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Lectures on ultrathin film ferromagnetism

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Pith's one-line read Quantum-confined ultrathin 3d metal films are effectively two-dimensional spin systems; the paper shows this explains dead layers, enhanced moments, oscillatory coupling, reorientation transitions, and Ising-like critical behavior.

desk verdict Solid, clearly-written lecture notes on ultrathin film magnetism; the visible derivations hold up, but the paper is incomplete and the claimed new insights live in the missing chapters. read the letter →

arxiv 2608.12189 v1 pith:SDXMJGBS submitted 2026-08-12 cond-mat.mes-hall

D. Pescia This is my paper · ORCID
classification cond-mat.mes-hall
keywords ultrathinfilmsferromagnetismtwo-dimensionalmagnetismquantumwellstatesmagneticanisotropyreorientationtransitioninterlayerexchangecouplingIsinguniversalityclass
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is a set of lecture notes arguing that the apparently complex magnetism of ultrathin Fe, Co, and Ni films reduces to a few universal principles. Because the films grow layer-by-layer, electrons are confined in quantum wells along the vertical direction, while the in-plane directions remain macroscopically extended; the spin system is therefore effectively two-dimensional. From this starting point the notes derive, with simple models, the existence of magnetically dead layers and enhanced moments, an oscillatory coupling between ferromagnetic films separated by a non-magnetic spacer, a perpendicular anisotropy arising from spin-orbit coupling in the lowered surface symmetry, and a reorientation transition where that perpendicular orientation turns in-plane. At finite temperature the same two-dimensionality explains why ferromagnetic order persists despite the theorem forbidding long-range order in two-dimensional isotropic spin systems, why the transition follows the two-dimensional Ising universality class, and why stripe domains of reversed perpendicular spins can form. The notes end by pointing out that these same principles transfer to the atomically thin magnets made by exfoliating bulk layered crystals, whose much flatter morphology realizes the idealized two-dimensionality more closely.

What carries the argument

The slab model: the film is a continuum medium of thickness d, laterally macroscopic, with the spin configuration strictly rigid along the vertical direction, so the spin field depends only on in-plane coordinates. This makes the exchange functional effectively two-dimensional; from the slab the notes derive the two-dimensional nonlinear sigma model, apply the Mermin-Wagner theorem and the Polyakov renormalization group, and compute the reorientation transition, stripe domains, and two-dimensional Ising critical behavior.

What would settle it

Measure the spin-wave dispersion of an ultrathin film such as Fe on W(110) as a function of thickness: if a vertical spin-wave branch with energy below the two-dimensional exchange stiffness appears already at a few monolayers, the rigidity assumption—and with it the strictly two-dimensional analysis—fails at that thickness. Alternatively, precision measurements of critical exponents showing a crossover from two-dimensional Ising to three-dimensional Heisenberg behavior with increasing thickness would settle the regime of validity.

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Extended reading notes

Core claim

The central claim is that vertical quantum confinement turns a few-monolayer 3d transition-metal film into a two-dimensional spin ensemble, and that this two-dimensionality, rather than material-specific chemistry, organizes the phenomenology. The ground-state moment is decided by the Stoner criterion, with a density of states modified by confinement: dead layers appear when the substrate broadens the d-levels, enhanced moments when reduced coordination narrows the bands. The interlayer coupling is an oscillatory spin-polarization exchange through the spacer. The competition between the perpendicular Néel anisotropy and the always-in-plane local dipolar term selects the spin orientation, and the reorientation transition follows from their competition. At finite temperature, ferromagnetic order is restored by small symmetry-breaking interactions within a two-dimensional nonlinear sigma model, analyzed with the Polyakov renormalization group, and the critical behavior follows the two-dimensional Ising universality class. The paper also claims the slab model—spin configuration rigid along the film normal—is the appropriate description, and acknowledges it is strictly valid only for sufficiently small thicknesses and that no model accounts for monoatomic steps and thickness fluctuations in real films.

