After thirty years discovery of high-T_c cuprate superconductor, their physical mechanism is still in suspense. As we know, the parent materials of high-T_c cuprates are widely accepted as Mott insulator in which all the electrons are localized by on-site Coulomb interaction U. When holes are doped into the two-dimensional CuO₂ plane, the Mott insulating phase is converted into a superconducting phase with appropriate hole concentration.^[1] However, besides superconducting order, the doped holes could have tendency to build up other symmetry-breaking state which competes with the d-wave superconducting order.^[2–7] In fact, there is indeed a very complicated doping dependent phase diagram in high-T_c cuprates^[8] as shown in Fig. 1. One thread to decode the high-T_c puzzle is whether the competition with a symmetry-breaking electronic state underlied high-T_c superconductivity plays a key role on the complexity of high-T_c cuprates. Several experiments are suggestive of electronic-liquid-crystal phases,^[9–19] of which an extreme version is ‘stripe order’: Charge filaments are embedded in a background of ordered spins as shown in Fig. 2. The first direct evidence of stripe order in a cuprate dates back to the discovery of stripes in La_1.48Nd_0.4Sr_0.12CuO₄.^[20] Later, the local evidence for the charge stripe order was found by scanning tunneling microscopy (STM) in Bi-based cuprates,^[21] in which the atomically flat surface enabled the real space image of charge stripe order in this system.^[21–23]

Fig. 1.

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	Fig. 1. Electronic phase diagram of high-Tc cuprates (from Ref. [8]).

Fig. 2.

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	Fig. 2. Stripe-like electronic order in cuprates (from Ref. [24]).

Although stripe order has been definitely confirmed in high-T_c cuprates,^{[20–23,25–33]} whether it is an intrinsic phenomenon to all of high-T_c cuprates is under strong debate. One critical argument against stripe is that static stripe order was just found in doped La₂CuO₄ family which is very disorder.^[1] Searching stripe order in other cleaner cuprate families is essential to clarify the role of stripe physics in high-T_c cuprates.

In 2007, the observation of quantum oscillation in high-quality underdoped YBa₂Cu₃O_6+x single crystal shed light on this issue.^[34–36] This breakthrough suggests that, at least under high magnetic field, certain symmetry broken order emerges to reconstruct the Fermi surface and produces closed Fermi pocket corresponding to quantum oscillation.^[14–16] It shows great potential to search stripe order in such very clean and “90 K” cuprate family. A new legend on stripe is coming. In 2011, stimulated by quantum oscillation experiment, high-field ⁶³Cu nuclear magnetic resonance (NMR) experiment successfully found stripe order in the underdoped YBa₂Cu₃O_6+x.^[37] It was found that while superconductivity is suppressed by external magnetic field, stripe order eventually emerges instead of superconducting order. In sharp contrast to stripe order observed in other cuprates, the stripe found in underdoped YBa₂Cu₃O_6+x is purely charge stripe without following spin order. Benefit from the absence of spin order, this is the first time to successfully detect charge stripe by high-field NMR technique in cuprates. Moreover, due to oxygen ordering structure, the filled CuO chain was found to be able to pin the charge-rich part of charge stripe, which allows ⁶³Cu and ¹⁷O NMR to distinguish one-dimensional (1D) charge stripe from other charge order patterns. In the following section, we would firstly explain how the NMR technique detects a charge modulation in a material.

In condensed matter physics, charge density wave (CDW) or charge order (CO) is a spatial modulation of electronic charge density with specific periodicity and directions (q-dependent) either commensurate or incommensurate with background lattice. As a spontaneous symmetry broken state, it generates a super-lattice structure which locally modifies the charge environment deviating from a uniform distribution. Such modulations on electronic degree of freedom are also seen by nuclei through electron–nucleus hyperfine interactions. Both static and dynamical properties of such phases can be sensitively detected by solid state nuclear magnetic/quadrupole resonance (NMR/NQR) measurements.

In NMR spectroscopy measurements, the frequency shift, line broadening, or line splitting of the NMR/NQR spectra are key quantities to characterize CDW/CO. Unlike the conventional diffraction methods, as a sensitive local probe, NMR/NQR could be done with extremely high energy and position resolutions when using high-quality single crystals. In NMR spectroscopy, Knight shift (K) measures the percent shift of the resonance frequency from an isolated nucleus in an external magnetic field (H_ext) with barely Zeeman splitting shown in Fig. 3 (I = 3/2 nuclei is taken as an example). Actually, Knight shift can be taken as an additional effective field experienced by the nuclei in crystals which is originated from the electron–nucleus hyperfine interaction. It contains detail information about the interaction. Generally, the magnetic hyperfine interaction between electron and nucleus is described as^[38] 1 The first term and second term describe the dipolar interactions between the nuclear spin and the electronic spin (with p-, d-, f-orbital characters) and the Fermi contact contributions (with s-orbital characters), respectively. The last term represents the orbital interaction between the nuclear spin I and the angular momentum L of the electron which usually contributes a temperature independent shift K_orb relative to Zeeman splitting. The first two terms contribute the spin part of the Knight shift K_spin which is usually temperature dependent and directly related to the electronic configurations near E_F and sensitively affected by even subtle reconstructions in the electronic structure. It can be rewritten as 2 where A_s is the hyperfine coupling constant, and χ_s is the uniform local spin susceptibility. In the CDW/CO state, the spatially modulated conduction electron density ρ(r) will consequently produce a local modulation of K_spin(r) by changing χ_s. Assuming a linear correlation between the CDW amplitude and the local resonance frequency ω_i at a nuclear site r_i, one obtains^[39] 3 where ω₀ = ω_ref(1 + K₀), K₀ is the Knight shift corresponding to uniform electron density. ω₁ is the modulation amplitude due to the conduction electron density of states which is assumed to be proportional to the CDW amplitude. q_j are the CDW wave vectors which can be commensurate or incommensurate with the lattice constants. ϕ_j are the CDW phases. This kind of NMR spectra analysis has been widely used to detect CDW states as shown in Fig. 4.

