Critical quantum liquids, and the cuprate high temperature superconductors

International Year of Quantum Science and Technology (IYQ)-2025 CQuERE (Centre for Quantum Engineering, Research and Education), TCG CREST, Kolkata, India September 19, 2025 Subir Sachdev HARVARD Critical quantum liquids, and the cuprate high temperature superconductors YBa2Cu3O6+x Cuprate high temperature superconductors Solid State Physics: Pre 1986: Mostly independent electrons (with pairing in BCS theory of superconductivity) Post 1986: Many-electron entanglement YBCO magnets allow for smaller, faster, and less expensive tokamaks for plasma fusion YBa2Cu3O6+x Cu YBa2Cu3O6+x Cu Antiferromagnetic insulator at a density of one electron per Cu YBa2Cu3O6+x P.W. Anderson and G. Baskaran (1988): The key to high temperature superconductivity is the formation of a “resonating valence bond state” (a type of quantum spin liquid) which entangles the electrons on Cu <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i <latexit sha1_base64="44EQSLd248yeLy8H80pBg3XRVUI=">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</latexit> |Gi=X D cD|Di D!dimer covering of lattice YBa2Cu3O6+x P.W. Anderson and G. Baskaran (1988): The key to high temperature superconductivity is the formation of a “resonating valence bond state” (a type of quantum spin liquid) which entangles the electrons on Cu <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i <latexit sha1_base64="44EQSLd248yeLy8H80pBg3XRVUI=">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</latexit> |Gi=X D cD|Di D!dimer covering of lattice YBa2Cu3O6+x P.W. Anderson and G. Baskaran (1988): The key to high temperature superconductivity is the formation of a “resonating valence bond state” (a type of quantum spin liquid) which entangles the electrons on Cu <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i <latexit sha1_base64="44EQSLd248yeLy8H80pBg3XRVUI=">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</latexit> |Gi=X D cD|Di D!dimer covering of lattice YBa2Cu3O6+x P.W. Anderson and G. Baskaran (1988): The key to high temperature superconductivity is the formation of a “resonating valence bond state” (a type of quantum spin liquid) which entangles the electrons on Cu <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i <latexit sha1_base64="44EQSLd248yeLy8H80pBg3XRVUI=">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</latexit> |Gi=X D cD|Di D!dimer covering of lattice YBa2Cu3O6+x <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i <latexit sha1_base64="Qg6+uUZpnKrnvU3xJ68xAGkZOmo=">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</latexit> Key feature: fractionalization. Excitations are particle-like, but cannot be created by local operators: they are classified under distinct superselection/anyon sectors. Kitaev: this is useful for quantum error correction. Spin S=1/2, charge neutral spinon YBa2Cu3O6+x <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i Spin S=1/2, charge neutral spinon <latexit sha1_base64="TBUA5XpurbDVZti3eNkukyz/+ZY=">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</latexit> •Theory of gapped spin liquids of insulators in two dimensions. ↭ The simplest, and closest to Anderson’s RVB state, is the Z2spin liquid (Read and S.S. 1991), with the same anyons as the toric code (Kitaev 1996). Implemented in Google’s fault tolerant qubit. We will use a critical spin liquid (a SU(2) gauge theory with massless Dirac and Higgs matter) and a critical charge liquid (the SYK model) which are gapless and strongly interacting, with no particle-like excitations Keimer, Kivelson, Norman, Uchida, and Zaanen, Nature 518, 179 (2015) YBa2Cu3O6+x Keimer, Kivelson, Norman, Uchida, and Zaanen, Nature 518, 179 (2015) tion (LDA), which are in good agreement with previous calculations [11,12], and a tight-binding fit of the experimentally determined FS. The spectra in Figs. 1(b) and 1(c) were measured along momentum space directions near the nodal and antinodal regions of the BZ, as indicated by the arrows in Fig. 1(a). Dispersive features are clearly observable, with a behavior which is ubiquitous among the cuprates [1]. Close to the nodal direction the QP peak exhibits a pronounced dispersion that can be followed over !250 meV below EF; near "!;0#, on the other hand, the band is much shallower with a van Hove singularity !39 meV below EF. By integrating over a $5 meV window about EFthe ARPES spectra normalized at high binding energies, one obtains an estimate for the normalstate FS [Fig. 1(d); the EF-intensity map across two BZs was downfolded to the reduced zone scheme and symmetrized with respect to the BZ diagonal, taking an average for equivalent kpoints, and then fourfolded]. As discussed later, at T%10 K ad-wave SC gap is open along the FS; thus this procedure returns the loci of minimum excitation energy across the gap, which, however, still correspond to the underlying normal-state FS crossings [1]. The FS of Tl2201-OD30 [Fig. 1(d)] consists of a large hole-pocket centered at "!;!#, which, as suggested by the low binding energy of the van Hove singularity [Fig. 1(c)], appears to be approaching a topological transition from hole to electronlike. The FS volume, counting holes, is 63 $2% of the BZ corresponding to a carrier concentration of 1:26 $0:04 hole=Cu atom, in very good agreement with Hall-coefficient [13] and AMRO [6] experiments, which found 1.30 and 1.24 itinerant holes, respectively, in slightly more overdoped samples. These measurements all indicate that the low-energy electronic structure of very overdoped Tl2201 is dominated by a single CuO band. In both ARPES and AMRO data there is no evidence for the TlO band that in LDA calculations crosses EFand gives rise to a small electron pocket centered at k%"0;0#for nonoxygenated (i.e., "%0) Tl2201 [Fig. 1(a), dashed FS]. This, however, is no surprise even within the independent particle picture. In fact, adjusting the chemical potential in the calculations