Load-bearing premise

The slab model requires the spin configuration to be strictly rigid along the film normal, making the film exactly two-dimensional; the notes state this is strictly true only if the film thickness is small enough, but they do not quantify the limit, and no model accounts for monoatomic steps and thickness fluctuations in real films.

Editorial extensions

If this is right

  • If the slab model is right, a one-monolayer difference in film thickness can flip the sign of the effective perpendicular anisotropy K = lambda a/d - Omega, moving the film through the reorientation transition.
  • The critical temperature of a symmetry-breaking two-dimensional Heisenberg film is set mainly by the exchange stiffness A d, with a logarithmic correction, so thicker films order at higher temperature.
  • The same oscillatory exchange mechanism that gives interlayer coupling predicts the measured oscillation periods from the spacer Fermi surface and explains the stripe pattern of parallel and antiparallel coupled domains in wedged multilayers.
  • The two-dimensional Ising universality class should describe the finite-temperature phase transition of epitaxial films, with critical exponents obtained from the renormalization-group analysis of the phi^4 Landau-Ginzburg-Wilson Hamiltonian.
  • The same universal principles should apply to exfoliated two-dimensional magnets, whose near-perfect flatness realizes the idealized two-dimensionality more closely than epitaxial films.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: because the slab model requires rigidity along the film normal, the predicted universality should break down once the film thickness exceeds the exchange length, where vertical spin waves become cheap; thickness-dependent measurements on Fe/Cu(100) could map where two-dimensional scaling crosses over to three-dimensional behavior.
  • Editorial inference: the dead-layer versus enhanced-moment dichotomy suggests a design rule: choose a substrate whose electron gas broadens the d-resonance just enough to sit on the magnetic side of the Stoner criterion; first-principles surveys of 3d overlayers on noble metals could rank substrates by predicted moment enhancement.
  • Editorial inference: since the reorientation transition is driven by the competition between lambda a/d and Omega, straining a film should shift the transition temperature; magneto-optical Kerr measurements under epitaxial strain could test this without new growth techniques.
  • Editorial inference: the stripe-disordering transition, which the notes say is not yet understood, may be fluctuation-driven and first-order along the lines of the Brazovskii instability discussed in the phi^4 chapter; checking whether stripe melting is discontinuous would connect two parts of the notes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript is a set of lecture notes on the magnetism of ultrathin 3d transition-metal films. It develops, from elementary quantum mechanics and magnetostatics, the standard hierarchy: local moments and Stoner band ferromagnetism; exchange, RKKY interlayer coupling, Néel and dipolar anisotropies; continuum Landau functionals and the Landau-Lifshitz equation; the nonlinear sigma model and Polyakov RG for finite-temperature order; Landau/Wilson RG for critical behavior; and topological stripe states. The central scientific claim is that sufficiently thin films behave as effectively two-dimensional spin ensembles, so that vertical quantum confinement plus two-dimensionality generates a common set of phenomena—dead/enhanced moments, oscillatory interlayer coupling, perpendicular anisotropy, reorientation transitions, quasi-Ising criticality, and stripe order—and that these lessons transfer to exfoliated two-dimensional magnets.

Significance. If the claims are taken at face value, the notes could serve as a valuable pedagogical synthesis. The strongest parts are the worked derivations: the Stoner free-energy balance in Appendix 3.E, the delta-function RKKY model in Appendix 4.A, the spin-wave/Landau-Lifshitz analysis in §5.4, and the magnetostatic kernel in §5.A are careful and self-contained. The paper explicitly distinguishes imported material parameters from derived constants, and it does not force data to a preferred set of free parameters. However, the original-science component is modest, and the advertised universality and transferability go beyond what is demonstrated; those claims need quantitative qualification.

