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arxiv: 1907.07696 · v1 · pith:RYOIETD4new · submitted 2019-07-17 · ❄️ cond-mat.str-el · cond-mat.mes-hall· cond-mat.mtrl-sci· hep-th· quant-ph

Topical Review on Skyrmions and Hall Transport

Bom Soo Kim This is my paper

Pith reviewed 2026-05-24 20:02 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hallcond-mat.mtrl-scihep-thquant-ph
keywords SkyrmionsHall viscosityWard identityparity symmetryHall transportspin torquethermo-electromagnetic effects
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0 comments X

The pith

Broken parity symmetry in Skyrmion systems requires Hall viscosity as an antisymmetric viscosity component.

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

The review applies a quantum field theory Ward identity to Skyrmion systems and shows that broken parity symmetry forces the existence of Hall viscosity. This new transport coefficient is an antisymmetric part of the viscosity tensor and appears alongside the familiar electric and thermal Hall conductivities. The same symmetry argument supplies a concrete experimental signature: the momentum dependence of the Hall conductivity should reveal the viscosity term. The paper also surveys spin torques and thermo-electromagnetic effects in both conducting and insulating magnets, providing background to place the Ward-identity result in context.

Core claim

Broken parity symmetry in the effective field theory of Skyrmions implies, via the quantum field theory Ward identity, the existence of Hall viscosity as an antisymmetric component of the viscosity tensor; this term is measurable through the momentum dependence of the Hall conductivity.

What carries the argument

Quantum field theory Ward identity applied to Skyrmion effective theories with broken parity symmetry.

If this is right

  • Hall viscosity must accompany electric and thermal Hall conductivities in any parity-broken Skyrmion system.
  • Momentum dependence of Hall conductivity provides a direct experimental probe of Hall viscosity.
  • The Ward-identity result holds for both insulating and conducting magnets.
  • Spin-torque and thermo-electromagnetic descriptions remain compatible with the symmetry-derived viscosity term.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Ward-identity logic may apply to other topological textures that break parity, such as merons or domain walls.
  • Momentum-resolved transport experiments on thin-film Skyrmion lattices could isolate the viscosity coefficient.
  • If Hall viscosity is confirmed, it supplies an additional hydrodynamic variable for modeling Skyrmion dynamics at long wavelengths.

Load-bearing premise

The Ward identity analysis assumes that the symmetries and conservation laws of the effective field theory directly produce a new independent transport coefficient without cancellations from microscopic dynamics.

What would settle it

A momentum-resolved Hall conductivity measurement that exhibits no antisymmetric viscosity contribution, or a microscopic Skyrmion calculation that cancels the predicted term.

Figures

Figures reproduced from arXiv: 1907.07696 by Bom Soo Kim.