in a rigid-band-like fashion to match the doping level of our Tl2201-OD30 sample (as determined by the total FS volume), the TlO band is emptied of its electrons and the LDA FS reduces to the single CuO pocket [Fig. 1(a), solid FS]. Since full depletion of the TlO band takes place for !EF’&0:159 eV, corresponding to the removal of 0.024 electrons from the TlO band (as well as 0.109 from the CuO band), already the deviation of the Tl3'and Cu2'content of our samples from the stoichiometric ratio 2:1, which contributes !0:14 hole=formula unit, would be sufficient to empty the TlO band even in the nonoxygenated "%0case. In this sense, the Tl-Cu nonstoichiometry and the presence of the TlO band cooperate in pushing the "%0system away from half filling, which may help explain why nonoxygenated Tl2201 is not a charge transfer insulator like undoped (i.e., x%0) LSCO [12]. As for the detailed shape of the FS, which in LDA calculations is more square than in ARPES and AMRO results, better agreement would require the inclusion in the calculations of correlation effects and/or O-doping beyond a rigid-band picture. Alternatively, the ARPES data can be modeled by the tightbinding dispersion #k%$'t1 2"coskx'cosky#'t2coskx( cosky't3 2"cos2kx'cos2ky#'t4 2"cos2kxcosky'coskx( cos2ky#'t5cos2kxcos2ky, as in Ref. [14] (setting a%1 for the lattice constant). With parameters $%0:2438, t1%&0:725,t2%0:302,t3%0:0159,t4%&0:0805, and t5%0:0034, all expressed in eV, this dispersion reproduces both the FS shape [Fig. 1(d)] and the QP energy at "0;0#and especially near "!;0#[Figs. 2(f) and 2(g)]. The analysis of the ARPES spectra in Fig. 2 indicates a SC gap consistent with a dx2&y2form. Because of the lack of normal-state data, the opening of the gap for this Tl2201-OD30 sample could not be followed via the shift of the leading edge midpoint (LEM) across Tc, as is commonly done (this was, however, possible in subsequent temperature dependent experiments on a less overdoped Tc%74 K sample). In the present case, the existence of a gap can be most easily visualized by the comparison of nodal and antinodal symmetrized spectra [15], in particular, by the presence of a peak at EFalong the nodal direction [signature of a FS crossing; bold line in Fig. 2(a)] and by the lack thereof along the antinodal [Fig. 2(b)]. For a more quantitative analysis, we performed a fit of the spectra along different k-space cuts intersecting the underlying normal-state FS [Fig. 2(d); as line shape we used a Lorentzian QP peak plus a steplike background identified by the ARPES intensity at k)kF, all multiplied by a Fermi function and convoluted with the instrumental en- (a) (π,π) I II (b) -200 0 I -200 0 II Bindin g Ener gy (meV) Tl2201-OD30 T=10 K hν=59 eV (c) (d) (π,π) (0,0) VFS=50% VFS=63% FIG. 1 (color online). (a) LDA FS for two different doping levels corresponding to a volume, counting holes, of 50% (cyan, dashed line) and 63% (blue, solid line) of the BZ. (b),(c) ARPES spectra taken at T%10 K on Tl2201-OD30 along the directions marked by arrows in (a). (d) ARPES FS of Tl2201-OD30 along with a tight-binding fit of the data (black lines). PRL 95, 077001 (2005) PHYSICAL REVIEW LETTERS week ending 12 AUGUST 2005 077001-2 Luttinger, 1960 Area enclosed by the Fermi surface is the same as that for free fermions with the same symmetry. <latexit sha1_base64="/uvxPHEAqR25KMhCGIq/4SSqZYU=">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</latexit> M. Plat´e et al.,PRL95, 077001 (2005) 1+p holes 1-p electrons Keimer, Kivelson, Norman, Uchida, and Zaanen, Nature 518, 179 (2015) <latexit sha1_base64="lH0Y4abNdfwNn+u2eYDoJmT0fy0=">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</latexit> Kyle M. Shen et al., Science 307, 901 (2005) Will explain with a Fractionalized Fermi Liquid (FL*) which evades the Luttinger constraint by a critical spin liquid T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003) Keimer, Kivelson, Norman, Uchida, and Zaanen, Nature 518, 179 (2015) Will explain with a local, two-dimensional extension of the Sachdev-Ye-Kitaev (SYK) model of mobile electrons, a critical charge liquid Aavishkar A. Patel, Haoyu Guo, Ilya Esterlis, S. Sachdev, Science 381, 790 (2023); Chenyuan Li, Aavishkar A. Patel, Haoyu Guo, Davide Valentinis, Jorg Schmalian, S.S., Ilya Esterlis, PRL 133, 186502 (2024) Experiments on the cuprate pseudogap phase “Fermi arcs” Kyle M. Shen, F. Ronning, D. H. Lu, F. Baumberger, N. J. C. Ingle, W. S. Lee, W. Meevasana, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, Z.-X. Shen, Science 307, 901 (2005) <latexit sha1_base64="KPRJJtUPMALE2+PyTCeepMqltTw=">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</latexit> Ca2-xNaxCuO2Cl2 at x = 0.10 Photoemission at small p Yawen Fang, Gaël Grissonnanche, Anaëlle Legros, Simon Verret, Francis Laliberté, Clément Collignon, Amirreza Ataei, Maxime Dion, Jianshi Zhou, David Graf, M. J. Lawler, Paul Goddard, Louis Taillefer, and B. J. Ramshaw, Nature Physics 18,!558 (2022) 7 FIG. 2. Fermi surface of Nd-LSCO at p=0.24 p=0.24 p=0.24.(a) Left panels: The angle-dependent magnetoresistance of Nd-LSCO at p=0.24 as a function of ✓for four di↵erent temperatures, T= 25, 20, 12, 6 K, and at B= 45 T. The grey bar near ✓= 90for T= 6 K and 12 K indicates the presence of superconductivity. Right panels: simulations obtained from the Chambers formula, using the tight-binding parameters from ARPES measurements [12], and in which the relaxation time ⌧(k) is the only free parameter. (b) The Fermi surface used for the calculation in panel (a), shown as cuts at kz= 0, ⇡/c, and 2⇡/c,wherecis the height of the body-centered-tetragonal unit cell (and c/2 is the distance between copper oxide layers). (c) The full 3D Fermi surface obtained from ADMR at p=0.24. The coloring corresponds to the vzcomponent of the Fermi velocity, with positive vzin teal, negative vzin purple, and vz= 0 in magenta. A single cyclotron orbit, perpendicular to the magnetic field, is drawn in