Rotational Susceptibilities Overview

• Rotational susceptibilities are response functions defined as derivatives with respect to rotation parameters (angle, angular velocity) that reveal how systems react to angular perturbations.
• They expose anisotropy and symmetry-dependent effects through tensor structures and channel mixing, playing a key role in fluctuation-dissipation relations and phase transition diagnostics.
• A variety of methods—including magnetic torque measurements, rotational anisotropy optics, and hadron resonance gas models—have been employed to quantify and analyze these susceptibilities in different physical contexts.

Rotational susceptibilities are response functions that quantify how a physical system changes under rotation, or under control parameters that couple to angular degrees of freedom. In the cited literature, the term covers several technically distinct objects: the curvature of the free energy with respect to a rotation angle in a magnetic field, $k(\theta)=d^{2}F/d\theta^{2}$; angular derivatives of harmonic-generation intensities in rotational-anisotropy optics; in-plane angular dependences of magnetic-susceptibility tensors; and derivatives of thermodynamic pressure with respect to angular velocity, $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$ (Shekhter et al., 2022, Torchinsky et al., 2014, Sahoo et al., 4 Jul 2025). Across these settings, the common structure is that anisotropy and symmetry determine which tensor components survive, which angular harmonics appear, and which response channels mix.

1. Definitions and taxonomic scope

The literature uses closely related but non-identical definitions. In magnetic thermodynamics, magnetotropic susceptibility is the second angular derivative of the free energy, and the corresponding first derivative is the magnetic torque (Shekhter et al., 2022). In nonlinear optics, rotational anisotropy experiments record harmonic intensity while the light scattering plane is rotated relative to crystal axes, so the angular pattern reflects the nonlinear optical susceptibility tensor (Torchinsky et al., 2014). In rotating hadronic matter, rotational susceptibility is defined by differentiation of pressure with respect to $\omega/T$, where $\omega$ plays the role of a rotational chemical potential (Sahoo et al., 4 Jul 2025). In rotating finite-size quark matter, susceptibility functions are integrated two-point correlation functions of fermion bilinears and become $r$-dependent because rigid rotation breaks radial translational invariance (Kawaguchi et al., 1 Jul 2025).

Context	Response quantity	Rotational control variable
Magnetically anisotropic thermodynamics	$k(\theta)=d^2F/d\theta^2$	Rotation angle $\theta$
Nonlinear optical rotational anisotropy	Harmonic intensity $I_{n\omega}(\phi)$	Scattering-plane angle $\phi$
In-plane magnetic anisotropy	$\chi_{\parallel}(\phi)$, $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$0	Field angle $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$1
Rotating QCD matter	$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$2	Angular velocity $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$3
Rotating finite cylinder	Bilinear two-point susceptibilities, moment of inertia	Angular velocity $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$4

A central implication is that “rotational susceptibility” is not a single scalar observable across all subfields. Rather, the phrase denotes a family of response coefficients whose common feature is differentiation with respect to angle, angular velocity, or a symmetry-equivalent rotational field.

2. Thermodynamic rotational susceptibility in magnetic systems

Magnetotropic susceptibility is defined for a sample in a uniform magnetic field whose free energy depends on the angle $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$5 between the field and the crystal axes. The torque is

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$6

and the magnetotropic susceptibility is

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$7

It is therefore the curvature of the free energy with respect to rotation angle, and the paper explicitly characterizes it as a “rotational susceptibility” because it measures how stiff the free energy is against infinitesimal rotations in a magnetic field (Shekhter et al., 2022).

The same work formulates a tensor version $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$8 through $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$9, where $\omega/T$0 specifies the rotation axis. A basic constraint is that $\omega/T$1 when $\omega/T$2, because rotation around the field direction does not change the magnetic free energy. In the linear-response regime, where $\omega/T$3 is field-independent, only differences of principal susceptibilities contribute. For rotation in the $\omega/T$4-plane about the $\omega/T$5-axis,

$\omega/T$6

so $\omega/T$7 directly measures anisotropy; an isotropic susceptibility tensor gives $\omega/T$8 (Shekhter et al., 2022).

