial expulsion with vortex penetration at lower fields, extending the superconducting regime.^[1]

Technological Uses

The Meissner effect plays a crucial role in magnetic levitation systems, particularly in maglev trains utilizing high-temperature superconductors such as yttrium barium copper oxide (YBCO). When cooled below its critical temperature, typically to 77 K using liquid nitrogen, a YBCO superconductor expels magnetic fields from permanent track magnets, enabling stable, frictionless suspension without energy consumption. This levitation is further stabilized by flux pinning, where magnetic flux lines are trapped within the superconductor, preventing lateral displacement and allowing high-speed travel with minimal noise and vibration.^[35]^[36] In magnetic resonance imaging (MRI) machines, the Meissner effect facilitates the generation of strong, uniform magnetic fields through persistent currents in superconducting solenoids. These currents, induced in niobium-titanium or similar type-II superconductors cooled to about 4 K, flow indefinitely without resistance due to the expulsion of external fields, maintaining fields up to 3 T or higher with no power loss after initial ramp-up. This efficiency reduces operational costs and heat generation, essential for continuous clinical use.^[37]^[38] Superconducting magnetic shields leverage the Meissner effect to protect sensitive instruments from stray magnetic fields. Enclosures made from lead or high-temperature superconductors, cooled below their transition temperatures, completely expel external fields, achieving shielding factors exceeding 10^6 in low-frequency regimes for applications like particle detectors and gravitational wave observatories. For instance, in MRI systems, self-shielded superconducting designs minimize fringe fields, enhancing patient safety and installation flexibility.^[39]^[40] In quantum computing, Josephson junctions exploit Meissner screening currents to realize superconducting qubits. These junctions, formed by thin insulating barriers between superconductors, allow tunable supercurrents that encode quantum information, with the underlying Meissner effect ensuring phase coherence and low dissipation in circuits operating at millikelvin temperatures. This enables scalable architectures, such as transmon qubits, with coherence times approaching microseconds.^[41] Superconducting quantum interference devices (SQUIDs) utilize the Meissner effect to achieve ultra-sensitive magnetic field measurements. These devices, based on Josephson junctions in a superconducting loop, detect minute changes in magnetic flux through interference patterns arising from screening currents that expel fields per the Meissner effect. Operating at cryogenic temperatures, SQUIDs enable applications in biomagnetism (e.g., magnetoencephalography), geophysical prospecting, and nondestructive materials testing, with sensitivities down to femtotesla levels.^[42]

Analogies in Physics

Higgs Mechanism

The Meissner effect in superconductors provided a crucial conceptual foundation for the development of the Higgs mechanism in particle physics. In 1963, Philip W. Anderson published a seminal paper demonstrating that the phenomenon of superconductivity involves a form of spontaneous gauge symmetry breaking, where the superconducting order parameter acquires a nonzero vacuum expectation value, leading to the apparent violation of gauge invariance and the generation of mass for the photon field within the superconductor.^[43] This insight directly inspired the independent works of François Englert and Robert Brout, Peter Higgs, and Gerald Guralnik, Carl Hagen, and Tom Kibble in 1964, who extended the idea to relativistic quantum field theories, proposing that a scalar field could break electroweak symmetry and endow gauge bosons with mass through a similar mechanism.^[44] A key parallel between the Meissner effect and the Higgs mechanism lies in spontaneous symmetry breaking, which results in the expulsion of magnetic fields from the superconductor's interior, analogous to how the Higgs field expels the would-be Goldstone modes and generates masses for gauge bosons in the Standard Model. In superconductivity, the complex order parameter \psi, representing the Cooper pair condensate, breaks the U(1) electromagnetic gauge symmetry, confining the magnetic field to a penetration depth \lambda and effectively giving the photon a mass proportional to $1/\lambda^2.^[45] This mass generation prevents the propagation of electromagnetic fields inside the material, mirroring the Higgs mechanism where the scalar field's vacuum expectation value breaks SU(2) × U(1) symmetry, providing masses to the W and Z bosons while leaving the photon massless.^[44] The Abelian Higgs model formalizes this analogy in a simplified U(1) gauge theory, consisting of a Lagrangian for a complex scalar field \phi coupled to a U(1) gauge field A_\mu, where the potential V(\phi) = \lambda (|\phi|^2 - v^2)^2 allows for spontaneous symmetry breaking when \phi acquires a vacuum expectation value v.^[44] In this model, the gauge field acquires a mass m_A = e v, leading to a Meissner-like expulsion of the gauge field, with topological defects such as vortices corresponding to magnetic flux tubes in type-II superconductors, where quantized flux is confined to line-like structures. This framework not only elucidates the Meissner effect but also serves as a paradigm for understanding mass generation and symmetry breaking in quantum field theories.