Figure 1
Figure 1. Figure 1: (a): Top left inset. Skyrmions crystal in the A-phase of the MnSi discovered in neutron scattering experiment [1]. (b): Bottom left inset. Illustration of the Skyrmion crystals revealed in a thin film of Fe0.5Co0.5Si using Lorentz transmission electron microscopy [2]. (c): Right. Rotation motion of Skyrmion crystals under the temperature gradient along with the current [3]. Reproduced with permission from … view at source ↗
Figure 2
Figure 2. Figure 2: (a): The direction of the shear viscosity, the black arrows, acts against that of the cylinder. (b): The Hall Viscosity acts perpendicular to the motion of the middle cylinder. Despite its name, it does not produce dissipation. Reproduced with permission from [37]. We offer a geometric description of the Hall viscosity in a more realistic setting. Let us imagine a finite size cylinder surrounded by a fluid… view at source ↗
Figure 3
Figure 3. Figure 3: Spontaneously generated angular momentum l(~x) = `/2 inside a square with length b. l(~x) = 0 outside. There is a momentum current going around along the boundary. Can we detect the angular momentum as the boundary is pushed to the infinity? The solution is independent of the choice of l(~x). Because the physical properties are independent of the details, we can choose l(~x) such that the computation becom… view at source ↗
Figure 4
Figure 4. Figure 4: Two in-equivalent area preserving shear transformations in 2 spatial dimensions. (a): Elongation of the square along the horizontal direction. (b): Elongation along the diagonal direction of the square. (c): A combination of these two shear transformations produces a rotation. Thus a certain combination of shear transformations can generate a rotation [17]. This already suggests that viscosities can be rel… view at source ↗
Figure 5
Figure 5. Figure 5: Skyrmion structures with varying m and γ. The arrows indicate the direction of the in-plane spin component, and the brightness indicates the normal component to the plane, with white denoting the up direction and black the down direction. All the structures of the anti-skyrmions (m = −1) are equivalent on rotation in the xy plane. Reproduced with permission from [49]. The energy momentum tensor can be comp… view at source ↗
Figure 6
Figure 6. Figure 6: Confirmation of the existence of the Hall viscosity. Intercept of thermal Hall conductivity κ (0) e is non-zero and is proportional to the Skyrmion charge density. Non-vanishing slope κ (2) e as a function of momentum squared q 2 confirms the existence of the Hall viscosity. 3.6. Topological WI with magnetic fields and electric currents The topological Ward identities (46) in abstract form and (51) with tr… view at source ↗
Figure 7
Figure 7. Figure 7: Confirmation of the existence of the Hall viscosity in the conducting magnetic materials. Intercept of electric Hall conductivity σ (0) e is non-zero and is proportional to the Skyrmion charge density. Non-vanishing slope σ (2) e as a function of momentum squared q 2 confirms the existence of the Hall viscosity. Let us consider the opposite limit Bb/ω → 0 after dividing equation (62) by the second equation… view at source ↗
Figure 8
Figure 8. Figure 8: (a) A simple classical model of an atom provides an intuitive picture for gyromagnetic ratio. (b) Spin motion due to the precession and damping terms given in the LLG equation (70). The definition of a torque for the angular momentum~L, and in tun for the dipole moment ~µ, is given by ~T = ∂~L ∂t = 1 γ0 ∂~µ ∂t . Thus, for a macroscopic volume that contains many atoms, the torque acting on a local magnetiza… view at source ↗
Figure 9
Figure 9. Figure 9: Illustration of the Domain Wall [(a): (left-handed) Neel DW and (b): Bloch DW] motion due to the effective field H~ e f f pointing out of the plane. Regardless of the DW structure, the domain with the magnetization pointing out of the plane expands due to the precession and damping terms in (70). This is explained by the damping term (represented by the red circles) and the associated precessional term (re… view at source ↗
Figure 10
Figure 10. Figure 10: Domain Wall (DW) motion due to the spin transfer torque from the spin-spin interaction with conduction electrons. The coefficient of the non-adiabatic term is positive. The net contribution of the spin transfer torque is along the direction of current ~J, opposite to the direction of electron flow. For the spin transfer torque, there is no difference between the Neel and Bloch DWs. The last two terms in (… view at source ↗
Figure 11
Figure 11. Figure 11: (a): Illustration of the Rashba interaction. Reproduced with permission from [82]. (b): The effect of effective field H~ sd on the DW structure. The field H~ sd does not generate a DW motion. 4.2. Spin-orbit torque Recently, a different way to produce non-equilibrium polarization of the conduction electron spins has been introduced using the spin-orbit coupling in materials with lacking inversion symmetry… view at source ↗
Figure 12
Figure 12. Figure 12: (a): Top. Left-handed DW motion under the influence of the spin-orbit torque due to the spin-orbit interaction with the conduction electron. The effective field for the field-like spin-orbit torque (SOT) is H~ FL SOT = M~ × H~ sd, which pushes DW against the direction of electron motion, while there is no DW motion in the presence of damping-like SOT. (b): Middle. Bloch type DW is not influenced by the SO… view at source ↗
Figure 13
Figure 13. Figure 13: Analogy between the charge Hall effect (a) and the spin Hall effect (b). The charge Hall effects in the presence of magnetic field are from the Lorentz force law and independent of the spin polarization. The spin Hall effects depend on the spins of the electrons without magnetic field. Spin polarization of an electron along zˆ behaves similar to the electron with magnetic field in the same direction. Thus… view at source ↗
Figure 14
Figure 14. Figure 14: A metal such as Pt is overlaid to provide a pathway for the conduction electrons to move to reduce the spin imbalance. Inverse spin Hall effect that results in measurable electric potential. In the literature the notion of the electromotive force ~ESHE is used to explain the spin Hall effects. Due to the motion of an electron the following electromotive force ~ESHE is generated depending on its polarizati… view at source ↗
Figure 15
Figure 15. Figure 15: The middle conducting ferromagnetic layer, with up direction magnetization, serves as a spin polarizer by the spin transport of the majority spins, up spins with arrow head. Accumulation of the minority spins, down spins with arrow tail, on the left side of the ferromagnetic layer and that of the majority spins on the right side. torques using 5 different layers that are composed of three normal metal lay… view at source ↗
Figure 16
Figure 16. Figure 16: DW motion under the influence of the spin Hall torque (85). The current ~JHM flows in the other layer (below the ferromagnet layer) and the blue arrows are the direction of zˆ ×~JHM. The red circles are the direction of H~ SL, which is similar to H~ e f f considered before, yet crucially different due to its dependence on the magnetization direction. (a): Neel DW. The left and middle domain expands to the… view at source ↗
Figure 17
Figure 17. Figure 17: Seebeck effect. Electric field is generated by a temperature gradient in (a). Illustration of the thermocouple in (b). Two different materials A and B with different thermoelectric powers, eA and eB are connected at two different points with different temperatures and generate different electric potential difference. The potential difference can be measured by the voltmeter V. In general, Seebeck effect i… view at source ↗
