Ferroaxial and nematic transitions in the charge density wave phase of 1T-TiSe$_2$

Abstract

Charge density waves (CDWs) with multi-component order parameters can break unexpected symmetries through the interplay of nearly degenerate instabilities. In the widely investigated material 1T-TiSe$_2$, a central question is whether the observed CDW has a chiral character, which would manifest as the spontaneous breaking of mirror and inversion symmetries. Previous experiments have reported conflicting results about the broken symmetries in the CDW phase of 1T-TiSe$_2$. Here, we resolve this controversy by identifying the bulk broken symmetry as ferroaxial, corresponding to the breaking of vertical mirrors while preserving inversion symmetry. Using symmetry-resolved elastoresistivity, we detect the spontaneous emergence of intrinsic off-diagonal elastoresistivity coefficients that satisfy an antisymmetric relation ($m_{xx-yy,xy} \approx -m_{xy,xx-yy}$), providing an unambiguous bulk transport signature of a macroscopic electric toroidal moment. Simultaneous elastocaloric measurements reveal that the onset of ferroaxial order occurs just below the CDW transition. As the temperature is lowered further, a diverging nematic susceptibility signals a distinct rotational symmetry-breaking instability inside the ferroaxial CDW state. Our findings demonstrate that the proposed ``chiral'' CDW in 1T-TiSe$_2$ is actually a centrosymmetric ferroaxial state, reconciling previous surface-sensitive observations with bulk symmetry constraints.

Figures (19)