black, with the Fermi velocity at di↵erent points around the orbit shown as gray arrows. The strong variation in vzaround the cyclotron orbits is what leads to ADMR. !cis the cyclotron frequency, m?is the cyclotron e↵ective mass, eis the electron charge, and 12 FIG. 4. Fermi surface reconstruction in Nd-LSCO at p=0.21 p=0.21 p=0.21.The top three panels show the Fermi surface for three di↵erent scenarios, and the bottom three panels show the resulting ADMR simulations. (b) Calculated ADMR using the same tight-binding and scattering rate parameters as in Figure 2a, but with the chemical potential shifted past the van Hove singularity (p⇡0.23) to p=0.21, for which the Fermi surface is shown in panel (a). (d) Calculated ADMR for a period-3 CDW reconstructed Fermi surface; the section of reconstructed Fermi surface used to calculate the ADMR is highlighted in orange in panel (c), the unreconstructed Fermi surface is shown with a blue dashed line. These are the small nodal electron pockets believed to result from CDW order in YBa2Cu3O6+xand are able to account for the ADMR in YBa2Cu3O6+xat p=0.11. (f) Calculated ADMR for reconstruction of the Fermi surface caused by a (⇡,⇡) order parameter, using the same tight-binding parameters as Figure 2, a gap of 58 kelvin, and a constant scattering rate; (e)The hole pockets used to simulate the ADMR in (f) are highlighted in orange. weakest along =45 . The gap magnitude (the strength of the potential associated with the FSR) that best reproduces the data is 58 kelvin — comparable to the onset temperature T?of the pseudogap phase at this doping [11, 29]. We find that a momentum-independent p>p cLarge Fermi surface <latexit sha1_base64="B7U4aMsZ31IpqRrsbqoLed/gmY8=">AAACBHicdVDJSgNBEO2JW4zbqMdcGhPBU5iJIYkXCQriwUMEs0AyhJ5OTdKkZ6G7RwhDDl78FS8eFPHqR3jzb+wsgoo+KHi8V0VVPTfiTCrL+jBSS8srq2vp9czG5tb2jrm715RhLCg0aMhD0XaJBM4CaCimOLQjAcR3ObTc0fnUb92CkCwMbtQ4Ascng4B5jBKlpZ6ZzUenUY/m8RURA8AXIHyGZSw8QqFn5qyCbZeLFQtbBWsGTcqVarF0jO2FkkML1Hvme7cf0tiHQFFOpOzYVqSchAjFKIdJphtLiAgdkQF0NA2ID9JJZk9M8KFW+tgLha5A4Zn6fSIhvpRj39WdPlFD+dubin95nVh5VSdhQRQrCOh8kRdzrEI8TQT3mQCq+FgTQgXTt2I6JIJQpXPL6BC+PsX/k2axYJcKJ9fFXO1sEUcaZdEBOkI2qqAaukR11EAU3aEH9ISejXvj0XgxXuetKWMxs49+wHj7BEAbl0E=</latexit> p<p cReconstructed Fermi surface <latexit sha1_base64="8Kmy3jX5BJ8GMwYlVlocSZlsgTM=">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</latexit> Fermi surface transformation at the pseudogap critical point of a cuprate superconductor <latexit sha1_base64="fpCc3x3Sb6DCDe08b+QTdjOn/3s=">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</latexit> Angle-dependent magnetoresistance (ADMR) of La1.6–xNd0.4SrxCuO4 <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">AAACV3icdVFNb9NAEN0YKMF8pXDksiJF4mTZKUnTWwUcOBaJtBV1FK3XY3vU9a61O6ZEVv4bf6N/oFf4B7BJ00MRjLTS7Jv3ZvbNZo1CR3F81Qvu3X+w87D/KHz85Omz54PdFyfOtFbCTBpl7FkmHCjUMCMkBWeNBVFnCk6ziw/r+uk3sA6N/kLLBua1KDUWKAV5aDH4mmZQou4kaAK7Cj+aBnXJheaoXas8a3MjLMBasxYD8UukildGgeOm4Dloh7Tke81emILOb3stBsM4Gh/Eyf6Ex1GcHE4P3/kk2R8l8ZQnUbyJIdvG8WJwneZGtrWXSyWcO0/ihuadsIRSwSpMWweNkBeihG7jfMXfeCjnhbH+aOIb9A5P1M4t68wza0GV+7u2Bv9VO2+pmM471E1LoOXNoKJVnAxfr5HnaEGSWvpESIv+hVxWwgrpnftODvxP6JKqLiX4TpeY+zndOBqjXoV+Mbfu+f+Tk1GUTKLJ59Hw6P12RX32ir1mb1nCDtgR+8SO2YxJ9oNds5/sV++q9zvYCfo31KC31bxkdyLY/QPn5LfE</latexit> Doping an insulating antiferromagnet with holes of density p AF metal Luttinger area. Broken symmetry <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 =(|"#i|#"i)/p2 Holon metal <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p Spin liquid with density p of spinless, charge +e “holons”. non-Luttinger area. Spin liquid <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007); R. K. Kaul, Y. B. Kim, S. S., T. Senthil, Nature Physics 4, 28 (2008) =(|"#i|#"i)/p2 Holon metal <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 Spin liquid with density p of spinless, charge +e “holons”. non-Luttinger area. Spin liquid T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007); R. K. Kaul, Y. B. Kim, S. S., T. Senthil, Nature Physics 4, 28 (2008) =(|"#i|#"i)/p2 Holon metal <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p Spin liquid with density p of spinless, charge +e “holons”. non-Luttinger area. Spin liquid <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007); R. K. Kaul, Y. B. Kim, S. S., T. Senthil, Nature Physics 4, 28 (2008) =(|"#i|#"i)/p2 Holon metal <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p Spin liquid with density p of spinless, charge +e “holons”. non-Luttinger area. Spin liquid <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007); R. K. Kaul, Y. B. Kim, S. S., T. Senthil, Nature Physics 4, 28 (2008) =(|"#i|#"i)/p2 Holon metal <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p Spin liquid with density p of spinless, charge +e “holons”. non-Luttinger area. Spin liquid <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007); R. K. Kaul, Y. B. Kim, S. S., T. Senthil, Nature Physics 4, 28 (2008) =(|"#i|#"i)/p2 Holon metal <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p Spin liquid with density p of spinless, charge +e “holons”. non-Luttinger area. Spin liquid <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007); R. K. Kaul, Y. B. Kim, S. S., T. Senthil, Nature Physics 4, 28 (2008) =(|"#i|#"i)/p2 Holon metal <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">AAACV3icdVFNb9NAEN0YKMF8pXDksiJF4mTZKUnTWwUcOBaJtBV1FK3XY3vU9a61O6ZEVv4bf6N/oFf4B7BJ00MRjLTS7Jv3ZvbNZo1CR3F81Qvu3X+w87D/KHz85Omz54PdFyfOtFbCTBpl7FkmHCjUMCMkBWeNBVFnCk6ziw/r+uk3sA6N/kLLBua1KDUWKAV5aDH4mmZQou4kaAK7Cj+aBnXJheaoXas8a3MjLMBasxYD8UukildGgeOm4Dloh7Tke81emILOb3stBsM4Gh/Eyf6Ex1GcHE4P3/kk2R8l8ZQnUbyJIdvG8WJwneZGtrWXSyWcO0/ihuadsIRSwSpMWweNkBeihG7jfMXfeCjnhbH+aOIb9A5P1M4t68wza0GV+7u2Bv9VO2+pmM471E1LoOXNoKJVnAxfr5HnaEGSWvpESIv+hVxWwgrpnftODvxP6JKqLiX4TpeY+zndOBqjXoV+Mbfu+f+Tk1GUTKLJ59Hw6P12RX32ir1mb1nCDtgR+8SO2YxJ9oNds5/sV++q9zvYCfo31KC31bxkdyLY/QPn5LfE</latexit> Doping an insulating antiferromagnet with holes of density p Spin