Phonon magnetic moment: a worked derivation

This is a reading note on “Phonon Magnetic Moment from Electronic Topological Magnetization” by Yafei Ren, Cong Xiao, Daniyar Saparov, and Qian Niu (2021) Ref. 1. The explanations and expanded derivations below accompany that paper.

The central idea is simple: a closed ionic orbit can pump an electronic loop current. The difficult part is separating that loop current from charge that merely moves back and forth. This guide does that algebra explicitly, then connects each term to a physical picture.

This note works through the connection between the paper’s local-current formula and its electronic magnetization, with the assumptions and intermediate algebra made explicit. Equation numbers such as Eq. (1) and Eq. (3) refer to the original article Ref. 1.

What are we trying to calculate?

We want the cycle-averaged electronic orbital magnetization caused by a prescribed phonon. This is not yet the total magnetic response, and it is not a derivation of the phonon’s dynamics.

The destination: Eq. (3), with the electronic integral abbreviated

There are two independent ingredients. The factor describes the ionic motion: its size, frequency, and handedness. The coefficient describes how the equilibrium electronic states respond to that motion. A large electronic coefficient cannot produce this contribution without an oriented phonon orbit.

Assumptions we will actually use

Adiabatic: electrons remain in the occupied subspace as the ion moves. A useful necessary scale separation is , although matrix elements and the driving amplitude also matter.

Small displacement: the electronic geometry changes little over the orbit. Near a small gap, “small compared with the lattice spacing” is not sufficient; the phonon-induced Hamiltonian change must also be small compared with the relevant gap.

Slow spatial variation: temporarily let . This lets us identify a curl current. The equilibrium crystal will be taken homogeneous.

Conventions and scope

; an electron has charge . We use . A counterclockwise orbit in the plane has positive ionic angular momentum.

For readability, the derivation uses a single isolated occupied band and Abelian curvature. Several occupied bands require the non-Abelian, occupied-subspace trace used later in the paper.

The semiclassical local-current formula, Eq. (1), is our input. We derive its connection to Eq. (3); we do not rederive the full wave-packet gradient expansion underlying Eq. (1).

KEY POINT

The problem is not “why is a moving charge magnetic?” It is “which part of the electronic response is a persistent loop current, and what determines its magnitude?”

Berry curvature measures how electronic states twist

How can an ionic displacement affect an electron if we never put an explicit classical electron trajectory into the Hamiltonian?

At every momentum and ionic configuration, diagonalize the electronic Hamiltonian:

Changing changes the periodic Bloch eigenstate, not just its energy. The state is defined only up to a phase. The Berry connection keeps track of this phase choice; the curvature is the gauge-invariant phase accumulated around an infinitesimal loop.

An oriented small rectangle in parameter space; no sum in this expression

The coordinates need not both be spatial. Here the useful parameter space is . Each curvature probes a different pair of directions:

: geometry within momentum space; it enters anomalous velocity and orbital magnetization.

: mixed geometry; it connects a change in ionic displacement to electronic polarization.

: geometry within displacement space; it gives the Berry phase accumulated during the phonon loop.

Why can curvature be large near a small gap?

Differentiate the eigenvalue equation and project onto a different band:

Substituting this into the curvature gives, for the connection convention above,

Small energy denominators make the eigenstates sensitive to parameter changes. This explains possible enhancement, not a universal divergence: the matrix elements, symmetry, and the full band structure still matter.

KEY POINT

Berry curvature is a response of the eigenstates. It is not a real magnetic field in the sample. Mixed momentum–displacement curvature is the bridge between ionic motion and electronic charge motion.

Why identifying magnetization requires a spatial gradient

Why does the paper introduce , only to say that the final phonon can be spatially uniform?

Polarization current: a changing dipole density moves charge locally. If the dipoles return to their starting values after one period, their ordinary time-derivative current has zero average:

We are considering small contractible cycles, not a quantized transport pump across a polarization branch. Magnetization current: neighboring microscopic loops cancel in the interior, leaving a net boundary current where the magnetization changes.