major comments (3)
  1. [§5.A, §5.4] The slab model's central rigidity assumption is never quantified. §5.A states that a strictly z-independent spin configuration is 'strictly true only if the film thickness is small enough that a rotation of the spin along the vertical direction costs too much exchange energy,' but no estimate of this thickness is given. This matters because the effective perpendicular anisotropy K = λ a/d − Ω in Eq. (5.44) changes sign at d/a ≈ λ/Ω, which with Table 1 values (λ ≈ 0.3–0.4 meV, Ω ≈ 0.28 meV) lies at 1–2 ML—the same thickness range as the films imaged in Chapter 2. A quantitative rigidity criterion, e.g. d ≪ sqrt(A/K) or a comparison of the vertical exchange energy with the anisotropy energies, is needed to support the claim that the 2D reduction is valid in the parameter window where the reorientation and stripe arguments are made.
  2. [Ch. 5 introduction; Ch. 2] The paper concedes in the introduction to Chapter 5 that 'there is no model that takes these defects into account' for monoatomic steps and thickness fluctuations, yet the predictions of a sharp reorientation transition (Ch. 8) and of stripe order (Ch. 11) rest on the homogeneous K in Eq. (5.44). The STM images in Chapter 2 show exactly the defects that are excluded: one-ML islands, voids, and terraces at 1–2 ML thickness. Since K varies linearly with d, these defects produce a lateral distribution of anisotropy energies; without an argument that the disorder is irrelevant at the relevant length scales, the clean 2D Ising universality and the reorientation transition are not established for the real epitaxial films discussed. Please add a quantitative discussion of lateral thickness fluctuations and their effect on the predicted phase behavior.
  3. [§4.2, Appendix 4.A] The interlayer coupling with 1/d^2 decay is a leitmotif of the notes, but the d^{-2} law is imported at the end of Appendix 4.A with only a reference, after the appendix derives the 1D decay ∝ 1/(k_F d) and notes the 3D point-decay ∝ 1/(k_F d)^3. The reader is not shown why the two-dimensional multilayer geometry changes the power to 2. Since the rest of the text is built on 'back of the envelope' derivations, either a derivation of the d^{-2} law or a precise statement of the model (e.g., planar array of dipoles versus quantum-well states) should be given.
minor comments (4)
  1. [Title/Abstract] The title page contains the typo 'ultrathin filmferromagnetism' and the abstract contains 'fromresearchon'; a careful proofreading pass is needed.
  2. [§5.3, Eq. (5.30)] Equation (5.30) appears to use −λ d/a, while §5.4 and Eq. (6.8) use −λ a/d (or the z-integrated form); please reconcile the notation, as the sign and the powers of d are essential for the reorientation balance.
  3. [§5.4, Table 1] Table 1 in the rendered version has no column headers and the footnotes are the only guide; a layout with explicit variable names, units, and a system column per row would make the table self-contained.
  4. [§5.A] The estimate of the nonlocal dipolar terms refers to an integral from the 'Supplemental Material' to Ref. [10], but with the bibliography not attached in the arXiv version the reader cannot retrieve the calculation; please give the integral explicitly or provide a complete citation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the slab-model and parameter-input limitations are acknowledged assumptions; the paper's central results are not re-statements of their inputs.