Other Theoretical Parallels

The Meissner effect, characterized by the expulsion of magnetic fields from the interior of a superconductor, finds parallels in cosmological models of the early universe, particularly during phase transitions where symmetry breaking leads to the formation of topological defects. In these scenarios, the transition from a symmetric high-temperature phase to a broken-symmetry phase can produce domain walls or cosmic strings, analogous to the Abrikosov vortices that form in Type II superconductors when magnetic fields penetrate beyond the lower critical field. Just as the Meissner effect enforces perfect diamagnetism in the superconducting state, the Higgs-like mechanism in cosmology expels gauge fields from regions of condensate, with cosmic strings acting as linear defects that confine flux in a manner similar to vortex lines carrying quantized magnetic flux in superconductors. This analogy highlights how phase transitions in the early universe could generate a network of defects whose evolution mirrors the dynamics of vortex lattices under applied fields. In condensed matter physics beyond traditional superconductors, the Meissner effect shares conceptual similarities with phenomena in topological insulators, where protected edge states enable dissipationless transport that mimics the screening surface currents responsible for field expulsion. In hybrid systems combining superconductors with topological insulators, such as superconductor/topological-insulator/superconductor junctions, the Meissner effect induces proximity superconductivity in the helical edge states, leading to gapless surface modes that carry supercurrents without bulk penetration, akin to the diamagnetic response at the superconductor boundary. These edge states, robust against backscattering due to time-reversal symmetry, effectively replicate the role of Cooper-pair currents in confining magnetic flux to the surface, providing a topological analog to the classical Meissner screening. Experiments with quantum spin Hall insulators, like those involving induced superconductivity, demonstrate how these chiral edge currents respond to magnetic fields in ways that parallel the expulsion mechanism, though without the full bulk gap closure of conventional superconductivity.^[46]^[47] A striking analogy exists in the realm of fluid dynamics, particularly with superfluid helium, where the expulsion of vorticity from the bulk mirrors the Meissner effect's rejection of magnetic fields. In superfluid ^4He below the lambda transition, the irrotational nature of the superfluid component prevents the formation of vorticity within the condensate, leading to a macroscopic quantum state where any applied rotation results in quantized vortex lines confined to the boundaries or defects, much like flux lines in Type II superconductors. This vorticity expulsion serves as the defining experimental signature of superfluidity, analogous to perfect diamagnetism in superconductors. The phenomenon is framed within the two-fluid model, which describes superfluid helium as a mixture of a viscous normal fluid and an inviscid superfluid component; the superfluid fraction alone enforces the irrotational flow, excluding vorticity in a manner that parallels how the superconducting condensate screens magnetic fields via persistent currents.^[48]^[49]