A uniform magnetization does not require a nonzero bulk curl. In the interior, ; the boundary still carries a loop current. So we first introduce a smooth texture to expose the coefficient of a spatial derivative, identify , and only then take a uniform interior.

There is some instantaneous freedom in partitioning bound current into polarization and magnetization. We use a periodic, local polarization and fix the phonon-induced magnetization to vanish at zero orbit amplitude. This is enough to identify the cycle-averaged contribution of interest.

KEY POINT

The temporary spatial texture is a measuring device for the magnetization coefficient. Setting all spatial derivatives to zero before extracting it would hide the magnetic response.

Four indices mean an antisymmetric pairing of two planes

What does actually contain?

To distinguish ordinary two-index curvature from the paper’s four-index object, write the latter as . This is only a change of notation:

The three terms are the three ways to pair four distinct directions. Their relative signs make the result fully antisymmetric. Exchanging two indices reverses its sign; repeated indices make it vanish. In the convention , this is the corresponding component of .

Eq. (1), with the spatial-index sum made explicit

The superscript labels the semiclassical gradient order. It does not mean “the current oscillates at twice the phonon frequency.” One time gradient and one spatial gradient are involved.

For an out-of-plane magnetization, choose the orientation . Define the scalar

There is no sum over in this definition. Reversing the momentum order changes the sign. Writing the scalar this way removes the ambiguous-looking from the paper’s final formula.

KEY POINT

The second Chern form is a local geometric response density. The integral here is over two momentum coordinates at fixed displacement; it is not generally an integer-valued second Chern number.

Symmetric products are time derivatives; antisymmetric products remember an orbit

What is hidden in “by symmetrizing the summation with respect to ”?

Temporarily abbreviate the integrated coefficient in Eq. (1) by , including the spatial-index sum. Taylor-expand its explicit displacement dependence:

With sums over repeated phonon indices, the two displayed orders in the paper become

At fixed, time-independent expansion coefficients, the first term is a total derivative. The second term is split by defining

The current index is suppressed only in this algebra

Symmetric part

The equality follows by differentiating both factors and exchanging the dummy indices in one term. For a monochromatic phonon, the quadratic polarization contains a constant part and a second harmonic. Its derivative has no constant part and averages to zero.

Antisymmetric part in two dimensions

The two off-diagonal terms combine because ; the diagonal terms vanish. Define

This quantity is an oriented area rate, not the derivative of a single-valued quadratic scalar.

KEY POINT — THE STEP STILL MISSING

Antisymmetry explains why a cycle average can survive. It does not by itself prove that the surviving local current is a curl. We still have to track the spatial derivative and all product-rule terms. The next two sections do this directly from Eq. (1).

Replace the hidden real-space index by a displacement gradient

Can we write the current in ordinary derivatives of and , so that no index manipulation is left implicit?

Assume the slowly varying real-space dependence of the local electronic state comes through . At fixed , the chain rule gives

Berry curvature transforms as a two-form. Consequently,

Insert these identities into all three products defining the four-form. Each contains the same gradient factor, so

In two dimensions the only nonzero displacement pairs are and . Fully antisymmetric momentum indices likewise contribute the orientation factor . Eq. (1) reduces to

G is a short name for the remaining displacement-gradient combination

Now scale the entire phonon texture by a small amplitude parameter . Both and scale as , so is already quadratic. To obtain the leading quadratic magnetization, evaluate the electronic coefficient at equilibrium:

KEY POINT

The electronic calculation is now contained in one equilibrium coefficient. What remains is ordinary calculus on the phonon displacement field. This explicit route reproduces the leading small-amplitude result under the stated homogeneous-equilibrium assumptions.

Derive the curl, including the term that must not be dropped

We have . How do we write it as a curl plus a time derivative?

Introduce the signed temporal area rate and its spatial analogue :

Differentiate the area rate in space

The first line is exactly . Call the second line . Thus .

Differentiate the spatial analogue in time

The first line is now , while the second is the same . Therefore .