Fig. 3.

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	Fig. 3. Sketch of the energy level and the NMR spectra of nucleus with I = 3/2. The Knight Shift aroused by electron–nucleus hyperfine interaction is involved while without consideration of quadrupolar interactions.

Fig. 4.

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	Fig. 4. The 93Nb NMR spectra of NbSe3 at 24 K, 13 K, and 9 K. The solid line represents the theoretical fitting, using the same equation with different physical cases (from Ref. [39]).

Besides the Knight shift modulated by CDW/CO, the quadrupole nuclei (nuclear spin I > 1/2) with non-cubic symmetrical environment (V_zz ≠ 0) is often taken as a particular local probe of such state since it reveals the sign, magnitude, symmetry, and orientation of the electron field gradient tensor (EFG).^[40] The quadrupolar Hamiltonian H_Q which describes the interactions between the quadrupole moment Q of the nucleus and the electric field gradient V is expressed as 4 where . V_zz can be chosen as the maximum component of diagonalized EFG elements along the principle axis (|V_zz| ≥ |V_yy| ≥ |V_xx|). η is the asymmetry parameter defined as (V_yy| – |V_xx|)/V_zz. For a nuclear spin I = 3/2 in zero magnetic field, the NQR frequency is 5 With axial symmetry (η = 0), ω_NQR = ω_q as sketched in Fig. 5. Through implementation of NQR measurements, the magnitude or periodicity of electronic density modulation and the asymmetry properties may be elucidated. When the CDW/CO sets in, beyond the lattice EFG experienced by the nuclei (Fig. 6), the modulated electron density generating q-dependent spatially non-equivalent sites other than the structural sites is the key of these experiments.

Fig. 5.

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	Fig. 5. Sketch of the energy level and the NQR spectra of nucleus with I = 3/2.

Fig. 6.

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	Fig. 6. Comparison between the observed and theoretical 75As NQR lineshapes in proustite (from Ref. [41]).

One can also detect the quadrupolar shift of the NMR spectra to extract more comprehensive parameters on the electronic charge distribution in CDW/CO states as illustrated in Fig. 4. The total Hamiltonian H_t of quadrupolar nuclei in an arbitrary external magnetic field is 6 where H₀ = H_z + H_hf, the intrinsic Knight shift is contained in , since the external magnetic field determines a specific direction for the nuclear spin system (I_z), the quadrupolar Hamiltonian will be angular dependent. With the definition of Euler angles, one can easily derive the angular dependent quadrupolar Hamiltonian which commutes with I_z^[42] 7 where θ is the polar angle of the external magnetic field in the principle axis of the EFG tensor, and ϕ is the azimuthal angle. Note that under the high magnetic field condition (H₀ ≫ H_Q), H_Q can be taken as a perturbation, the first order correction will give the generation of satellite peaks. e.g., the case of I = 3/2 as shown in Fig. 7. 8 Since the shift brought to the central-transition line only exists in higher order corrections, through detection of the shift or linewidth of the satellites and comparing it to the central-transition line, one can identify the quadrupolar contributions from the magnetic one. By measuring the angular dependent spectra of high quality single crystals, one can get the detail parameters of the CDW/CO phases such as the EFG tensors, the degree of homogeneity/inhomogeneity, the magnetic shift contributions, and so on.

Fig. 7.

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	Fig. 7. Sketch of the energy level and the NMR spectra of nucleus with I = 3/2 with only consideration of quadrupolar interactions.

In this section, we will briefly review the high-field NMR progress on charge stripe in underdoped YBCO.

As we mentioned above, stimulated by quantum oscillation experiment in 2007, searching symmetry-breaking state under high magnetic field becomes an urgent task in high-T_c community. However, many important techniques are not available under high magnetic field, such as neutron scattering and angle resolved photoemission spectroscopy (ARPES). Therefore, NMR technique becomes a unique probe to detect symmetry-breaking state under high magnetic field. At the end of 2009, the Grenoble NMR group started the exploration on symmetry-breaking state in underdoped YBCO with ortho-II ordering structure at Grenoble high magnetic field lab. As shown in Fig. 8, the ortho-II ordering structure leads to two distinct planar Cu NMR sites: Cu2F are those Cu located below oxygen-filled chains and Cu2E those below oxygen-empty chains.^[43] As shown in Fig. 9, these two distinct planar Cu sites could be easily resolved on quadrupole satellite of ⁶³Cu spectra. Such site-selective NMR gives us a possibility for spatial resolution of the following charge ordering state. The main finding in the first high magnetic field NMR experiment is shown in Fig. 9. The satellite for Cu2F site splits into two peaks below T_c under the high magnetic field along c-axis. At the same time, there is also a Knight shift splitting on the central line of Cu2F. Based on a detailed NMR analysis, both of quadrupole and Knight shift splittings are ascribed to the emergence of charge ordering state as we learned above. If we used the quadrupole splitting as an order parameter, its temperature dependent evolution looks like a second order phase transition. On the other hand, the above splittings on both Knight shift and quadrupole frequency are only found at Cu2F site but absent at Cu2E site as shown in Fig. 9.

