Vortex Matter, Pinning, and Flux Flow

A superconducting vortex becomes vortex matter when its position, distortion, history, and motion are collective variables rather than details of one static core. An ideal static mean-field type-II model gives Abrikosov order. Thermal fluctuations and activation are intrinsic at nonzero temperature, while real specimens and protocols add surfaces, quenched disorder, anisotropy, finite observation time, and drive. Together these ingredients create irreversible magnetic response, creep, plastic channels, flux-flow voltage, and sometimes genuine phase transitions. They also make the word critical dangerous unless the current, field, geometry, time window, and criterion are all stated.

This page begins after Ginzburg–Landau Theory has supplied the equilibrium mixed state, a static core, the flux quantum, critical fields, and the ideal Abrikosov lattice. It owns the next layer: elastic vortex matter, bulk and surface pinning, critical states, thermal and quantum-creep qualifications, driven motion, and the joint interpretation of transport, magnetometry, microwave, and imaging records. It does not rederive the isolated vortex or use a voltage onset as a universal phase boundary.

Required background. Ginzburg–Landau Theory supplies the amplitude-resolving core, $B_{c1}$, $B_{c2}$, the type-II criterion, and the ideal static lattice.

Helpful background. London Theory supplies fluxoid, penetration, thin-film, and fixed-amplitude electrodynamic conventions. Disorder in Quantum Matter supplies the disorder ensemble and length-scale ledger. Transport Measurements and Magnetic Susceptibility own acquisition, geometry, and calibration. Driven Many-Body Systems supplies protocol, energy-balance, and observation-window discipline.

Declare the Vortex-Matter Problem

No vortex claim is complete without a state-and-measurement ledger.

1. Material and state. State composition, temperature, pressure or doping, transition temperature, anisotropy, layering, granularity, coherent charge, and the validity window of London or Ginzburg–Landau parameters.
2. Field and geometry. Distinguish applied field, internal auxiliary field, and internal induction. Give orientation, magnitude, demagnetization, sample geometry, zero- or field-cooling, sweep direction, previous extrema, waiting time, and thermal and current history.
3. Vortex object. State species, winding multiplicity, flux orientation, density normalization, and whether the objects are lines, pancakes, stacks, bundles, a lattice, a glass, a liquid, or driven channels.
4. Scales. Record coherence, penetration, film, intervortex, sample, elastic, pinning, thermal, drive, and observation-time scales.
5. Disorder. Name point, correlated, extended, surface, edge, or patterned defects; their density, correlation length, strength distribution, and whether they are quenched on the measurement timescale.
6. Drive. Give amplitude, direction, waveform, ramp rate, dwell time, pulse width, duty cycle, frequency, waiting time, and the thermal path to the bath.
7. Claimed regime. Declare whether the proposed regime is elastic, pinned, a critical state, creep, plastic flow, free flow, liquid, glass, or an unresolved crossover.
8. Observable and forward model. Identify electric field, magnetization, susceptibility, complex impedance, image, scattering record, or spectrum; include contacts, demagnetization, current crowding, surface sensitivity, convolution, background, pickup, resolution, and numerical criteria.
9. Evidence and alternatives. State which observations support the claim and compare equilibrium transition, dynamic crossover, depinning, creep, plasticity, surface entry, heating, loss of pairing, and instrumental floor.
10. Uncertainty and stop rule. Propagate field, geometry, current, temperature, timing, calibration, and model uncertainty. Name the next discriminating test and the strongest claim licensed now.

We use SI units, $e>0$, and a signed coherent-field charge $q^\ast$. For an ordinary Cooper-pair condensate,

$$q^\ast=-2e, \qquad \Phi_0=\frac{h}{|q^\ast|}>0.$$

Applied field $\mathbf H_{\mathrm{appl}}$, internal auxiliary field $\mathbf H$, magnetization $\mathbf M$, and induction $\mathbf B$ are not interchangeable. Macroscopically in SI,

$$\mathbf B = \mu_0(\mathbf H+\mathbf M),$$

while sample shape and demagnetization connect $\mathbf H$ to $\mathbf H_{\mathrm{appl}}$. Let the internal induction define

$$\mathbf B = B\,\hat{\mathbf t}, \qquad B>0,$$

where $\hat{\mathbf t}$ points along positive magnetic flux. We assign one positive flux vector $\boldsymbol\Phi_v=\Phi_0\hat{\mathbf t}$ to an aligned singly quantized line. With $q^\ast=-2e$, this positive-flux orientation has the opposite signed phase winding in the Ginzburg–Landau convention $\Phi=(h/q^\ast)N$.

For explicit force and voltage examples we choose $\mathbf J=J\hat{\mathbf x}$ and $\mathbf B=B\hat{\mathbf z}$. Positive longitudinal voltage means $E_x>0$; positive transverse voltage means $E_y>0$. Other axis or lead conventions must be transformed rather than silently compared.