Subtract; the unwanted mixed derivatives cancel

That is the origin of the factor . It is not an extra factor inserted because the phonon is circular.

Identify polarization and magnetization

Here is independent of position and time for the homogeneous equilibrium crystal. Comparing with the constitutive equation identifies a valid local decomposition:

The second expression is the polarization contribution accompanying this gradient-order current—not the complete ordinary Born-charge polarization. Its time derivative has zero cycle average for a periodic texture.

Average over one cycle

Eq. (3), with its orientation and time averaging explicit

KEY POINT

We did not discard the derivative acting on the phonon. We kept it and identified it as polarization current. The surviving cycle average is the magnetic loop-current contribution.

How this relates to the Bianchi identity mentioned in the earlier version

For a smooth Abelian curvature, , hence . Set , , , , and . Closure gives

Integrating over the periodic Brillouin zone removes the total momentum derivatives. Reordering the remaining indices yields

This is the geometric identity behind the coefficient-based derivation. But multiplying it by a spatially varying phonon and pulling everything into a spatial derivative still requires the product rule. The worked calculation above makes that bookkeeping visible instead of hiding it.

A closed orbit can remember its handedness after time averaging

Does a current that oscillates in time really produce a nonzero average magnetic moment?

Take a monochromatic ellipse, with handedness for counterclockwise and for clockwise:

Differentiate each component and form the antisymmetric and symmetric products:

The antisymmetric product is constant for this ellipse. The symmetric product changes sign twice per cycle and averages to zero. They are not two approximations to the same motion; they extract different information from it.

The two plotted combinations and the readout are normalized by .

The signed enclosed area is . Therefore

KEY POINT

Magnetization measures oriented area swept per unit time. Flatten the ellipse and the effect vanishes; reverse the traversal and the effect changes sign. The sign of the actual magnetization also contains the electronic coefficient, so counterclockwise motion does not universally mean positive magnetization.

Mixed curvature maps an ionic ellipse into an electronic ellipse

How does the paper’s statement give exactly the first two terms of the second Chern form?

At a fixed momentum, define a two-by-two mixed-curvature matrix . This new symbol avoids confusing a matrix with the full curvature two-form:

The wave packet contributes a dipole change . Since its charge is , its center shift must satisfy , so

The second relation uses evaluated at equilibrium; varying along the orbit contributes beyond the leading order retained here. A general linear map takes a circle to an ellipse, not necessarily another circle.

Expand the cross product once

The terms proportional to and cancel. Geometrically, a determinant is the signed area-magnification factor of a linear map.

Use the moment of a charge circulating around a loop

These are precisely the first two terms, including their relative signs and the electron-charge sign. The equality is between magnetic moments after multiplying the curvature combination by , not between a magnetic moment and a bare curvature.

KEY POINT

The first two terms are a signed-area effect of the electronic center-of-mass orbit. The negative electron charge contributes an additional minus sign. Setting the determinant to zero collapses this mapped orbit and kills this contribution, even if individual mixed curvatures remain nonzero.

Average the products—not the charges first

If mixed Berry curvature gives the Born effective charge, why does the usual Born-charge estimate miss the enhancement?

In the paper’s normalization, the electronic polarization response is

A conventional per-cell Born effective charge also involves the cell-volume or cell-area convention and the bare ionic contribution. Here we keep the paper’s electronic, polarization-density convention separate from those additions.

The wave-packet magnetic coefficient instead involves

In general, an integral of a product cannot be reconstructed from the separate integrals of its factors. To isolate that elementary issue, use a normalized one-dimensional average and a toy matrix:

What this plot does and does not claim: it demonstrates the difference between first and second moments. It is not a specified Bloch Hamiltonian, not a prediction for a material, and not a claim that all curvature components can be varied independently in a physical band.

In the neutral, rigid atomic limit, an electron cloud following its ion gives an electronic loop moment that cancels the corresponding positive ionic loop moment. That cancellation is a limiting physical picture, not a general identity in a covalent or small-gap crystal.