liquid with density p of spinless, charge +e “holons”. non-Luttinger area. Spin liquid <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007); R. K. Kaul, Y. B. Kim, S. S., T. Senthil, Nature Physics 4, 28 (2008) =(|"#i|#"i)/p2 Holon metal <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p Spin liquid with density p of spinless, charge +e “holons”. non-Luttinger area. Spin liquid <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007); R. K. Kaul, Y. B. Kim, S. S., T. Senthil, Nature Physics 4, 28 (2008) =(|"#i|#"i)/p2 Spin liquid with density p of spinless, charge +e “holons” and charge 0 spin-1/2 “spinons”. Holon metal <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p non-Luttinger area. Spin liquid <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 =(|"#i|#"i)/p2 Spin liquid with density p of spinless, charge +e “holons” and charge 0 spin-1/2 “spinons”. Holon metal <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">AAACV3icdVFNb9NAEN0YKMF8pXDksiJF4mTZKUnTWwUcOBaJtBV1FK3XY3vU9a61O6ZEVv4bf6N/oFf4B7BJ00MRjLTS7Jv3ZvbNZo1CR3F81Qvu3X+w87D/KHz85Omz54PdFyfOtFbCTBpl7FkmHCjUMCMkBWeNBVFnCk6ziw/r+uk3sA6N/kLLBua1KDUWKAV5aDH4mmZQou4kaAK7Cj+aBnXJheaoXas8a3MjLMBasxYD8UukildGgeOm4Dloh7Tke81emILOb3stBsM4Gh/Eyf6Ex1GcHE4P3/kk2R8l8ZQnUbyJIdvG8WJwneZGtrWXSyWcO0/ihuadsIRSwSpMWweNkBeihG7jfMXfeCjnhbH+aOIb9A5P1M4t68wza0GV+7u2Bv9VO2+pmM471E1LoOXNoKJVnAxfr5HnaEGSWvpESIv+hVxWwgrpnftODvxP6JKqLiX4TpeY+zndOBqjXoV+Mbfu+f+Tk1GUTKLJ59Hw6P12RX32ir1mb1nCDtgR+8SO2YxJ9oNds5/sV++q9zvYCfo31KC31bxkdyLY/QPn5LfE</latexit> Doping an insulating antiferromagnet with holes of density p non-Luttinger area. Spin liquid <latexit sha1_base64="qkFOnwbv6K3xGDG3MWFsRxs5Y/s=">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</latexit> Area p/4 =(|"#i|#"i)/p2 =(|" i+| "i)/p2 Each green “dimer” is a bound state (a “magnetic polaron”) of a vacancy and a free spin FL* <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p <latexit sha1_base64="q00wo1Rpr955orrCD/xnoH2Rb98=">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</latexit> Area p/8 T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007) M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015) non-Luttinger area. Spin liquid =(|"#i|#"i)/p2 =(|" i+| "i)/p2 FL* <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p Metal with density p of spin-1/2, charge +e “holes” (or “magnetic polarons”) and charge 0 spin-1/2 “spinons”. T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007) M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015) non-Luttinger area. Spin liquid <latexit sha1_base64="q00wo1Rpr955orrCD/xnoH2Rb98=">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</latexit> Area p/8 =(|"#i|#"i)/p2 T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007) M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015) non-Luttinger area. Spin liquid Metal with density p of spin-1/2, charge +e “holes” (or “magnetic polarons”) and charge 0 spin-1/2 “spinons”. FL* <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p =(|" i+| "i)/p2 <latexit sha1_base64="q00wo1Rpr955orrCD/xnoH2Rb98=">AAACLHicdVDLTuMwFHWA4VFmoMCSjUWZ0awySVBb2PHYsASJAlJTVY5z21o4TmTfAFXUP+BbWLCFz2CDEFu2/AJuKRKD4EiWjs65x9c+USaFQc97cCYmp35Mz8zOleZ//lpYLC8tH5s01xwaPJWpPo2YASkUNFCghNNMA0siCSfR2d7QPzkHbUSqjrCfQSthXSU6gjO0Urv8J4ygK1TBQSHoQWnHhul69m9zvRSCit/1drniuV59K9gIqCXVDT+oWlIPar5fpb7rjVAhYxy0yy9hnPI8sXEumTFN38uwVTCNgksYlMLcQMb4GetC01LFEjCtYvSfAf1tlZh2Um2PQjpSPyYKlhjTTyI7mTDsmc/eUPzKa+bY2WwVQmU5guJvizq5pJjSYTk0Fho4yr4ljGth30p5j2nGbQf2JgO2X9XFXhEiXOKFiO2ewndrQg0beq+Bfk+OA9evubXDoLK9O+5qlqySNfKX+KROtsk+OSANwskVuSG35M65du6dR+fpbXTCGWdWyH9wnl8BebyolA==</latexit> Area p/8 =(|"#i|#"i)/p2 =(|" i+| "i)/p2 FL* <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007) M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015) non-Luttinger area. Spin liquid FL* <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p Metal with density p of spin-1/2, charge +e “holes” (or “magnetic polarons”) and charge 0 spin-1/2 “spinons”. <latexit sha1_base64="q00wo1Rpr955orrCD/xnoH2Rb98=">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</latexit> Area p/8 =(|"#i|#"i)/p2 =(|" i+| "i)/p2 FL* <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007) M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015) non-Luttinger area. Spin liquid Metal with density p of spin-1/2, charge +e “holes” (or “magnetic polarons”) and charge 0 spin-1/2 “spinons”. <latexit sha1_base64="q00wo1Rpr955orrCD/xnoH2Rb98=">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</latexit> Area p/8 =(|"#i|#"i)/p2 =(|" i+| "i)/p2 FL* <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007) M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015) non-Luttinger area. Spin liquid Metal with density p of spin-1/2, charge +e “holes” (or “magnetic polarons”) and charge 0 spin-1/2 “spinons”. <latexit sha1_base64="q00wo1Rpr955orrCD/xnoH2Rb98=">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</latexit> Area p/8 =(|"#i|#"i)/p2 =(|" i+| "i)/p2 FL* <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007) M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015) non-Luttinger area. Spin liquid Metal with density p of spin-1/2, charge +e “holes” (or “magnetic polarons”) and charge 0 spin-1/2 “spinons”. <latexit sha1_base64="q00wo1Rpr955orrCD/xnoH2Rb98=">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</latexit> Area p/8 =(|"#i|#"i)/p2 =(|" i+| "i)/p2 FL* <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007) M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015) non-Luttinger area. Spin liquid Metal with density p of spin-1/2, charge +e “holes” (or “magnetic polarons”) and charge 0 spin-1/2 “spinons”. <latexit sha1_base64="q00wo1Rpr955orrCD/xnoH2Rb98=">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</latexit> Area p/8 =(|"#i|#"i)/p2 =(|" i+| "i)/p2 FL* <latexit sha1_base64="xkFr0GtJK09blGK5cQ6+sHvTfxc=">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</latexit> Doping an insulating antiferromagnet