Fig. 8.

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	Fig. 8. The crystal structure of ortho-II YBCO (from Ref. [43]).

Fig. 9.

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	Fig. 9. High-field NMR spectra of YBa2Cu3O6.54 (from Ref. [37]).

This result could be used to constrain the charge ordering pattern in space. As shown in Fig. 10, there are two kinds of possible charge ordering patterns that could fit the NMR results. However, the first one seems unphysical in real case. Then, the only plausible choice is the second one which is the same charge ordering pattern with a uniaxial 4a period as the previous stripe order found in LNSCO.^[20] But, in YBCO, there is no spin order coupled with charge order. This result has a strong implication on the quantum oscillation experiment, suggesting that the charge order reconstructs the Fermi surface and creates Fermi pockets in YBCO. Since the splitting due to charge ordering only happens at high magnetic field, it strongly suggests that the charge order is a field-induced phenomenon and would be hidden in the superconducting dome without applying any external field. Although such charge order only appears under high magnetic field, it is not simply induced by applying a magnetic field within CuO₂ plane as shown in Fig. 9. Because the upper critical field for the superconducting state is highly anisotropic in cuprates, the superconducting state is quite robust under magnetic field within CuO₂ plane.

Fig. 10.

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	Fig. 10. Charge density modulations compatible with NMR spectra (from Ref. [37]).

Therefore, the field-induced charge order is actually caused by the suppression of superconducting state under the external magnetic field. It suggests that the charge order is a competing order of the superconducting state. A new phase diagram is proposed for YBCO as shown in Fig. 11, in which a field-induced charge order is peaked around 1/8 hole concentration and beneath the superconducting dome. This is a definite evidence for a long-range charge order in a clean cuprate superconductor family. It strongly suggests that charge order is an intrinsic phenomenon in high-T_c cuprates and it shows a strong correlation with the superconducting state.

Fig. 11.

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	Fig. 11. Phase diagram of underdoped YBa2Cu3Oy (from Ref. [37]).

Following the pioneering ⁶³Cu NMR experiment, the field-induced charge order was further verified by ¹⁷O NMR experiment.^[44] As shown in Fig. 12, charge order induced quadrupole splitting is also observed on the satellite of ¹⁷O NMR spectra. Due to more remarkable splitting, the precise field-dependent order parameter of charge order could be extracted from the field dependent splitting of the ¹⁷O NMR spectra. As shown in Fig. 13, the critical field of the field-induced charge ordering state shows a V-shape doping dependence corresponding to the doping dependent upper critical field of the superconducting state, which also suggests a close correlation between superconducting state and charge ordering state. In a short summary, both ⁶³Cu and ¹⁷O NMR spectra exhibit unambiguous and consistent evidence for a field-induced charge order in underdoped YBCO. Such field-induced charge order is a dominating competing order for superconductivity around 1/8 hole concentration and is hidden beneath the superconducting dome. When the superconducting state is killed by the external magnetic field, the competing charge order would appear through a second-order quantum phase transition, which is driven by the high magnetic field (Fig. 13).

Fig. 12.

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	Fig. 12. The 17O NMR evidence of charge order in YBa2Cu3O6.56 (from Ref. [44]).

Fig. 13.

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	Fig. 13. Quantum phase transition to the charge-ordered state (from Ref. [44]).

Following the discovery of NMR works, a fluctuating CDW has also been found far above T_c in (Y, Nd)Ba₂Cu₃O_6+x by resonant soft x-ray scattering even without applying external magnetic field.^[45] Two resonant peaks corresponding to CDW signal were observed at (0.31, 0) and (0, 0.31) of the in-plane reciprocal space. They are different from the commensurate value of 1/3 so imply an incommensurate CDW. The temperature-dependent evolution of CDW signal in YBa₂Cu₃O_6.6 compound is shown in Fig. 14. The CDW signal starts to increase and becomes narrow upon cooling below 150 K until T_c. The correlation length of CDW can be roughly estimated by the reciprocal of the linewidth, which reaches the maximal value of 16a at T_c. The authors claimed that the above behavior of correlation length is more relative to a fluctuating CDW rather than a static CDW. Below T_c, a sudden suppression of intensity happens, which implies a competition between CDW and superconductivity. An interesting detail is that the intensities of the above two CDW signals are unequal. This brings about a question of whether the CDW is a single one with bi-axis or two uniaxial CDWs in two orthorhombic domains.

Fig. 14.

