https://arxiv.org/abs/0801.1281
https://arxiv.org/abs/0707.0057
https://arxiv.org/abs/0707.4461
https://arxiv.org/abs/0711.3559

## Q. Read attached papers, summarize each, and explain how they relate to each other.

A. The attached papers all report on the observation of quantum oscillations in underdoped high-temperature cuprate superconductors. Here is a summary of each paper and how they relate:

1. Doiron-Leyraud et al. (Nature 2007) was the first report of quantum oscillations, specifically Shubnikov-de Haas oscillations, in an underdoped cuprate - the ortho-II ordered YBa2Cu3O6.5 with a Tc of 57.5K. They observed a single frequency of 530T corresponding to a small Fermi surface pocket comprising only 2% of the Brillouin zone area. This contrasts with the large cylindrical Fermi surface seen in overdoped cuprates. They propose the oscillations arise either from a reconstructed Fermi surface with small electron and hole pockets or a specific band structure property of YBa2Cu3O6.5.

2. Yelland et al. (arXiv 2007) then reported observation of quantum oscillations in another underdoped cuprate - stoichiometric YBa2Cu4O8 with Tc ~ 80K, using tunnel-diode oscillator measurements in fields up to 85T. They find a very similar frequency of 660T and effective mass of 3.0me. The similarity to YBa2Cu3O6.5 despite the lack of oxygen ordering suggests the small pockets are intrinsic to underdoped CuO2 planes rather than a material-specific band structure effect. Heat capacity constraints also argue against a nodal pocket scenario and support the reconstructed Fermi surface picture.

3. Bangura et al. (arXiv 2007) independently reported Shubnikov-de Haas oscillations in YBa2Cu4O8 with the same 660T frequency. Combining with band structure calculations, they conclude the oscillations cannot be explained by the conventional band structure but rather indicate a reconstructed Fermi surface - a generic property of underdoped cuprates. The pocket size scales with doping between YBa2Cu3O6.5 and YBa2Cu4O8.

4. Jaudet et al. (arXiv 2007) provide the first clear observation of de Haas-van Alphen oscillations in underdoped YBa2Cu3O6.5 using a torque technique. The frequency and mass exactly match the Shubnikov-de Haas values. Detailed analysis shows the oscillations obey the standard Lifshitz-Kosevich theory for a 2D Fermi liquid. Angular dependence confirms a 2D cylindrical Fermi surface. This thermodynamic evidence unambiguously confirms the existence of coherent quasiparticles and a well-defined closed Fermi surface in the pseudogap state.

In summary, these studies collectively established the existence of small Fermi surface pockets in the underdoped pseudogap regime of hole-doped cuprates, in contrast to the large Fermi surface in the overdoped regime. This appears to be a generic cuprate property rather than a material-specific band structure effect, and is consistent with proposals for Fermi surface reconstruction due to density-wave order. The Lifshitz-Kosevich analysis indicates a Fermi liquid ground state, challenging some models of the pseudogap state.

## Q. Make a table comparing papers in materials studied, experimental techniques used (SdH, dHvA, etc.), and effective mass measured.

A. Here is a table comparing the key aspects of the four papers:

| Paper | Material | Experimental Techniques | Effective Mass (m*) |
| --- | --- | --- | --- |
| Doiron-Leyraud et al. (2007) | YBa2Cu3O6.5 (ortho-II, Tc = 57.5K) | Shubnikov-de Haas (SdH) oscillations in resistance | 1.9 ± 0.1 m0 |
| Yelland et al. (2007) | YBa2Cu4O8 (Tc ~ 80K) | Tunnel-diode oscillator (TDO) measurements of magnetic susceptibility | 3.0 ± 0.3 m0 |
| Bangura et al. (2007) | YBa2Cu4O8 (Tc ~ 80K) | Shubnikov-de Haas (SdH) oscillations in resistance and Hall resistance | 2.7 ± 0.3 m0 |
| Jaudet et al. (2007) | YBa2Cu3O6.5 (Tc = 57.5K) | de Haas-van Alphen (dHvA) oscillations in torque magnetometry | 1.76 ± 0.07 m0 |

Note: m0 represents the free electron mass.

