The crystal system and point group of a material are encoded in the structure of its second- and higher-rank optical susceptibility tensors8, which can be determined from the anisotropy of its linear and nonlinear optical responses. The second-harmonic (SH) response is particularly sensitive to the presence of global inversion symmetry because unlike the linear electric-dipole susceptibility tensor χijED, which is allowed in all crystal systems, the SH electric-dipole susceptibility tensor χijkED vanishes in centrosymmetric point groups, leaving typically the far weaker electric-quadrupole term χijklEQ as the primary bulk radiation source9,10. For these reasons, it has been proposed that optical SH generation may be an effective probe of inversion symmetry breaking in the cuprates11.

To fully resolve the spatial symmetries underlying the pseudogap and its neighbouring regions, we performed both linear and SH optical rotational anisotropy (RA) measurements on de-twinned single crystals of YBa2Cu3Oy as a function of oxygen content and temperature. The RA measurements track variations in the intensities of light reflected at the fundamental (Iω) and SH frequencies (I2ω) of an obliquely incident laser beam, which is resonant with the O 2p to Cu 3d charge transfer energy (ω = 1.5 eV), as the scattering plane is rotated about the c axis (Fig. 1a). The use of a recently developed rotating optical grating-based technique12 made it possible to perform full 360 ° sweeps of the scattering plane angle (φ) over large temperature ranges while keeping the incident beam spot (100 μm) fixed to the same location on the crystal to within a few micrometres.

Figure 1: Layout of optical anisotropy experiment and predicted response of YBa2Cu3Oy.
figure 1

a, The intensity of light reflected at the fundamental (Iω) and SH (I2ω) frequencies of an obliquely incident beam is measured as a function of the angle (φ) between the scattering plane and the ac plane of YBa2 Cu3Oy. The Cu–O chains run along the b axis. The incident photon energy (1.5 eV) is resonant with the O 2p to Cu 3d charge transfer transition (inset). The polarization of the incident (in) and reflected (out) beams can be selected to lie either parallel (P) or perpendicular (S) to the scattering plane. b, Rotational anisotropy of the electric-dipole-induced linear response and electric-quadrupole-induced SH response simulated for the orthorhombic crystal system and the mmm point group respectively in SinSout geometry. The crystallographic a and b axes are aligned respectively along the laboratory x and y axes.

We first simulate the RA patterns expected from hole-doped YBa2Cu3Oy (y > 6) based on its reported orthorhombic mmm crystallographic point group13, which is inversion symmetric and consists of three generators mac, mbc and mab that denote mirror symmetries across the a–c, b–c and a–b planes. Unlike its tetragonal parent (y = 6) compound14, the point group of hole-doped YBa2Cu3Oy endows the crystal only with two-fold (C2) rather than four-fold (C4) rotational symmetry about the c axis because the excess oxygen atoms form chains that run along the b axis (Fig. 1a). Figure 1b shows representative linear and SH RA patterns computed in the electric-dipole and electric-quadrupole approximations respectively, where I0 is the incident beam intensity and both the incident (in) and reflected (out) polarizations and are selected to be perpendicular (SinSout) to the scattering plane (see Supplementary Section 1 for other polarization geometries). The orthorhombic crystal system is identifiable by an ovular Iω(φ) pattern with maxima and minima aligned strictly along the a and b axes15 while the mmm point group is identifiable by a four-lobed I2ω(φ) pattern that is mirror symmetric about the a and b axes.

Figure 2 shows linear and SH RA data from YBa2Cu3Oy with hole-doping levels (p) of 0.125 (y = 6.67; underdoped; Tc = 65 K), 0.135 (y = 6.75; underdoped; Tc = 75 K), 0.165 (y = 6.92; optimal doped; Tc = 92 K) and 0.190 (y = 7.0; overdoped; Tc = 86 K) measured at room temperature (T > T) in SinSout geometry (see Supplementary Section 2 for all other geometries). In the linear RA data, we see that the anisotropy (χxxEDχyyED) becomes more pronounced with hole doping as expected due to the filling of Cu–O chains. However, in contrast to Fig. 1b, the intensity maxima and minima are rotated away from the a and b axes, indicating an absence of mac and mbc symmetries consistent with a monoclinic distortion. In fact, by using the structure of χijED for a monoclinic crystal system in the expression for Iω(φ), excellent fits to the data are obtained (Fig. 2). In the SH RA data, we also find clear violations of the mmm point group symmetries at all doping levels. In particular, the alternation in lobe magnitude as a function of φ and the rotation of the lobe bisectors away from the a and b axes indicate an absence of mac and mbc symmetries consistent with the linear RA data. We find excellent agreement of the data with an electric-quadrupole-induced SH response from the centrosymmetric 2/m monoclinic point group (Fig. 2), which consists of two generators 2 and m that denote C2 and mab symmetries. On the contrary, fits to electric-dipole-induced SH from the non-centrosymmetric 2 and m monoclinic point groups, as well as to magnetic-dipole-induced11 or surface electric-dipole-induced SH from the 2/m point group, do not adequately describe the data (see Supplementary Section 3).