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	Fig. 14. Temperature dependence of the CDW signal obtained by resonant soft x-ray scattering in YBa2Cu3O6.6 (from Ref. [45]).

Meanwhile, a similar result was obtained in another work with hard x-ray technique under external magnetic field up to 17 T. They found that the CDW under zero field can be enhanced by the magnetic field below the superconducting transition temperature.^[46] The results of the hard x-ray experiment in three-dimensional reciprocal space are shown in Fig. 15. Two diffraction peaks due to CDW appear at the same in-plane wave vectors as the previous work. The rather wide lineshape in reciprocal space means a short-range modulation along c-axis with 0.6 lattice unit, which suggests a 2D nature for CDW. Above T_c, there is no any magnetic field effect on CDW signal while below T_c the CDW signal shows a remarkable enhancement with the application of the magnetic field. This is a direct evidence for the competition between CDW and superconductivity. Under the magnetic field of 17 T, the in-plane correlation length reaches ∼95 Å (∼25 lattice units) at 2 K, which is still a short-range CDW. Here, it should be noted that the onset temperature of CDW observed by x-ray technique is much higher than that determined by NMR technique. One plausible reason is different timescales. Usually, the timescale for x-ray technique is much shorter than that for NMR technique. Therefore, although a fluctuating CDW might be absent in a NMR timescale, it can be still observed by the x-ray technique. In contrast, another opinion on this issue is that there are two different CDW phases.^[47] Next, we will discuss this possibility with more details.

Fig. 15.

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	Fig. 15. Incommensurate charge-density-wave order (from Ref. [46]).

Fig. 16.

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	Fig. 16. Competition between charge-density-wave order and superconductivity (from Ref. [46]).

Wu et al. carefully studied the low-field NMR spectra and found evidence for another short-range CDW different from the one under high magnetic field.^[48] As shown in Fig. 17, the broadening effects of satellites and central line show a similar temperature-dependent behavior but with distinct magnitudes upon cooling below T_onset ≈ 140–170 K. From the previous section, we have learnt that the satellites are broadened by both electric quadrupole and magnetic contributions while the central line is only affected by the magnetic contribution. When a short-range CDW appears, although both the quadrupole broadening δν_quad and the magnetic broadening δν_magn should be taken into account for the linewidth, only the magnetic contribution appears on both the central line and the satellites with the same magnitude. In this case, the magnitude of the broadening effect for satellites and central line should be different, just as what we see in Fig. 17. Similar situation is also observed for NMR spectra splitting at the long-range CDW state under high magnetic field. It should be noted that in contrast to that the splitting only appearing at Cu2F sites,^[37] the above broadening exists at both Cu2E and Cu2F sites, which also supports a 2D nature. Based on these results, the difference between x-ray and NMR results strongly suggests two distinct CDW phases in the H–T phase diagram. The field-induced 3D CDW phase under high magnetic field causes a remarkable splitting of NMR lines. The other 2D CDW phase under zero magnetic field is related to the significant quadrupole broadening of the NMR statellites. These two CDW phases have different charge modulation patterns. Therefore, it is quite clear that there are two different CDW phases in the H–T phase diagram. The following part will discuss the relationship between these two distinct CDW phases. In addition, since the broadenings of O sites along different Cu–O bonds show different values as shown in Figs. 17(a) and 17(b), the above NMR results due to a short-range CDW phase also suggest a possible intra-unit-cell nematic order, which is consistent with the proposal of biaxial CDW debated in previous x-ray work.^[45]

Fig. 17.

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	Fig. 17. Qualitative evidence of static charge modulation (from Ref. [48]).

Gerber et al. presented a pretty work with magnetic field up to 28 T by method of x-ray scattering at the x-ray free electron laser (FEL) whose high brilliance enabled detection of weak CDW signals.^[49] The previous work was reproduced that the short-range correlated incommensurate 2D CDW order develops to maximum at T_c without external field. Thanks to the synchronization of millisecond pulsed magnetic field and femtosecond x-ray FEL pulses in their measurements, an intriguing low temperature 3D CDW order is revealed under high fields (> 15 T). The comparison of the temperature dependent evolution between these two CDW features is shown in Fig. 18. Instead of centering at l ∼ 0.5, the new revealed 3D CDW locates in l ∼ 1 in reciprocal space and is significantly enhanced with increasing magnetic field. As shown in Fig. 18(c), the temperature dependent intensity of this 3D CDW is consistent with the early high field NMR measurements, suggesting a same origination.^[37] The two CDWs coexist at high field and have the same incommensurate in-plane components of the Q-vector (Fig. 19). While, the correlation length of the field induced one is over ∼180 Å alone b-axis and ∼50 Å (4–5 lattice units) alone c-axis at 28 T (Fig. 19). Such long correlation length indicates its 3D nature and rules out the possible origination of this CDW as a result of field aligned CDW regions associated with vortices.

Fig. 18.

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	Fig. 18. Temperature dependence of the CDW order at H = 0, 20 T (from Ref. [49]).

Fig. 19.

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	Fig. 19. Field dependence of the CDW order at T = 10 K (from Ref. [49]).

