Abstract
The phase diagram of cuprate hightemperature superconductors features an enigmatic pseudogap region that is characterized by a partial suppression of lowenergy electronic excitations^{1}. Polarized neutron diffraction^{2,3,4}, Nernst effect^{5}, terahertz polarimetry^{6} and ultrasound measurements^{7} on YBa_{2}Cu_{3}O_{y} suggest that the pseudogap onset below a temperature T^{∗} coincides with a bona fide thermodynamic phase transition that breaks timereversal, fourfold rotation and mirror symmetries respectively. However, the full point group above and below T^{∗} has not been resolved and the fate of this transition as T^{∗} approaches the superconducting critical temperature T_{c} is poorly understood. Here we reveal the point group of YBa_{2}Cu_{3}O_{y} inside its pseudogap and neighbouring regions using highsensitivity linear and secondharmonic optical anisotropy measurements. We show that spatial inversion and twofold rotational symmetries are broken below T^{∗} while mirror symmetries perpendicular to the Cu–O plane are absent at all temperatures. This transition occurs over a wide doping range and persists inside the superconducting dome, with no detectable coupling to either charge ordering or superconductivity. These results suggest that the pseudogap region coincides with an oddparity order that does not arise from a competing Fermi surface instability and exhibits a quantum phase transition inside the superconducting dome.
Main
The crystal system and point group of a material are encoded in the structure of its second and higherrank optical susceptibility tensors^{8}, which can be determined from the anisotropy of its linear and nonlinear optical responses. The secondharmonic (SH) response is particularly sensitive to the presence of global inversion symmetry because unlike the linear electricdipole susceptibility tensor χ_{ij}^{ED}, which is allowed in all crystal systems, the SH electricdipole susceptibility tensor χ_{ijk}^{ED} vanishes in centrosymmetric point groups, leaving typically the far weaker electricquadrupole term χ_{ijkl}^{EQ} as the primary bulk radiation source^{9,10}. For these reasons, it has been proposed that optical SH generation may be an effective probe of inversion symmetry breaking in the cuprates^{11}.
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 detwinned single crystals of YBa_{2}Cu_{3}O_{y} as a function of oxygen content and temperature. The RA measurements track variations in the intensities of light reflected at the fundamental (I^{ω}) and SH frequencies (I^{2ω}) of an obliquely incident laser beam, which is resonant with the O 2p to Cu 3d charge transfer energy (ℏω = 1.5 eV), as the scattering plane is rotated about the c axis (Fig. 1a). The use of a recently developed rotating optical gratingbased technique^{12} 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.
We first simulate the RA patterns expected from holedoped YBa_{2}Cu_{3}O_{y} (y > 6) based on its reported orthorhombic mmm crystallographic point group^{13}, which is inversion symmetric and consists of three generators m_{ac}, m_{bc} and m_{ab} that denote mirror symmetries across the a–c, b–c and a–b planes. Unlike its tetragonal parent (y = 6) compound^{14}, the point group of holedoped YBa_{2}Cu_{3}O_{y} endows the crystal only with twofold (C_{2}) rather than fourfold (C_{4}) 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 electricdipole and electricquadrupole approximations respectively, where I_{0} is the incident beam intensity and both the incident (in) and reflected (out) polarizations and are selected to be perpendicular (S_{in}–S_{out}) to the scattering plane (see Supplementary Section 1 for other polarization geometries). The orthorhombic crystal system is identifiable by an ovular I^{ω}(φ) pattern with maxima and minima aligned strictly along the a and b axes^{15} while the mmm point group is identifiable by a fourlobed I^{2ω}(φ) pattern that is mirror symmetric about the a and b axes.
Figure 2 shows linear and SH RA data from YBa_{2}Cu_{3}O_{y} with holedoping levels (p) of 0.125 (y = 6.67; underdoped; T_{c} = 65 K), 0.135 (y = 6.75; underdoped; T_{c} = 75 K), 0.165 (y = 6.92; optimal doped; T_{c} = 92 K) and 0.190 (y = 7.0; overdoped; T_{c} = 86 K) measured at room temperature (T > T^{∗}) in S_{in}–S_{out} geometry (see Supplementary Section 2 for all other geometries). In the linear RA data, we see that the anisotropy (χ_{xx}^{ED} − χ_{yy}^{ED}) 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 m_{ac} and m_{bc} symmetries consistent with a monoclinic distortion. In fact, by using the structure of χ_{ij}^{ED} for a monoclinic crystal system in the expression for I^{ω}(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 m_{ac} and m_{bc} symmetries consistent with the linear RA data. We find excellent agreement of the data with an electricquadrupoleinduced SH response from the centrosymmetric 2/m monoclinic point group (Fig. 2), which consists of two generators 2 and m that denote C_{2} and m_{ab} symmetries. On the contrary, fits to electricdipoleinduced SH from the noncentrosymmetric 2 and m monoclinic point groups, as well as to magneticdipoleinduced^{11} or surface electricdipoleinduced SH from the 2/m point group, do not adequately describe the data (see Supplementary Section 3).