full rationale

The paper is a review that assembles standard models (Stoner band magnetism, Heisenberg/RKKY exchange, Néel spin-orbit anisotropy, dipole magnetostatics, NLSM/Polyakov RG) from textbook derivations and imports material constants from external first-principles calculations and experiments: e.g., Gay–Richter λ≈0.3–0.4 meV, Small–Heine J≈46 meV, spin-wave stiffness D, and the computed dipolar constant Ω≈0.28 meV. The claimed phenomena (enhanced/dead moments, interlayer oscillations, perpendicular anisotropy, reorientation, Ising criticality, stripe order) are derived from these models, not fitted back into them. The reorientation criterion K = λa/d − Ω is a construction that combines independently obtained λ and Ω and then predicts a thickness-dependent sign change; no equation is shown to be equal to another by construction, and no fitted parameter is renamed as a prediction. The slab model's rigid-spin-in-z assumption is an acknowledged idealization: Appendix 5.A states it is 'strictly true only if the film thickness is small enough that a rotation of the spin along the vertical direction costs too much exchange energy,' and Chapter 5's introduction expressly admits that 'there is no model that takes these defects into account' for steps and thickness fluctuations. These are limitations of scope and evidence, not circular reductions; the paper does not use the phenomena it aims to explain as inputs to the derivations. Self-citations, if any, are not load-bearing because the central quantitative inputs are externally computed or measured, and no uniqueness theorem is imported from the author's own prior work. Accordingly, the derivation chain is self-contained as a review exposition, and no circular step is identified.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The ledger records the material constants and modeling assumptions that the lecture notes import from prior literature. Most of the physics is built on standard theorems (Mermin-Wagner, Onsager, Landau) and on domain approximations (Stoner, RKKY, slab model, classical spins). No new particles, forces, or conserved quantities are introduced. The paper's contribution is pedagogical synthesis, not a new parameterization.

free parameters (6)
  • Interatomic exchange coupling J = ~46 meV for bcc Fe (Ref. [75])
    Input to the exchange functional and spin wave stiffness; taken from prior spin wave stiffness calculation, not derived in this review.
  • Néel anisotropy constant lambda = ~0.38 meV per surface unit cell for Fe (Ref. [24])
    Sets the perpendicular anisotropy term; imported from first-principles calculation by Gay and Richter.
  • Dipolar coupling constant Omega = ~0.28 meV per bulk unit cell for Fe
    Derived in the text from M0 and lattice constant; value depends on magnetization magnitude.
  • Spin wave stiffness D / exchange stiffness A = D ~350 meV·Å^2 for Fe (Ref. [75])
    Connects J and S; quoted from literature and used in the spin wave dispersion.
  • Spin length S = ~1.1 for bulk Fe (Ref. [75])
    Classical spin magnitude; input from band structure results.
  • Film thickness d = 1 to 3 monolayers
    Key control variable in the slab model; not computed from first principles.
assumptions (7)
  • standard math Mermin-Wagner theorem: continuous symmetries cannot break spontaneously in two dimensions at finite temperature
    Invoked in Chapter 6 to establish absence of magnetic order in isotropic 2D Heisenberg model.
  • standard math Onsager's exact solution of the 2D Ising model
    Used as the reference universality class for the observed phase transition.
  • standard math Landau theory of phase transitions and the Maxwell construction (Lebowitz-Penrose)
    Underlies the definition of spontaneous magnetization in Chapter 1.
  • domain assumption Stoner mean-field model of itinerant ferromagnetism
    Used in Chapter 3 to explain moment formation, dead layers and enhanced moments.
  • domain assumption RKKY coupling mediated by free-electron spin polarization
    Used in Chapter 4 for interlayer exchange coupling; the delta-function model is an approximation.
  • domain assumption Slab model with rigid spin distribution along the film normal
    Reduces the film to a 2D system; the text notes it holds only for small thickness and ignores steps and defects.
  • domain assumption Classical vector spin representation
    Used for the Heisenberg Hamiltonian; strictly justified only for small deviations from the ferromagnetic state.