KEY POINT

A moderate macroscopic effective charge does not imply a small magnetic response. Momentum-dependent contributions can cancel in the first moment while adding strongly in the quadratic combinations.

A Berry phase changes the energy entering the boundary current

Where does come from, if it is not the center-of-mass orbit just discussed?

The phonon loop gives a geometric phase

During one small orbit in displacement space, the state accumulates a dynamical phase and a Berry phase. With curvature evaluated at equilibrium,

Matching the phases on the branch continuous at zero phonon amplitude gives

This is an effective adiabatic phase-derived energy correction. It should not be confused with an ordinary static change in the instantaneous band energy along the orbit.

Insert that shift into the orbital-magnetization boundary term

The modern orbital-magnetization expression contains

Keeping this contribution to leading order, the change due to the phase-derived energy shift is

The two minus signs cancel. One factor says how much geometric phase the phonon loop produces; the other says how an energy shift modifies the Berry-curvature boundary-current contribution.

KEY POINT

Two distinct physical mechanisms combine: motion of the wave-packet center supplies the mixed-curvature determinant, and a phase-induced shift of the boundary-current contribution supplies the product of pure momentum and pure displacement curvatures.

Put the pieces together—and check what the formula really says

Combining the two mechanisms gives the complete Abelian second-Chern contribution:

Why a nonmagnetic equilibrium crystal is allowed

Under time reversal, momentum changes sign while static displacement does not. For a time-reversal-related band or the appropriate occupied-subspace formulation,

The determinant of even mixed curvatures is even; the product of the two odd curvatures is also even. Opposite momenta therefore need not cancel. There is no contradiction: the chosen chiral phonon motion itself breaks time reversal. Reversing the motion changes and the magnetization sign.

Units: magnetization density is not yet a per-phonon moment

With a wave vector and a length, the mixed curvature is dimensionless. The four-index combination is dimensionless, while has units of inverse area. Hence

This is the two-dimensional magnetization (moment per area). Multiplying by sample area gives a magnetic moment in . A moment per phonon additionally requires the mode normalization and the occupation-dependent ionic angular momentum; do not insert for the representative-ion angular momentum without that normalization.

Small gaps enhance geometry, but cannot justify an infinite prediction

In the paper’s particular gapped-graphene effective model, a contribution scales as . The plot illustrates that model scaling, not a universal law. Its gap is ; the shaded region fails even the rough gap-versus-phonon-energy test.

The horizontal coordinate is ; the plotted normalized response is . The boundary is not a guarantee of accuracy just outside it. A quantitatively controlled adiabatic calculation needs a stronger separation, and a small-amplitude expansion needs a sufficiently small electron–phonon perturbation.

THE WHOLE NARRATIVE IN ONE SENTENCE

A phonon sweeps oriented area in displacement space; electronic Berry geometry converts that area rate into a bound loop current; separating the product-rule polarization remainder identifies its cycle-averaged magnetization.

Three checks you should now be able to do:

Set : the area and this magnetization vanish. Reverse the orbit: the sign flips. Keep the same macroscopic Born charge but change its momentum dependence: the quadratic electronic coefficient can change.

What remains outside this derivation?

The derivation starts from the semiclassical topological local current. It does not include the paper’s additional nontopological electronic contribution, arbitrary metals, nonadiabatic driving, or the full multi-ion phonon normalization. Several occupied bands must be treated with the non-Abelian occupied-subspace expressions, not by applying a single-band orbit picture indiscriminately inside a degenerate subspace.

The note above focuses on Eqs. (1)–(5) and the discussion of Fig. 1 in Ref. Ref. 1. The expanded product-rule calculation is an explicit reconstruction under the assumptions stated here, not a quotation from the supplement.

Reference

[1] Yafei Ren, Cong Xiao, Daniyar Saparov, and Qian Niu, “Phonon Magnetic Moment from Electronic Topological Magnetization,” Physical Review Letters 127, 186403 (2021). DOI: 10.1103/PhysRevLett.127.186403. Author manuscript: arXiv:2103.05786.