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 m_{ac} and m_{bc} symmetries at all measured temperatures above T^{∗} (see Supplementary Section 4) originates from vacancyinduced monoclinic distortions of the oxygen sublattice, which are also known to be present in La_{2−x}Sr_{x}CuO_{4} (ref. 16). This interpretation is further supported by recent SH RA measurements performed near the charge transfer resonance of layered perovskite iridates^{9,10}, which revealed subtle oxygen sublattice distortions that are difficult to resolve using diffractionbased probes.
Having established the point group of holedoped YBa_{2}Cu_{3}O_{y} above T^{∗}, we proceed to search for changes in symmetry across the strange metal to pseudogap boundary. Previous infrared conductivity experiments^{17} have shown that optical transition rates at frequencies well above the pseudogap energy scale (ℏω ≳ 0.3 eV) do not exhibit any measurable temperature dependence across T^{∗}. As a corollary, any temperaturedependent change in magnitude of χ_{ij}^{ED} or χ_{ijkl}^{EQ} at the frequencies used in this study (ℏω = 1.5 eV and 2ℏω = 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 S_{in}–S_{out} 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 orderparameterlike upturn below a dopingdependent critical temperature T_{Ω}. This dichotomy between the linear and SH responses can naturally be reconciled if bulk inversion symmetry is broken below T_{Ω}, which would turn on a new and stronger source of electricdipoleinduced SH radiation on top of the already existing electricquadrupole contribution.
By plotting the doping dependence of T_{Ω} atop the phase diagram of YBa_{2}Cu_{3}O_{y} (Fig. 3e), we find that the observed onset of inversion symmetry breaking coincides very well with the pseudogap phase boundary T^{∗} defined by spinpolarized neutron diffraction^{2,3,4}, Nernst anisotropy^{5}, terahertz polarimetry^{6} and resonant ultrasound^{7} 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 vanish^{18}. Interestingly, the temperature dependence of the SH intensity shows no measurable anomalies after crossing either the superconducting or the charge density wave ordering^{19,20} temperatures (Fig. 3e) and does not rapidly diminish at lower temperatures like the Nernst anisotropy^{5}, which may be caused by charge density wave ordering^{21}. This shows that unlike the superconducting and chargeordered phases, which are competing Fermi surface instabilities^{22}, the observed inversionsymmetrybroken 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 T_{Ω}. Figure 4a shows RA patterns measured at 295 K and 30 K in S_{in}–S_{out} geometry for the optimal doped sample (T_{Ω} ∼ 110 K), which is representative of all other doping levels studied (see Supplementary Section 5). Below T_{Ω} we see that the SH intensity is enhanced in a φdependent manner that preserves the twofold rotational symmetry of the pattern, which ostensibly implies that C_{2} is preserved in the crystal. However, this interpretation can be ruled out because the inferred above electricdipoleinduced SH radiation is strictly forbidden by symmetry in S_{in}–S_{out} geometry for any point group (nonmagnetic or magnetic) that contains C_{2} (see Supplementary Section 3). The only alternative interpretation is that the order parameter below T_{Ω} breaks C_{2} 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 C_{2} breaking below T^{∗} in Nernst effect^{5} and terahertz polarimetry^{6} data as well as the need for multiple magnetic domains to refine the polarized neutron diffraction data^{2,3,4,23}, which are all integrated over even larger areas of the crystal compared with our measurements.
The set of all noncentrosymmetric subgroups of 2/m that do not contain C_{2} consists of the two independent magnetic point groups 2′/m and m1′ and their associated magnetic and nonmagnetic subgroups (see Supplementary Section 6), where the generators 2′, m and 1′ denote C_{2} combined with timereversal, m_{ab} and the identity operation combined with timereversal respectively. Using the twodomain (α = 1, 2) averaged expression , we were able to reproduce all features of the lowtemperature SH RA data, including the small peaks around the b axis (Fig. 4a), by applying the structure of χ_{ijk, α}^{ED} for either the 2′/m or m1′ point group. The decomposition of the fit into its two singledomain components is shown in Fig. 4b to explicitly illustrate the loss of C_{2}. An equally good fit to the data can naturally be achieved using any magnetic or nonmagnetic subgroup of 2′/m or m1′ because they necessarily allow the same or more nonzero independent χ_{ijk, α}^{ED} tensor elements. Therefore, while further removal of symmetry elements from 2′/m or m1′ is not necessary to explain the lowtemperature data, it cannot be completely ruled out.