The thermodynamic formulation also separates a “kinematic” or “reactive” term from the fluctuation term. With $\omega/T$9 the rotated Hamiltonian,

$\omega$0

and

$\omega$1

The last term is a torque-torque correlation function. This makes magnetotropic susceptibility more than a geometric second derivative of a macroscopic free-energy profile; it is also a thermodynamic-response coefficient with a fluctuation-dissipation component (Shekhter et al., 2022).

The dynamic generalization $\omega$2 becomes relevant in oscillating cantilever experiments. There,

$\omega$3

and the sample contributes an effective spring constant that shifts the resonance frequency according to

$\omega$4

in the weak-coupling limit $\omega$5. The real part produces the dispersive frequency shift, while the imaginary part produces dissipation and linewidth broadening; by causality, the two are linked by Kramers–Kronig relations. Near a second-order phase transition, the paper further gives an Ehrenfest-like relation,

$\omega$6

establishing $\omega$7 as a bona fide thermodynamic coefficient (Shekhter et al., 2022).

3. Angular magnetic susceptibilities in anisotropic spin systems

In single-crystal $\omega$8-RuCl$\omega$9, rotational magnetic susceptibility is realized as the angle dependence of the in-plane longitudinal and transverse susceptibilities under a $r$0 sample rotation in a fixed magnetic field. The longitudinal response $r$1 has $r$2-periodicity,

$r$3

with maxima at $r$4 and minima at $r$5. The transverse susceptibility $r$6 is $r$7-periodic, with maxima at $r$8, minima at $r$9, and nodes at $k(\theta)=d^2F/d\theta^2$0. These oscillations persist well above the zigzag ordering temperature $k(\theta)=d^2F/d\theta^2$1 K, and the paper interprets them as a direct fingerprint of bond anisotropy in a bond-anisotropic Kitaev-Heisenberg-$k(\theta)=d^2F/d\theta^2$2 Hamiltonian (Lampen-Kelley et al., 2018).

The same analysis identifies the strong difference between the mean in-plane longitudinal susceptibility $k(\theta)=d^2F/d\theta^2$3 and the out-of-plane susceptibility $k(\theta)=d^2F/d\theta^2$4 with symmetric off-diagonal $k(\theta)=d^2F/d\theta^2$5 exchange,

$k(\theta)=d^2F/d\theta^2$6

rather than a simple $k(\theta)=d^2F/d\theta^2$7-factor effect. The in-plane oscillation amplitude is controlled by

$k(\theta)=d^2F/d\theta^2$8

so the oscillation vanishes if the bonds are equivalent. Using Curie-Weiss intercepts extracted from $k(\theta)=d^2F/d\theta^2$9 for longitudinal data and $\theta$0 for transverse data, the paper reports

$\theta$1

and, with $\theta$2,

$\theta$3

The extracted couplings indicate large $\theta$4, $\theta$5 terms and notable bond anisotropy in $\theta$6 (Lampen-Kelley et al., 2018).

A complementary rotational decomposition appears in uniaxial paramagnets and superparamagnets. For

$\theta$7

linear response separates into

$\theta$8

where the longitudinal susceptibility is

$\theta$9

and the transverse susceptibility is

$I_{n\omega}(\phi)$0

The transverse response contains the Kubo factor because the perturbation does not commute with $I_{n\omega}(\phi)$1, so it is sensitive to level spacings and level mixing rather than only to population redistribution (0907.3181).

The finite-$I_{n\omega}(\phi)$2 temperature dependence is qualitatively distinct in the two channels. For $I_{n\omega}(\phi)$3, $I_{n\omega}(\phi)$4 at low $I_{n\omega}(\phi)$5, while $I_{n\omega}(\phi)$6 and no pronounced peak appears. For $I_{n\omega}(\phi)$7, $I_{n\omega}(\phi)$8 develops a small broad maximum around $I_{n\omega}(\phi)$9, then tends to $\phi$0. For general $\phi$1, high $\phi$2 gives Curie behavior, low $\phi$3 gives an Ising-like longitudinal regime, and for $\phi$4 the transverse susceptibility exhibits a broad maximum that survives into the classical superparamagnetic limit. For randomly oriented anisotropy axes, the exact orientational average is

$\phi$5

and the paper derives a low-temperature approximation whose crossover is controlled by $\phi$6 (0907.3181).