with holes of density p T. Senthil, S. S., M. Vojta, PRL 90, 216403 (2003); R. K. Kaul, A. Kolezhuk, M. Levin, S.S., T. Senthil, PRB 75, 235122 (2007) M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015) non-Luttinger area. Spin liquid Metal with density p of spin-1/2, charge +e “holes” (or “magnetic polarons”) and charge 0 spin-1/2 “spinons”. <latexit sha1_base64="q00wo1Rpr955orrCD/xnoH2Rb98=">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</latexit> Area p/8 5 0 15 30 45 60 75 90 3 (o) 0 0.1 0.2 0.3 0.4 0.5 /;c (+.cm) 85 K B = 72 T ? = 0o ? = 23o ? = 45o 0 15 30 45 60 75 90 3 (o) 0 0.1 0.2 0.3 0.4 0.5 /;c (+.cm) Calculation ? = 0o ? = 23o ? = 45o 0 1 0 1kk ky (r.l.u.) kx (r.l.u.) ? a b FIG. 2. Comparison of measured angle dependent magnetoresistivity with a Boltzmann transport model.a, Magnetoresistivity ωεcat 72 T and 85 K as a function of ϑfor ϖ=0 →, 23o and 45o. Full underlying data is shown in Supplementary Fig. S1. b, Equivalent simulated ωεc(ϑ) (see Methods). Inset: Fermi-surface used for simulations. relation, ωc(ε)→↑vc(ε)vc(↓ε)↔= ↑ ! 0 dt exp{iεt}↑vc(t)vc(0)↔k. The time-delayed velocity correlation ↑vc(t)vc(0)↔k, is averaged over all possible starting points kon the Fermi surface [12, 13]. The resistivity ϑcis then obtained by inverting the conductivity ωc(ε)and taking the zero-frequency limit. The more e!ectively vcaverages to zero around a cyclotron orbit, the smaller is the c-axis conductivity, and therefore the larger is the c-axis resistivity. Such a physical picture enables a qualitative understanding of the results in Fig. 1. For very long relaxation times, εcϖ↗1, electrons can traverse the cyclotron orbit multiple times and the c-axis resistivity approaches the ‘clean’ limit. In this limit, the conductivity corresponds to the average velocity of a single complete orbit. As the scattering becomes more intense, and εcϖdecreases, the time-delayed velocity correlation is suppressed at long times such that the magnetic field dependence of the c-axis resistivity weakens and eventually disappears in the ‘dirty limit’, εcϖ↘1 When Bis aligned along the c-axis, the c-axis component of the velocity does not change around any given orbit for the simply warped Fermi-surface of Hg1201, (ϱ=0 →in Fig. 1c). Hence, the conductivity is the same as that at B=0suchthatςϑc≃0atϱ=0 →,as observed (Fig. 1e). A very weak magnetoresistivity for Balong symmetry directions is well arXiv:2411.10631 (p/8 also in YRZ ansatz, Peter Johnson photoemission, and Jenny Hoffman and Seamus Davis STMs; Stanescu-Kotliar?) 3 0.05 0.1 0.15 Hole doping p 0 100 Temperature (K) Tc pseudogap HgBa2CuO4+/ a b ? 3 c B 2: c B 3 = 0o B 10-1 3 = 3Yamaji 0 30 60 90 0 0.2 0.4 /;c (+.cm) 85 K ? = 0o # 72 T 50 T 25 T 0 30 60 90 3 (o) 0 0.1 0.2 0.3 /;c [72 T - 50 T] (+.cm) ? = 0o# 73 K 85 K 0 30 60 90 3 (o) 0 0.1 0.2 /;c [72 T - 50 T] (+.cm) # # ? = 0o ? = 45o 85 K 3Yamaji 3Yamaji 2kcal ? = 45o 2kcal ? = 0o a b c d e f g vc(a:u:) FIG. 1. Observation of the Yamaji e!ect. a, Phase diagram of superconducting temperature versus doping of Hg1201 [28]. Red dot marks the focus of this work. b, Schematic of the polar (ω) and azimuthal (ε) angles. a, b and care crystallographic directions. c, Schematic example orbit on a quasi-2D Fermi-surface for B→c. Arrows indicate the instantaneous velocity on the orbit. The c-axis component of the velocity, vc, does not change on the orbit. d, Schematic of example orbit for ω=ωYama ji .vcoscillates around the orbit and averages to zero. e, Polar magnetic field orientation-dependent curves of the magnetoresistivity ϑϖcat three di!erent fields, and at 85 K and ε=0 →. The arrow highlights the Yamaji peak, which disappears into the background at lower Bdue to the lower ϱcς.f, The same for two di!erent temperatures. The peak is more pronounced at lower temperature due to a higher ϱcς. To emphasize the Yamaji peak and to avoid the e!ect of superconductivity at lower temperature, we plot the di!erence ϖc(72 T) ↑ϖc(50 T). g, Azimuthal angle-dependence of the Yamaji peak at ε=0 →and 45→. The inset schematically shows the relationship between caliper radius kcal(ε) and εfor an elliptical pocket. The caliper radius is determined directly from the Yamaji angle by kcal(ε)=3φ/4ctan[ωYama ji (ε)] where cis the c↑axis lattice parameter (see Methods) [12, 13, 29]. We find kcal(ε=0 →)=0.12 ±0.01 ˚ A↑1 and kcal(ε= 45→)=0.16 ±0.02 ˚ A↑1, respectively. i.e. a significant fraction of electrons must complete a cyclotron orbit without scattering. <latexit sha1_base64="D2ztSsr44Ygn+v1rrdRb+1CsKFA=">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</latexit> “The small size of the pockets determined from the Yamaji e!ect is . . . approximately 1.3% of the Brillouin zone area” FL* pocket fraction = p/8=1.25% ! Fluctuating AF metal fraction = p/4=2.5%. <latexit sha1_base64="Gsg6H8uaO+rR2kZnm3MK7lqaxdc=">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</latexit> Doping p=0.1 5 0 15 30 45 60 75 90 3 (o) 0 0.1 0.2 0.3 0.4 0.5 /;c (+.cm) 85 K B = 72 T ? = 0o ? = 23o ? = 45o 0 15 30 45 60 75 90 3 (o) 0 0.1 0.2 0.3 0.4 0.5 /;c (+.cm) Calculation ? = 0o ? = 23o ? = 45o 0 1 0 1kk ky (r.l.u.) kx (r.l.u.) ? a b FIG. 2. Comparison of measured angle dependent magnetoresistivity with a Boltzmann transport model.a, Magnetoresistivity ωεcat 72 T and 85 K as a function of ϑfor ϖ=0 →, 23o and 45o. Full underlying data is shown in Supplementary Fig. S1. b, Equivalent simulated ωεc(ϑ) (see Methods). Inset: Fermi-surface used for simulations. relation, ωc(ε)→↑vc(ε)vc(↓ε)↔= ↑ ! 