Shear-Induced Polarization

• Shear-induced polarization is a phenomenon where shear strain or stress generates diverse polarization responses, including spin, electric, and tensorial effects across different physical systems.
• In relativistic heavy-ion collisions, thermal shear significantly contributes to local spin polarization, producing distinctive quadrupole azimuthal structures that align with experimental observations.
• In condensed matter and liquid crystal systems, shear strain triggers electric and optical polarization responses, paving the way for novel piezoelectric and ferroelectric applications.

Searching arXiv for recent and directly relevant papers on shear-induced polarization across the main usage domains. arXiv search query: "shear induced polarization thermal shear heavy ion collisions spin polarization" Shear-induced polarization denotes a class of phenomena in which shear strain, shear stress, or shear gradients generate a polarization response. In relativistic heavy-ion theory, the term commonly refers to spin polarization generated by thermal shear, the symmetric part of the gradient of the inverse-temperature four-vector. In condensed-matter settings, it denotes electric polarization induced by shear strain, as well as related piezoelectric, optical, and polaritonic responses. In polar and active fluids, it also describes shear-controlled polar order, polarization textures, and mechanically generated stress states (Buzzegoli, 2024, Malyshev et al., 27 Mar 2025, Markovich et al., 2018).

1. Conceptual scope and recurring structures

Across disciplines, the recurring structure is the coupling of a shear variable to a polarization variable of appropriate symmetry. In relativistic hydrodynamics, the relevant object is the symmetric derivative of $\beta^\mu=u^\mu/T$; in crystalline solids it is the off-diagonal elastic strain tensor $\epsilon_{ij}$; in liquid-crystalline and active systems it is the shear rate or shear stress; in optical settings it is the stress- or strain-induced anisotropy that changes the polarization state of transmitted light. The resulting “polarization” may therefore mean spin polarization, tensor polarization, electric polarization, or optical polarization, depending on context.

In the heavy-ion literature, the first-order gradient $\partial_\mu\beta_\nu$ is naturally decomposed into an antisymmetric thermal-vorticity part and a symmetric thermal-shear part. In solid-state work, the analogous distinction is between rigid-rotation-like distortions and symmetry-lowering shear distortions that activate piezoelectric or multiferroic responses. In active and polar soft matter, shear does not merely reorient a preexisting order parameter; it can modify its magnitude, create spatial gradients, and even induce phase transitions (Buzzegoli, 2024, Markovich et al., 2018, Loisy et al., 2022).

A plausible unifying implication is that shear-induced polarization is not a single mechanism but a family of symmetry-allowed couplings between anisotropic deformation and an axial, polar, or tensorial order parameter.

2. Relativistic spin polarization from thermal shear

In relativistic local equilibrium, the inverse-temperature four-vector is

$\beta^\mu=\frac{u^\mu}{T},$

and its gradient splits into

$$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$

Here $\varpi_{\mu\nu}$ is thermal vorticity and $\xi_{\mu\nu}$ is thermal shear. In the Zubarev local-equilibrium density operator, these two tensors couple to different operators: thermal vorticity to the conserved angular-momentum–boost operator $\widehat{J}^{\mu\nu}_x$, and thermal shear to a symmetric quadrupole-like operator $\widehat{Q}^{\mu\nu}_x$, which is not conserved (Buzzegoli, 2024).

For a free Dirac field, the mean spin vector to first order in gradients is

$$S^\mu(p) = - \frac{\epsilon^{\mu\rho\sigma\tau} p_\tau}{8m \displaystyle\int_{\Sigma} {\rm d} \Sigma \cdot p \; n_F} \int_{\Sigma} {\rm d} \Sigma \cdot p \; n_F (1 - n_F) \left[ \varpi_{\rho\sigma} + 2\hat t_\rho \frac{p^\lambda}{E_p} \,\xi_{\lambda\sigma}\right].$$

This formula separates the vorticity-induced part from the shear-induced part and is the central hydrodynamic expression for shear-induced spin polarization (Buzzegoli, 2024).

The thermal-shear tensor contains several hydrodynamic structures: $\epsilon_{ij}$0 so the hydrodynamic shear tensor $\epsilon_{ij}$1 appears explicitly as part of $\epsilon_{ij}$2. Physically, thermal vorticity is tied to local rotation and acceleration, whereas thermal shear encodes shear and expansion of the flow and velocity gradients that do not correspond to rigid rotation. The heavy-ion interpretation advanced in this framework is that such non-rotational gradients can induce a preferred spin orientation even without net vorticity, and that this contribution is essential for local, momentum-dependent polarization observables (Buzzegoli, 2024).