For straight, singly quantized lines aligned with $\mathbf B$, the areal line density is

$$n_v = \frac{B}{\Phi_0}.$$

This is a coarse-grained density statement, not proof that every imaged object carries one quantum. Multiquanta vortices, tilted lines, strong layering, domains, and finite fields of view require an explicit counting model.

From a Static Vortex to Elastic Matter

For an ideal triangular lattice of straight singly quantized lines, flux per unit cell gives

$$\frac{\sqrt{3}}{2}a_0^2 B = \Phi_0,$$

so the nearest-neighbor spacing is

$$a_0 = \left( \frac{2\Phi_0}{\sqrt{3}B} \right)^{1/2}.$$

The relation tests average density. Crystal anisotropy, nonlocal response, and multicomponent order can change lattice shape, while disorder and finite resolution broaden or split the positional peaks.

Write a flux line as a displacement from a reference lattice,

$$\mathbf R_i(z) = \mathbf R_i^{(0)} + \mathbf u(\mathbf R_i^{(0)},z).$$

For small, slowly varying distortions with no free dislocations, a local isotropic elastic approximation is

$$F_{\mathrm{el}} = \frac{1}{2} \int \mathrm d^3r \left[ c_{11} (\boldsymbol{\nabla}_{\perp}\!\cdot\mathbf u)^2 + c_{66} (\boldsymbol{\nabla}_{\perp}\!\times\mathbf u)^2 + c_{44} |\partial_z\mathbf u|^2 \right].$$

The compression, shear, and tilt moduli are $c_{11}$, $c_{66}$, and $c_{44}$. In real superconductors they can depend on field, temperature, direction, and wavevector. Compression and tilt response are often strongly nonlocal. A single fitted constant is therefore an operational parameter over a declared window, not an exact material constant.

Elasticity fails when displacements are not small, dislocations proliferate, lines cut or reconnect, layers decouple, or a driven state forms separate channels. The failure is physical information: it marks the handoff from an elastic solid to plastic or more strongly fluctuating vortex matter.

Pinning, Barriers, and Critical States

A defect changes vortex energy because it changes condensation, core, magnetic, elastic, or interface energy. A pinning potential per unit length $U_p(\mathbf u)$ produces

$$\mathbf f_p = -\boldsymbol{\nabla}_{\mathbf u}U_p,$$

where $\mathbf f_p$ is force per unit vortex length. Point disorder need not pin vortices independently. When elastic energy couples many lines or many defects, the moving object is a correlated bundle, and its effective barrier depends on bundle size and drive.

Three mechanisms must be separated.

• Surface and geometrical barriers impede entry or exit even in a clean sample. They can make first penetration differ from equilibrium $B_{c1}$ and can produce asymmetric field history.
• Strong individual pinning is controlled by rare defects whose force can trap a line or small segment nonperturbatively.
• Weak collective pinning balances random forces over a correlation volume. The Larkin scale is the distance beyond which cumulative disorder destroys the accuracy of a chosen undistorted reference lattice; it is not a universal microscopic defect spacing.

Near a stable pin and for sufficiently small ac displacement,

$$U_p(u) \simeq U_p(0) + \frac{\alpha_p}{2}u^2.$$

Here $\alpha_p$ is a Labusch curvature per unit vortex length. In a simple uniform bulk geometry the associated Campbell length obeys

$$\lambda_C^2 = \frac{B\Phi_0}{\mu_0\alpha_p}.$$

This probes the local curvature of occupied wells, not their full depth or a unique dc depinning current. History and creep can change which wells are occupied.

At macroscopic scale a pinned state supports an induction gradient. When every free, screening, and magnetization current is included in $\mathbf J_{\mathrm{tot}}$, the microscopic Maxwell equation is

$$\boldsymbol{\nabla}\times\mathbf B = \mu_0\mathbf J_{\mathrm{tot}}.$$

In macroscopic matter, by contrast, $\boldsymbol{\nabla}\times\mathbf H=\mathbf J_{\mathrm{free}}$, with bound currents represented through $\mathbf M$. For an ideal infinite slab with one-dimensional penetration, a field-independent critical persistent-current density, and negligible reversible-magnetization gradients, the Bean approximation gives

$$\left| \frac{\mathrm dB}{\mathrm dx} \right| = \mu_0J_c.$$

This is the simplest Bean critical-state model. Its $J_c$ is an operational history-dependent persistent-current scale, not the equilibrium depairing current. Converting a magnetic-loop width into $J_c$ requires the correct sample shape, current path, demagnetization correction, and field dependence.