Chang et al. have carried out systematic high field hard x-ray scattering experiments on YBa₂Cu₃O_6+x systems which put forward the study of the detail structure of the CDW order and explored the doping, field, and temperature dependent phase diagram of these systems.^[50] As shown in Fig. 20, they confirmed the sudden appearance of an additional 3D CDW order at high fields in YBa₂Cu₃O_6.67 (hole doping ∼ 0.123). They also found that the effects of applying magnetic fields are very different on the two in-plane components (q_a, q_b) of the CDW. With increasing magnetic field, the q_a correlation simply becomes stronger without any extra modulations. While for the q_b correlations, significant enhancement of the correlation length ξ_b is observed and accompanied with an l∼1 scattering peak emerged when the magnetic field is beyond 10 T. As shown in Fig. 21(a), the 3D CDW breaks the mirror symmetry of an individual CuO₂ bilayer which is the same as the lower field one, but the atomic displacements in adjacent bilayers are in phase. As sketched in Figs. 21(b)–21(d), the field induced CDW can be weakly pinned by impurities in some regions of the sample to form fully 3D coherence and static with time scale > 0.1 ms as inferred from the NMR experiments.^[37,48] The identified new field induced anisotropy in the CDW also answers the question mentioned above, that there are two in-plane uniaxial orthorhombic domains at high fields. All these suggest a field induced phase transition in YBCO, which implies that the magnetic field would enlarge the in-plane anisotropies of the CDW through enhancement of the correlations alone b-axis and in turn modify the coupling between the CuO₂ bilayers (Fig. 22(a)). Besides, from the doping–magnetic-field phase diagram (Fig. 22(b)), the 3D CDW order is most easily stabilized for doping around p = 0.11–0.12. Most recently, a quantum critical point (QCP) at p = 0.08 where the CDW can be formed with applying magnetic fields was revealed as shown in Fig. 23.^[51]

Fig. 20.

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	Fig. 20. Charge density wave correlations induced by a magnetic field in YBa2Cu3O6.67 (from Ref. [50]).

Fig. 21.

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	Fig. 21. Magnetic field effect on c-axis correlations in YBa2Cu3O6.67 (from Ref. [50]).

Fig. 22.

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	Fig. 22. Phase diagram of YBa2Cu3O6+x (from Ref. [50]).

Fig. 23.

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	Fig. 23. A sketch of the YBCO phase diagram in the heavily underdoped regime with several other measurements superimposed (from Ref. [51]).

At this stage, we shall draw a short summary about H–T phase diagram of YBCO.^[52] As shown in Fig. 24, at high temperatures, only short-range 2D CDW correlation has been observed. Under low field region, the superconductivity overcomes the 2D CDW order at T_c ∼ 60 K. As superconductivity is gradually suppressed by magnetic field, a sharp phase boundary emerges (∼15 T) where a 3D CDW order sets in. Under higher fields, the 3D CDW takes place of the superconductivity as the new ground state of the system. Intriguingly, the two kinds of CDW cannot simply convert into one another. They coexist with each other in different domains under high fields and low temperatures. CDW has been known to occur in many low-dimensional metals since last century. But how exactly the wave is triggered by cooperation between different degrees-of-freedom is still elusive and hot debated, especially in 2D systems. Cuprates are not only 2D metals, they are particularly complex ones with strongly correlated electrons and multi-order competing phase diagrams. To figure out the origin of the CDW is a hard theoretical task, much less its connection to superconductivity (though all experiments point to an evident competition between superconductivity and CDW order). Nevertheless, instructive insights from the experimental side are still expected. Some outstanding puzzles and effects are left to be explored, such as, what exactly is the role played by the inter-layer chain-oxygen defects? (only help to forming antiphase charge oscillations in two consecutive CuO₂ bilayers?). How about the pinning effects of the disorder? (stabilizing the 2D CDW fluctuating correlation into short-range order and help to establish fully 3D coherence of the 3D CDW order?). How the electronic structure and Fermi surface topology will evolve with CDW setting in? (reconstruction, new gap opening, and loss of density of states?). What will happen at even higher fields? (will two kinds of CDWs still coexist with each other or the 3D CDW finally overwhelms the 2D CDW even at high temperatures?). What is the relation between the CDW and the pseudogap? Can similar phenomena take place in other cuprate systems? With these thoughts in mind, next we will move on to the discussions of specific cases and gather some newest experimental progresses on part of these interesting issues.

Fig. 24.

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	Fig. 24. The H–T phase diagram of YBCO (from Ref. [52]).