Figure 2: Crystal system and point group of YBa2Cu3Oy above T.
figure 2

ad, Polar plots of Iω (φ) and I2ω (φ) measured at T = 295 K in SinSout geometry from YBa2Cu3Oy with y = 6.67 (a), 6.75 (b), 6.92 (c) and 7.0 (d). The Iω(φ) and I2ω(φ) data sets, which have intensity error bars of approximately ±1% and ±10% respectively, are each plotted on the same intensity scale with a scaling ratio of Iω(φ) : I2ω(φ) ≈ 1 : 3 × 10−11. The former are fitted to the electric-dipole-induced linear response of the monoclinic crystal system (red lines) and the latter are fitted to the electric-quadrupole-induced SH response of the 2/m point group (blue lines). The angular deviations of the maxima of Iω(φ) and lobe bisectors of I2ω(φ) away from the a axis are shaded red and blue respectively for clarity.

The degree of monoclinicity, which can be qualitatively tracked via the angular deviation of the intensity maximum (lobe bisector) in the linear (SH) RA data away from the a axis, decreases monotonically between y = 6.67 and y = 7 as shown in Fig. 2. This suggests that the absence of mac and mbc symmetries at all measured temperatures above T (see Supplementary Section 4) originates from vacancy-induced monoclinic distortions of the oxygen sublattice, which are also known to be present in La2−xSrxCuO4 (ref. 16). This interpretation is further supported by recent SH RA measurements performed near the charge transfer resonance of layered perovskite iridates9,10, which revealed subtle oxygen sublattice distortions that are difficult to resolve using diffraction-based probes.

Having established the point group of hole-doped YBa2Cu3Oy above T, we proceed to search for changes in symmetry across the strange metal to pseudogap boundary. Previous infrared conductivity experiments17 have shown that optical transition rates at frequencies well above the pseudogap energy scale (ω 0.3 eV) do not exhibit any measurable temperature dependence across T. As a corollary, any temperature-dependent change in magnitude of χijED or χijklEQ at the frequencies used in this study (ω = 1.5 eV and 2ω = 3 eV) should be correspondingly weak. Figure 3a–d shows the temperature dependences of both the linear and SH intensities measured at a fixed value of φ in Sin–Sout geometry. For all four doping levels studied in Fig. 2, we observe no change in the linear response as a function of temperature as expected. Surprisingly however, all of the SH responses exhibit a significant order-parameter-like upturn below a doping-dependent critical temperature TΩ. This dichotomy between the linear and SH responses can naturally be reconciled if bulk inversion symmetry is broken below TΩ, which would turn on a new and stronger source of electric-dipole-induced SH radiation on top of the already existing electric-quadrupole contribution.

Figure 3: Inversion-symmetry-breaking transition across the phase diagram of YBa2Cu3Oy.
figure 3

ad, Temperature dependence of the linear and SH response of YBa2 Cu3 Oy with y = 6.67 (a), 6.75 (b), 6.92 (c) and 7.0 (d) normalized to their room-temperature values. Data were taken in SinSout geometry at angles φ corresponding to the smaller SH lobe maxima at T = 295 K (see Fig. 2). The error bars represent the standard deviation of the intensity over 60 independent measurements. Blue curves overlaid on the data are guides to the eye. The inversion-symmetry-breaking transition temperatures TΩ determined from the SH response are marked by the dashed lines. The width of the shaded grey intervals represents the uncertainty in TΩ. e, Temperature versus doping phase diagram of YBa2Cu3Oy. The values of TΩ (red circles) coincide with the onset of the pseudogap as defined by spin-polarized neutron diffraction2,3,4 (blue squares), Nernst effect5 (open blue squares), terahertz polarimetry6 (purple diamonds) and resonant ultrasound7 (orange circles). The onset of a polar Kerr effect33 (green diamonds) and short-range charge order as determined by X-ray diffraction19,20 (green squares) occur at temperatures below T. Error bars denote the uncertainty in the onset temperatures. The superconducting transition temperatures of the samples used in our work are denoted by grey circles.