Thermal Activation and Creep

A finite barrier produces a hierarchy of observation times. For an activated event with attempt time $\tau_0$,

$$\tau \sim \tau_0 \exp\!\left( \frac{U}{k_B T} \right).$$

A state may look pinned for $10^{-3}\ \mathrm s$ and relax over $10^4\ \mathrm s$ without changing its equilibrium classification. Conversely, a voltage below the instrumental floor is an upper bound on motion, not proof of an infinite barrier.

A common magnitude parameterization, with $E_0>0$ and $U>0$, is

$$|E|(|J|,B,T) = E_0(|J|,B,T) \exp\!\left[ -\frac{U(|J|,B,T)}{k_B T} \right].$$

The electric-field sign follows the declared current and voltage convention; the prefactor, barrier, and even the moving object are model dependent. The Anderson–Kim form uses a barrier that decreases approximately linearly toward zero near a fitted current scale. Collective-creep descriptions instead often use barriers that grow as a power of $J_c/J$ at small drive. Neither form is a universal interpolation across single-vortex, bundle, plastic, and free-flow regimes.

Magnetic relaxation is often summarized by

$$S \equiv -\frac{\mathrm d\ln|M_{\mathrm{irr}}|} {\mathrm d\ln t}.$$

Only when irreversible magnetization tracks the persistent current and one slowly varying effective barrier controls the window may one estimate

$$U^\ast \simeq \frac{k_B T}{S}.$$

The result is an effective time-window barrier, not automatically a defect binding energy. A plateau in low-temperature relaxation also does not by itself prove quantum tunneling. Temperature gradients, instrumental drift, surface barriers, a distribution of classical barriers, and a changing current profile must be excluded before a quantum-creep model is warranted.

Driven Motion, Flux Flow, and Hall Response

For a transport current perpendicular to a straight vortex, define the force per unit length

$$\mathbf f_L = \mathbf J\times (\Phi_0\hat{\mathbf t}).$$

With pinning force $\mathbf f_p=-\boldsymbol{\nabla}U_p$, a minimal overdamped force ledger is

$$\eta\mathbf v_L + \alpha_H \hat{\mathbf t}\times\mathbf v_L = \mathbf f_L + \mathbf f_p + \mathbf f_{\mathrm{th}}.$$

The drag coefficient $\eta$ is per unit line length, $\alpha_H$ is a transverse coefficient, and $\mathbf f_{\mathrm{th}}$ represents fluctuations. Inertia, memory, nonlocality, line tension, and interactions are omitted from this local equation. The sign of $\alpha_H$ depends on the declared vorticity and voltage conventions.

Moving flux produces the coarse-grained electric field

$$\mathbf E = \mathbf B\times\mathbf v_L = -\mathbf v_L\times\mathbf B.$$

Thus a measured longitudinal voltage is a velocity measurement only after current distribution, internal induction, and moving fraction are known. A stationary pinned population can coexist with rapidly moving channels, so a single mean velocity may hide broad spatial heterogeneity.

Ignoring pinning and the Hall term gives

$$\rho_{\mathrm{ff}} = \frac{B\Phi_0}{\eta}.$$

The Bardeen–Stephen normal-core benchmark is

$$\rho_{\mathrm{ff}} \simeq \rho_n \frac{B}{B_{c2}}.$$

It is a useful dirty, conventional, local-core estimate—not a definition of flux flow. Clean cores, nodal quasiparticles, multiband structure, strong anisotropy, pinning backflow, and nonequilibrium occupations can change both longitudinal and transverse response. A mixed-state Hall sign change is not, by itself, a measurement of the carrier sign inside a normal core. The general response tensor remains with the Hall Effect owner. Hall Measurements owns transverse-lead geometry, field/current symmetrization, contact mixing, and acquisition signs.

When the Hall term is retained but pinning is absent, the longitudinal part is

$$\rho_{xx} = \frac{B\Phi_0\eta} {\eta^2+\alpha_H^2}.$$

For the declared $+\hat{\mathbf x}$ current and $+\hat{\mathbf z}$ induction,

$$\rho_{yx} = -\frac{B\Phi_0\alpha_H} {\eta^2+\alpha_H^2}.$$

The transverse sign follows from the declared axes, $\hat{\mathbf t}$, and the sign assigned to $\alpha_H$. Reporting only a scalar “vortex viscosity” from $\rho_{xx}$ silently assumes the Hall contribution is negligible.

Disorder, Melting, and Glass Claims

Vortex-lattice, Bragg-glass, vortex-glass, Bose-glass, and vortex-liquid labels refer to different observables and limits. They are not synonyms for ordered, hysteretic, immobile, or resistive.