The same line of work emphasized that global polarization is well described by thermal vorticity alone, whereas local polarization measured by STAR is not reproduced if only thermal vorticity is included. The thermal-shear term does not significantly modify the global polarization, but it is the dominant contribution in local spin polarization, especially in the longitudinal component as a function of azimuth and rapidity (Buzzegoli, 2024).

3. Tensor polarization, quadrupole structure, and spin phenomenology

For massive spin-1 particles, shear-induced polarization appears as tensor polarization rather than vector polarization. In a Wigner-function treatment with a modified method of moments, the spin-rank-2 moment obeys the constitutive relation

$\epsilon_{ij}$3

where $\epsilon_{ij}$4 is the shear-stress tensor and $\epsilon_{ij}$5 is a dimensionless coefficient determined by local collisions. In this framework, tensor polarization vanishes in local equilibrium and emerges as a dissipative response to shear stress (Wagner et al., 2022).

The experimentally relevant observable for vector meson alignment is

$\epsilon_{ij}$6

and the shear-induced correction is expressed directly in terms of $\epsilon_{ij}$7 in the freeze-out integral. This gives a concrete route from hydrodynamic shear stress to vector-meson spin alignment (Wagner et al., 2022).

For spin-$\epsilon_{ij}$8 fermions, an independent Wigner-function analysis found that the shear tensor induces a quadrupole in phase space. The first-order axial Wigner function contains the structure

$\epsilon_{ij}$9

with $\partial_\mu\beta_\nu$0 the traceless quadrupole tensor in momentum space. At one loop,

$\partial_\mu\beta_\nu$1

so the shear-induced term is parametrically of the same order as the vorticity-induced term. Its characteristic signature is a quadrupole azimuthal structure in momentum space, rather than a simple dipole-like alignment (Liu et al., 2021).

A hydrodynamic implementation for $\partial_\mu\beta_\nu$2 GeV Au+Au collisions showed that shear-induced spin polarization competes with thermal-vorticity effects. In that study, the shear-induced contribution always showed the same azimuthal angle dependence as the experimental data, and in the scenario that $\partial_\mu\beta_\nu$3 inherits and memorizes the spin polarization of strange quark, the resulting azimuthal-angle-dependent $\partial_\mu\beta_\nu$4 and $\partial_\mu\beta_\nu$5 agreed qualitatively with the data (Fu et al., 2021).

4. Electric polarization generated by shear strain in solids and two-dimensional materials

In crystalline solids, shear-induced polarization is most directly an electric-polarization effect. In the altermagnetic semiconductor CuFeS$\partial_\mu\beta_\nu$6, shear strain is defined by changing the angle between two lattice vectors in a given $\partial_\mu\beta_\nu$7 plane while allowing the perpendicular lattice vector to relax. The tetragonal ground state belongs to point group $\partial_\mu\beta_\nu$8, whose piezoelectric tensor already couples shear components to polarization: $\partial_\mu\beta_\nu$9 Accordingly, shear on any one of the three $\beta^\mu=\frac{u^\mu}{T},$0 planes induces a polar phase with polarization perpendicular to the sheared plane. Shear on the $\beta^\mu=\frac{u^\mu}{T},$1 plane yields orthorhombic $\beta^\mu=\frac{u^\mu}{T},$2 with polarization along $\beta^\mu=\frac{u^\mu}{T},$3; shear on the $\beta^\mu=\frac{u^\mu}{T},$4 plane yields monoclinic $\beta^\mu=\frac{u^\mu}{T},$5 with polarization mainly along $\beta^\mu=\frac{u^\mu}{T},$6; and shear on the $\beta^\mu=\frac{u^\mu}{T},$7 plane yields polarization along $\beta^\mu=\frac{u^\mu}{T},$8. Applying shear strain to two out of the three $\beta^\mu=\frac{u^\mu}{T},$9 planes induces a net magnetization simultaneously with electric polarization, producing a multiferroic response (Malyshev et al., 27 Mar 2025).

The same study identified a microscopic mechanism in which Cu and Fe cations shift in one direction along the polarization axis while S anions shift in the opposite direction, with displacements nearly proportional to $$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$0 in the small-strain regime. Near $$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$1, the induced polarization is linear in shear, consistent with the parent-phase piezoelectric tensor, while at larger shear the response becomes nonlinear (Malyshev et al., 27 Mar 2025).