## Q. What are recent developments since these four papers from 2007?

A. Since these groundbreaking discoveries in 2007, there have been several important developments and extensions of the quantum oscillation measurements in underdoped cuprates:

1. Quantum oscillations have been observed in several other hole-doped cuprate families beyond YBCO, including HgBa2CuO4+δ (Hg1201), La2-xSrxCuO4 (LSCO), and (Y,Nd)Ba2Cu3Oy. This establishes that small Fermi pockets are a generic property of the pseudogap regime.

2. Quantum oscillation measurements have been extended to lower doping levels, closer to the Mott insulating state. The observed Fermi pocket size continues to shrink with decreasing doping, consistent with a density-wave order scenario.

3. Evidence for the coexistence of a large Fermi surface and small pockets has been reported in some studies, suggesting a possible coexistence of competing orders.

4. The detailed shape and topology of the Fermi pockets has been mapped out using angle-dependent quantum oscillation measurements. Most studies are consistent with a single electron pocket per CuO2 plane, possibly arising from bidirectional charge-density wave reconstruction.

5. The magnitude and sign of the Hall coefficient and Seebeck coefficient at low temperatures have been shown to be consistent with the small electron pocket scenario inferred from quantum oscillations.

6. Quantum oscillation measurements have been extended to higher temperatures and lower magnetic fields, providing insights into the evolution of the Fermi surface with temperature and the relationship to the pseudogap and superconducting states.

7. Theoretical models have been developed to explain the origin of the reconstructed Fermi surface, including proposals based on charge, spin, or electron nematic density waves. However, the exact nature of the ordering remains an open question.

Despite these advances, many aspects of the pseudogap state and the transition to the large Fermi surface in the overdoped regime remain unresolved and an active area of research. The discovery of quantum oscillations has provided a powerful new tool to probe the electronic structure of underdoped cuprates and constrain theoretical models.

## Q. Cite main papers to read for all of your seven points.

A. Sure, here are some key papers that cover the developments I mentioned:

1. Quantum oscillations in other cuprate families:
- HgBa2CuO4+δ: Barisic et al., Nat. Phys. 9, 761-764 (2013)
- La2-xSrxCuO4: Rourke et al., New J. Phys. 12, 105009 (2010)
- (Y,Nd)Ba2Cu3Oy: Helm et al., Phys. Rev. Lett. 103, 157002 (2009)

2. Lower doping levels:
- Sebastian et al., Phys. Rev. B 81, 214524 (2010)
- Ramshaw et al., Science 348, 317-320 (2015)

3. Coexistence of large and small Fermi surfaces:
- Doiron-Leyraud et al., Phys. Rev. X 3, 021019 (2013)
- Maharaj et al., Phys. Rev. B 93, 094503 (2016)

4. Fermi pocket shape and topology:
- Sebastian et al., Phys. Rev. Lett. 108, 196403 (2012)
- Harrison and Sebastian, Phys. Rev. Lett. 106, 226402 (2011)

5. Consistency with transport measurements:
- Laliberté et al., Nat. Commun. 2, 432 (2011)
- Badoux et al., Nature 531, 210-214 (2016)

6. High-temperature and low-field measurements:
- Ramshaw et al., Nat. Phys. 7, 234-238 (2011)
- Chan et al., Phys. Rev. Lett. 113, 177005 (2014)

7. Theoretical models:
- Chakravarty and Kee, Proc. Natl. Acad. Sci. U.S.A. 105, 8835-8839 (2008)
- Allais et al., Phys. Rev. B 90, 155114 (2014)
- Sachdev and La Placa, Phys. Rev. Lett. 111, 027202 (2013)

These papers represent some of the key experimental and theoretical advances, but there have been many other important contributions. The references within these papers can guide you to further relevant literature.