• Figure 1: Experimental signatures of symmetry breaking in 1T-TiSe$_2$.a. Crystal structure of 1T-TiSe$_2$, showing the trigonal unit cell with point group D$_{3d}$. b. A top-down view of a single quasi-2D layer. Dashed lines indicate the dihedral (vertical) mirror planes c. Schematics distinguishing ferroaxial order (characterized by an electric toroidal dipole moment $\mathbf{G}$) from chiral order (proposed for this material as a helical charge ordering Ishioka2010). d. 3D Brillouin zone showing the three charge density wave vectors ($Q_1, Q_2, Q_3$) connecting $\Gamma$ to the $L$ points. e. Schematics of the AC strain experimental setup. A bar-shaped single crystal is suspended between two piezoelectric stacks. A type-E thermocouple (Constantan/Chromel) and electrical contacts (gold) allow for simultaneous measurements of temperature oscillations (elastocaloric) and resistivity (elastoresistivity) while combined DC strain offset ($\varepsilon_{DC}$) and AC strain oscillation ($\varepsilon_{AC}$) are applied. f--j. Transport and thermodynamic responses measured at zero DC strain ($\varepsilon_{DC} = 0$). f. Longitudinal resistivity ($\rho_{xx}$) as a function of temperature, showing the characteristic broad hump that onsets at $T_{CDW}$ (vertical dashed line). g. Elastocaloric coefficient $\partial T / \partial \varepsilon$ (red curve). Note the fine structure (a shoulder) near the transition, indicative of the two distinct phase boundaries discussed in the text. The shoulder is marked with a red arrow. h. Longitudinal elastoresistivity $\partial\tilde{\rho}_{xx} / \partial \varepsilon_{xx}$ (yellow curve), exhibiting a giant step-like increase at the transition. i. Transverse resistivity ($\rho_{xy}$) as a function of temperature measured in zero magnetic field. The spontaneous nonzero value indicates symmetry breaking. j. Off-diagonal elastoresistivity $\partial\tilde{\rho}_{xy} / \partial \varepsilon_{xx}$ (green curve). This coefficient is symmetry-forbidden in the high-temperature phase but turns on at $T_{CDW}$.
• Figure 2: Ferroaxial order revealed by off-diagonal elastoresistivity.a. The elastoresistivity tensor projected onto the subspace of the $E_g$ irreducible representation relates resistivity anisotropy components to symmetry-matched strains. In the parent $D_{3d}$ point group, diagonal elements ($m_{11}, m_{22}$) probe the nematic susceptibility, while off-diagonal elements ($m_{12}, m_{21}$) are symmetry-forbidden. Their nonzero values indicate the breaking of vertical mirror planes either by ferroaxial order ($m_{12}-m_{21}$) DayRoberts2025 or by the second component of the nematic order parameter ($m_{12}+m_{21}$). Diagrams illustrate the specific sample geometries used to isolate $m_{12}$ (rotated Montgomery) and $m_{21}$ (Hall bar). b. Induced change in resistivity anisotropy $\Delta(\tilde{\rho}_{xx} - \tilde{\rho}_{yy})$ versus shear strain $2\varepsilon_{xy}$, showing a linear response ($m_{12}$). c. Induced change in transverse resistivity $\Delta\tilde{\rho}_{xy}$ versus anisotropic strain $\varepsilon_{xx} - \varepsilon_{yy}$, showing a linear response ($m_{21}$). d. Temperature dependence of the extracted off-diagonal coefficients. The two signals display a striking antisymmetric relationship ($m_{12} \approx -m_{21}$) emerging near $T_{CDW}$ (vertical dashed line). Upper inset: Experimental configuration showing the square sample rotated by $45^\circ$ relative to the poling direction of the piezo stack (applying shear strain) and the Hall bar aligned with the axis (to apply anisotropic strain). e. Symmetry decomposition of the off-diagonal response. The difference ($m_{12} - m_{21}$, yellow) tracks the ferroaxial order parameter and onsets at $T_{CDW}$. The sum ($m_{12} + m_{21}$, purple), which tracks the shear component (i.e., the second component) of the nematic order parameter, remains suppressed until $T < 170$ K, confirming the separation of the two symmetry-breaking scales.
• Figure 3: Diverging nematic susceptibility deep within the CDW phase.a.Top: The diagonal block of the elastoresistivity tensor relating the nematic order parameter (resistivity anisotropy) to the conjugate symmetry-breaking strain. Bottom: Schematics of the experimental geometry used to measure $m_{11}$. A square single crystal is aligned with the principal strain axes of the piezoelectric stack, measuring the longitudinal resistivity anisotropy. b. Isothermal curves of the induced resistivity anisotropy $\tilde{\rho}_{xx} - \tilde{\rho}_{yy}$ versus anisotropic strain $\varepsilon_{xx} - \varepsilon_{yy}$. The response is linear over the accessible strain range, with a slope ($m_{11}$) that changes dramatically with temperature. c. Temperature dependence of the diagonal element in the $E_g$ elastoresistivity block, $m_{11}$ ($\chi_{E_g}$). The response exhibits a divergent-like behavior peaking well below the CDW transition (vertical dashed line). The solid blue line represents a Curie-Weiss fit, indicating a bare nematic temperature $T^* \approx 170$ K. d. Temperature dependence of the isotropic elastoresistivity coefficient $m_{A_{1g}}$. In comparison to the $E_g$ channel, this response is small and non-divergent above the nematic transition, confirming that the critical fluctuations correspond to an order parameter that breaks rotational symmetry. e. Normalized resistivity anisotropy $(\rho_{xx} - \rho_{yy})/(\rho_{xx} + \rho_{yy})$ versus temperature for various fixed piezo voltage values. The color bar shows the anisotropic strain ($\varepsilon_{xx} - \varepsilon_{yy}$) values measured at $150$ K. The "fan-like" spread demonstrates the high tunability of the electronic anisotropy near $170$ K.
• Figure 4: Strain-temperature phase diagram and the hierarchy of broken symmetries.a. Elastocaloric coefficient ($\partial T/\partial \varepsilon_{xx}$) versus temperature for various DC strain offsets ranging from compressive (light) to tensile (dark). Curves are offset vertically for clarity. b. Temperature derivative of the elastocaloric coefficient, $\frac{\partial}{\partial T}(\partial T/\partial \varepsilon_{xx})$, with curves offset vertically. The distinct peaks and anomalies, indicated with circular markers, clearly resolve the primary CDW onset, the subsequent ferroaxial transition, and the strain-split $1Q/2Q \rightarrow 3Q$ boundaries. c. Color map of the temperature derivative of the elastocaloric coefficient in the strain-temperature plane. The topological structure of the bright and dark bands precisely maps the underlying phase boundaries. d. Off-diagonal elastoresistivity ($\partial\tilde{\rho}_{xy}/\partial\varepsilon_{xx}$) versus temperature. Inset: Color map of the same data, mirroring the thermodynamic boundaries established by the ECE. e. Longitudinal elastoresistivity ($-\partial\tilde{\rho}_{xx}/\partial\varepsilon_{xx}$) versus temperature, serving as a proxy for the nematic susceptibility. Inset: Color map of the longitudinal response. f. Calculated temperature derivative of the ECE derived from a Landau free energy expansion. The model incorporates the bilinear coupling of the conjugate uniaxial strain field to the multi-component $3Q$ CDW order parameter. g. The comprehensive strain-temperature phase diagram constructed from the aggregated thermodynamic and transport data. The diagram illustrates the primary CDW transition, the secondary ferroaxial (FA) transition, the strain-split $1Q$ and $2Q$ phases, and the low-temperature spontaneous nematic transition within the $3Q$ phase.
• Figure A1: 1T-TiSe$_2$ crystals.