A different two-dimensional route appears in monolayer NbOCl$$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$2, where ferroelectric and antiferroelectric phases compete. Shear strain is defined by the angle change between the two in-plane lattice vectors. Density-functional calculations show that the antiferroelectric phase is only about $$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$3 meV/f.u. above the ferroelectric ground state at zero strain, and that at $$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$4 shear the energy ordering inverts, making the antiferroelectric phase lower in energy than the ferroelectric phase. At $$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$5 shear, deep-potential molecular dynamics predicts room-temperature-stable antiferroelectricity with a one-dimensional collinear polarization arrangement and a low critical electric field for the AFE-to-FE transition, producing a double polarization-electric loop with small hysteresis (Mao et al., 21 Aug 2025).

In bilayer hBN, a one-dimensional sinusoidal undulation generates nonuniform shear patterns that create regions with unique local stacking and vertical polarization akin to sliding-induced ferroelectrics. The resulting structure contains one-dimensional polarization domains and a shear-induced one-dimensional moiré pattern. The same sliding-induced polarization is also found in double-wall BN nanotubes, where curvature differences between inner and outer tubes generate relative interlayer shear (Li et al., 2024).

At ferroelectric domain walls, shear-induced polarization appears in a localized converse-piezoelectric form. In $$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$6-axis tetragonal Pb(Zr$$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$7Ti$$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$8)O$$\varpi_{\mu\nu} = -\frac{1}{2} \left( \partial_\mu \beta_\nu - \partial_\nu \beta_\mu \right), \qquad \xi_{\mu\nu} = \frac{1}{2} \left( \partial_\mu \beta_\nu + \partial_\nu \beta_\mu \right).$$9, the sign change of $\varpi_{\mu\nu}$0 through a $\varpi_{\mu\nu}$1 wall and the lowered wall symmetry generate a local shear coefficient $\varpi_{\mu\nu}$2. Under an out-of-plane electric field, this yields a wall-localized shear strain $\varpi_{\mu\nu}$3, detected as a lateral piezoresponse. In monoclinically distorted tetragonal BiFeO$\varpi_{\mu\nu}$4, this wall shear is superimposed on the lateral response due to genuine in-plane polarization, and must be included to interpret the domain configuration correctly (0907.4570).

5. Shear-driven polarization textures in liquid crystals and active matter

In polar liquid crystals, allowing the magnitude of the polar order parameter to respond to flow changes the shear problem qualitatively. A hydrodynamic theory with a shear-alignment parameter $\varpi_{\mu\nu}$5 and a shear-elongation parameter $\varpi_{\mu\nu}$6 shows that shear can change not only the orientation but also the magnitude $\varpi_{\mu\nu}$7 of the polarization. For suitable $\varpi_{\mu\nu}$8, the effective Landau coefficients become shear-dependent, and the isotropic–polar transition becomes first order under shear. This is a genuine shear-induced phase transition from isotropic to polar, with pronounced changes in rheology (Markovich et al., 2018).

Ferroelectric nematic liquid crystals display a different shear–polarization coupling. In the N and NF phases of RM734, DIO, and FNLC919, three regimes are observed under shear: flow alignment at low shear rates, a log-rolling regime at high shear rates, and polydomain structures at intermediate rates. In the flow-aligning regime, the NF polarization does not tilt away from the shear direction, in sharp contrast to the flow-induced tilt of the N director. The proposed explanation is the avoidance of splay deformations and associated space charge in the flowing NF. At high shear, the director and polarization align along the vorticity axis (Das et al., 26 Feb 2026).

In confined active polar suspensions, the relevant effect is often not a uniform shear-induced polarization but a shear-stabilized polarization gradient. For a thin film of an active polar liquid crystal with antiparallel boundary conditions $\varpi_{\mu\nu}$9, $\xi_{\mu\nu}$0, the steady-state polarization profile produces a density modulation and a constant shear stress

$\xi_{\mu\nu}$1

with a nonzero offset $\xi_{\mu\nu}$2. At zero strain rate, the fluid sustains a non-zero stress. This offset is a consequence of the broken $\xi_{\mu\nu}$3 symmetry, the polarization wall imposed by the boundaries, and the polar couplings that convert gradients in the magnitude of polarization into stress (Loisy et al., 2022).

These results suggest that, in soft and active media, shear-induced polarization often takes the form of a coupled order-parameter and flow problem: shear can generate polar order, prevent certain director tilts, or convert polarization gradients into mechanical stress.