Figure 18
Figure 18. Figure 18: Peltier effect is generated when two conducting materials are connected with isothermal junction with electric current. The Peltier effect is inverse of the Seebeck effect. Temperature gradient can be generated by the electric field due to the accumulation of the charge carriers. This happens because charge carriers are also heat carriers. In particular, the Peltier effect is the evolution of heat accompa… view at source ↗
Figure 19
Figure 19. Figure 19: Thompson effect. (a): heat current without electric current. (b): heat and electric currents. The Thompson effect concerns the heat absorbed per unit electric current and per unit temperature gradient. Let us consider a setup more clear [92]. Consider the figure 19 (a) that there is a heat current without electric current due to the temperature difference dT = δT at two ends A and B of a conducting materi… view at source ↗
Figure 20
Figure 20. Figure 20: The first Kelvin relation among Seebeck, Peltier, and Thompson effects. There exist an interesting relation between the three coefficients, Seebeck, Peltier and Thompson coefficients. By taking a derivative of the Peltier coefficient, we get dπAB dT + τA − τB = eA − eB. This can be demonstrated by the thermocouple with the Voltmeter replaced by a battery that negates the Seebeck voltage so that there is n… view at source ↗
Figure 21
Figure 21. Figure 21: Setup for the thermo-electromagnetic transport coefficients in the presence of the Magnetic Field H~ as well as the charge ~Je = e~JN and heat Q~ = ~JQ currents [PITH_FULL_IMAGE:figures/full_fig_p044_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Illustration of the Nernet effect. Temperature gradient produce the electric field and current when connected to a closed circuit. In the presence of magnetic field, there is a transverse electric field and the corresponding transverse current when connected to a circuit. This can be understood as the Seebeck effect in the presence of magnetic field. We can think about the inverse of the Nernst effect, wh… view at source ↗
Figure 23
Figure 23. Figure 23: Spin Seebeck effect. (a): Electrons with spin up (red arrow head) and spin down (blue arrow tail) components have different Seebeck coefficients in a metallic magnet under a temperature gradient. (b): Illustration of the calculated distributions of electrochemical potentials for spin-up and spin-down electrons. Reproduced from [95]. Similar to the Seebeck effect that two different conducting materials hav… view at source ↗
Figure 24
Figure 24. Figure 24: Measurement of the spin Seebeck effect using inverse spin Hall effect. (a): The Ni81Fe19 film has Pt wires attached to the ends of the film. (b): Demonstration of spin Seebeck effect by the electromotive forces in the Pt wires. Reproduced from [95]. in-film electromotive force (83) due to the spin Hall effect vanishes because the current ~JS is parallel to the spin polarization. This is demonstrated by th… view at source ↗
Figure 25
Figure 25. Figure 25: Illustration of the spin Peltier effect. A spin current is push through a nonmagnetic metal and ferromagnetic metal interface. The Peltier heat current vanishes In the nonmagnetic metal while it is non-zero in the ferromagnetic metal because the Peltier coefficients are different for the majority and minority spins. better conductivity for the majority spin (up-spin depicted as a red arrow head). Thus the… view at source ↗
Figure 26
Figure 26. Figure 26: Excitations of ferromagnet. (a): Top configuration representing the ground state of a magnet. (b): Middle spin configuration with a spin flip that represents a high energy excitation. (c): Bottom spins with a spin wave or quantized Magnon that has a lower energy than (b). The spins are precessing around their equilibrium positions. Magnons are low energy excited states in magnetic materials, and can be th… view at source ↗
Figure 27
Figure 27. Figure 27: Two different spin currents. (a): Conduction electron spin current that is carried by the electron diffusion. (b): Spin wave spin current that is carried by collective magnetic-moment precession. (c): Spin pumping of the spin wave spin current generates electric potential through the inverse spin Hall effect. Reproduced with permission [100]. ferromagnetic materials. The spin wave spin current (Magnon cur… view at source ↗
Figure 28
Figure 28. Figure 28: Thermal Hall conductivity κxy measurement of the Lu2V2O7 at the low temperature insulating phase. Reproduced with permission [101] [PITH_FULL_IMAGE:figures/full_fig_p052_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: A schematic figure that explains basic elements of the Skyrmions motion under the influence of electric currents. Reproduced with permission [49]. 6.1. Topological Hall effect Mobile spins interact with the localized spins and vice versa. The extended spin configuration of the Skyrmions trades non-trivial physical effects with the conduction electrons through a coupling with the spins of the conduction el… view at source ↗
Figure 30
Figure 30. Figure 30: Experimental data for the topological Hall effects. A box function-like contribution in the left figure exists only in the Skyrmion phase (a). This extra contribution to the electric resistivity confirms the existence of the emergent magnetic field~b and thus the existence of the Skyrmions. (b) shows more systematic results of the box function-like contribution to the resistivity from the~b. The ranges of… view at source ↗
Figure 31
Figure 31. Figure 31: (a): An experimental verification of the Skyrmion Hall effect through the accumulation of the Skyrmions in one side of the film. c – d illustrate the Hall effect for holes with a unit electronic charge of +e, that accumulate at the opposite edges of a device upon reversal of the magnetic field directions. e – f illustrate the skyrmion Hall effect, for which the reversal of the magnetic field direction rev… view at source ↗
Figure 32
Figure 32. Figure 32: Numerical studies for the Skyrmion motion in the insulating magnets. (a): Skyrmions move from the cold to the hot region, while Magnons diffuse from the hot to the cold region. The Skyrmions exhibit the transverse motion in addition to the longitudinal motion. Reproduced with permission from [106]. (b): An additional non-magnetic layer, such as Pt, can be used to detect the spin pumping resulting from Sky… view at source ↗
Figure 33
Figure 33. Figure 33: (a): Observed Lorentz Transmission electron microscopy image of the Skyrmion rotation motion in the chiral-lattice magnets MNSi. Skyrmions show clockwise rotation under the temperature gradient along with the magnetic field pointing into the page. (b): Simulation of the thermally driven rotational motion of the Skyrmion microcrystal. Reproduced with permission from [60]. where the magnetization ~n is sepa… view at source ↗
Figure 34
Figure 34. Figure 34: Illustration of possible measurement of the Hall viscosity of Skyrmion system in contact with a rotating cylinder, with neither electric current nor temperature gradient. (a): Initial setup. (b): the rotation of the cylinder contacting with the ferromagnet will produce the radially outward motion of Skyrmions. This produce the spin imbalance that can be measured by an overlaid non-magnet metal through inv… view at source ↗
read the original abstract