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Cite this review

Pith. "Pith review of Lectures on ultrathin film ferromagnetism." pith.science (2026). https://pith.science/paper/SDXMJGBS

@misc{pith2026260812189,
  author       = {Pith},
  title        = {Pith review of: Lectures on ultrathin film ferromagnetism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SDXMJGBS}},
  note         = {Machine review of arXiv:2608.12189}
}
read the original abstract

In these Lecture Notes we review some of the fundamental principles that have emerged from research on the ferromagnetism of ultrathin films consisting of 3d transition-metal overlayers. Their growth is often layer-by-layer. This growth mode produces quantum wells along the vertical direction that profoundly impact any physical property of the materials. In addition, the vertical confinement establishes spin ensembles that extend to macroscopic distances along the in-plane directions and are finite along the vertical (perpendicular) direction, i.e. they are two-dimensional. Accordingly, they display ground state properties that originate from the two-dimensionality, such as ``dead'' magnetic layers or ``enhanced magnetic moments'', an oscillatory interlayer magnetic coupling and an anomalous perpendicular versus in-plane magnetic anisotropy that produces, in some specific situations, a perpendicular collective orientation of the spins. At finite temperatures, ferromagnetic order is observed to persist and an analysis of the magnetic order of ultrathin films in terms of the renormalization group provides a suitable framework for explaining this observation. The ferromagnetic order is lost at a phase transition which follows closely the two-dimensional Ising universality class, as shown by an accurate analysis of data in the vicinity of the critical point. The perpendicular spin orientation is often observed to turn in-plane by a reorientation phase transition which is also properly described by a renormalization group argument. Finally, the perpendicular spin orientation introduces topological excitations of the ferromagnetic order, consisting of stripes of reversed perpendicular spin direction. The stripe order undergoes a phase transition to the paramagnetic state that is not yet completely understood.

Figures

Figures reproduced from arXiv: 2608.12189 by the authors.