4. Rotational anisotropy optics and ferro-rotational order

In nonlinear optics, rotational susceptibility is encoded in the angle dependence of harmonic generation under rotation of the scattering geometry. The nonlinear harmonic generation rotational anisotropy spectrometer measures second- or third-harmonic intensity as the light scattering plane is rotated relative to the crystal axes, while the crystal remains fixed. With monochromatic input at frequency $\phi$7, the emitted intensity at $\phi$8 or $\phi$9 is measured as a function of

$\chi_{\parallel}(\phi)$0

the angle between the crystal axes and the light scattering plane. The key rationale is that the harmonic signal is governed by the nonlinear optical susceptibility tensor, and the angular dependence projects different tensor components into the laboratory frame (Torchinsky et al., 2014).

The construction uses oblique-incidence reflection geometry, with a pulsed laser focused through a reflective objective at an incidence angle of about $\chi_{\parallel}(\phi)$1. Reflection is necessary because bulk crystals are often opaque or too thick for transmission, and oblique incidence gives access to field components normal to the surface, making more susceptibility elements observable. Rotating the scattering plane rather than the sample minimizes beam walk, keeps the surface normal aligned to the rotation axis, preserves the incidence angle, and permits operation in ultra-low temperature cryostats, high magnetic fields, and high pressure or strain environments. The paper emphasizes that these features are what make the instrument suitable for small bulk single crystals and harsh sample environments (Torchinsky et al., 2014).

The formal structure is

$\chi_{\parallel}(\phi)$2

where $\chi_{\parallel}(\phi)$3 rotates tensor elements from the crystal frame into the laboratory frame. Different polarization combinations such as PP and PS probe different tensor-component combinations, and the observed lobes and nodes repeat according to the crystallographic point group. In Sr$\chi_{\parallel}(\phi)$4IrO$\chi_{\parallel}(\phi)$5, room-temperature $\chi_{\parallel}(\phi)$6 SHG in PP geometry is attributed to the bulk electric quadrupolar response $\chi_{\parallel}(\phi)$7 and shows four-fold rotational symmetry; at $\chi_{\parallel}(\phi)$8 K, $\chi_{\parallel}(\phi)$9 THG in PS geometry is attributed to the bulk electric dipole response $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$00 and again reproduces the expected crystal symmetry (Torchinsky et al., 2014).

A more specialized use of rotational-anisotropy SHG appears in the optical detection of ferro-rotational order in RbFe(MoO$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$01)$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$02. There the order parameter is an axial vector that is time-reversal even and spatial-inversion even, so ordinary electric-dipole SHG is symmetry-forbidden and the relevant signal must arise from the electric quadrupole channel,

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$03

The room-temperature patterns are consistent with point group $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$04, while below $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$05 K the low-temperature phase is identified as point group $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$06: the threefold axis remains, the three mirror planes are lost, and inversion symmetry is retained (Jin et al., 2019).

The low-temperature state contains two ferro-rotational domains with opposite rotational vectors. The measured intensity is fit by

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$07

where $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$08 is the domain population weight. The nonzero background in the RA pattern follows from mixed-domain superposition, and the inferred domain size is much smaller than the $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$09 beam spot. The azimuthal rotation angle of the single-domain pattern,

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$10

jumps from $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$11 to about $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$12 at $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$13 and grows toward $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$14 at lower temperature. Because $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$15 is forbidden above $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$16 and allowed below it, its discontinuous onset is used to identify the transition as weakly first order. The same work shows that a specific nonlinear optical field combination,

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$17

has the symmetry of the ferro-rotational order parameter and functions as its conjugate coupling field (Jin et al., 2019).

5. Rotational susceptibilities in rotating QCD matter

In hot and dense hadronic matter, rotational susceptibility is defined by direct analogy with conserved-charge susceptibilities. Starting from the rotating thermodynamic relation

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$18

the $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$19-th order rotational susceptibility is

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$20

and the first derivative gives the angular-momentum density,

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$21

The same paper also gives a fluctuation interpretation,

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$22

making $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$23 the rotational analogue of magnetic susceptibility (Sahoo et al., 4 Jul 2025).

The calculation is performed in the hadron resonance gas model and in the interacting van der Waals hadron resonance gas model. For the ideal rotating HRG, the single-particle distribution is

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$24

with $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$25 and the causality condition $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$26. The interacting extension uses the van der Waals equation of state

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$27

with $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$28 representing repulsive excluded-volume effects and $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$29 representing attractive interactions. The parameters used are $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$30 fm for mesons, $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$31 fm for baryons and antibaryons, and $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$32 (Sahoo et al., 4 Jul 2025).