We review recent progresses towards an understanding of the Skyrmion Hall transport in insulating as well as conducting materials. First, we consider a theoretical breakthrough based on the quantum field theory Ward identity, a first principle analysis, relying on symmetries and conservation laws. Broken parity (inversion) symmetry plays a crucial role in Skyrmion Hall transport. In addition to the well known thermal and electric Hall conductivities, our analysis has led us to the discovery of a new and unforeseen physical quantity, Hall viscosity - an anti-symmetric part of the viscosity tensor. We propose a simple way to confirm the existence of Hall viscosity in the measurements of Hall conductivity as a function of momentum. We provide various background materials to assist the readers to understand the quantum field theory Ward identity. In the second part, we review recent theoretical and experimental advancements of the Skyrmion Hall effects and the topological (Magnon) Hall effects for conducting (insulting) magnets. For this purpose, we consider two enveloping themes: spin torque and thermo-electromagnetic effect. First, we overview various spin torques, such as spin transfer torque, spin-orbit torque, and spin Hall torque, and generalized Landau-Lifshitz-Gilbert equations and Thiele equations using a phenomenological approach. Second, we consider irreversible thermodynamics to survey possible thermo-electromagnetic effects, such as Seebeck, Peltier and Thompson effects in the presence of the electric currents, along with the Hall effects in the presence of a background magnetic field. Recently developed spin Seebeck effects are also a significant part of the survey. We also accommodate extensive background materials to make this review self-contained. Finally, we revisit the Skyrmion Hall transport from the Ward identity view point.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. This topical review covers Skyrmion Hall transport in insulating and conducting magnets. The central claim is that broken parity symmetry, analyzed via quantum field theory Ward identities relying on symmetries and conservation laws, implies the existence of a new independent transport coefficient—Hall viscosity, the antisymmetric component of the viscosity tensor—in addition to known thermal and electric Hall conductivities; this is proposed to be measurable via momentum dependence of the Hall conductivity. The second part surveys spin torques (including spin-transfer, spin-orbit, and spin-Hall torques), generalized LLG and Thiele equations, thermo-electromagnetic effects (Seebeck, Peltier, spin Seebeck), and revisits Skyrmion Hall transport, with extensive background material provided throughout.