• Elastic lattice or solid. Specify translational correlations, reciprocal peaks, displacement growth, dislocations, dimensionality, and equilibrium window. One locally triangular image establishes local order only.
• Bragg glass. Within a dislocation-free elastic window, quenched disorder can produce quasi-long-range translational order and singular broadened Bragg correlations without a perfect lattice. This is a finite-scale correlation claim with explicit defect and resolution tests, not a synonym for a vortex glass.
• Melting. A thermodynamic melting claim needs an equilibrium singularity or discontinuity and reversibility controls. A Lindemann displacement criterion is a useful estimate, not a theorem or independent measurement.
• Vortex liquid. A liquid lacks static shear rigidity on the stated scales. Its transport can still be nonlinear, viscous, anisotropic, and affected by short-range correlations.
• Vortex glass. A glass claim requires a specified disorder ensemble, the $J\to0$ linear-response limit, and an explicit order for $L\to\infty$ and $\omega\to0$ or $t_{\mathrm{obs}}\to\infty$. It also needs a controlled scaling window and evidence beyond slow relaxation or hysteresis. A finite nonlinear $E$–$J$ curve over a few decades does not uniquely prove a glass transition.
• Bose glass. Correlated line defects define a distinct anisotropic pinning problem. The name is not licensed merely because the underlying Cooper pairs are bosonic.

Quenched disorder can destroy long-range positional order without producing a single universal glass. Thermal fluctuations can round, shift, or preempt elastic transitions. Layered materials add pancake misalignment, line cutting, and decoupling. Finite specimens replace sharp thermodynamic statements by scale- and time-dependent claims unless a controlled extrapolation is shown.

High-Drive Dynamics and Stopping Rules

Increasing current does not move a specimen through a universal sequence. Possible regimes include activated creep, elastic depinning, plastic channels, dynamic reordering, viscous flux flow, phase-slip structures, and abrupt instability. Their boundaries depend on disorder, field, temperature, geometry, and protocol.

Joule power density is

$$p = \mathbf J\cdot\mathbf E.$$

For spatially uniform steady motion in the represented vortex channel,

$$p = n_v\eta v_L^2.$$

The Hall force is perpendicular to $\mathbf v_L$ and therefore does no work, so this identity remains valid for finite $\alpha_H$. It is a vortex-channel energy-balance check only: additional quasiparticle, contact, or inhomogeneous dissipation must be added separately. It does not guarantee the electrons, phonons, substrate, and thermometer share one temperature. At high drive, quasiparticle nonequilibrium, vortex-core shrinkage, current crowding, thermal runaway, phase slips, and contact heating can all generate voltage jumps.

Stop an intrinsic-dynamics interpretation when any of the following occurs:

• the inferred electron or local lattice temperature is not bounded;
• voltage depends on pulse width or duty cycle without a thermal model;
• current density is computed from a nominal cross-section despite strong crowding or filamentary flow;
• sweep direction or waiting time changes the branch but history is omitted;
• the same voltage law fits creep, plastic channels, and heating over the available dynamic range;
• a local probe and a bulk transport probe demonstrably sample different vortex populations;
• an instability is called a phase transition without an equilibrium control.

How Experiments See Vortex Matter

Each probe measures a forward-model-weighted projection.

Transport. Four-terminal voltage samples the line integral of electric field between contacts. Convert $V$ to $E$ and applied current to local $\mathbf J$ only with a geometry model. Reverse current, reverse field, vary pulse width, and monitor heating. During field sweeps, subtract or model inductive voltage before interpreting flux motion. The acquisition contract is owned by Transport Measurements.

Magnetometry and ac susceptibility. An $M(H)$ loop measures a global, history-dependent current distribution. Relaxation measures redistribution over a time window. Small-amplitude ac response can probe a local pinning-well curvature; larger drives sample nonlinear depinning and higher harmonics. Demagnetization, holder subtraction, sweep rate, and field uniformity remain part of the inference. See Magnetic Susceptibility.

Microwave and terahertz response. Complex impedance mixes condensate, quasiparticle, and vortex contributions. Pinning frequency, drag, creep, skin depth, multilayer optics, and field inhomogeneity can be correlated in a fit. Frequency and field sweeps are needed before assigning one fitted parameter to one microscopic process. Terahertz and Infrared Probes owns the electromagnetic inversion.

Imaging. Decoration, magneto-optical, Hall, SQUID, Lorentz, and tunneling methods have different height kernels, spatial resolution, time resolution, field ranges, and surface sensitivities. A vortex coordinate map must report the localization algorithm and field of view. Scanning Tunneling Microscopy and Spectroscopy owns vortex-core spectral imaging; it does not directly measure the bulk line configuration.

Small-angle neutron scattering. Reciprocal-space vortex peaks probe a bulk-weighted field modulation, but their width and intensity convolve finite domain size, mosaic spread, line wandering, form factor, instrumental resolution, and motion during exposure. Peak loss is therefore not uniquely melting. Neutron Scattering owns beam geometry, resolution, background, and structure-factor inversion; this page owns the resulting vortex-order claim.