Recently, pieces of important information about the microscopic nature of the CDW have been revealed by Zhou et al.^[53] Through meticulous analysis of the ¹⁷O NMR spectra in several YBa₂Cu₃O_y samples with different hole doping levels, they found an anomalous distribution of local density of states (DOS) on in-plane oxygen sites which results in asymmetric NMR spectra as shown in Fig. 25. By using bi-Gaussian fittings on the skewed peaks, they figured out that the asymmetry arises from the broadening of the right (high frequency) part of the line (ω_R in Figs. 26(a) and 26(b)). Meanwhile considering the continuous aspect of the line shapes, they inferred that ¹⁷K_spin is maximal at relatively few locations and decays over a typical distance much larger than a lattice step. To some extent, such Knight shift distribution is rare. For instance, the system is far away from any magnetic order and the vortex lattice if any is negligible at such high fields. Besides, staggered magnetization produced by oxygen vacancies in the chains or impurities in the planes only leads to symmetric broadenings. In fact, the interference between CDW and Friedel oscillations would yield an asymmetric charge-density distribution. While if so, an accompanied prominent quadrupolar effect which influences differently on the satellite and central lines should be observed. On the other hand, locally promoted DOS at the Fermi level would provide an explanation to this result. But, unlike the case in the presence of Anderson localization, the locally enhanced DOS in YBCO systems most likely originated from the bounded (in-gap) electronic states. After ruling out various types of alternative candidates such as the Andreev bound states, short-range fluctuating pairing-density-wave (PDW) states (none of these are static at NMR time scale), and the defect-induced bound states in the chains (inconsistent onset fields and missing in optimally-doped samples), the author finally ascribed the skewness of the peaks to the defect-induced bound states in the CuO₂ planes. Another remarkable character of this spatial distribution of local fields is that it seems arises only in conjunction with long-range CDW order as they share similar field and temperature dependent behavior which can be scaled with each other (Figs. 26(c) and 26(d)). As shown in Fig. 27, this kind of bound state widely exists in YBCO samples with different doping levels. Since the defects in the Cu–O chains constitute the main source of electronic scattering ubiquitously in oxygen-ordered YBCO, the bound states should be intimately related to it. While, from above analysis we know that the quasiparticle is not directly scattered by defects in the Cu–O chains, which leaves a paradox. Clarifying which aspect of the CDW is crucial in the formation of the bound states should be informative on its microscopic nature. An indication is that the c-axis coherence of the high filed 3D CDW order seems irrelevant to the formation of such bound states. Interestingly, the author pointed out that disorder induced bound states are widely observed among different sample systems (iron-based superconductors, heavy fermion systems, surface of topological insulators, etc) and they have been argued to be a generic property of metals with Dirac-type electronic dispersion.^[54–58] The presence of a Dirac cone in the band structure would then provide clues on the reconstructed Fermi surface in high fields.^[59]

Fig. 25.

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	Fig. 25. The 17O NMR spectrum of YBa2Cu3O6.56 at T = 3 K and H ≃ 28.5 T (from Ref. [53]).

Fig. 26.

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	Fig. 26. Field and temperature dependence of ωR, ωL, and calculated asymmetry (from Ref. [53]).

Fig. 27.

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	Fig. 27. O(2) and O(3) first low-frequency quadrupole satellites showing similar asymmetric profiles at T ≃ 2 K for four different samples (from Ref. [53]).

Another excellent work done by Zhou et al. is the microscopic determination of the upper critical field H_c2 of the YBCO systems.^[60] Using the ¹⁷O nuclei as a local probe, they can detect the field dependent intrinsic spin susceptibility χ_spin (mainly contributed from the N(E_F) (DOS at the Fermi level) at low temperature) through measurements of the Knight shift at varied fields. As shown in Fig. 28, a clear saturation of K_spin is observed at high fields and the kinks signal the arrival of the upper critical filed for each sample. Plotting the determined H_c2 data obtaining from different experimental methods on one graph shows a good consistence that H_c2 in YBa₂Cu₃O_y has a pronounced depression around p = 0.11–0.12 where it reaches values as low as ∼20 T (Fig. 29). After subtraction of orbital shift K_orb and diamagnetic shift K_dia, a linearly field dependent K_spin seems more physically to fit their data below H_c2. Following their analysis will lead to identification of a sizable residual K_spin: K₀ (K₀/K(H_c2) ∼ 30% ± 1%) at zero field which is to say a residual N(E_F). As shown in Fig. 30, the residual N(E_F) is positively related to the concentration of disorder. While, extrapolating this relation to zero-width limit (free of disorder) still yields nonzero N(E_F). This suggests an intrinsic N(E_F) preserved in the superconducting state, which agrees with the results of specific heat measurement.^[61] Note that, is supposed to be the general case in cuprates at T ≪ T_c as expected for the Doppler shift of the quasiparticles outside the vortex cores in d-wave superconductors. So the linearity between H and K_spin is elusive which asks for further studies. One possible explanation they proposed is that Zeeman energy of the nodal quasiparticles could exceed the Doppler energy when the high field induces changes in the Fermi surface. In their experiments, they did not see any sign of the existence of vortex solids at high fields (> 30 T), suggesting that vortex melting transition detected in other experiments may arises from some extrinsic effects, e.g., disorder induced heterogeneous regions with anomalously high H_c2. Intriguingly, they also estimated the relative magnitudes of the saturated K_spin at high fields and the K_spin at high temperature of a nearly optimally doped sample. The small DOS remaining in high fields (∼18%) demonstrates that the nonsuperconducting ground state has a large pseudogap separating from the superconducting gap. In addition, they also found that K_spin does not undergo any significant change at either (10–20 T) or (5–10 T higher than ). The absence of a decrease of K_spin at the CDW transition indicates that long-range CDW order does not produce substantial gapping of quasiparticle excitations. This means either that the Fermi surface reconstruction is totally unrelated to the CDW or that the short-range CDW modulations presenting at any field already deplete N(E_F) before the transition toward long-range order.

Fig. 28.