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May 11, 2022 · In this Letter, using small-angle neutron scattering (SANS) as a bulk probe, we find a clear signature of DMS in single-crystal samples of the ...Abstract · Article Text · ACKNOWLEDGMENTSMissing: verification | Show results with:verification
[25]
The Meissner effect in a strongly underdoped cuprate above its ...
Apr 12, 2011 · The local probe technique we have chosen for this study (low-energy (LE) muon spin rotation (μSR)) in fact fulfils all these requirements.
[26]
The 2003 Nobel Prize in Physics - Popular information
Abrikosov was able to tie up the theory by showing that type-II superconductors had precisely these values. Our knowledge of superconductivity has led to ...
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Interaction between moving Abrikosov vortices in type-II ...
Jul 6, 2020 · The London equations express the basic Meissner effect and can be used at any temperature for problems where vortex cores are irrelevant.
[28]
On the temperature dependence of the London penetration depth in ...
Journal of Superconductivity; Article. On the temperature dependence of the London penetration depth in a superconductor. Articles; Published: October 1991.
[29]
Electrodynamics of high- T c superconductors
Aug 26, 2005 · This review covers the electromagnetic response of high-Tc superconductors using terahertz, infrared, and optical spectroscopies, focusing on ...Abstract · Article Text · Electrodynamics in the Normal... · Infrared Signatures of...
[30]
Stability limit of the Meissner state in thin superconducting films
The local stability of Meissner states in thin superconducting films is determined by studying the effect of small fluctuations on solutions of the ...
[31]
Measurement of the nonlinear Meissner effect in superconducting ...
We report observation of the nonlinear Meissner effect (NLME) in Nb films by measuring the resonance frequency of a planar superconducting cavity as a ...
[32]
[PDF] phenomenology of superconductivity - Physics Courses
Later on below, we shall relate the penetration depth λL to a stiffness parameter in the Ginzburg-Landau theory. ... Meissner effect, h0 = −4πm. In the ...Missing: paper | Show results with:paper
[33]
[PDF] Phenomenology of Superconductivity (I) Contents
2.2 Thermodynamic critical field Hc. We define the thermodynamic critical field Hc as related to the condensation energy,. H2 c. 8π. = α2. 2|β|. ⇒. H2 c. 4π. =.
[34]
[PDF] Superconductivity - Physics 122
We saw that the Meissner effect implies a magnetic susceptibility X = -1/4~ in CGS in the superconducting state or, in SI, X = - 1. Can we modify a consti-.
[35]
High temperature superconducting magnetic levitation train
The magnetic suspension behaviour described relies on the Meissner effect. At 77 K, samples may be magnetically levitated without consuming any power and ...
[36]
(PDF) The Effect of Critical Parameters of High Temperature ...
Aug 15, 2024 · To study the efficiency and improvement of magnetic levitation train, high temperature superconductor YBCO samples were prepared by using the ...
[37]
[PDF] Superconductivity: The Meissner Effect, Persistent Currents and the ...
BCS theory. The London equation, Eq.(4), is not a fundamental theory of superconductivity. It is an ad hoc restriction on classical electrodynamics ...
[38]
[PDF] Superconductivity - UCSB MRL
▫ The superconducting elements. ▫ The Meissner effect and superconducting levitation. ▫ Type I and type II superconductors. ▫ Superconducting magnets and MRI.Missing: persistent | Show results with:persistent
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[PDF] SUPERCONDUCTING SHIELDS FOR MAGNETIC FLUX ...
As the cylinder is cooled through the transition region, an almost perfect Meissner effect will be achieved because the effective resistivity of the metal ...
[40]
Superconducting Self-Shielded and Zero-Boil-Off ... - NIH
Sep 18, 2024 · A high-temperature superconducting magnetic shield (HTSMS) uses the perfect antimagnetic property of superconductors (Meissner effect) by ...
[41]
[PDF] Thermal Activation of Superconducting Josephson Junctions
May 18, 2007 · In 1933,. Meissner and Ochsenfeld demonstrated that superconductors also actively expel all external magnetic fields from their interior [2], an ...<|control11|><|separator|>
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Plasmons, Gauge Invariance, and Mass | Phys. Rev.
We show that the theory of plasma oscillations is a simple nonrelativistic example exhibiting all of the features of Schwinger's idea.Missing: symmetry | Show results with:symmetry
[43]
Broken Symmetries and the Masses of Gauge Bosons
Oct 11, 2013 · Phys. Rev. Lett. 13, 508 (1964) ... The 2013 Nobel Prize in Physics has been awarded to two of the theorists who formulated the Higgs mechanism, ...
[44]
Superconductivity, broken gauge symmetry, and the Higgs mechanism
Jun 1, 2019 · ... let us consider the Lorentz invariant analogue of a superconductor: the Abelian-Higgs model, defined by the Lagrangian density26. μ μ ...
[45]
Majorana Zero Modes Induced by the Meissner Effect at Small ...
In this Letter, we propose that the Meissner effect can bring the topological superconducting phase to a superconductor/topological-insulator/superconductor ( ...
[46]
Induced superconductivity in the quantum spin Hall edge - Nature
Aug 3, 2014 · Topological superconductors, like topological insulators, possess a bulk energy gap and gapless surface states. ... Meissner effect. The ...
[47]
[PDF] Vortex Nucleation in a Superfluid - UBC Library Open Collections
This expulsion of vorticity, analogous to the Meissner effect, provides a funda- mental experimental definition of superfluidity. As a consequence ...
[48]
[PDF] Superfluid States of Matter
Nov 8, 2022 · ... superfluids. First is the Meissner effect of expulsion of magnetic field from the superconductor bulk, and then its twin effect, the expulsion ...