In conclusion, our results show that the pseudogap region in YBa_{2}Cu_{3}O_{y} is bounded by a line of phase transitions associated with the loss of global inversion and C_{2} symmetries. Although previous terahertz polarimetry measurements on holedoped YBa_{2}Cu_{3}O_{y} thin films reported the onset of a linear dichroic response near T^{∗} that breaks m_{ac} and m_{bc} symmetries^{6}, 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 stripe^{24} or nematic^{5} type orders alone, which have also been reported to develop below T^{∗}. Instead it suggests the presence of an oddparity magnetic order parameter, which is consistent with theoretical proposals involving a ferroic ordering of current loops circulating within the Cu–O octahedra^{11,25,26,27}, local Cusite magnetic quadrupoles^{28}, Osite moments^{29} or magnetoelectric multipoles generated dynamically through spin–phonon coupling^{30}. 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 T_{c} and quasiparticle mass^{31} observed in its vicinity. Interestingly, a similar oddparity magnetic phase has also recently been found in the pseudogap region of a 5d transition metal analogue of the cuprates^{32}, which hints at a possibly more robust connection between the pseudogap and this unusual form of broken symmetry.
Methods
Material growth.
YBa_{2}Cu_{3}O_{y} single crystals were grown in nonreactive BaZrO_{3} crucibles using a selfflux technique. The Cu–Ochain oxygen content was set to y = 6.67, 6.75, 6.92 and 7.0 by annealing in a flowing O_{2}/N_{2} 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 detwinned and aligned to high accuracy by Xray Laue diffraction.
Optical RA measurements.
The RA measurements were performed using a rotating optical gratingbased technique^{12} 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 laserinduced 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 hightemperature (T > T_{Ω}) linear optical anisotropy data were fitted to the expression and the high (T > T_{Ω}) and lowtemperature (T < T_{Ω}) 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, χ_{ij}^{ED} is the linear electricdipole susceptibility tensor, χ_{ijkl}^{EQ} is the SH electricquadrupole susceptibility tensor and χ_{ijk, α}^{ED} is the SH electricdipole susceptibility tensor. The α = 1 and α = 2 versions of χ_{ijk, α}^{ED} are related by 180° rotation about the c axis. The nonzero independent elements of these tensors in the frame of the crystal are determined by applying the monoclinic crystal symmetries to χ_{ij}^{ED} and the appropriate point group symmetries (2/m for χ_{ijkl}^{EQ} and either 2′/m or m1′ for χ_{ijk, α}^{ED}) and degenerate SH permutation symmetries to χ_{ijkl}^{EQ} and χ_{ijk, α}^{ED}. These operations reduce χ_{ij}^{ED} to 5 nonzero independent elements (xx, xz, yy, zx, zz), χ_{ijkl}^{EQ} to 28 nonzero 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 χ_{ijk, α}^{ED} to 10 nonzero independent elements (xxx, xyx = xxy, xyy, xzz, yxx, yyx = yxy, yyy, yzz, zzx = zxz, zzy = zyz). Values of R^{2} > 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.
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Acknowledgements
We thank D. N. Basov, P. Bourges, B. Keimer, S. A. Kivelson, P. A. Lee, J. W. Lynn, J. Orenstein, S. Raghu, B. Ramshaw, C. Varma and N.C. Yeh for valuable discussions. This work was supported by ARO Grant W911NF1310059. Instrumentation for the RA measurements was partially supported by ARO DURIP Award W911NF1310293. D.H. acknowledges funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (PHY1125565) with support of the Gordon and Betty Moore Foundation through Grant GBMF1250. N.P.A. acknowledges support from ARO Grant W911NF1510560. Work at the University of British Columbia was supported by the Canadian Institute for Advanced Research and the Natural Science and Engineering Research Council.
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L.Z., D.H. and N.P.A. planned the experiment. L.Z. and C.A.B. performed the RA measurements and N.P.A. determined the crystal alignment. L.Z., D.H. and N.P.A. analysed the data. R.L., D.A.B. and W.N.H. prepared and characterized the samples. L.Z. and D.H. wrote the manuscript.
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Correspondence to D. Hsieh.
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Zhao, L., Belvin, C., Liang, R. et al. A global inversionsymmetrybroken phase inside the pseudogap region of YBa_{2}Cu_{3}O_{y}. Nature Phys 13, 250–254 (2017). https://doi.org/10.1038/nphys3962
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