0 dt exp{iεt}↑vc(t)vc(0)↔k. The time-delayed velocity correlation ↑vc(t)vc(0)↔k, is averaged over all possible starting points kon the Fermi surface [12, 13]. The resistivity ϑcis then obtained by inverting the conductivity ωc(ε)and taking the zero-frequency limit. The more e!ectively vcaverages to zero around a cyclotron orbit, the smaller is the c-axis conductivity, and therefore the larger is the c-axis resistivity. Such a physical picture enables a qualitative understanding of the results in Fig. 1. For very long relaxation times, εcϖ↗1, electrons can traverse the cyclotron orbit multiple times and the c-axis resistivity approaches the ‘clean’ limit. In this limit, the conductivity corresponds to the average velocity of a single complete orbit. As the scattering becomes more intense, and εcϖdecreases, the time-delayed velocity correlation is suppressed at long times such that the magnetic field dependence of the c-axis resistivity weakens and eventually disappears in the ‘dirty limit’, εcϖ↘1 When Bis aligned along the c-axis, the c-axis component of the velocity does not change around any given orbit for the simply warped Fermi-surface of Hg1201, (ϱ=0 →in Fig. 1c). Hence, the conductivity is the same as that at B=0suchthatςϑc≃0atϱ=0 →,as observed (Fig. 1e). A very weak magnetoresistivity for Balong symmetry directions is well arXiv:2411.10631 (p/8 also in YRZ ansatz, Peter Johnson photoemission, and Jenny Hoffman and Seamus Davis STMs; Stanescu-Kotliar?) 3 0.05 0.1 0.15 Hole doping p 0 100 Temperature (K) Tc pseudogap HgBa2CuO4+/ a b ? 3 c B 2: c B 3 = 0o B 10-1 3 = 3Yamaji 0 30 60 90 0 0.2 0.4 /;c (+.cm) 85 K ? = 0o # 72 T 50 T 25 T 0 30 60 90 3 (o) 0 0.1 0.2 0.3 /;c [72 T - 50 T] (+.cm) ? = 0o# 73 K 85 K 0 30 60 90 3 (o) 0 0.1 0.2 /;c [72 T - 50 T] (+.cm) # # ? = 0o ? = 45o 85 K 3Yamaji 3Yamaji 2kcal ? = 45o 2kcal ? = 0o a b c d e f g vc(a:u:) FIG. 1. Observation of the Yamaji e!ect. a, Phase diagram of superconducting temperature versus doping of Hg1201 [28]. Red dot marks the focus of this work. b, Schematic of the polar (ω) and azimuthal (ε) angles. a, b and care crystallographic directions. c, Schematic example orbit on a quasi-2D Fermi-surface for B→c. Arrows indicate the instantaneous velocity on the orbit. The c-axis component of the velocity, vc, does not change on the orbit. d, Schematic of example orbit for ω=ωYama ji .vcoscillates around the orbit and averages to zero. e, Polar magnetic field orientation-dependent curves of the magnetoresistivity ϑϖcat three di!erent fields, and at 85 K and ε=0 →. The arrow highlights the Yamaji peak, which disappears into the background at lower Bdue to the lower ϱcς.f, The same for two di!erent temperatures. The peak is more pronounced at lower temperature due to a higher ϱcς. To emphasize the Yamaji peak and to avoid the e!ect of superconductivity at lower temperature, we plot the di!erence ϖc(72 T) ↑ϖc(50 T). g, Azimuthal angle-dependence of the Yamaji peak at ε=0 →and 45→. The inset schematically shows the relationship between caliper radius kcal(ε) and εfor an elliptical pocket. The caliper radius is determined directly from the Yamaji angle by kcal(ε)=3φ/4ctan[ωYama ji (ε)] where cis the c↑axis lattice parameter (see Methods) [12, 13, 29]. We find kcal(ε=0 →)=0.12 ±0.01 ˚ A↑1 and kcal(ε= 45→)=0.16 ±0.02 ˚ A↑1, respectively. i.e. a significant fraction of electrons must complete a cyclotron orbit without scattering. <latexit sha1_base64="D2ztSsr44Ygn+v1rrdRb+1CsKFA=">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</latexit> “The small size of the pockets determined from the Yamaji e!ect is . . . approximately 1.3% of the Brillouin zone area” FL* pocket fraction = p/8=1.25% ! Fluctuating AF metal fraction = p/4=2.5%. <latexit sha1_base64="Gsg6H8uaO+rR2kZnm3MK7lqaxdc=">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</latexit> Doping p=0.1 Best evidence yet for fractionalization ! in the cuprates FL* theory: hole pockets in magnetotransport and Fermi arcs in photoemission <latexit sha1_base64="A8Segu7dH19/rktDRKgCUx8ClPY=">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</latexit> Plan: •Identification of spin liquid: critical spin liquid without quasiparticles. One description is a SU(2) gauge theory with Nf= 2 massless Dirac spinons. •Dope spin liquid with holes, not holons. The Ancilla Layer Model (ALM) enables a theory of FL* hole pockets for a general spin liquid. •Theory of pseudogap (and its low Tinstabilities): Hole pockets coupled to critical spin liquid. <latexit sha1_base64="kQOPyGjVnpwnop1JLFjywvOfxM0=">AAACB3icdVDLSgNBEJz1GeMr6tHLYBA8LbuS1zHoxWNEY4QkhNlJJxkyO7vM9GrCkg/w4FU/w5t49TP8Cn/BSYygogUNRVU33V1BLIVBz3tzFhaXlldWM2vZ9Y3Nre3czu6ViRLNoc4jGenrgBmQQkEdBUq4jjWwMJDQCIanU79xA9qISF3iOIZ2yPpK9ARnaKWLYWfUyeU9t1jyi5UC9VxvhikpemXPp/5cyZM5ap3ce6sb8SQEhVwyY5q+F2M7ZRoFlzDJthIDMeND1oempYqFYNrp7NQJPbRKl/YibUshnanfJ1IWGjMOA9sZMhyY395U/MtrJtirtFOh4gRB8c9FvURSjOj0b9oVGjjKsSWMa2FvpXzANONo08m2DNjoVB8HaQthhLeia/ekvlsRamIT+oqB/k+ujl2/5JbOC/nqyTyrDNknB+SI+KRMquSM1EidcNIn9+SBPDp3zpPz7Lx8ti4485k98gPO6weKX5qG</latexit> kx <latexit sha1_base64="6mkTlCx96wLaVfXUBEQ0WMVgxHY=">AAACB3icdVDLSgNBEJz1bXxFPXoZDIKnZVfM4yh68RjRxEASwuykkwyZnV1metVlyQd48Kqf4U28+hl+hb/g5CGoaEFDUdVNd1cQS2HQ896dufmFxaXlldXc2vrG5lZ+e6duokRzqPFIRroRMANSKKihQAmNWAMLAwnXwfBs7F/fgDYiUleYxtAOWV+JnuAMrXQ57KSdfMFziyW/WDmmnutNMCZFr+z51J8pBTJDtZP/aHUjnoSgkEtmTNP3YmxnTKPgEka5VmIgZnzI+tC0VLEQTDubnDqiB1bp0l6kbSmkE/X7RMZCY9IwsJ0hw4H57Y3Fv7xmgr1KOxMqThAUny7qJZJiRMd/067QwFGmljCuhb2V8gHTjKNNJ9cyYKNTfRxkLYQ7vBVduyfz3YpQI5vQVwz0f1I/cv2SW7o4LpyczrJaIXtknxwSn5TJCTknVVIjnPTJA3kkT8698+y8OK/T1jlnNrNLfsB5+wSMAJqH</latexit> ky <latexit sha1_base64="OO8U2BOKijMCtCBmNNSpvz01AqY=">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</latexit> Massless Dirac fermionic spinons I. Affleck and J.B. Marston, PRB 37, 3774 (1988) N. Read and S. Sachdev, PRL 62, 1694 (1989) C. Wang, A. Nahum, M. A. Metlitski, C. Xu, T. Senthil, Phys. Rev. X 7, 031051 (2017) Zheng Zhou, Liangdong Hu, Wei Zhu, Yin-Chen He, PRX 14, 021044 (2024) Ya-Hui Zhang and S. S., PRR 2, 023172 (2020) E. Mascot, A. Nikolaenko, M. Tikhanovskaya, Ya-Hui Zhang, D. K. Morr, S. S., PRB 105, 075146 (2022) <latexit sha1_base64="A8Segu7dH19/rktDRKgCUx8ClPY=">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</latexit> Plan: •Identification of spin liquid: critical spin liquid without quasiparticles. One description is a SU(2) gauge theory with Nf= 2 