Figure 1.1
Figure 1.1. (a) Sketch of the Helmholtz free energy as a function of M. The thick continuous curve is the equilibrium free energy: it has a flat portion joining the two values ±Ms of the spontaneous magnetization (path a), see Ref. [9]. Within Landau theory the interval [−Ms , +Ms ] admits, in addition, the higher lying dashed curve (path b), along which a barrier separates the two minima. (b) Sketch of the graph of M as a func… view at source ↗
Figure 1.2
Figure 1.2. A state of the two-dimensional Ising model, of the type used by Peierls in his proof of the existence of a spontaneous magnetization[14]: within a sea of aligned spins (black), droplets of overturned spins (red) are enclosed by closed boundaries. Along each unit of boundary length two antiparallel spins meet, so that a droplet costs an exchange energy proportional to the length of its boundary. (a) and (b)) is, howe… view at source ↗
Figure 2.1
Figure 2.1. View from above of a section of a 2-3 ML thick Fe stripe (dark) on W(110) (brighter), imaged by Scanning Electron Microscopy. In this specific exper￾iment, the primary SEM-beam has an energy of about 10 keV. When the electron beam hits the surface, it produces a set of scattered electrons. The scattering cross section is material dependent, producing a contrast between the intensity of the electrons emitted from Fe … view at source ↗
Figures from the paper (29 more)
Figure 2.2
Figure 2.2. Figure 2.2: Images of a section of the boundary between Fe and W. The nominal thick￾ness of the Fe film increases from left to right and from the top image to the bottom image, from the submonolayer range on the left-hand side of the top image to ≈ 2 ML on the right-hand side of…
Figure 2
Figure 2. Figure 2: , where the actual Fe film starts, small, one ML-thick Fe elements are visible, [PITH_FULL_IMAGE:figures/full_fig_p015_2.png]
Figure 2.3
Figure 2.3. Figure 2.3: Left: The free electron band along the ∆ direction in Cu. The confinement produces quantum well states (black dots). Their position changes (open dots) when the thickness of the Cu-overlayer is varied; the arrows indicate the direction in which the states move as the…
Figure 3.1
Figure 3.1. Figure 3.1: Schematic energy diagram of an atom, resulting from single electron numer￾ical computations. The (n, l)-shell, resulting from the solution of the eigenvalue problem of the single￾electron configuration Hamiltonian, is (2l + 1) · 2-times essentially degenerate. The (2…
Figure 3
Figure 3. Figure 3: summarizes, schematically, the typical sequence of energy levels that develop in [PITH_FULL_IMAGE:figures/full_fig_p026_3.png]
Figure 3.2
Figure 3.2. Figure 3.2: Schematic energy diagram summarizing the atomic levels: the hierarchy of interactions, from the configuration Hamiltonian on the left to the Zeeman sublevels on the right, each interaction lifting part of the degeneracy left by the previous one and on a smaller energ…
Figure 3.3
Figure 3.3. Figure 3.3: Average magnetic moment µz (in units of µB) per ion as a function of Bz/T (in units of 10−1 Tesla/K) for the three paramagnetic ions Gd3+ (caged into a salt GdSO4 · 8H2O), F e3+ (caged into NH4F e(SO4)2 · 12H2O) and Cr3+ (caged into KCr(SO4)2 · 12H2O). Adapted from R…
Figure 3.4
Figure 3.4. Figure 3.4: a: Selected bands of ferromagnetic Fe at the Γ point and along the ∆ di￾rection, drawn from [73]. b: Intensity of energy-analyzed photoemitted electrons, excited along the normal to the (100)-surface at a photon energy of 60 eV, as a function of their energy with res…
Figure 3.5
Figure 3.5. Figure 3.5: a: Spin resolved density of states for a Ni impurity in Ag. (b): Spin re￾solved density of states for a Fe impurity in Ag. (c): Computed magnetic moment per atom (circles) and exchange splitting (asterisks) for impurities of the 3d transition metals (horizontal scale…
Figure 3.6
Figure 3.6. Figure 3.6: Schematic energy diagram summarizing the triplet-singlet splitting. The sign of the intra-atomic exchange integral is positive so that the triplet has a lower energy than the singlet (the positivity of J is the first Hund rule). There is an intuitive explanation for …
Figure 3
Figure 3. Figure 3: illustrates a back-of-the-envelope model of the physical principles spelled out by [PITH_FULL_IMAGE:figures/full_fig_p043_3.png]
Figure 3.7
Figure 3.7. Figure 3.7: illustrates a back-of-the-envelope model of the physical principles spelled out by Friedel. In (a) the solid is illustrated by an energy diagram. The horizontal direction is some coordinate across the solid. The solid terminates at a boundary at which the potential, …
Figure 4.1
Figure 4.1. Figure 4.1: (a): Schematic stacked structure used for coupling experiments. The grey rectangles are the magnetically active films, in the specific case Co thin films[92]. The spacer is made of Cu(100). The black arrows indicate the orientation of the magnetization vector. (b): T…