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	Fig. 28. Magnetic-field dependence of (the spin part of) 17O(3) Knight shift in four different samples (from Ref. [60]).

Fig. 29.

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	Fig. 29. Comparison of the determined Hc2 data obtained from different experimental methods (from Ref. [60]).

Fig. 30.

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	Fig. 30. Plotting the residual Kspin in the H = 0 limit (K0) with varied parameters (from Ref. [60]).

To take a step on further understanding of the electronic structure at the Fermi level in YBCO systems at high fields, Kačmarčík et al. made comprehensive NMR and specific heat investigations on under-doped YBa₂Cu₃O_y samples.^[62] As expected, their experimental results confirm the existence of a saturated DOS above a characteristic field H_DOS at low temperatures (Fig. 31). Furthermore, an important feature about H_DOS is that its temperature dependence H_DOS(T) displays a clear inflection when the field-induced long-range CDW order develops. As shown in Fig. 32, H_DOS(T) first increases with temperature lowering, then after a subsequent plateau it sharply increases below ∼10 K, tending towards H_c2 for T → 0, resulting in an S shape. At the same temperature range (< 10 K), an overshoot of specific heat (T/T_c ∼ 1/30) is observed which is very unusual. As mentioned above, the vortex melting field is significantly lower than H_DOS, which indicates that the saturation of the DOS cannot be related to the melting of the vortex solid or an accidental compensation of the decrement from promoting CDW and increment from suppression of superconductivity. In principle, the constant DOS could be due to entering into a new type of gapless superconducting state, but the author failed to find any theoretical supports. It is also worth noting that the phase diagrams of Fig. 32(b) and 32(c) leave little doubt that the unusual S shape of H_DOS(T) directly results from the influence of the 3D CDW order on superconductivity. The upswing of the H_DOS(T) line below ∼10 K suggests that superconductivity eventually finds a way to accommodate the presence of the 3D long-range CDW order. One would image that, high magnetic fields may help to establish a PDW state in which spatial variations of the superconducting- and CDW-order parameters are intertwined.^[63] Since it has been proposed that the 3D CDW is actually a consequence of a primary PDW order,^[64] the author claimed that the above-proposed interpretation of their data seems to fit more naturally with the view that the PDW order is stabilized only in the coexistence region by the presence of an independent 3D CDW order.

Fig. 31.

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	Fig. 31. Magnetic field dependence of the electronic specific heat, Ces(T, H)/T, and Kspin (from Ref. [62]).

Fig. 32.

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	Fig. 32. The HDOS versus temperature for the indicated doping contents (from Ref. [62]).

In the early days, dense studies on charge stripe order were focus on another high-T_c cuprates family — Bi-based cuprates.^[21–23,47] In fact, real-space imagination of CO by STM were firstly realized on Bi-based cuprates.^[21] Through applying an external magnetic field (5 T perpendicular to the CuO₂ plane), fourfold, bidirectional modulations of the local DOS can be clearly imaged in the vortex cores (Fig. 33).^[47] The observed CO is confirmed to be non-dispersive over an extended energy range and exists above 100 K which rules out the possible origination as a quasi-particle interference (QPI) pattern.^[22,23] Latter STM experiments on Na doped Ca₂CuO₂Cl₂ (Na-CCOC) revealed similar features (nearly period-4) of the CO in Bi-based cuprates.^[65] In recent years, the detail structure and doping dependent phase diagram of Bi2212 and Na-CCOC have been intensively studied.^[66–68] Evidence for charge order in single-layered Bi-based compounds, Bi_2–yPb_ySr_2–zLa_zCuO_6+δ (Bi2201), was also reported.^[69] Real and reciprocal space studies on the temperature evolution of CDW/CO in Bi2212 and Bi2201 systems have revealed possible connections between charge order and the pseudogap state.^[11,70] The doping dependent CDW wave vector seems to intimate relate to the hot spots of the Fermi surface by comparison between RXS and ARPES results.^[71,72] Previous Knight shift measurements also detect residual DOS in the pseudogap states at high fields and it shows a non-trivial carrier concentration dependence (an abrupt decrease when it is close to Mott transition which suggests a symmetry breaking without magnetic ordering).^[73] The underlying physics of the normal states are still unclear. Special density modulations of the CDW/CO are predicted in the PDW scenario.^[8,74]

Fig. 33.

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	Fig. 33. STM explorations of charge order in Bi-based cuprates (from Refs. [21] and [23]).