Significance. If the Ward-identity derivation rigorously establishes an independent Hall viscosity without absorption into existing coefficients or cancellation by Skyrmion-specific dynamics, the result would add a falsifiable, first-principles transport coefficient to the literature on parity-broken topological magnets. The review's self-contained background sections on Ward identities, spin torques, and irreversible thermodynamics are a clear strength for accessibility.

major comments (2)
  1. [Ward identity / theoretical breakthrough section] Section on the Ward-identity analysis (theoretical breakthrough part): the claim that broken parity plus conservation laws directly produces an independent Hall viscosity does not address whether Skyrmion-specific features (topological charge, spin texture, or lattice effects) could induce cancellations that force the antisymmetric viscosity coefficient to zero or absorb it into the symmetric viscosity or Hall-conductivity terms already present in the EFT.
  2. [Measurement proposal paragraph] Proposal for experimental confirmation via momentum-dependent Hall conductivity: the manuscript does not supply the explicit relation (or error estimate) connecting the Hall-viscosity coefficient to the measurable momentum dependence, leaving open whether the predicted signature is distinguishable from ordinary dispersive corrections already present in the conductivity tensor.
minor comments (2)
  1. [Abstract] Abstract: 'insulting magnets' is a typographical error and should read 'insulating magnets'.
  2. [Theoretical background] Notation for the viscosity tensor and its antisymmetric component is introduced without an explicit equation defining the decomposition; adding this would improve clarity for readers unfamiliar with the Hall-viscosity literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report on our topical review. The comments highlight important points for clarification regarding the generality of the Ward-identity derivation and the experimental proposal. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: Section on the Ward-identity analysis (theoretical breakthrough part): the claim that broken parity plus conservation laws directly produces an independent Hall viscosity does not address whether Skyrmion-specific features (topological charge, spin texture, or lattice effects) could induce cancellations that force the antisymmetric viscosity coefficient to zero or absorb it into the symmetric viscosity or Hall-conductivity terms already present in the EFT.

    Authors: The Ward identities follow directly from the symmetries and conservation laws assumed in the effective theory for parity-broken magnets; these are general and independent of the specific microscopic realization. Skyrmion topological charge and spin texture enter the EFT parameters but do not modify the tensor structure of the allowed transport coefficients. The antisymmetric viscosity coefficient remains independent under the stated assumptions and is not absorbed into symmetric viscosity or Hall conductivity terms. To make this explicit, we will add a clarifying paragraph in the revised manuscript discussing why Skyrmion-specific features do not induce the cancellations suggested. revision: yes

  2. Referee: Proposal for experimental confirmation via momentum-dependent Hall conductivity: the manuscript does not supply the explicit relation (or error estimate) connecting the Hall-viscosity coefficient to the measurable momentum dependence, leaving open whether the predicted signature is distinguishable from ordinary dispersive corrections already present in the conductivity tensor.

    Authors: We agree that an explicit relation would improve the proposal. In the revision we will supply the leading-order relation between the Hall-viscosity coefficient and the momentum dependence of the Hall conductivity, derived from the hydrodynamic expansion, together with a brief discussion of the momentum scales at which the Hall-viscosity signature can be distinguished from ordinary dispersive corrections, including a rough error estimate based on typical material parameters. revision: yes

Circularity Check

0 steps flagged

Ward identity derivation of Hall viscosity stands as independent first-principles result

full rationale

The paper frames Hall viscosity as a new quantity emerging directly from broken parity symmetry via quantum field theory Ward identities applied to the Skyrmion effective theory, relying on standard conservation laws and symmetries rather than any fitted parameters, self-defined relations, or load-bearing self-citations that reduce the claim to its own inputs. No equations or steps in the provided abstract or description exhibit a prediction that collapses by construction to a prior fit or renaming; the analysis is presented as additive to existing Hall conductivities and is supported by background QFT materials. The review structure incorporates external advancements in spin torques and thermo-electromagnetic effects without circular reduction in the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The review relies on the applicability of quantum field theory Ward identities to effective Skyrmion models and on standard conservation laws; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Quantum field theory Ward identities apply directly to the effective description of Skyrmion transport
    Invoked for the first-principle analysis based on symmetries and conservation laws.
invented entities (1)
  • Hall viscosity no independent evidence
    purpose: Antisymmetric component of the viscosity tensor arising from broken parity
    Presented as a newly identified physical quantity from the Ward-identity analysis.

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