Figure 4.2
Figure 4.2. Figure 4.2: Scheme of the energy levels for a d 1 -electron configuration (a) under octahe￾dral symmetry (b), tetragonal elongation along the axis z (c) and rectangu￾lar distortion in the xy-plane (d). The distortions away from the octahedral geometry are driven by the Jahn-Tell…
Figure 4.3
Figure 4.3. Figure 4.3: Plot of the graph of ρ(kF · x) at kF = π and κ kF = 0.2. Some particular limits are worked out now. For small x we have (E1(z) ≈ −γ − ln z) ρ(x) ≈ kF π + κ π [PITH_FULL_IMAGE:figures/full_fig_p063_4_3.png]
Figure 5.1
Figure 5.1. Figure 5.1: Spin wave frequency as a function of the applied magnetic field for an ul￾trathin film with perpendicular spontaneous magnetization, drawn from Eqs. (5.38), (5.41) and (5.52) with K = λ a d − Ω. Continuous line: field applied along the normal, Eq. (5.38); its interce…
Figure 7.1
Figure 7.1. Figure 7.1: Sketch of the geometrical issues in the Polyakov RG of the elastic Heisenberg Lagrangian. We expect that, for a given ν, the set ⃗ea can be chosen so that ∂ν⃗ea = −cνa · ⃗n0 (7.A9) The same set, however, cannot fulfill this condition for a different ν. In general, th…
Figure 8.1
Figure 8.1. Figure 8.1: Sketch of the transition lines in the temperature (T)-thickness (d) parameter plane, as expected from experimental[135] results and theoretical works[136, 108]. In this Chapter, we apply the RG protocol to an ultrathin slab with the aim of computing the approximate t…
Figure 9.1
Figure 9.1. Figure 9.1: Sketch of g(r, b) for |r| sufficiently far from the critical point (the empty interval |r| < 1 in the figure). Computed with u = 1 and µ = 1; g, r, b are then in units of u. The cusp along b = 0 for r < 0 is the signature of the spontaneous magnetization: ∂g ∂b jumps…
Figure 9.2
Figure 9.2. Figure 9.2: The scaling function in Griffiths’ variables, as Landau theory predicts it: the straight line h = x + u, here with u = 1. (a) linear; (b) double logarithmic, where its large-x slope, which is the critical exponent γ, is seen to be 1. The dashed line of slope 7 4 is w…
Figure 9.3
Figure 9.3. Figure 9.3: The scaling function in the variables suggested by Milošević and Stanley, as Landau theory predicts it (µ = u = 1). The measured counterpart of this curve is [PITH_FULL_IMAGE:figures/full_fig_p131_9_3.png]
Figure 9.4
Figure 9.4. Figure 9.4: Experimental m(r) data at different applied magnetic fields, r = T −Tc Tc , m in arbitrary units, the field in Gauss given by the colour code of the legend. The black points are the remanence, i.e. B = 0; they are the only ones that vanish identically above the trans…
Figure 9.5
Figure 9.5. Figure 9.5: Collapsing of the experimental data of Fig.9.4 in the variables of Milošević and Stanley, m B1/δ against r B1/(βδ) , with the 2D Ising values β = 1 8 and δ = 15. The continuous line is the theoretical curve; the data are the eleven fields of Fig.9.4, not connected. M…
Figure 9.6
Figure 9.6. Figure 9.6: Collapsing of the experimental data of Fig.9.4 in the canonical variables proposed by Griffiths[142]: B Mδ against r M1/β , with β = 1 8 , δ = 15. Main panel: double logarithmic, branch x > 0; the collapse holds over twelve decades and the measured slope is 1.7500, i…
Figure 10.1
Figure 10.1. Figure 10.1: a. The graph corresponding to the term S⃗k1 · S⃗k2 · S⃗k3 · S⃗k4 · δ⃗k1+⃗k2+⃗k3+⃗k4,0 has only external lines. The magnitude of the vertex is u 4 · 1 · 1. b. The graph corresponding to the term S⃗k1 · S⃗k2 · D σ⃗q1 · σ⃗q2 E · δ⃗k1+⃗k2+⃗q1+⃗q2,0 has two external line…
Figure 10.2
Figure 10.2. Figure 10.2: Some second order graphs. For a comment see the bulk of the section. The graph of the type in ”a” on the left is designated as being disconnected: the two vertices are not ”connected” by any lines. The graphs in ”b” and ”c” are connected and topologically equivalent…
Figure 10.3
Figure 10.3. Figure 10.3: Second order connected graphs, their magnitude and strength estimate. we summarize the second order connected diagrams, their magnitude and make a statement about whether they are allowed or not with respect to momentum conservation (Kronecker-delta). On the left of…
Figure 11.1
Figure 11.1. Figure 11.1: The model used for the computation of the topological instability. a: the film is filled with ”up” spins. b: a wall along x separates domains with opposite perpendicular spin polarization. In Fig.11.1a, this state is rendered with a white color and the spin vector i…
Figure 11.2
Figure 11.2. Figure 11.2: Sketch of the stripe structure computed in this section. The profile runs along x, the stripes along y; 2L is the period, W the width of the minority stripe, d the thickness of the slab. In zero applied field W = L. The details of the computation of the total magnet…

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Pith tools

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