A strong inference aligns preparation, temperature, internal field, current, waiting time, and field of view across probes. A beautiful image taken after field cooling cannot validate a transport transition measured on a different history without a state-matching argument.

Worked Audit: Depinning to Flux Flow

The record is synthetic, so its purpose is reproducibility rather than a claim about a named compound.

1. Material and state. A dirty conventional type-II film is measured at $T=10.00\ \mathrm K$, well below its declared $T_c$. A separately validated normal-state resistivity is

   $$\rho_n = 1.00\times10^{-6}\ \Omega\,\mathrm m,$$

   and $B_{c2}=20.0\ \mathrm T$ at the same temperature.

2. Field and geometry. A $100\ \mathrm{nm}$-thick, $10.0\ \mu\mathrm m$- wide bridge has voltage taps separated by $100\ \mu\mathrm m$. It is field cooled into a locally calibrated internal induction $\mathbf B=2.00\ \mathrm T\,\hat{\mathbf z}$ before current is applied along $+\hat{\mathbf x}$.

3. Vortex object. The ordinary charge-$2e$ condensate, calibrated mean induction, and mixed-state window motivate a singly quantized-line working model with positive flux orientation. Its average-density benchmark gives $a_0=34.5\ \mathrm{nm}$. The available coarse imaging cannot resolve winding line by line; it only shows dislocations and moving channels on larger scales, so multiply quantized or bundled objects are not excluded by that image alone.

4. Scales. The GL upper-critical-field estimate gives $\xi=[\Phi_0/(2\pi B_{c2})]^{1/2}=4.06\ \mathrm{nm}$. The hierarchy $\xi\ll a_0$ is consistent with distinct cores. The voltage and image frames are synchronized to each pulse.

5. Disorder. Weak point disorder is distributed through the film, with a sparse population of stronger defects. Its full force distribution is not known, so no unique microscopic pin potential is fitted.

6. Drive. $10\ \mu\mathrm s$ current pulses have duty cycle $10^{-3}$. Current and field are both reversed. A calibrated local thermometer bounds the pulse-window temperature rise by $0.03\ \mathrm K$.

7. Claimed regime. The low-drive branch is tested for creep and plastic flow; the high-drive branch is tested against homogeneous free flux flow.

8. Observable and forward model. Four-terminal voltage is converted to $E$ with the tap spacing and to $J$ with the patterned cross-section. The longitudinal high-drive slope is

   $$\frac{E_x}{J_x} = (1.02\pm0.05) \times10^{-7}\ \Omega\,\mathrm m.$$

   The Bardeen–Stephen benchmark predicts

   $$\rho_n\frac{B}{B_{c2}} = 1.00\times10^{-7}\ \Omega\,\mathrm m.$$

   Lead mixing is removed by current and field reversal. A separately measured quasiparticle Hall background is propagated through a two-channel fit. Under the stated high-drive pure-vortex model—where the residual pinning backflow and other transverse channels are below that fit uncertainty—the residual tensor bounds $|\rho^{(v)}_{yx}/\rho^{(v)}_{xx}|<0.10$ at 95% confidence. Let $r_H\equiv\alpha_H/\eta=-\rho^{(v)}_{yx}/\rho^{(v)}_{xx}$ and define the longitudinal-only estimate

   $$\eta_0 = \frac{B\Phi_0}{\rho_{xx}} = (4.06\pm0.20) \times10^{-8}\ \mathrm{N\,s\,m^{-2}}.$$

   The Hall bound gives $\eta=\eta_0/(1+r_H^2)$, a correction below 1%, smaller than the quoted slope uncertainty. At $E_x=1.00\ \mathrm{V\,m^{-1}}$, the induction-weighted Lorentz-direction velocity component is

   $$|v_y| = \frac{E_x}{B} = 0.500\ \mathrm{m\,s^{-1}}.$$

   The imaging point-spread width is $100\ \mathrm{nm}$, so it resolves channels but not individual $35\ \mathrm{nm}$-spaced vortices or their instantaneous core displacements. The total speed differs from $|v_y|$ by less than 0.5% under the Hall bound.

   At that point $J_x=E_x/\rho_{xx}=9.80\times10^6\ \mathrm{A\,m^{-2}}$, so the measured vortex-channel power ledger is

   $$p = J_xE_x = 9.80\times10^6\ \mathrm{W\,m^{-3}} = n_v\eta_0|v_y|^2.$$

   With the full Hall parameters the last expression is equivalently $n_v\eta|\mathbf v_L|^2$; the transverse force contributes no work.