By plotting the doping dependence of TΩ atop the phase diagram of YBa2Cu3Oy (Fig. 3e), we find that the observed onset of inversion symmetry breaking coincides very well with the pseudogap phase boundary T defined by spin-polarized neutron diffraction2,3,4, Nernst anisotropy5, terahertz polarimetry6 and resonant ultrasound7 measurements in the optimal and underdoped regions, suggesting a common underlying mechanism. Moreover, we detect an onset of inversion symmetry breaking even inside the superconducting dome in the overdoped region, which may imply a quantum phase transition slightly beyond p = 0.20 near where the pseudogap energy scale has also been extrapolated to vanish18. Interestingly, the temperature dependence of the SH intensity shows no measurable anomalies after crossing either the superconducting or the charge density wave ordering19,20 temperatures (Fig. 3e) and does not rapidly diminish at lower temperatures like the Nernst anisotropy5, which may be caused by charge density wave ordering21. This shows that unlike the superconducting and charge-ordered phases, which are competing Fermi surface instabilities22, the observed inversion-symmetry-broken phase is independent of and coexistent with both of them.

To determine which symmetries in addition to inversion are removed from the 2/m point group below the pseudogap temperature, we performed a comparative study of the SH RA data measured above and below TΩ. Figure 4a shows RA patterns measured at 295 K and 30 K in SinSout geometry for the optimal doped sample (TΩ 110 K), which is representative of all other doping levels studied (see Supplementary Section 5). Below TΩ we see that the SH intensity is enhanced in a φ-dependent manner that preserves the two-fold rotational symmetry of the pattern, which ostensibly implies that C2 is preserved in the crystal. However, this interpretation can be ruled out because the inferred above electric-dipole-induced SH radiation is strictly forbidden by symmetry in SinSout geometry for any point group (non-magnetic or magnetic) that contains C2 (see Supplementary Section 3). The only alternative interpretation is that the order parameter below TΩ breaks C2 symmetry, but is obscured in the RA patterns because of spatial averaging over domains of two degenerate orientations of the order parameter that are related by 180° rotation about the c axis. This would imply a characteristic domain length scale that is much smaller than our laser spot size and consistently explains the absence of any signatures of C2 breaking below T in Nernst effect5 and terahertz polarimetry6 data as well as the need for multiple magnetic domains to refine the polarized neutron diffraction data2,3,4,23, which are all integrated over even larger areas of the crystal compared with our measurements.

Figure 4: Point group symmetry of YBa2Cu3Oy below T.
figure 4

a, Polar plot of I2ω (φ) measured at T = 30 K in SinSout geometry from YBa2Cu3O6.92 (squares). The intensity error bars are approximately ±10%. The blue curve is a best fit to the average of two 180°-rotated domains with 2′/m point group symmetry as described in the text. The fit to the T = 295 K data (pink shaded area) reproduced from Fig. 2c is overlaid for comparison. b, Decomposition of the fit to the T = 30 K data (shaded blue area) into its individual domain contributions (shaded orange and purple areas).

The set of all non-centrosymmetric subgroups of 2/m that do not contain C2 consists of the two independent magnetic point groups 2′/m and m1′ and their associated magnetic and non-magnetic subgroups (see Supplementary Section 6), where the generators 2′, m and 1′ denote C2 combined with time-reversal, mab and the identity operation combined with time-reversal respectively. Using the two-domain (α = 1, 2) averaged expression , we were able to reproduce all features of the low-temperature SH RA data, including the small peaks around the b axis (Fig. 4a), by applying the structure of χijk, αED for either the 2′/m or m1′ point group. The decomposition of the fit into its two single-domain components is shown in Fig. 4b to explicitly illustrate the loss of C2. An equally good fit to the data can naturally be achieved using any magnetic or non-magnetic subgroup of 2′/m or m1′ because they necessarily allow the same or more non-zero independent χijk, αED tensor elements. Therefore, while further removal of symmetry elements from 2′/m or m1′ is not necessary to explain the low-temperature data, it cannot be completely ruled out.