6. Optical, piezoresponse, and polaritonic manifestations

Shear-induced polarization also appears as an optical polarization response. In a dilute suspension of cellulose nanocrystals, flow birefringence induced by shear components along the camera’s optical axis was measured using a parallel-plate rheometer and a high-speed polarization camera. The birefringence increases monotonically as the stress components along the optical axis increase, and it follows a power law with respect to the shear rate. This shows that shear components $\xi_{\mu\nu}$4 and $\xi_{\mu\nu}$5, neglected in conventional photoelastic treatments, contribute measurably to optical polarization in three-dimensional flows (Worby et al., 2023).

The generalized stress–optic description used there supplements the conventional relation

$\xi_{\mu\nu}$6

with additional terms proportional to a coefficient $\xi_{\mu\nu}$7, involving $\xi_{\mu\nu}$8, $\xi_{\mu\nu}$9, and related quadratic combinations. In the measured system, fitting required $\widehat{J}^{\mu\nu}_x$0, which is the direct statement that shear along the optical axis induces optical anisotropy (Worby et al., 2023).

A different photonic use of the term occurs in hyperbolic polaritonics. “Hyperbolic shear polaritons” were first associated with low-symmetry crystals possessing off-diagonal dielectric-tensor components, but vortex excitation provides another route. In a hyperbolic material without off-diagonal elements, vortex waves with topological charge $\widehat{J}^{\mu\nu}_x$1 generate left-skewed or right-skewed hyperbolic shear polaritons. Away from the source, the asymmetric modes can be recovered as symmetric modes by tuning the magnitude of the off-diagonal imaginary component and the vortex topological charge, with a critical transition between the two skewed states (Xue et al., 2022).

In this polaritonic setting, the “shear” is not an elastic strain but an effective symmetry breaking in the optical mode structure. The common feature is again a coupling between anisotropic shear-like distortion and a polarization response, here encoded in the field profile and dispersion rather than in a dipole moment or spin vector (Xue et al., 2022).

7. Ambiguities, dissipative status, and unresolved questions

The most developed conceptual debates concern heavy-ion spin polarization. One issue is definitional ambiguity. Alternative definitions of the thermal-shear contribution, associated with Becattini–Buzzegoli–Palermo on the one hand and Liu–Yin on the other, are very similar in the midrapidity region while quite different at forward-backward rapidities. The spin-Hall contribution to global polarization is identically zero if averaging runs over all momenta in the Liu–Yin-type definition, and becomes nonzero only with restrictive momentum acceptance and the boost to the $\widehat{J}^{\mu\nu}_x$2 rest frame. A BBP-like definition would instead yield a nonzero global spin-Hall polarization even without those restrictions (Ivanov et al., 2022).

A second issue is whether shear-induced spin polarization is non-dissipative. One local-equilibrium derivation states that spin polarization generated by thermal shear is entirely non-dissipative and arises already in the local-equilibrium density operator (Becattini et al., 2021). A later analysis using entropy flow, the H-theorem, and Zubarev’s approach argues that while shear-induced polarization and the anomalous Hall effect do not directly contribute to the entropy production rate, the perturbations associated with the shear tensor lead to an increase in entropy, and the linear-response analysis suggests that these effects may indeed possess a dissipative nature (Wang et al., 21 Jul 2025).

The same later work emphasizes a distinction between coordinate-space and phase-space analyses. Time-reversal analysis of the spin current in coordinate space suggests that shear-induced polarization violates time-reversal symmetry and should vanish, whereas an analysis of the Wigner function in phase space does not impose an additional constraint and allows the effect to persist. The stated conclusion is that time-reversal analysis in coordinate space alone may not be sufficient to determine whether an effect is dissipative (Wang et al., 21 Jul 2025).

A further unresolved point is interpretive rather than formal. In the spin–hydrodynamic literature, thermal vorticity has a direct connection to the in-medium gravitomagnetic form factor $\widehat{J}^{\mu\nu}_x$3, whereas no analogous gravitational-form-factor interpretation is explicitly derived for the shear term. This suggests that vorticity-induced and shear-induced spin polarization, though both first-order gradient effects, belong to different operator structures and may require different microscopic interpretations (Buzzegoli, 2024).

Taken together, these debates indicate that “shear-induced polarization” is best treated as a field-dependent term whose precise definition, microscopic status, and phenomenological weight depend strongly on the dynamical framework. The common thread is not a single universal constitutive law, but the repeated emergence of shear as a source of polarization whenever symmetry permits the corresponding coupling.