The reported qualitative behavior is model dependent. In the ideal HRG, rotational susceptibilities grow smoothly with temperature and $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$33. In the interacting VDWHRG, the susceptibilities are generally suppressed relative to the ideal gas, but the interaction terms can generate phase-transition-like structure absent in the ideal model. At $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$34 GeV, a noticeable bump appears at intermediate temperature. The paper studies ratios

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$35

arguing that volume factors cancel in these combinations and that nonmonotonicity may signal the QCD phase transition. It also reports the orderings

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$36

The second-order susceptibility is strongly spin dependent, with the ordering spin-0 $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$37 spin-1 $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$38 spin-3/2 $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$39 spin-1/2 in the studied setup (Sahoo et al., 4 Jul 2025).

A field-theoretic treatment of rotating quark matter inside a finite cylinder uses a different but related notion of susceptibility. Because rigid rotation breaks radial translational invariance, the Dirac propagator must be constructed in the Fourier-Bessel basis rather than the plane-wave basis, with radial momenta $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$40 set by Bessel zeros and the causality constraint $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$41. The propagator exhibits the rotational energy shift

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$42

showing explicit coupling to total angular momentum $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$43 (Kawaguchi et al., 1 Jul 2025).

Within the two-flavor NJL model and the local density approximation, the paper derives resummed $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$44-dependent meson susceptibilities,

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$45

with the chiral Ward-Takahashi identity

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$46

It also defines the baryon number susceptibility and the moment of inertia through two-point functions; at one loop,

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$47

while

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$48

The numerical results show stronger thermal suppression of meson, topological, and baryon susceptibilities under rotation, especially near the cylinder boundary, and identify the moment of inertia as another indicator of the chiral transition (Kawaguchi et al., 1 Jul 2025).

6. Tensor structure, channel mixing, and nonstandard rotational response

A recurring theme in the broader susceptibility literature is that broken rotational invariance replaces scalar response coefficients with matrices or more elaborate tensor structures. In anisotropic Fermi liquids, where the Fermi surface lacks continuous rotational invariance, the generalized static susceptibilities are defined by

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$49

with interaction decomposed as

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$50

The resulting response is matrix valued, and the natural anisotropic Landau parameters are

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$51

The paper emphasizes that anisotropic Fermi liquids do not admit a simple scalar Landau-parameter description in general: different symmetry channels mix through $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$52, and instabilities occur when $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$53 loses invertibility (Rodríguez-Ponte et al., 2014).

An even more nonstandard case appears in the Hamiltonian mean-field model with asymmetric zero-mean momentum distributions. There the susceptibility tensor relating $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$54 to $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$55 takes the form

$$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$56

with $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$57 for asymmetric distributions. The tensor is therefore not symmetric and not diagonalizable over the reals; its eigenvalues are $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$58. The paper further states that the tensor has no divergence even at the stability threshold, in contrast to the symmetric case where divergence occurs as $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$59 (Yamaguchi, 2015).

Rotational dynamics under thermal agitation produces a different departure from textbook response. For long-range interacting dipoles obeying forced rotational diffusion, the complex susceptibility spectrum depends sharply on the Kirkwood correlation factor $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$60. When $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$61, parallel correlations create an effective double-well-like landscape, and the imaginary part of the susceptibility develops two features: a low-frequency peak associated with collective thermally activated reversal and a Debye-like peak near $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$62. When $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$63, the normalized spectrum remains practically unaltered with respect to the ideal-gas phase and stays essentially single-peaked. The smallest nonzero eigenvalue obeys an Arrhenius-Kramers form for large interaction strength in the $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$64 regime, whereas no true barrier-crossing form emerges for $$\chi_\omega^n=\partial^n[P/T^4]/\partial(\omega/T)^n$$65 (Déjardin et al., 2018).

Taken together, these examples show that rotational susceptibilities need not be symmetric, diagonal, static, or even reducible to a single channel. A plausible implication is that rotational response is often a particularly sensitive diagnostic of anisotropy, because angular perturbations directly expose channel mixing, off-diagonal couplings, and the geometry imposed by boundaries or by broken point-group symmetry.