9. Evidence and alternatives. The high-drive slope agrees with the normal-core benchmark over the measured window. Below it, synchronized images show intermittent channels, so $E/B$ is not every vortex’s speed. Pulse-width invariance and the thermometer bound disfavor bulk heating, but current crowding and a nonuniform moving fraction remain. The Hall correction is conditional on the background and backflow separation; if that separation fails, only the effective longitudinal $\eta_0$ may be reported.

10. Uncertainty and stop rule. Resistivity-slope uncertainty dominates the quoted $\eta_0$ error; field, geometry, and $B_{c2}$ systematics must be added for a final material value. The licensed claim is a crossover from plastic channels toward an approximately homogeneous branch consistent with Bardeen–Stephen flow. Stop before claiming a unique depinning current or a microscopic drag mechanism.

Worked Audit: Magnetic Relaxation

This second synthetic record asks a different question: do slow relaxation and finite positional correlations license a melting or glass claim?

1. Material and state. A $1.00\times0.50\times0.050\ \mathrm{mm^3}$ platelet with $T_c=35\ \mathrm K$ is field cooled from $40\ \mathrm K$ to $20.0\ \mathrm K$ and allowed to equilibrate thermally for $300\ \mathrm s$.

2. Field and geometry. The applied field is then held fixed along the short platelet axis. A calibrated mean internal induction is $1.00\ \mathrm T$; the local induction profile remains nonuniform and may relax.

3. Vortex object. The net density is consistent with singly quantized lines. SANS shows broadened sixfold peaks. A stated line-shape model returns the effective in-plane scale $\xi_{\mathrm{eff}}=(12\pm2)a_0$; it is not yet a unique thermodynamic correlation length.

4. Scales. The observation window is $10^2$–$10^4\ \mathrm s$. The SANS exposure is $60\ \mathrm s$, shorter than the full relaxation window but not instantaneous. Sample dimensions, $a_0$, and instrumental reciprocal-space resolution are included in the fit.

5. Disorder. Independent microscopy finds dilute point and edge defects. Whether the observed relaxation is dominated by bulk disorder or surface exit is unresolved.

6. Drive. No transport current is applied. Persistent screening currents generated by field cooling supply the changing Lorentz drive. The magnetic moment, SANS exposure, and waiting-time origin use the same preparation.

7. Claimed regime. The immediate hypothesis is pinned vortex matter with creep. Equilibrium melting and a vortex-glass transition are competing claims to be tested, not assumed.

8. Observable and forward model. The irreversible magnetization changes from

   $$|M_{\mathrm{irr}}(10^2\ \mathrm s)| = 20.0\ \mathrm{kA\,m^{-1}}$$

   to

   $$|M_{\mathrm{irr}}(10^4\ \mathrm s)| = 17.0\ \mathrm{kA\,m^{-1}}.$$

   The log-window estimate is

   $$S = -\frac{\ln(17.0/20.0)}{\ln(10^4/10^2)} = 0.0353.$$

   Under the single-effective-barrier approximation,

   $$U^\ast \simeq \frac{k_BT}{S} = 48.8\ \mathrm{meV}.$$

   The magnetization-to-current conversion uses a platelet critical-state model. The SANS fit convolves instrumental resolution, the measured induction and $a_0$ distribution, mosaic spread, finite domains, line wandering, and motion during the $60\ \mathrm s$ exposure. Those components remain partly degenerate, so $\xi_{\mathrm{eff}}$ is a model-dependent finite-correlation scale.

9. Evidence and alternatives. Relaxation plus a broadened finite-correlation record supports pinned, disordered vortex matter with motion on the observation timescale. It does not distinguish a crossover from a thermodynamic glass, and one broadened SANS pattern does not establish melting. Surface exit, bundle-size distributions, line wandering, mosaic, motion blur, and a changing internal current profile can all contribute.

10. Uncertainty and stop rule. Repeat several temperatures, fields, waiting-time origins, sizes, and aspect ratios; add matched calorimetry or another equilibrium test and current-window scaling before a phase claim. The present result is a model-limited $U^\ast$ and a pinned regime with an effective finite-correlation scale. Stop before calling $48.8\ \mathrm{meV}$ a microscopic pin depth, $\xi_{\mathrm{eff}}$ a unique thermodynamic length, the broadened peaks a melted lattice, or the relaxation a vortex glass.