In conclusion, our results show that the pseudogap region in YBa2Cu3Oy is bounded by a line of phase transitions associated with the loss of global inversion and C2 symmetries. Although previous terahertz polarimetry measurements on hole-doped YBa2Cu3Oy thin films reported the onset of a linear dichroic response near T that breaks mac and mbc symmetries6, we find that these symmetries are already broken in the crystallographic structure above T and are thus necessarily absent in any tensor response that turns on below T. The low symmetry of the point group (probably 2′/m or m1′) underlying the pseudogap region cannot be explained by stripe24 or nematic5 type orders alone, which have also been reported to develop below T. Instead it suggests the presence of an odd-parity magnetic order parameter, which is consistent with theoretical proposals involving a ferroic ordering of current loops circulating within the Cu–O octahedra11,25,26,27, local Cu-site magnetic quadrupoles28, O-site moments29 or magneto-electric multipoles generated dynamically through spin–phonon coupling30. Regardless of microscopic origin, our results suggest that this order undergoes a quantum phase transition inside the superconducting dome slightly beyond optimal doping, which may be responsible for the enhanced Tc and quasiparticle mass31 observed in its vicinity. Interestingly, a similar odd-parity magnetic phase has also recently been found in the pseudogap region of a 5d transition metal analogue of the cuprates32, which hints at a possibly more robust connection between the pseudogap and this unusual form of broken symmetry.


Material growth.

YBa2Cu3Oy single crystals were grown in non-reactive BaZrO3 crucibles using a self-flux technique. The Cu–O-chain oxygen content was set to y = 6.67, 6.75, 6.92 and 7.0 by annealing in a flowing O2/N2 mixture and homogenized by further annealing in a sealed quartz ampoule, together with ceramic at the same oxygen content. The absolute oxygen content (y) is accurate to ±0.01 based on iodometric titration. The crystals used in our experiments were de-twinned and aligned to high accuracy by X-ray Laue diffraction.

Optical RA measurements.

The RA measurements were performed using a rotating optical grating-based technique12 with ultrashort (80 fs) optical pulses produced from a regeneratively amplified Ti:sapphire laser operating at a 10 kHz repetition rate. The angle of incidence was 30° and the incident fluence was maintained below 1 mJ cm−2 to ensure no laser-induced changes to the samples (see Supplementary Section 7). Alignment of the optical axis to the crystallographic c axis was determined to better than 0.1° accuracy as described in Supplementary Section 8. Reflected fundamental (λ = 800 nm) and SH light (λ = 400 nm) were collected using photodiodes and photomultiplier tubes respectively. Crystals were sealed in a dry environment during transportation and immediately pumped down to pressures <5 × 10−6 torr for measurements.

Fitting procedure.

The high-temperature (T > TΩ) linear optical anisotropy data were fitted to the expression and the high- (T > TΩ) and low-temperature (T < TΩ) SH anisotropy data were fitted to the expressions  and respectively. Here A is a constant determined by the experimental geometry, and are the polarizations of the incoming and outgoing light in the frame of the crystal, χijED is the linear electric-dipole susceptibility tensor, χijklEQ is the SH electric-quadrupole susceptibility tensor and χijk, αED is the SH electric-dipole susceptibility tensor. The α = 1 and α = 2 versions of χijk, αED are related by 180° rotation about the c axis. The non-zero independent elements of these tensors in the frame of the crystal are determined by applying the monoclinic crystal symmetries to χijED and the appropriate point group symmetries (2/m for χijklEQ and either 2′/m or m1′ for χijk, αED) and degenerate SH permutation symmetries to χijklEQ and χijk, αED. These operations reduce χijED to 5 non-zero independent elements (xx, xz, yy, zx, zz), χijklEQ to 28 non-zero independent elements (xxxx, yyyy, zzzz, xxyy = xyyx, yyxx = yxxy, xyxy, yxyx, xxzz = xzzx, zzxx = zxxz, xzxz, zxzx, yyzz = yzzy, zzyy = zyyz, yzyz, zyzy, xzyy = xyyz, xyzy, yxzy = yyzx, yxyz = yzyx, yyxz = yzxy, zxyy = zyyx, zyxy, zxxx, xxzx, xxxz = xzxx, xzzz, zzxz, zzzx = zxzz), and χijk, αED to 10 non-zero independent elements (xxx, xyx = xxy, xyy, xzz, yxx, yyx = yxy, yyy, yzz, zzx = zxz, zzy = zyz). Values of R2 > 0.95 were achieved for all fits shown. However, the fitted values of the susceptibility tensor elements are not unique. Therefore, the fits serve only to address the qualitative question of whether a certain point group can or cannot reproduce the data. They are not intended to convey any quantitative information about the magnitudes of various tensor elements.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.