massless Dirac spinons. •Dope spin liquid with holes, not holons. The Ancilla Layer Model (ALM) enables a theory of FL* hole pockets for a general spin liquid. •Theory of pseudogap (and its low Tinstabilities): Hole pockets coupled to critical spin liquid. <latexit sha1_base64="9CiTc++8h3cC/4nDV4K6K7okomY=">AAAB8XicdVDLSgMxFL3js9ZX1aWbYCu4GmdarXVXdeOygn1gO5RMmrahmcyQZIQy9C/cuFDErX/jzr8xbUdQ0QOBwznnknuPH3GmtON8WAuLS8srq5m17PrG5tZ2bme3ocJYElonIQ9ly8eKciZoXTPNaSuSFAc+p01/dDX1m/dUKhaKWz2OqBfggWB9RrA20t2FyaJCdFwpdHN5xy6XTkqOixzbmcEQt3juVk6Rmyp5SFHr5t47vZDEARWacKxU23Ui7SVYakY4nWQ7saIRJiM8oG1DBQ6o8pLZxhN0aJQe6ofSPKHRTP0+keBAqXHgm2SA9VD99qbiX1471v2KlzARxZoKMv+oH3OkQzQ9H/WYpETzsSGYSGZ2RWSIJSbalJQ1JXxdiv4njaLtlu3yTTFfvUzryMA+HMARuHAGVbiGGtSBgIAHeIJnS1mP1ov1Oo8uWOnMHvyA9fYJ6qePyg==</latexit> Area p/8 <latexit sha1_base64="i2gFDyhdC2uJHD+P5TXY3BpKuJ0=">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</latexit> Hole pockets and decoupled spin liquid Ya-Hui Zhang and S. S., PRR 2, 023172 (2020) E. Mascot, A. Nikolaenko, M. Tikhanovskaya, Ya-Hui Zhang, D. K. Morr, S. S., PRB 105, 075146 (2022) <latexit sha1_base64="cWRGP3yXjd7Is4MNJ1o/KlOK26o=">AAACzHicdVLLjtNAEByb1xJeAY5cWiRI2YvlZJckcFrBJSe0iGR3pTisxuO208p4xnjGLMHKlW/jF/gI/oFxEhAgaGmkUnWVulU9cSHJ2DD85vnXrt+4eevgduvO3Xv3H7QfPjozuioFzoSWuryIuUFJCmeWrMSLokSexxLP49Xrpn/+EUtDWk3tusBFzjNFKQluHXXZ/hrFmJGqyWJOn3HTihoE0yXqcg06hcJgleiMF9DjKgGyBqS+gu60C6SM5TFJsoTm8GUUwURLhEKLFTqZ0FUhMQGrQZROI7gEU5ACSR8qSiBeA3ceseRlhtDFLkwoywykhDJpRtslNq/MnfHdrDc4hIxX2ZZzywWtCFXya/HLdicMnh+NRy9GEAbhthowPjoeDqG/ZzpsX6eX7e9RokWVo7JCcmPm/bCwi5qXblPZBFEZLLhY8QznDiqeo1nU28Q38MwxCaS6dE9Z2LK/O2qeG7POY6fMuV2av3sN+a/evLLpeFGTKiqLSuwGpZVsQmzOBwmVKKxcO8B3qW4D5MK6I7cig+4HqMwu68jiJ3tFiZtTH5Nq8vkZAvwfnA2C/jAYvh10Tl7tkzpgT9hT1mN9NmInbMJO2YwJL/Cm3sJ777/xrV/7m53U9/aex+yP8r/8AKhB30Y=</latexit> •Theory of pseudogap (and its low Tinstabilities): Hole pockets coupled to critical spin liquid by a charge eHiggs field of the thermal SU(2) gauge theory. H. Pandey, M. Christos, P.M. Bonetti, R. Shanker, S. Sharma, S.S., arXiv:2507.05336 <latexit sha1_base64="cWRGP3yXjd7Is4MNJ1o/KlOK26o=">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</latexit> •Theory of pseudogap (and its low Tinstabilities): Hole pockets coupled to critical spin liquid by a charge eHiggs field of the thermal SU(2) gauge theory. H. Pandey, M. Christos, P.M. Bonetti, R. Shanker, S. Sharma, S.S., arXiv:2507.05336 <latexit sha1_base64="cWRGP3yXjd7Is4MNJ1o/KlOK26o=">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</latexit> •Theory of pseudogap (and its low Tinstabilities): Hole pockets coupled to critical spin liquid by a charge eHiggs field of the thermal SU(2) gauge theory. <latexit sha1_base64="h1cVbI+Zyc5AyC1d77dxjfRPr3w=">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</latexit> Kyle M. Shen, . . . Z.-X. Shen, Science 307, 901 (2005) 100 200 300 400 f0/H 0 2 4 6 8 10 12 14 Ω a) 0.00 0.05 0.10 0.15 0.20 0.25 f/f0 0 2 4 6 8 10 12 14 16 |F[Ω]| b) Bi=0 r/T =1.2 r/T =0.40 r/T =0.12 100 200 0 2 100 200 300 400 f0/H 0 2 4 6 8 10 12 14 Ω a) 0.00 0.05 0.10 0.15 0.20 0.25 f/f0 0 2 4 6 8 10 12 14 16 |F[Ω]| b) Bi=0 r/T =1.2 r/T =0.40 r/T =0.12 100 200 0 2 °1 0 1 kx/º °1.0 °0.5 0.0 0.5 1.0 ky/º a) r/T =1.2 °1 0 1 kx/º °1.0 °0.5 0.0 0.5 1.0 ky/º b) r/T =0.40 °1 0 1 kx/º °1.0 °0.5 0.0 0.5 1.0 ky/º c) r/T =0.12 0.0 0.2 0.4 0.6 0.8 1.0 Ac(!=0,k)/A0 <latexit sha1_base64="E0MXHmzzizGmLRx67wZBlFOYcWQ=">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</latexit> Fermi arcs co-exist with quantum oscillations of hole pockets of area p/8. H. Pandey, M. Christos, P.M. Bonetti, R. Shanker, S. Sharma, S.S., arXiv:2507.05336 <latexit sha1_base64="cWRGP3yXjd7Is4MNJ1o/KlOK26o=">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</latexit> •Theory of pseudogap (and its low Tinstabilities): Hole pockets coupled to critical spin liquid by a charge eHiggs field of the thermal SU(2) gauge theory. Critical quantum charge liquids: From SYK to strange metals The SYK model Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="IKWLBVaOopL04SJMr/BCf3dhiuA=">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</latexit> U4,5;11,18 The SYK model Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="rsJF7k7so/HPd5gsUXHjJYWvZ8A=">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</latexit> U14,19;1,13 The SYK model Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="rsJF7k7so/HPd5gsUXHjJYWvZ8A=">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</latexit> U14,19;1,13 The SYK model Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="Utq9HnZ9cYobM/K6ortiIQYaw9k=">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</latexit> U9,18;5,15 The SYK model Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="Utq9HnZ9cYobM/K6ortiIQYaw9k=">AAACLXicbZDLSsNAFIYn3q13XboJFsFFCYmmXnAjunGpYFVoQ5lMTtuhk0mcOVFL6HO41SfwaVwI4tbXcFKz8HZg4OP/z+HM+cNUcI2u+2qNjU9MTk3PzFbm5hcWl5ZXVi91kikGDZaIRF2HVIPgEhrIUcB1qoDGoYCrsH9S+Fe3oDRP5AUOUghi2pW8wxlFIwWNdn5Q8/YP6zWvPmwvV13HHZXtOr5X93cKKBWvhCop66y9YlVaUcKyGCQyQbVuem6KQU4VciZgWGllGlLK+rQLTYOSxqBr0S1P9QiDfHTA0N40ZmR3EmWeRHukfh/Oaaz1IA5NZ0yxp397hfif18ywsx/kXKYZgmRfizqZsDGxizTsiCtgKAYGKFPcfNtmPaooQ5NZpaXBBCq72MtbCPd4xyOzJ/edOpc/jsvDuEjP+53VX7jcdrxdxz/3q0fHZY4zZJ1skC3ikT1yRE7JGWkQRm7IA3kkT9az9WK9We9frWNWObNGfpT18QkIfKde</latexit> U9,18;5,15 The SYK model Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="kyfUy4iLmDHmZ1iLj3Dez4cecBc=">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</latexit> U6,8;4,14 The SYK model Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="kyfUy4iLmDHmZ1iLj3Dez4cecBc=">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</latexit> U6,8;4,14 A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015) S. Sachdev and J. Ye, PRL 70, 3339 (1993) (See also: the “2-Body Random Ensemble” in nuclear physics; did not obtain the large N limit; T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, and S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981)) U↵; are independent random