Common Failure Modes

• Calling first penetration $B_{c1}$ without excluding surface and geometrical barriers.
• Calling magnetic irreversibility an equilibrium phase boundary without a rate, waiting-time, and thermodynamic check.
• Treating transport $J_c$, Bean-model $J_c$, depinning current, and depairing current as the same quantity.
• Inferring one pin depth from a Campbell curvature or one relaxation slope.
• Fitting an Arrhenius line without allowing a current-dependent barrier and temperature-dependent prefactor.
• Calling a low-temperature relaxation plateau quantum creep before bounding heating, noise floor, drift, and classical barrier distributions.
• Using $\rho_nB/B_{c2}$ as a universal law rather than a benchmark.
• Interpreting $E/B$ as every vortex velocity in a plastic state.
• Calling a locally triangular snapshot an equilibrium vortex lattice.
• Calling slow, hysteretic response a vortex glass without thermodynamic and zero-frequency scaling.
• Calling a voltage jump an intrinsic instability without a pulse-width and thermal-impedance audit.
• Combining transport, magnetometry, and imaging taken on unmatched histories.

Exercises

1. Density and triangular spacing

Derive $n_v=B/\Phi_0$ and the triangular spacing $a_0$. Evaluate $a_0$ at $B=1.00\ \mathrm T$ using $\Phi_0=2.068\times10^{-15}\ \mathrm{Wb}$. What calibrated imaging or scattering result could falsify the singly quantized-line density model?

Solution

Each singly quantized line carries $\Phi_0$, so $BA=N_v\Phi_0$ and $n_v=N_v/A=B/\Phi_0$. A triangular primitive cell has area $\sqrt{3}a_0^2/2$, hence

$$a_0 = \left( \frac{2\Phi_0}{\sqrt{3}B} \right)^{1/2}.$$

At $1.00\ \mathrm T$,

$$a_0 = 48.9\ \mathrm{nm}.$$

The result is an average-density benchmark; disorder and anisotropy can change the observed neighbor distribution. A calibrated real-space count or reciprocal-cell area that remains inconsistent with $B/\Phi_0$ after correcting for tilt, finite field of view, point-spread resolution, domains, and multiquanta objects falsifies the singly quantized-line model over that window.

2. Read the elastic modes

For the quadratic elastic functional, identify a displacement pattern that isolates compression, shear, and tilt. Why can a constant-modulus fit fail?

Solution

A longitudinal in-plane wave with $\mathbf u\parallel\mathbf q_\perp$ has $\boldsymbol{\nabla}_\perp\cdot\mathbf u\ne0$ and tests $c_{11}$. A transverse in-plane wave has nonzero in-plane curl and tests $c_{66}$. A displacement that varies along the line direction has $\partial_z\mathbf u\ne0$ and tests $c_{44}$. Electromagnetic interactions, anisotropy, layering, and proximity to critical fields make the moduli dispersive and tensorial. Dislocations or plastic flow invalidate the single-valued small-displacement field itself.

3. Bean slab versus equilibrium

An ideal slab has half-width $a$, constant $J_c$, and one-dimensional flux penetration. Integrate $|\mathrm dB/\mathrm dx|=\mu_0J_c$. What does the result establish, and what does it not establish?

Solution

On a monotonic penetrated branch,

$$B(x) = B(x_0) \pm\mu_0J_c(x-x_0)$$

where the branch sign is chosen from the current direction and field history. The full-penetration field scale is $\mu_0J_ca$ in this ideal geometry. This constructs a critical state and its magnetic hysteresis. It does not give an equilibrium flux profile, a depairing current, or a geometry-independent conversion from loop width to $J_c$. Field-dependent $J_c$, creep, demagnetization, and surface barriers must be added when relevant.

4. A relaxation-window barrier

At $T=30.0\ \mathrm K$, $|M_{\mathrm{irr}}|$ falls from $12.0$ to $10.2\ \mathrm{kA\,m^{-1}}$ between $10^2$ and $10^5\ \mathrm s$. Estimate $S$ and $U^\ast$, then state two reasons not to call $U^\ast$ a defect energy.

Solution

The normalized rate is

$$S = -\frac{\ln(10.2/12.0)}{\ln(10^5/10^2)} = 0.0235.$$

Using $k_BT=2.585\ \mathrm{meV}$ gives

$$U^\ast \simeq 110\ \mathrm{meV}.$$

It is a window-dependent effective barrier. The current profile changes during relaxation, and a distribution of bundle sizes, surface barriers, or pin strengths can generate the same slope. The approximation also assumes $M_{\mathrm{irr}}\propto J$ and a slowly varying dominant barrier.

5. Force, velocity, and flux-flow resistivity

Take $\mathbf J=J\hat{\mathbf x}$ and $\mathbf B=B\hat{\mathbf z}$. Neglect pinning and Hall force. Find the vortex velocity direction, electric-field direction, and $\rho_{\mathrm{ff}}$.