Besides, two major questions about the CDW in YBCO systems have been remained until now: one is that only out-of-plane field can induce the 3D CDW which leads to speculations that it is due to incipient CDW in the vortex cores that becomes overlapped as the field gets stronger, the other is the exact role played by the Cu–O chains. In order to get some insights on these questions, the high field investigations on CDW/CO in Bi2201 are highly needed. Kawasaki et al. made great experimental efforts on this subject. They have conducted NMR study on a couple of hole doped chainless systems — Bi2201 with in-plane magnetic fields up to 42.5 T.^[75] As shown in Fig. 34, comparing with YBCO, similar NMR features have been observed in under-doped Bi2201 samples above a characteristic field H_CDW (∼10 T), which demonstrate a field induced CDW (FICDW). The H_CDW is slightly lower than that in YBCO, suggesting that CDW has a similar energy scale across different class of cuprates. As shown in Fig. 35, with doping concentration, field, and temperature as tuning parameters, they obtained comprehensive phase diagrams of the Bi2201 system. There are several unique characters of the CDW in Bi2201 system. One remarkable point is that the onset temperature T_CDW takes over the antiferromagnetic order temperature T_N beyond a critical doping level at which superconductivity starts to emerge, and scales with the pseudogap temperature T*. Since no vortex cores are created in CuO₂ planes under in-plane fields, the H_CDW and T_CDW of Bi2201 are more related with the doping concentration, which is different from the case in YBCO where H_CDW scales with H_c2. Considering the single CuO₂ layered structure without any Cu–O chains in Bi2201, the observed CDW is inherently 2D and a larger CDW amplitude (around twice of that of YBCO) is realized. Interestingly, in their experiments, the short-range CDW sets in right at T* and the FICDW emerges far above the SC dome and finally coexists with it. Based on this, they proposed that the CDW in Bi2201 is closely related to the pseudogap (or to say the pseudogap is the incipient CDW in Bi2201).^[76,77] At the moment, there still remain many elusive experimental results. But, without any doubt that the CDW order is another outstanding quantum phenomenon that should be addressed on the same footing as the AF spin order.

Fig. 34.

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	Fig. 34. Field and temperature evolution of NMR satellite line for under-doped Bi2201 (from Ref. [75]).

Fig. 35.

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	Fig. 35. Comprehensive phase diagrams of Bi2201 system with doping concentration, field, and temperature as tuning parameters (from Ref. [75]).

Iron-based superconductors (IBS) are recognized as the second family of high-T_c superconductors.^[78–82] Although the IBS exhibits a similar phase diagram as cuprates, the parent compounds of IBS are found to be multi-band correlated metals with striped AFM ground state^[83] and the electronic correlation is strongly influenced by the Hund’s coupling.^[84] It would be very interesting to explore similar charge ordering phenomenon as cuprates in IBS, which would be important to build up a universal connection between high-T_c superconductivity and charge ordering.

Until now, although there are many efforts to explore charge ordering in IBS, the definitive evidence for charge ordering is still elusive. Initially, possible existence of CO states in slightly electron doped LaOFeAs_1−xF_x, SmOFeAs_1−xF_x (1111 systems), and optimally hole doped Ba_1−xK_xFe₂As₂ (122 system) was claimed.^[85–87] Lang et al. have conducted NQR study on series electron doped 1111 systems.^[85] They proposed a nanoscale electronic order on the Fe sites stimulated by observing a clear two-peak structure in ⁷⁵As NQR spectra of slightly doped LaOFeAs_1−xF_x and SmOFeAs_1−xF_x. Subsequently, Jasek et al. got similar conclusion through ⁵⁷Fe Mossbauer spectroscopy.^[86,87] In recent years, more evidences for possible CO state have also been accumulated in heavily hole doped IBS — AFe₂As₂ (A = K, Rb, Cs) systems.^[88–90] Wang et al. found an NMR line splitting with in-plane external field in pressurized KFe₂As₂ with P = 2.42 GPa, suggesting a pressure induced checkboard-type CO.^[88] Civardi et al. also found a possible CO state in RbFe₂As₂ under ambient pressure. Interestingly, in contrast to pressurized KFe₂As₂ and RbFe₂As₂, Li et al. found an NMR evidence for local B_2g nematic order in CsFe₂As₂,^[90] which might be also related to possible CO state. However, all these phenomena in AF₂As₂ family are very sensitive on sample quality and further careful study is needed.^[91] Lately, high-resolution ARPES study on hole doped 122 system (Na doped BaFe₂As₂) revealed a possible emergent charge order in the C₄ re-entrant phase.^[92] The band splitting at the X point only in the C₄ re-entrant phase suggests a checkerboard CO which is compatible and strong coupled with the double-Q magnetic order (DQMO) in the C₄ re-entrant phase. The results are reminiscent of the intertwined electronic stripe order appearing at 1/8 doping in the La_2−xBa_xCuO₄ cuprate system where the CO strongly couples to the spin order. In a short summary, although there are many intriguing phenomena relevant to the possible CO state in IBS, whether these phenomena are intrinsic or universal for IBS is still elusive. Further experiments are urgently needed to figure out the above issue in near future.

In this short review, we have introduced the recent high-field NMR progress on charge order in YBCO. Based on high-field NMR results, a field-induced charge order is discovered which is actually different from the following short-range charge order discovered by low-field x-ray scattering experiment. The implications of NMR results go beyond the microscopic explanation of quantum oscillation experiments. In correlated electrons system, quantum melting of electronic crystalline phase often gives rise to many novel electronic phases. In cuprate superconductors, melting the Mott insulating phase with carrier doping is believed to lead a quantum version of liquid crystal phase, the electronic nematicity, which breaks the rotational symmetry and exhibits a tight twist with high-T_c superconductivity. These NMR findings would be an important experimental evidence to support such physical scenario for high-T_c superconductivity. On the other hand, it would be an interesting work to search similar charge order in high-T_c Fe-based superconductors, which would be very important to unify both high-T_c families.