variables with U↵; = 0 and |U↵;|2=U2 N!1yields critical strange metal. <latexit sha1_base64="yQteaZaLoEYlztplY68/SkIu+Hc=">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</latexit> The Sachdev-Ye-Kitaev (SYK) model <latexit sha1_base64="5HvB9s0savM8oVuT5ii1CiV2awU=">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</latexit> H=1 (2N)3/2 N X ↵,,,=1 U↵; c† ↵c† ccµX ↵ c† ↵c↵ c↵c+cc↵=0 ,c ↵c† +c† c↵=↵ Q=1 NX ↵ c† ↵c↵;[H,Q]=0; 0Q1 ⌃= <latexit 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sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit> <latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit> G <latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit> <latexit 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sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit> <latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit> S. Sachdev and J. Ye, PRL 70, 3339 (1993) The Sachdev-Ye-Kitaev (SYK) model <latexit sha1_base64="HvhMYNOwEGd7P72O1PevqvdHvd8=">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</latexit> Feynman graph expansion in U↵;, and graph-by-graph average, yields exact equations in the large Nlimit: G(i!)= 1 i!+µ⌃(i!),⌃(⌧)=U2G2(⌧)G(⌧) G(⌧=0 )=Q. U U <latexit sha1_base64="cXZAxo9CgsFaThlpD5UPLC7PCAQ=">AAAB/XicdVBNS8NAEN34WetX1aOXxSJ4KkltU70VvXisYD+gDWWy3bRLN5uwuxFKKP4Gr3r2Jl79LR79J27bCFb0wcDjvRlm5vkxZ0rb9oe1srq2vrGZ28pv7+zu7RcODlsqSiShTRLxSHZ8UJQzQZuaaU47saQQ+py2/fH1zG/fU6lYJO70JKZeCEPBAkZAG6ndAx6PIN8vFO2S7dacSgUbMoch7uV5tVrGTqYUUYZGv/DZG0QkCanQhINSXceOtZeC1IxwOs33EkVjIGMY0q6hAkKqvHR+7hSfGmWAg0iaEhrP1Z8TKYRKTULfdIagR+q3NxP/8rqJDi68lIk40VSQxaIg4VhHePY7HjBJieYTQ4BIZm7FZAQSiDYJLW3xw6nJ5Ptx/D9plUuOW3JvK8X6VZZODh2jE3SGHFRDdXSDGqiJCBqjR/SEnq0H68V6td4WrStWNnOElmC9fwHWUpYB</latexit> ↵ <latexit 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sha1_base64="p/C/hQZxV/o1a574fBqft9yuY5s=">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</latexit> X  U2 N3 =U2 U U <latexit sha1_base64="cXZAxo9CgsFaThlpD5UPLC7PCAQ=">AAAB/XicdVBNS8NAEN34WetX1aOXxSJ4KkltU70VvXisYD+gDWWy3bRLN5uwuxFKKP4Gr3r2Jl79LR79J27bCFb0wcDjvRlm5vkxZ0rb9oe1srq2vrGZ28pv7+zu7RcODlsqSiShTRLxSHZ8UJQzQZuaaU47saQQ+py2/fH1zG/fU6lYJO70JKZeCEPBAkZAG6ndAx6PIN8vFO2S7dacSgUbMoch7uV5tVrGTqYUUYZGv/DZG0QkCanQhINSXceOtZeC1IxwOs33EkVjIGMY0q6hAkKqvHR+7hSfGmWAg0iaEhrP1Z8TKYRKTULfdIagR+q3NxP/8rqJDi68lIk40VSQxaIg4VhHePY7HjBJieYTQ4BIZm7FZAQSiDYJLW3xw6nJ5Ptx/D9plUuOW3JvK8X6VZZODh2jE3SGHFRDdXSDGqiJCBqjR/SEnq0H68V6td4WrStWNnOElmC9fwHWUpYB</latexit> ↵ <latexit sha1_base64="cXZAxo9CgsFaThlpD5UPLC7PCAQ=">AAAB/XicdVBNS8NAEN34WetX1aOXxSJ4KkltU70VvXisYD+gDWWy3bRLN5uwuxFKKP4Gr3r2Jl79LR79J27bCFb0wcDjvRlm5vkxZ0rb9oe1srq2vrGZ28pv7+zu7RcODlsqSiShTRLxSHZ8UJQzQZuaaU47saQQ+py2/fH1zG/fU6lYJO70JKZeCEPBAkZAG6ndAx6PIN8vFO2S7dacSgUbMoch7uV5tVrGTqYUUYZGv/DZG0QkCanQhINSXceOtZeC1IxwOs33EkVjIGMY0q6hAkKqvHR+7hSfGmWAg0iaEhrP1Z8TKYRKTULfdIagR+q3NxP/8rqJDi68lIk40VSQxaIg4VhHePY7HjBJieYTQ4BIZm7FZAQSiDYJLW3xw6nJ5Ptx/D9plUuOW3JvK8X6VZZODh2jE3SGHFRDdXSDGqiJCBqjR/SEnq0H68V6td4WrStWNnOElmC9fwHWUpYB</latexit> ↵ <latexit sha1_base64="S8xq3N16jWfNMBSZ0JPj+VRMPKs=">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</latexit> X  U2 N3 = U2 N <latexit sha1_base64="E9IqEksPcc1jJSN1pS0nh3hcFxg=">AAAB/HicdVBNS8NAEN3Ur1q/qh69LBbBU0lqm+qt6MVjBfsBbSib7bRdu5uE3Y1QQv0NXvXsTbz6Xzz6T9y2Eazog4HHezPMzPMjzpS27Q8rs7K6tr6R3cxtbe/s7uX3D5oqjCWFBg15KNs+UcBZAA3NNId2JIEIn0PLH1/N/NY9SMXC4FZPIvAEGQZswCjRRmp2fdAk18sX7KLtVp1yGRsyhyHuxVmlUsJOqhRQinov/9nthzQWEGjKiVIdx460lxCpGeUwzXVjBRGhYzKEjqEBEaC8ZH7tFJ8YpY8HoTQVaDxXf04kRCg1Eb7pFESP1G9vJv7ldWI9OPcSFkSxhoAuFg1ijnWIZ6/jPpNANZ8YQqhk5lZMR0QSqk1AS1t8MTWZfD+O/yfNUtFxi+5NuVC7TNPJoiN0jE6Rg6qohq5RHTUQRXfoET2hZ+vBerFerbdFa8ZKZw7REqz3LwgFlY0=</latexit>  <latexit sha1_base64="YjjIvi2FoLEKwW3E5nEcpijRQDk=">AAAB/XicdVBNS8NAEN34WetX1aOXxSJ4KkltU70VvXisYD+gDWWy3bRLd5OwuxFKKP4Gr3r2Jl79LR79J27bCFb0wcDjvRlm5vkxZ0rb9oe1srq2vrGZ28pv7+zu7RcODlsqSiShTRLxSHZ8UJSzkDY105x2YklB+Jy2/fH1zG/fU6lYFN7pSUw9AcOQBYyANlK7NwQhIN8vFO2S7dacSgUbMoch7uV5tVrGTqYUUYZGv/DZG0QkETTUhINSXceOtZeC1IxwOs33EkVjIGMY0q6hIQiqvHR+7hSfGmWAg0iaCjWeqz8nUhBKTYRvOgXokfrtzcS/vG6igwsvZWGcaBqSxaIg4VhHePY7HjBJieYTQ4BIZm7FZAQSiDYJLW3xxdRk8v04/p+0yiXHLbm3lWL9Kksnh47RCTpDDqqhOrpBDdREBI3RI3pCz9aD9WK9Wm+L1hUrmzlCS7DevwDRjJX+</latexit>  <latexit sha1_base64="cXZAxo9CgsFaThlpD5UPLC7PCAQ=">AAAB/XicdVBNS8NAEN34WetX1aOXxSJ4KkltU70VvXisYD+gDWWy3bRLN5uwuxFKKP4Gr3r2Jl79LR79J27bCFb0wcDjvRlm5vkxZ0rb9oe1srq2vrGZ28pv7+zu7RcODlsqSiShTRLxSHZ8UJQzQZuaaU47saQQ+py2/fH1zG/fU6lYJO70JKZeCEPBAkZAG6ndAx6PIN8vFO2S7dacSgUbMoch7uV5tVrGTqYUUYZGv/DZG0QkCanQhINSXceOtZeC1IxwOs33EkVjIGMY0q6hAkKqvHR+7hSfGmWAg0iaEhrP1Z8TKYRKTULfdIagR+q3NxP/8rqJDi68lIk40VSQxaIg4VhHePY7HjBJieYTQ4BIZm7FZAQSiDYJLW3xw6nJ5Ptx/D9plUuOW3JvK8X6VZZODh2jE3SGHFRDdXSDGqiJCBqjR/SEnq0H68V6td4WrStWNnOElmC9fwHWUpYB</latexit> ↵ <latexit sha1_base64="Z/cvfpAhkfKXwaMKh/vpS17zjwo=">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</latexit> Solution G(ω)→T→1/2F(⊋ω/kBT) A. Georges and O. Parcollet PRB 59, 5341 (1999) A solvable model of multi-particle quantum entanglement. Yields a quantum state whose excitations are not particle-like i.e. no bosons, fermions, anyons…. Current is carried by an “entangled quantum soup” The Sachdev-Ye-Kitaev (SYK) model Sachdev, Ye (1993); Kitaev (2015) S. Sachdev, arXiv:0907.0008 (2009) A critical spin liquid (SU(2) gauge theory with massless Dirac and Higgs matter) and a critical charge liquid (the 2d-Yukawa-SYK model). Gapless and strongly interacting, many-body system with no particle-like excitations S. Sachdev, arXiv:0907.0008 (2009) A critical spin liquid (SU(2) gauge theory with massless Dirac and Higgs matter) and a critical charge liquid (the 2d-Yukawa-SYK model). Gapless and strongly interacting, many-body systems with no particle-like excitations S. Sachdev, arXiv:0907.0008 (2009) A critical spin liquid (SU(2) gauge theory with massless Dirac and Higgs matter) and a critical charge liquid (the 2d-Yukawa-SYK model). Gapless and strongly interacting, many-body systems with no particle-like excitations