Solution

The force per unit length is

$$\mathbf f_L = J\hat{\mathbf x} \times \Phi_0\hat{\mathbf z} = -J\Phi_0\hat{\mathbf y}.$$

Thus $\mathbf v_L=-(J\Phi_0/\eta)\hat{\mathbf y}$. The induced field is

$$\mathbf E = \mathbf B\times\mathbf v_L = \frac{BJ\Phi_0}{\eta} \hat{\mathbf x},$$

so $\rho_{\mathrm{ff}}=B\Phi_0/\eta$. Reversing $B$ reverses the vortex orientation and force but leaves the longitudinal resistivity positive. Moreover,

$$\mathbf J\cdot\mathbf E = \frac{B\Phi_0J^2}{\eta} = n_v\eta|\mathbf v_L|^2 > 0,$$

so the longitudinal flux-flow channel has positive dissipation.

6. Include the transverse force

With no pinning, solve $\eta\mathbf v+\alpha_H\hat{\mathbf z}\times\mathbf v =\mathbf J\times\Phi_0\hat{\mathbf z}$ for the longitudinal resistivity. Why must a transverse sign be reported with conventions?

Solution

Inverting the two-dimensional operator gives a longitudinal electric field

$$E_{\parallel} = J \frac{B\Phi_0\eta} {\eta^2+\alpha_H^2},$$

hence

$$\rho_{xx} = \frac{B\Phi_0\eta} {\eta^2+\alpha_H^2}.$$

With $\mathbf J=J_x\hat{\mathbf x}$, $\mathbf B=B\hat{\mathbf z}$, and $E_i=\rho_{ij}J_j$, the same inversion gives

$$\rho_{yx} = \frac{E_y}{J_x} = -\frac{B\Phi_0\alpha_H} {\eta^2+\alpha_H^2}.$$

This sign changes if the voltage axes, flux orientation, or definition of $\alpha_H$ is reversed. Pinning backflow and quasiparticle Hall response can also modify the measured tensor, so a scalar drag extraction requires a negligible-Hall check.

7. Audit a melting or glass claim

A transport curve becomes nonlinear, a magnetic loop opens, and one image shows sixfold local coordination below $T_\star$. Does this establish a vortex glass, a vortex lattice, or melting at $T_\star$?

Solution

No unique phase follows. Nonlinearity and hysteresis can arise from finite-time pinning or creep. Sixfold local coordination establishes only local order in one surface-sensitive field of view. A lattice claim needs positional correlations, dislocation and finite-size analysis, and state matching. A melting claim needs equilibrium thermodynamic or controlled structural evidence with rate and hysteresis tests. A glass claim needs a declared disorder ensemble, the $J\to0$ linear-response limit, and a controlled order of $L\to\infty$ and $\omega\to0$ or $t_{\mathrm{obs}}\to\infty$. The correct present conclusion is a pinned, correlated vortex regime with phase classification unresolved.

8. Build a joint evidence record

Design the minimum transport, magnetometry, and imaging record needed to test whether a voltage onset is elastic depinning rather than surface entry, plastic flow, creep, or heating.

Solution

Use one specimen and match field-cooling or sweep history, temperature, internal field, waiting time, and current protocol. Transport should include current and field reversal, pulse-width and duty-cycle scans, local-temperature bounds, contact geometry, and a declared voltage floor. Magnetometry should measure loop and relaxation behavior at the same field and waiting times, with demagnetization and surface-barrier checks. Time-resolved or stroboscopic imaging is required to test whether motion is coherent, bundled, edge initiated, or channeled, with the localization kernel and field of view stated. Before/after images show net rearrangement but cannot exclude transient plastic channels. Elastic depinning is supported only if a collective threshold survives thermal and surface controls and the moving configuration remains elastically correlated. Otherwise stop at the more limited dynamic description.

Canonical Boundaries

• Ginzburg–Landau Theory owns the static core, equilibrium critical fields, ideal type criterion, and Abrikosov solution.
• London Theory owns fixed-amplitude screening, fluxoid quantization, and the Pearl-scale electrodynamics outside a core.
• This page owns collective vortex elasticity, pinning, creep, critical-state and driven-motion ledgers, and phase/evidence qualifications.
• Disorder in Quantum Matter owns the generic disorder ensemble, scattering, and statistical contract.
• The Transport, Response, and Optics gateway owns general response- regime selection; the Hall owner keeps non-vortex transverse mechanisms.
• How Quantum Matter Is Measured and its probe leaves own acquisition, calibration, and inversion.
• Data Interpretation and Pitfalls owns the general evidence and uncertainty workflow.
• Vortex-core bound states, Majorana claims, and BdG invariants belong to Topological Superconductors, not to a pinning or flux-flow analysis.

References

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• Y. Yeshurun, A. P. Malozemoff, and A. Shaulov, “Magnetic Relaxation in High-Temperature Superconductors,” Reviews of Modern Physics 68, 911–949 (1996), doi:10.1103/RevModPhys.68.911.